$\phi_{\mathrm{tos}}^{+}(\eta_{1},\eta_{2})=2\log\Big{|}\frac{\sinh\tfrac{1}{2}\big{(}\eta_{1}-\eta_{2}\big{)}}{\sinh\tfrac{1}{2}\big{(}\eta_{1}+\eta_{2}\big{)}}\Big{|}$
(10.39)

for $\eta_{1}\eta_{2}>0$ and

$\phi_{\mathrm{tos}}^{-}(\eta,\eta^{\prime})=2\log\Big{|}\frac{\cosh\tfrac{1}{2}\big{(}\eta_{1}-\eta_{2}\big{)}}{\cosh\tfrac{1}{2}\big{(}\eta_{1}+\eta_{2}\big{)}}\Big{|}$
(10.40)

for $\eta_{1}\eta_{2}<0$. Clearly, on their domain of definition,
$\phi_{\mathrm{tos}}^{+}<0$ as hard core particles with $\mathsfit{a}>0$ and
$\phi_{\mathrm{tos}}^{-}(\eta,\eta^{\prime})>0$ as hard core particles with
$\mathsfit{a}<0$.

According to the collision rate ansatz, the effective velocity for Toda
solitons is

$v^{\mathrm{eff}}(\eta)=|\eta|^{-1}\Big{(}\sinh\eta+\int_{\mathbb{R}}\mathrm{d}\eta\phi_{\mathrm{tos}}(\eta,\eta^{\prime})f(\eta^{\prime})\big{(}v^{\mathrm{eff}}(\eta^{\prime})-v^{\mathrm{eff}}(\eta)\big{)}\Big{)}.$
(10.41)

In terms of Eq. (10.28) one infers that

$P(\eta)=\mathrm{sgn}(\eta)\eta^{2}/2,\quad E(\eta)=\cosh\eta.$ (10.42)

Without any changes we can now follow the blueprint of the KdV equation, in
particularly the soliton-based hydrodynamic equations are given by (10.25)
with $\phi_{\mathrm{kv}}$ substituted by $\phi_{\mathrm{tos}}$.

For the interpretation of the soliton counting function, we have to work out
the analogue of (10.27). As can be seen from (10.35), in the Flaschka
variables the soliton does not decay to $0$. Therefore one has to subtract the
constant at infinity as

$Q^{[n]\infty}=\mathrm{tr}[L^{n}-(L_{\infty})^{n}],$ (10.43)

$n\geq 1$, with $(L_{\infty})_{i,j}=1$ for $i=j\pm 1$ and $0$ otherwise. We
denote by $K_{n}$ the conserved field $Q^{[n]\infty}$ evaluated at the one-
soliton solution written in (10.35), (10.36). These coefficients can be
obtained from the inverse scattering transform (IST). The result is not as
explicit as the one in (10.27) and reads

$\sum_{n=1}^{\infty}\frac{1}{n}\big{(}\frac{z}{1+z^{2}}\big{)}^{n}K_{n}(\eta)=\sum_{n=1}^{\infty}\frac{2}{n}z^{n}\sinh(n\eta)$
(10.44)

for $\eta\geq 0$. To extend to negative $\eta$ one notes the symmetry
$K_{n}(\eta)=(-1)^{n}K_{n}(-\eta)$, which follows from the random walk
expansion of the right hand side of (10.43). Thus to lowest order,

$Q^{[1]\infty}=2\sinh\eta,\qquad Q^{[2]\infty}=2|\sinh(2\eta)|.$ (10.45)

Higher orders are determined recursively, at least in principle. But the basis
functions are no longer polynomials as in the case of KdV.

###  10.3 Comparing soliton- and particle-based hydrodynamics

This seems to be a good moment for a broader perspective, for which purpose
various integrable many-body systems are listed, some them only to be
discussed in later chapters.

model | soliton-based | particle-based | method
---|---|---|---
hard rods | no | yes | model specific
box-ball | yes | no | model specific
KdV | yes | no | IST
Toda (classical) | yes | yes | IST, scattering coordinates, RMT
Calogero (classical) | (yes) | yes | scattering coordinates
continuum NLS | yes | no | IST
Ablowitz-Ladik | (yes) | yes | IST, RMT
$\delta$-Bose gas | no | yes | Bethe ansatz

Table: IST stands for inverse scattering transform and RMT for random matrix
theory.

In the context of classical models, IST is the key tool for soliton-based
hydrodynamics. Clearly hard rods do not possess solitons. Box-ball is a
spacetime discrete version of KdV. Configurations are infinite sequences of
$0$’s and $1$’s. For a finite number of particles, the update from time $t$ to
$t+1$ is given by the following rule: an empty carrier with unbounded capacity
starts from the very left, moves to the right, and either picks up or drops a
particle whenever such step is possible. After the carrier has moved to the
very right the time $t+1$ configuration is completed. An $N$-soliton is simply
a stretch of $N$ consecutive particles, isolated and with no holes. The
soliton speed equals $N$. Soliton-based dynamics is confirmed both through
mathematical proofs and numerical simulations. Most remarkably the collision
rate ansatz remains valid. Based on our discussions in Section 10.1, KdV is
soliton based and so is the nonlinear Schrödinger equation (NLS).

Less clear-cut is the Calogero model with $1/\sinh^{2}$ interaction potential.
Since the Toda lattice is a low density approximation to the Calogero model,
one might have expected at least the analytic form of one-solitons. This is
not the case, but promising numerical results are recorded in the literature.
For the Ablowitz-Ladik system, to be discussed in Chapter 12, $N$-soliton
solutions are available through IST and also the two-soliton scattering shift.
But apparently soliton-based hydrodynamics has been left out. Presumably
efforts have been focussed more strongly on the study of the hydrodynamic
scale of the KdV and continuum nonlinear Schrödinger equation because of their
relevance for experimental realizations.

For particle-based hydrodynamics, a prerequisite is the availability of a Lax
matrix. But then two distinct methods are used. One is the transformation to
scattering coordinates, as explained already for the Toda fluid. An
alternative method emerges from the observation that under GGE the Lax matrix
becomes a random matrix. Thereby tools from random matrix theory (RMT) become
available. This method works well for the Toda lattice and also for the
Ablowitz-Ladik model. In the latter case the eigenvalues of the Lax matrix are
on the unit circle. In fact, the Ablowitz-Ladik model plays a peculiar role.
In view of RMT, it is clearly particle-based. From the perspective of RMT it
is obviously particle-based. But in contrast to the Toda lattice and Calogero
fluid, the true particle trajectories are hidden.

In deriving GHD for quantum many-body systems the very first step towards an
analysis is always the Bethe ansatz, which determines the eigenvalues of the
many-body system, compare with Chapter 13. Thereby generalized thermodynamics
can be constructed. Since in the respective Bethe equations the two-particle
phase shift appears explicitly, it is not so surprising that this scattering
shift reappears in the generalized free energy. Here two-particle phase shift
refers to the dynamics in the two-particle subspace. In our models this would
be the Hilbert space $L^{2}(\mathbb{R}^{2},\mathrm{d}x_{1}\mathrm{d}x_{2})$
and quantum wave function is of the form $\psi(x_{1},x_{2},t)$. Thus, clearly,
for integrable quantum many-body systems GHD is particle-based.

In analogy to IST, the Faddeev school developed the quantum inverse scattering
method (QISM). This is a very powerful tool from which a unifying view of
integrability emerged, encompassing also integrable models from two-
dimensional statistical mechanics. Apparently, the method was never used to
construct a soliton gas, say like the one for the KdV equation. There is
interesting work around the notion of quantum solitons. Whether these objects
are really solitons in the sense of scattering theory remains to be explored.

Notes and references

Section 10.1

The integrability of the KdV equation has been established in the late 60ies
in a series of contributions by Gardner et al. [127], Miura et al.[210], and
Kruskal et al. [179]. The hamiltonian structure is discussed in Gardner [126].
An important step was the development of inverse scattering transform, which
was then applied to a variety of other models. We refer to the review by
Ablowitz et al. [3] and the books by Ablowitz and Segur [6] and Ablowitz et
al. [7]. Scattering theory of the KdV equation is studied by Tanaka [283,
284]. Whitham [296] developed his modulation theory in the context of KdV, see
also Flaschka et al. [116]. Soliton gases were first introduced by Zakharov
[299], see also Zakharov [300]. He used a low density approximation, for which
in the collision rate asnsatz (10.26) the term $v^{\mathrm{eff}}$ on the right
hand side is replaced by its bare value $4\eta^{2}$. The hydrodynamic scale
was pioneered by Gennady El in his seminal contributions El [101] and El and
Kamchatnov [103]. El uses the Whitham modulation theory in a limit where the
spectral bands of $L(t)$ become very flat, see El [102] for a very instructive
review. In fact, the collision rate ansatz appears only indirectly through the
coupled equations

$-Tf(\eta)+\sigma(\eta)f(\eta)=\eta,\qquad-
Tv(\eta)+\sigma(\eta)f(\eta)=4\eta^{3},$ (10.46)

where $v=v^{\mathrm{eff}}f$ is the spectral flux density. Eliminating $\sigma$
yields Eq. (10.26) for $v^{\mathrm{eff}}$. A further confirmation comes from
numerical studies of a tracer soliton and from matching with hydrodynamic
reductions, see Carbone et al. [51]. The latter mean that the initial
$f(x;\eta)$ is piecewise constant and takes only two or three values. Then the
Euler equations simplify to make explicit solutions available, which can be
compared with numerical solutions of KdV. The initial conditions have spatial
randomness, but very little noise in the $\eta$ dependence. A further, this
time mathematical confirmation is accomplished by Girotti et al. [133]. The
initial state is a long periodic arrangement of one-solitons, in other words
the cnoidal wave with spatial cut-off. In addition a tracer soliton is
incoming from the left with a sufficiently high velocity when compared to the
building blocks of the cnoidal wave. The initial data are not random. Still in
simulations, the tracer motion fluctuates random-like relative to a constant
drift. The outgoing tracer velocity is given by (10.26) with $f$ referring to
the cnoidal background.

A more detailed comparison between GHD and KdV soliton-based hydrodynamics can
be found in the article Bonnemain et al. [29]. Bidirectional solitons are
studied by Congy et al. [61] and soliton condensates by Congy et al.[62].
Experimental realizations are reported by Redor et al. [243].

Section 10.2

In part II of his famous papers on the Toda lattice, Flaschka [114] developed
the inverse scattering transform for the Toda lattice. Hirota [154]
constructed the $N$-soliton solution. A very readable account is the article
by Toda [289], where the derivation of scattering shift is explained and the
similarity with KdV outlined. Scattering theory is studied by Krüger and
Teschl [180]. For the infinite lattice one considers initial conditions such
that

$\sum_{j\in\mathbb{Z}}(1+|j|)\big{(}|a_{j}-1|+|p_{j}|\big{)}<\infty.$ (10.47)

In the long time limit, both future and past, the solution converges to $N$
freely propagating one-solitons with scattering shifts determined by (10.39),
(10.40). There seems to be no concrete algorithm to determine $N$ and the
respective soliton velocities for a given initial configuration.

Section 10.3

The clear distinction between particle-based and soliton-based hydrodynamics
seems to be a novel insight. In principle, the $N$-soliton solutions of the
Calogero fluid can be constructed through a Bäcklund transformation, also
known as dual particles, see Philip [231]. Numerical solutions show
convincingly the motion of a single soliton in a trap and collision between
two solitons as reported by Gon and Kulkarni [135]. Named as “soliton cellular
automaton” an ultra-discrete wave equation was introduced by Takahashi and
Satsuma [280]. Now box-ball is the universally accepted label referring to the
picture that the carrier picks up balls from filled boxes, resp. drops balls
to empty boxes. The hydrodynamic scale of the box-ball system has been studied
in considerable detail by Ferrari et al. [111], Croydon and Sasada [64], and
Kuniba et al. [184, 185].

Korepin et al. [173] is the standard monograph on QISM. Faddeev [106] provides
an illuminating account on the history. His 1996 Les Houches lectures, Faddeev
[107], still stand out as a masterful introduction.

##  11 Calogero models

Based on the experience with the Toda lattice, a physically natural goal is to
advance towards one-dimensional classical fluids. The standard hamiltonian
reads

$H_{\mathrm{mec},N}=\sum_{j=1}^{N}\tfrac{1}{2}p_{j}^{2}+\tfrac{1}{2}\sum_{i\neq
j=1}^{N}V_{\mathrm{mec}}(q_{i}-q_{j}),$ (11.1)

using units such that the particle mass equals $1$. The particles move on the
real line, with possibly some confining mechanism to be introduced later. The
interaction potential, $V_{\mathrm{mec}}$, is even and hence the hamiltonian
is invariant under relabelling of particles. A proper hydrodynamic scale
assumes a short range potential, say with exponential decay. The borderline
for a liquid-gas phase transition would be
$V_{\mathrm{mec}}(x)\approx-|x|^{-2}$ for large $|x|$. But to our knowledge,
the minimal decay for hydrodynamic behavior has not been investigated
systematically. At least, the free energy has to be well-defined which
essentially is ensured by the stability property

$\sum_{i\neq j=1}^{N}V_{\mathrm{mec}}(q_{i}-q_{j})\geq-\frac{c_{0}}{N}$ (11.2)

with some constant $c_{0}\geq 0$ independent of $N$. For a purely attractive
potential such a condition is violated and typically particles would clump
together.

For a generic choice of $V_{\mathrm{mec}}$, one expects number, momentum, and
energy to be the only locally conserved fields. As a consequence the conserved
fields are expected to be governed by the conventional hydrodynamic equations,
which usually are written as

$\displaystyle\hskip
10.0pt\partial_{t}\rho_{\mathsf{f}}+\partial_{x}(\rho_{\mathsf{f}}v)=0\,,$
$\displaystyle\hskip
10.0pt\partial_{t}(\rho_{\mathsf{f}}v)+\partial_{x}(\rho_{\mathsf{f}}v^{2}+P_{\mathsf{f}})=0\,,$
$\displaystyle\hskip
10.0pt\partial_{t}(\rho_{\mathsf{f}}\mathfrak{e}_{\mathrm{tot}})+\partial_{x}(\rho_{\mathsf{f}}\mathfrak{e}_{\mathrm{tot}}v+P_{\mathsf{f}}v)=0\,.$
(11.3)

Here $\rho_{\mathsf{f}}$ is the particle density per unit length,
$\rho_{\mathsf{f}}v$ the momentum density, and
$\rho_{\mathsf{f}}\mathfrak{e}_{\mathrm{tot}}$ the total energy density.
$P_{\mathsf{f}}$ is the thermodynamic pressure that depends on
$\rho_{\mathsf{f}}$ and the internal energy, the latter being obtained from
$\mathfrak{e}_{\mathrm{tot}}=\mathfrak{e}_{\mathrm{int}}+\tfrac{1}{2}v^{2}$.
These equations are similar to Euler type equations of a non-integrable chain.
Only, since the potential is a sum over all pairs of particles, it is
considerably more difficult to compute the pressure $P_{\mathsf{f}}$.

A priori, there might no choice of $V_{\mathrm{mec}}$ which makes the model
integrable. We discussed already a fluid with hard core potential, which is
integrable but relies on having only contact interactions. In 1975 Francesco
Calogero investigated the issue and discovered exactly two interaction
potentials for which the $N$-particle dynamics is integrable: the long ranged
$1/x^{2}$ potential, called rational Calogero-Moser model, and the short range
repulsive $1/\sinh^{2}(x)$ potential, which we will refer to as Calogero
fluid, resp. hyperbolic Calogero model.

$\blackdiamond\blackdiamond$ Two distinct Lax matrices. For a hamiltonian as
in (11.1), to check integrability Calogero assumes the existence of a Lax
pair, $L,M$, of the general form

$\displaystyle L_{i,j}=\delta_{ij}p_{j}+(1-\delta_{ij})\alpha(q_{i}-q_{j}),$
$\displaystyle M_{i,j}=\delta_{ij}\sum_{k=1,k\neq
j}^{N}\beta(q_{j}-q_{k})+(1-\delta_{ij})\gamma(q_{i}-q_{j})$ (11.4)

with yet arbitrary functions $\alpha,\beta,\gamma$. The commutator condition
for the Lax pair then yields functional relations for $\alpha,\beta,\gamma$
and the potential $V_{\mathrm{mec}}$. These equations can be analysed with the
result $\gamma=\alpha^{\prime}$, $\beta=\alpha^{\prime\prime}/(2\alpha)$, and
$V_{\mathrm{mec}}(x)=\alpha(x)\alpha(-x)$. Now for $\alpha$ there are
essentially only three choices: (i) $\alpha(x)=\mathrm{i}g/x$, (ii)
$\alpha(x)=\mathrm{i}ga/\sinh(ax)$, and (iii)
$\alpha(x)=\mathrm{i}ga\coth(ax)$. Here $a,g$ are real parameters. Case (i)
yields the rational Calogero model, which will be discussed in Section 11.5,
while (ii) and (iii) yield the hyperbolic Calogero model of the section below.
In addition there are solutions corresponding to Lax pairs which result from
periodizing over a ring of given size. This then generates the interaction
potential $1/\sin x$ in case (i) and particular elliptic functions in cases
(ii) and (iii).

It is surprising to have two Lax matrices, say $L^{[2]},L^{[3]}$ for the same
hamiltonian. Depending on the problem, either one of them might be of
advantage. There seems to be no simple relation between $L^{[2]}$ and
$L^{[3]}$. But since their eigenvalues are conserved when evolved by the same
dynamics, there must be a functional relation between either set of
eigenvalues.

For chains one can carry out a similar analysis. Since a smooth potential is
assumed, only the exponential potential qualifies as being integrable. In fact
there are also two distinct Lax matrices, but they are related by a simple
similarity transform. $\blackdiamond\blackdiamond$

Figure 11: Displayed are trajectories of 12 Calogero particles in a periodic
box of size $\ell=12$ and at temperature $\beta=1$. Particles repel each other
and cannot cross. There is one fast quasiparticle which can be distinguished
from the background. Courtesy from M. Kulkarni.

###  11.1 Hyperbolic Calogero model, charges and currents

The interaction potential of the Calogero fluid is given by

$V_{\mathrm{ca}}(x)=\frac{1}{(2\sinh(\tfrac{1}{2}x))^{2}}.$ (11.5)

More commonly, one introduces length and time scales through

$V_{\mathrm{ca}}(x)=\frac{g^{2}a^{2}}{\sinh^{2}(ax)},$ (11.6)

which would make the formulas more lengthy. If needed, these parameters can be
introduced by a linear change of spacetime coordinates. The choice
$g=1,a=\tfrac{1}{2}$ is dictated by recovering the Toda lattice at low
density. For $N$ particles on the entire real line, Newton equations of motion
are

$\frac{d^{2}}{dt^{2}}q_{j}=\sum_{i=1,i\neq
j}^{N}\tfrac{1}{4}\cosh(\tfrac{1}{2}(q_{j}-q_{i}))\big{(}\sinh(\tfrac{1}{2}(q_{j}-q_{i}))\big{)}^{-3}$
(11.7)

for $j=1,\ldots,N$. At short distances the potential has the repulsive
$1/|x|^{2}$ singularity, which implies that the spatial ordering of particles
is maintained throughout time. Momentarily, we will introduce a Lax matrix
which is the avenue towards the particle-based scheme. The scattering shift
will refer to the motion of two particles.

The dynamics admits a Lax pair, $L,M$, through

$\displaystyle
L_{i,j}=\delta_{ij}p_{j}+\mathrm{i}(1-\delta_{ij})\big{(}2\sinh(\tfrac{1}{2}(q_{i}-q_{j}))\big{)}^{-1},$
(11.8) $\displaystyle M_{i,j}=\mathrm{i}\delta_{ij}\sum_{k=1,k\neq
j}^{N}\big{(}2\sinh(\tfrac{1}{2}(q_{j}-q_{k}))\big{)}^{-2}-\mathrm{i}(1-\delta_{ij})\cosh(\tfrac{1}{2}(q_{i}-q_{j}))\big{(}2\sinh(\tfrac{1}{2}(q_{i}-q_{j}))\big{)}^{-2}$

for $i,j=1,\ldots,N$. Referring to the Insert above, our choice is $L^{[2]}$,
since $L^{[3]}$ would lead yo nonlocal conservation laws. For a lighter
notation we omit the index $N$ that was used for the Toda lattice. The Lax
matrix is hermitian, hence has real eigenvalues, while the partner matrix is
anti-hermitian. Then, with $L,M$ evaluated along trajectories of (11.7),

$\frac{d}{dt}L(t)=[L(t),M(t)].$ (11.9)

Merely having a commutator implies that

$\frac{d}{dt}\mathrm{tr}[L(t)^{n}]=0.$ (11.10)

The eigenvalues of $L(t)$ do not change in time. The conserved charges are

$Q^{[n]}=\mathrm{tr}[L^{n}]$ (11.11)

and their density is given by

$Q^{[n]}(x)=\sum_{j=1}^{N}\delta(x-q_{j})(L^{n})_{j,j}$ (11.12)

with $x\in\mathbb{R}$. The total momentum corresponds to $Q^{[1]}$ and total
energy to $\tfrac{1}{2}Q^{[2]}$. As a general fact for Calogero type models,
under the standard Poisson bracket,

$\\{Q^{[m]},Q^{[n]}\\}=0.$ (11.13)

Thus we have obtained a set of $N$ conserved fields in involution. Setting
$n=0$ our labelling includes the particle number,

$\mathrm{tr}[L^{0}]=N=Q^{[0]},\quad Q^{[0]}(x)=\sum_{j=1}^{N}\delta(x-q_{j}).$
(11.14)

In the sense of hamiltonian systems the constant function is a trivial
conservation law. But it must be included in a hydrodynamic context, since the
particle density is a central physical observable.

To obtain the currents one considers

$\frac{\mathrm{d}}{\mathrm{d}t}Q^{[n]}(f)=\frac{\mathrm{d}}{\mathrm{d}t}\sum_{j=1}^{N}f(x-q_{j})(L^{n})_{j,j}=\sum_{j=1}^{N}f(x-q_{j})p_{j}(L^{n})_{j,j}+\sum_{i\neq
j=1}^{N}\big{(}f(q_{j})-f(q_{i})\big{)}M_{i,j}(L^{n})_{j,i},$ (11.15)

which implies

$J^{[n]}(x)=\sum_{j=1}^{N}\big{(}\delta(q_{j}-x)p_{j}(L^{n})_{j,j}+\sum_{i=1,i\neq
j}^{N}\big{(}\theta(q_{j}-x)-\theta(q_{i}-x)\big{)}M_{i,j}(L^{n})_{j,i}\big{)}.$
(11.16)

In particular,

$\partial_{t}Q^{[n]}(x,t)+\partial_{x}J^{[n]}(x,t)=0.$ (11.17)

Most naturally, the system is confined by having particles move in a periodic
box $[0,\ell]$, $\ell>0$. Since the interaction potential has a tail, one has
to include the potential from all image particles. To introduce a notation,
for some rapidly decreasing function, $f$, we define its $\ell$-periodized
version through

$f_{\mathrm{per},\ell}(x)=\sum_{m\in\mathbb{Z}}f(x+m\ell).$ (11.18)

By construction $f_{\mathrm{per},\ell}$ is $\ell$-periodic and its domain can
be restricted to the box $[0,\ell]$. The periodized interaction potential can
be written in terms of the elliptic (double periodic) Weierstrass function,
$\wp(z)$. Historically its two periodicities are denoted by
$2\omega_{1},2\omega_{2}$ and

$\wp(z|\omega_{1},\omega_{2})=\frac{1}{z^{2}}+\sum_{m,n\in\mathbb{Z}}{{}^{\prime}}\hskip
2.0pt\Big{(}\big{(}z+2\omega_{1}m+2\omega_{2}n\big{)}^{-2}-\big{(}2\omega_{1}m+2\omega_{2}n\big{)}^{-2}\Big{)},$
(11.19)

where $\sum{{}^{\prime}}$ means that the origin, $(m,n)=0$, has to be omitted
in the sum. One notes the derivative

$\wp(z|\omega_{1},\omega_{2})^{\prime}=-\sum_{m,n\in\mathbb{Z}}2\big{(}z+2\omega_{1}m+2\omega_{2}n\big{)}^{-3}$
(11.20)

and the identities

$\sum_{n\in\mathbb{Z}}\big{(}z+\mathrm{i}\pi
n\big{)}^{-2}=\sinh^{-2}(z),\qquad-\sum_{n\in\mathbb{Z}}2\big{(}z+\mathrm{i}\pi
n\big{)}^{-3}=(\sinh^{-2}(z))^{\prime}.$ (11.21)

We now set $2\omega_{1}=\ell$ and $2\omega_{2}=\mathrm{i}\pi$ and conclude
from (11.20) and (11.21) that

$(\sinh^{-2})_{\mathrm{per},\ell}(x)=\wp(x|\tfrac{1}{2}\ell,\mathrm{i}\tfrac{1}{2}\pi)-\tfrac{1}{12}.$
(11.22)

The constant $-\tfrac{1}{12}$ has to be fixed by a separate argument and drops
out when studying the dynamics.

The equations of motion for the Calogero particles in a periodic box are still
given by Eq. (11.7) with the force replaced by the periodized force. However
the Lax matrix for the ring is no longer simply the periodized Lax matrix. By
the method indicated in the Insert above, one can still construct a Lax pair
for the periodized Calogero model in terms of particular elliptic functions.
As for the Toda chain, the Calogero model remains integrable on the ring.

$\blackdiamond\blackdiamond$ Dilute limit. If particles are far apart the
interaction potential can be approximated as $\mathrm{e}^{-|x|}$. Labelling
particles in increasing order, the force on the $j$-th particle is
exponentially dominated by the force from its two neighbors. Thus, in
approximation,

$\frac{d^{2}}{dt^{2}}q_{j}=\mathrm{e}^{-(q_{j}-q_{j-1})}-\mathrm{e}^{-(q_{j+1}-q_{j})},$
(11.23)

which is recognized as the Toda dynamics.

One would expect that in the dilute limit also the Lax pair converges to the
Lax matrix of the Toda lattice. In this approximation

$\displaystyle L_{j,j}=p_{j},\quad L_{j,j+1}=-\mathrm{i}a_{j},\quad
L_{j+1,j}=\mathrm{i}a_{j},$ $\displaystyle M_{j,j}=0,\quad
M_{j,j+1}=M_{j+1,j}=-\tfrac{1}{2}\mathrm{i}a_{j},$ (11.24)

where, as to recalled, the Flaschka variables are
$a_{j}=\exp[-(q_{j+1}-q_{j})/2]$. For the open system all other matrix
elements vanish, while for the closed system $L_{1,N}=\mathrm{i}a_{N}$ and
$L_{N,1}=-\mathrm{i}a_{N}$. One checks that for the closed system there is a
diagonal unitary matrix, $\tilde{U}$, such that
$\tilde{U}L\tilde{U}^{*}=L_{N}$ and and
$\tilde{U}M\tilde{U}^{*}=-B_{N}^{\diamond}$, while for the open system there
is a diagonal unitary matrix, $\tilde{U}^{\prime}$, such that
$\tilde{U}^{\prime}M\tilde{U}^{\prime*}=-B_{N}$ and
$\tilde{U}^{\prime}L\tilde{U}^{\prime*}=L_{N}^{\diamond}$ and
$\tilde{U}^{\prime}M\tilde{U}^{\prime*}=-B_{N}^{\diamond}$, compare with
(2.12) and (2.14). $\blackdiamond\blackdiamond$

###  11.2 Scattering coordinates

To compute the generalized free energy we will use action-angle variables, as
in Section 9.2 for the Toda fluid. This time, the map is less explicit and
some stretch of linear algebra is in demand. In fact, we do start with the Lax
matrix $L$ in (11.8) for particles moving on the entire real line. Since
particles cannot cross we order them in increasing index. Then the particle
phase space is
$(q,p)\in\Gamma_{\\!N}^{\triangleright}=\mathbb{W}_{N}\times\mathbb{R}^{N}$
with the Weyl chamber $\mathbb{W}_{N}=\\{q_{1}<\ldots<q_{N}\\}$, while the
phase space of scattering coordinates is
$(\lambda,\phi)\in\Gamma_{\\!N}^{\triangleright}$ with ordered eigenvalues.
The eigenvalue problem for $L$ reads

$L\psi_{\alpha}=\lambda_{\alpha}\psi_{\alpha},$ (11.25)

$\alpha=1,\ldots,N$. Since $L=L^{*}$, there is a unitary matrix, $U$, such
that

$ULU^{*}=\mathrm{diag}(\lambda_{1},\ldots,\lambda_{N}).$ (11.26)

We also introduce

$A=\mathrm{diag}(\mathrm{e}^{q_{1}},\ldots,\mathrm{e}^{q_{N}}),\qquad
d_{j}=\mathrm{e}^{q_{j}/2},$ (11.27)

and correspondingly

$\tilde{A}=\mathrm{diag}(\mathrm{e}^{-q_{1}},\ldots,\mathrm{e}^{-q_{N}}),\qquad\tilde{d}_{j}=\mathrm{e}^{-q_{j}/2}.$
(11.28)

Clearly $A\tilde{A}=1$. Now

$[A,L]=\mathrm{i}\big{(}|d\rangle\langle
d|-A\big{)},\qquad[\tilde{A},L]=-\mathrm{i}\big{(}|\tilde{d}\rangle\langle\tilde{d}|-\tilde{A}\big{)}.$
(11.29)

Setting

$Q=UAU^{*},\quad|e\rangle=U|d\rangle,\qquad\tilde{Q}=U\tilde{A}U^{*},\quad|\tilde{e}\rangle=U|\tilde{d}\rangle,$
(11.30)

conjugating the identities (11.29) with $U$, and taking the $i,j$ matrix
elements results in

$Q_{i,j}=e_{i}\frac{\mathrm{i}}{\lambda_{j}-\lambda_{i}+\mathrm{i}}e_{j}^{*},\qquad\tilde{Q}_{i,j}=\tilde{e}_{i}\frac{\mathrm{i}}{\lambda_{i}-\lambda_{j}+\mathrm{i}}\tilde{e}_{j}^{*}.$
(11.31)

Since $Q\tilde{Q}=1$, from the diagonal matrix elements $(Q\tilde{Q})_{j,j}=1$
one concludes

$-e_{j}\tilde{e}_{j}^{*}\sum_{m=1}^{N}\frac{e_{m}^{*}\tilde{e}_{m}}{\big{(}\lambda_{m}-\lambda_{j}+\mathrm{i}\big{)}^{2}}=1.$
(11.32)

In particular, the expansion coefficients $e_{j},\tilde{e}_{j}$ cannot vanish.
From the off-diagonal matrix elements, $(Q\tilde{Q})_{i,j}=0,i\neq j$, one
infers

$0=\sum_{m=1}^{N}\frac{e_{m}^{*}\tilde{e}_{m}(\lambda_{i}-\lambda_{j})}{\big{(}\lambda_{m}-\lambda_{i}+\mathrm{i}\big{)}\big{(}\lambda_{m}-\lambda_{j}+\mathrm{i}\big{)}}=\sum_{m=1}^{N}\frac{e_{m}^{*}\tilde{e}_{m}}{\big{(}\lambda_{m}-\lambda_{i}+\mathrm{i}\big{)}}-\sum_{m=1}^{N}\frac{e_{m}^{*}\tilde{e}_{m}}{\big{(}\lambda_{m}-\lambda_{j}+\mathrm{i}\big{)}}.$
(11.33)

Hence the latter sum equals some constant, $c$, independent of $j$,

$\sum_{m=1}^{N}\frac{e_{m}^{*}\tilde{e}_{m}}{\big{(}\lambda_{m}-\lambda_{j}+\mathrm{i}\big{)}}=c.$
(11.34)

From the Cauchy-type identity in (11.44) below, it follows that

$e_{j}^{*}\tilde{e}_{j}=\mathrm{i}cV_{j}$ (11.35)

with

$V_{j}=\prod_{m=1,m\neq
j}^{N}\Big{(}1+\frac{\mathrm{i}}{\lambda_{j}-\lambda_{m}}\Big{)},$ (11.36)

while the identity (11.45) below implies

$\langle e,\tilde{e}\rangle=\mathrm{i}cN.$ (11.37)

On the other hand $\langle e,\tilde{e}\rangle=N$ from the definition (11.30)
and $c=-\mathrm{i}$ follows. Hence

$e_{j}^{*}\tilde{e}_{j}=V_{j}$ (11.38)

and using (11.46) below one checks that Eq. (11.32) is satisfied. Together
with (11.31) the final result reads

$Q_{j,j}=|e_{j}|^{2},\qquad\tilde{Q}_{j,j}=|\tilde{e}_{j}|^{2},\qquad|e_{j}|^{2}|\tilde{e}_{j}|^{2}=|V_{j}|^{2}.$
(11.39)

One defines a real-valued $\phi_{j}$ through

$Q_{j,j}=|V_{j}|\mathrm{e}^{\phi_{j}},$ (11.40)

which indeed turns out to be equal to the scattering shift $\phi_{j}^{+}$ of
the Calogero model, compare with (2.43). Somewhat implicitly, we have obtained
the scattering map $\Phi:(\lambda,\phi)\mapsto(q,p)$.

Since $Q_{j,j}=|e_{j}|^{2}$, the overall phase of the eigenvectors of $L$ can
be chosen such that

$e_{j}=|V_{j}|^{1/2}\mathrm{e}^{\phi_{j}/2}.$ (11.41)

From the invariance of the trace under unitary transformations one concludes

$\mathrm{tr}[A]=\sum_{j=1}^{N}\mathrm{e}^{q_{j}}=\mathrm{tr}[Q]=\sum_{j=1}^{N}|V_{j}|\mathrm{e}^{\phi_{j}},\qquad\mathrm{tr}[\tilde{A}]=\sum_{j=1}^{N}\mathrm{e}^{-q_{j}}=\mathrm{tr}[\tilde{Q}]=\sum_{j=1}^{N}|V_{j}|\mathrm{e}^{-\phi_{j}}.$
(11.42)

$\blackdiamond\blackdiamond$ Cauchy-type identities. We consider coefficients
$b_{j},j=1,\ldots,N$, satisfying the linear equations

$\sum_{j=1}^{N}\frac{b_{j}}{x_{j}-y_{i}}=1,$ (11.43)

$i=1,\ldots,N$, for some given set of coefficients
$\\{x_{j},y_{j},j=1,\ldots,N\\}$. Then the following identities hold:

$b_{j}=\frac{\prod_{m=1}^{N}(x_{j}-y_{m})}{\prod_{m=1,m\neq
j}^{N}(x_{j}-x_{m})},$ (11.44)
$\sum_{j=1}^{N}b_{j}=\sum_{j=1}^{N}(x_{j}-y_{j}),$ (11.45)

$\sum_{j=1}^{N}\frac{b_{j}}{(x_{j}-y_{i})^{2}}=-\frac{\prod_{m=1,m\neq
i}^{N}(y_{i}-y_{m})}{\prod_{m=1}^{N}(y_{i}-x_{m})}.$ (11.46)

$\blackdiamond\blackdiamond$

###  11.3 Generalized free energy

While the periodized system is still integrable, there seems to be no analogue
of the Dumitriu-Edelman change of volume elements. On the other hand,
scattering coordinates and a carefully chosen box potential can still be
implemented.

To the GGE potential we add the external potential given by

$V_{\mathrm{ex}}(q)=\sum_{j=1}^{N}\mathrm{e}^{-\ell/2}\cosh(q_{j}),$ (11.47)

which has a very flat bottom over the interval $[-\ell/2,\ell/2]$ and then
increases exponentially with rate $1$. In fact, for this particular choice the
Calogero fluid remains integrable. While this property is not used directly,
it is reflected by having a simple formula for the transformed
$V_{\mathrm{ex}}$, perhaps

$V_{\mathrm{ex}}(q)\circ\Phi=\sum_{j=1}^{N}\mathrm{e}^{-\ell/2}Y_{j}\cosh(\phi_{j})$
(11.48)

with

$Y_{j}=|V_{j}|=\prod_{m=1,m\neq
j}^{N}\Big{(}1+\frac{1}{(\lambda_{m}-\lambda_{j})^{2}}\Big{)}^{1/2}.$ (11.49)

Using the symmetry in $q$, the partition function reads

$\displaystyle
Z_{\mathrm{ca},N}(\ell,V)=\int_{\Gamma_{\\!N}^{\triangleright}}\mathrm{d}^{N}q\,\mathrm{d}^{N}p\exp\Big{(}-\mathrm{tr}\big{[}V(L)\big{]}-\sum_{j=1}^{N}\mathrm{e}^{-\ell/2}\cosh(q_{j})\Big{)}$
$\displaystyle\hskip
20.0pt=\int_{\Gamma_{\\!N}^{\triangleright}}\mathrm{d}^{N}\lambda\,\mathrm{d}^{N}\phi\exp\Big{(}-\sum_{j=1}^{N}\big{(}V(\lambda_{j})+\mathrm{e}^{-\ell/2}Y_{j}\cosh\phi_{j}\big{)}\Big{)}.$
(11.50)

Since the integrand is symmetric in $\lambda$, at the expense of a factor
$1/N!$, the $\lambda$-integration is extended to $\mathbb{R}^{N}$. The
$\phi_{j}$-integration yields the order zero modified Bessel function of the
second kind $K_{0}$, see (9.35), resulting in

$Z_{\mathrm{ca},N}(\ell,V)=\frac{1}{N!}\int_{\mathbb{R}^{N}}\mathrm{d}^{N}\lambda\prod_{j=1}\mathrm{e}^{-V(\lambda_{j})}\prod_{j=1}^{N}2K_{0}\big{(}2\mathrm{e}^{-\ell/2}Y_{j}\big{)}.$
(11.51)

As a consequence, the joint distribution of eigenvalues of the Lax matrix
(11.8) under the normalized measure

$Z_{\mathrm{ca},N}(\ell,V)^{-1}\exp\Big{(}-\mathrm{tr}[V(L)]-\sum_{j=1}^{N}\mathrm{e}^{-\ell/2}\cosh(q_{j})\Big{)}\mathrm{d}^{N}p\,\mathrm{d}^{N}q$
(11.52)

is given by

$Z_{\mathrm{ca},N}(\ell,V)^{-1}\exp\Big{[}-\sum_{j=1}^{N}V(\lambda_{j})+\sum_{j=1}^{N}\log\Big{(}2K_{0}\big{(}2\exp\big{[}-\tfrac{1}{2}N\big{(}\nu-\frac{2}{N}\sum_{i=1,i\neq
j}^{N}\log Y_{j}\big{)}\big{]}\big{)}\Big{)}\Big{]}$ (11.53)

relative to $(1/N!)\mathrm{d}^{N}\lambda$. Note that

$-\frac{2}{N}\log Y_{j}=\frac{1}{N}\sum_{i=1,i\neq
j}^{N}\phi_{\mathrm{ca}}(\lambda_{j}-\lambda_{i})$ (11.54)

with

$\phi_{\mathrm{ca}}(w)=-\log\Big{(}1+\frac{1}{w^{2}}\Big{)}.$ (11.55)

As discussed in the inset, $\phi_{\mathrm{ca}}$ is indeed the scattering shift
of two Calogero particles.

We arrived at an expression very similar to the Toda lattice, except for a
distinct $Y_{j}$. Following the previous steps with

$\Upsilon_{j}=\nu+\frac{1}{N}\sum_{i=1,i\neq
j}^{N}\phi_{\mathrm{ca}}(\lambda_{j}-\lambda_{i}),$ (11.56)

the free energy functional becomes

$\mathcal{F}_{\mathrm{ca}}^{\circ}(\varrho)=\nu^{-1}\int_{\mathbb{R}}\mathrm{d}w\varrho(w)\Big{(}V(w)-1+\log\varrho(w)-\log\big{(}\nu+\int_{\mathbb{R}}\mathrm{d}w^{\prime}\varrho(w^{\prime})\phi_{\mathrm{ca}}(w-w^{\prime})\big{)}\Big{)}$
(11.57)

in case

$\nu+\int_{\mathbb{R}}\mathrm{d}w^{\prime}\varrho(w^{\prime})\phi_{\mathrm{ca}}(w-w^{\prime})>0$
(11.58)

and $\mathcal{F}_{\mathrm{ca}}^{\circ}(\varrho)=\infty$ otherwise. This
infinite-dimensional constraint can be ignored, just as for the Toda fluid.
The variation is over all $\varrho$ such that $\varrho>0$ and
$\langle\varrho\rangle=1$. Substituting $\nu^{-1}\varrho=\rho$, the free
energy per unit length of the Calogero fluid is determined by the free energy
functional

$\mathcal{F}_{\mathrm{ca}}(\rho)=\int_{\mathbb{R}}\mathrm{d}w\rho(w)\Big{(}V(w)-1+\log\rho(w)-\log\big{(}1+\int_{\mathbb{R}}\mathrm{d}w^{\prime}\rho(w^{\prime})\phi_{\mathrm{ca}}(w-w^{\prime})\big{)}\Big{)}.$
(11.59)

Supporting the generic structure of GHD, $\mathcal{F}_{\mathrm{ca}}$ agrees
with (9.42) upon substituting the scattering shift $\phi_{\mathrm{to}}$ by
$\phi_{\mathrm{ca}}$. Therefore the steps from (9.43) to (9.50) can be
repeated, in particularly leading to the TBA equation for the Calogero fluid.

$\blackdiamond\blackdiamond$ Two-particle Calogero scattering shift. For two
particles the equations of motion are

$\ddot{q}_{1}(t)=\tfrac{1}{4}\cosh(\tfrac{1}{2}(q_{1}-q_{2}))\sinh^{-3}(\tfrac{1}{2}(q_{1}-q_{2})),\quad\ddot{q}_{2}(t)=-\tfrac{1}{4}\cosh(\tfrac{1}{2}(q_{1}-q_{2}))\sinh^{-3}(\tfrac{1}{2}(q_{1}-q_{2})),$
(11.60)

which have to be solved with the asymptotic condition $p_{1}(-\infty)=p_{1}$,
$p_{2}(-\infty)=p_{2}$, and $p_{2}<p_{1}$. Hence $q_{1}(t)<q_{2}(t)$. The
solution can be worked out explicitly with the result

$\displaystyle q_{1}(t)=\tfrac{1}{2}(p_{1}+p_{2})t-\tfrac{1}{2}q(t),\qquad
q_{2}(t)=\tfrac{1}{2}(p_{1}+p_{2})t+\tfrac{1}{2}q(t),$ $\displaystyle
q(t)=2\log\Big{[}b\cosh(\gamma t)+\big{(}b^{2}\cosh^{2}(\gamma
t)-1\big{)}^{1/2}\Big{]},\quad b^{2}=1+(2\gamma)^{-2}$ (11.61)

with $\gamma=\tfrac{1}{2}(p_{1}-p_{2})>0$. The large time asymptotics is given
by

$q_{1}(t)=\begin{cases}p_{1}t-\log b,\\\\[4.30554pt] p_{2}t-\log
b,\end{cases}\quad q_{2}(t)=\begin{cases}p_{2}t+\log b,&\qquad
t\to-\infty,\\\\[4.30554pt] p_{1}t+\log b,&\qquad t\to\infty.\end{cases}$
(11.62)

Since quasiparticle 1 is shifted by $-2\log b$ and quasiparticle 2 by $2\log
b$, we conclude that the two-particle scattering shift is given by

$\phi_{\mathrm{ca}}(p_{1}-p_{2})=-\log\Big{(}1+\frac{1}{(p_{1}-p_{2})^{2}}\Big{)},$
(11.63)

compare with the discussion in Section 2.3. For small momentum transfer, one
recovers the Toda two-particle scattering shift $2\log|p_{1}-p_{2}|$. Since
particles do not cross each other, $\phi_{\mathrm{ca}}<0$, consistent with the
sign for hard rods. $\blackdiamond\blackdiamond$

###  11.4 Hydrodynamic equations

This discussion can be very brief. The density current is momentum, which
itself is conserved. Therefore Eq. (9.20) is still valid. In the limit
$\mu\to-\infty$ the density, and hence also the GGE averaged currents, vanish.
Eq. (9.21) still holds and as a consequence

$v^{\mathrm{eff}}=\frac{\varsigma_{1}^{\mathrm{dr}}}{\varsigma_{0}^{\mathrm{dr}}}\,.$
(11.64)

Of course, the dressing transformation is now defined through the scattering
shift (11.63). As for the Toda fluid, in terms of the properly normalized
density of states the hydrodynamic equations read

$\partial_{t}\rho_{\mathsf{p}}(x,t;v)+\partial_{x}\big{(}v^{\mathrm{eff}}(x,t;v)\rho_{\mathsf{p}}(x,t;v)\big{)}=0$
(11.65)

in correspondence with Eq. (9.25).

###  11.5 Classical Bethe equations

Bethe equations determine the eigenvalues of a quantum hamiltonian whose
eigenfunctions are obtained from employing the Bethe ansatz. In Chapter 13 we
will discuss such method using the Lieb-Liniger $\delta$-Bose gas as prime
example. A similar structure appears for other integrable quantum many-body
systems. An even wider class of integrable quantum models can be handled by
the asymptotic Bethe ansatz, meaning that the Bethe ansatz holds only for
large separation of particles. An example is the quantum Toda lattice covered
in Chapter 14. In this section we argue that Bethe equations can be written
also for the Calogero fluid. Of course, this is not a microscopic foundation.
Of interest is merely the structural similarity. From the Bethe equations one
deduces again the validity of TBA and thereby the DOS previously recorded.

Formally one introduces the classical “phase shift” of the Calogero fluid as

$\theta_{\mathrm{ca}}(w)=-w\log(1+w^{-2})-2\arctan
w,\qquad\theta_{\mathrm{ca}}^{\prime}=\phi_{\mathrm{ca}},\quad\theta_{\mathrm{ca}}(0)=0.$
(11.66)

The Bethe equations are defined through

$u_{j}=\nu
N\lambda_{j}+\sum_{i=1}^{N}\theta_{\mathrm{ca}}(\lambda_{j}-\lambda_{i})$
(11.67)

for $j=1,\ldots,N$. The input is $u\in\mathbb{W}_{N}$ and
$(\lambda_{1},\ldots,\lambda_{N})$ is the output. The input is weighted
uniformly by the Lebesgue measure $\mathrm{d}^{N}u$ and the induced output is
assumed to have the probability

$Z^{-1}\exp\Big{(}-\sum_{j=1}^{N}V(\lambda_{j})\Big{)}\mathrm{d}^{N}u.$
(11.68)

To define a probability measure, this expression has to be restricted to the
domain for which the Jacobian matrix $\\{\partial
u_{j}/\partial\lambda_{i}\\}$, $i,j=1,\ldots,N$, is strictly positive. To find
out we differentiate (11.67) with the result

$\frac{\partial u_{j}}{\partial\lambda_{i}}=\delta_{ij}\Big{(}\nu
N+\sum_{i=1}^{N}\phi_{\mathrm{ca}}(\lambda_{j}-\lambda_{i})\Big{)}-(1-\delta_{ij})\phi_{\mathrm{ca}}(\lambda_{j}-\lambda_{i})=N\big{(}\Lambda^{\circ}_{i,j}+N^{-1}\Lambda^{\bot}_{i,j}\big{)},$
(11.69)

where $\Lambda^{\circ}$ is the diagonal part and $\Lambda^{\bot}$ the off-
diagonal one. Then (11.68) transforms to

$\frac{1}{Z}\exp\Big{(}-\sum_{j=1}^{N}V(\lambda_{j})\Big{)}\exp\big{(}N\log
N+\mathrm{tr}\big{[}\log(\Lambda^{\circ}+N^{-1}\Lambda^{\bot})\big{]}\big{)}\mathrm{d}^{N}\lambda.$
(11.70)

For the exponent we write

$\mathrm{tr}\big{[}\log\big{(}\Lambda^{\circ}\big{(}1+N^{-1}(\Lambda^{\circ})^{-1}\Lambda^{\bot}\big{)}\big{)}\big{]}=\mathrm{tr}\big{[}\log\Lambda^{\circ}\big{]}+\mathrm{tr}\big{[}\log\big{(}1+N^{-1}(\Lambda^{\circ})^{-1}\Lambda^{\bot}\big{)}\big{]}.$
(11.71)

Expanding the latter logarithm yields

$\frac{1}{N}\mathrm{tr}\big{[}(\Lambda^{\circ})^{-1}\Lambda^{\bot}\big{]}-\frac{1}{2N^{2}}\mathrm{tr}\big{[}\big{(}(\Lambda^{\circ})^{-1}\Lambda^{\bot}\big{)}^{2}\big{]}.$
(11.72)

Since $(\Lambda^{\circ})^{-1}$ is diagonal, the first summand vanishes, while
the second one is of order 1, since there are $N^{2}$ summands each one of
order 1. The leading term is of order $N$ relative to which the remainder is
small. Within this approximation one obtains

$\frac{1}{Z}\exp\Big{(}-\sum_{j=1}^{N}V(\lambda_{j})+\sum_{j=1}^{N}\log\Big{(}\nu+N^{-1}\sum_{i=1}^{N}\phi_{\mathrm{ca}}(\lambda_{j}-\lambda_{i})\Big{)}\Big{)}\mathrm{d}^{N}\lambda,$
(11.73)

an expression which we have encountered already. For large $N$ the positivity
of the Jacobian is ensured by

$\nu+N^{-1}\sum_{i=1}^{N}\phi_{\mathrm{ca}}(\lambda_{j}-\lambda_{i})>0$
(11.74)

and agreement with (11.57) is accomplished. To fill in the remaining steps
requires more efforts. One hurdle is the positivity of the Jacobian at fixed
$N$, which is connected with having a unique solution to the classical Bethe
equations (11.68).

For quantum systems solvable by Bethe ansatz, upon substituting the
appropriate phase shift, Eq. (11.67) stays intact. Only the input Lebesgue
measure is replaced by the counting measure over integers
$I_{1}<\ldots<I_{N}$, which result from quantizing the momenta of the model
with zero interaction.

###  11.6 Trigonometric Calogero-Moser model

For a high density of particles the potential
$V_{\mathrm{ca}}(x)=(2\sinh(x/2))^{-2}$ can be approximated by
$V_{\mathrm{cm}}(x)=1/x^{2}$, which leads to a fluid called rational Calogero-
Moser model. Its hamiltonian reads

$H_{\mathrm{cm},N}=\sum_{j=1}^{N}\tfrac{1}{2}p_{j}^{2}+\tfrac{1}{2}\sum_{i,j=1,i\neq
j}^{N}\frac{1}{(q_{i}-q_{j})^{2}}.$ (11.75)

This model is still integrable with Lax pair

$L_{i,j}=\delta_{ij}p_{j}+\mathrm{i}(1-\delta_{ij})\frac{1}{q_{i}-q_{j}},\quad
M_{i,j}=\mathrm{i}\delta_{ij}\sum_{m=1,m\neq
j}^{N}\frac{1}{(q_{j}-q_{m})^{2}}-\mathrm{i}(1-\delta_{ij})\frac{1}{(q_{i}-q_{j})^{2}}.$
(11.76)

For systems in one dimension, equilibrium correlations generically have the
same decay as the potential. Thus the much emphasized locality of conservation
laws can no longer be maintained. Nevertheless, as to be discussed, the
Calogero-Moser model has a structure rather similar to the Toda and Calogero
fluid. So to speak, integrability dominates long range interactions.

To start with a low key, we study the two-particle scattering. The relative
motion $q(t)=q_{1}(t)-q_{2}(t)$ is governed by

$\frac{d^{2}}{dt^{2}}q(t)=4q(t)^{-3}.$ (11.77)

For $v>0$ we impose $\dot{q}(0)=0$ and $q(0)=2/v$, which matches with
$\dot{q}(\pm\infty)=\pm v$. Then the solution consists of two branches, one
for $t>0$ and one for $t<0$, with

$q(t)=\pm\sqrt{(vt)^{2}+4v^{-2}}\simeq\pm vt\big{(}1+2v^{-4}t^{-2}\big{)}$
(11.78)

for large $|t|$. Since the first order deviation from the linear motion decays
as $|t|^{-1}$, the scattering shift of the rational model vanishes,

$\phi_{\mathrm{cm}}(p_{1}-p_{2})=0.$ (11.79)

If GHD is still applicable, then the hyperbolic conservation laws governing
the motion of the conserved fields would decouple and the Euler type dynamics
cannot be distinguished from the one of an ideal gas. Of course, in a local
spacetime patch the conserved fields are highly correlated and very far from a
Poisson distribution.

To check whether such predictions are valid, one has to compute the
generalized free energy. Because of long range interactions, the notion of
finite volume has some level of arbitrariness. One physically natural
procedure is to start from an infinitely extended periodic configuration of
spatial period $\ell$, $\ell>0$, and study the limit when $\ell\to\infty$.
Since particles cannot cross, the order $q_{j}<q_{j+1}$ is maintained. The
periodicity condition thus reads

$q_{j+N}=q_{j}+\ell,\quad p_{j+N}=p_{j}$ (11.80)

for all $j\in\mathbb{Z}$. The interaction potential becomes then the
periodized rational potential

$\sum_{m\in\mathbb{Z}}\frac{1}{(x-\ell
m)^{2}}=\frac{\pi^{2}}{\ell^{2}\sin^{2}(\pi x/\ell)}=V_{\mathrm{tcm}}(x).$
(11.81)

Particles with interaction potential $V_{\mathrm{tcm}}$ are called
trigonometric Calogero-Moser model. Each cell $[m\ell,(m+1)\ell]$ contains $N$
particles. A single cell can be viewed as a ring of length $\ell$,
$\\{z\in\mathbb{C}\big{|}|z|=\ell/2\pi\\}$. If $q_{i},q_{j}$ are the locations
of particles on the ring, then their interaction potential is
$1/d(q_{i},q_{j})^{2}$, where $d(q_{i},q_{j})$ denotes their cord distance.

The phase space of this model can be taken as
$\Gamma_{{\\!N,\ell}}^{\triangleright}=\mathbb{W}_{N,\ell}\times\mathbb{R}^{N}$
with the Weyl chamber $\mathbb{W}_{N,\ell}=\\{0\leq
q_{1}<\ldots<q_{N}<\ell\\}$. The trigonometric Calogero-Moser model is still
integrable with Lax matrix

$L_{i,j}=p_{j}\delta_{ij}+\mathrm{i}(1-\delta_{ij})\frac{\pi}{\ell\sin(\pi(q_{i}-q_{j})/\ell)},$
(11.82)

which depends on $\ell$, $L=L(\ell)$. Through the canonical transformation
$\check{p}_{j}=\ell p_{j}$, $\check{q}_{j}=q_{j}/\ell$ one obtains the scaling
with $\ell$ as

$\ell L(\ell)=L(1).$ (11.83)

As a side remark, for the Calogero fluid the interaction is
$V_{\mathrm{ca}}(x)=g^{2}(\mu/2)^{2}(\sinh(x/2\mu))^{-2}$. Thus the periodized
potential (11.81) corresponds to $g=1$ and $\mu=\mathrm{i}2\pi/\ell$, which is
merely the analytic continuation of the potential $V_{\mathrm{ca}}$ to purely
imaginary arguments.

Since particles move on the circle, rather than scattering coordinates, the
conventional action-angle variables come into use. Phase space is foliated
into invariant tori and the trajectories are quasi-periodic in time. The
action-angle variables are constructed by an algebra very similar to the one
in Section 11.2. Under the action-angle map $\Phi$, the angle variables vary
as $\phi_{j}\in[0,\ell]$. In comparison to the hyperbolic model, the
transformation to the eigenvalues is much more explicit, in the sense that the
eigenvalues are merely constrained as

$\lambda_{j+1}-\lambda_{j}\geq 2\pi/\ell,$ (11.84)

which defines the set $\Omega_{N,\ell}$. The Boltzmann weight then transforms
as

$\mathrm{e}^{-\mathrm{tr}[V(L(\ell))]}d^{N}qd^{N}p=\exp\Big{(}-\sum_{j=1}^{N}V(\lambda_{j})\Big{)}\chi(\Omega_{N,\ell})d^{N}\lambda\mathrm{d}^{N}\phi,$
(11.85)

where $\chi(\Omega_{N,\ell})$ is the characteristic function of the set
$\Omega_{N,\ell}$.

We first consider the partition function, in fact its version with the
additional constraint $\lambda_{N}\leq w$. The set of admissible eigenvalues
is then denoted by

$\Omega_{N,\ell}(w)=\\{\lambda\in\mathbb{R}^{N}|\lambda_{j+1}-\lambda_{j}\geq(2\pi/\ell),j=1,\ldots,N-1,\lambda_{N}\leq
w\\}$ (11.86)

with the convention $\Omega_{N,\ell}=\Omega_{N,\ell}(\infty)$. The constrained
partition function is given by

$Z_{N,\ell}(V,w)=\int_{\mathbb{W}_{N,\ell}}\mathrm{d}^{N}q\int_{\mathbb{R}^{N}}\mathrm{d}^{N}p\exp\big{(}-\mathrm{tr}(V(L))\big{)}=\ell^{N}\int_{\Omega_{N,\ell}(w)}\mathrm{d}^{N}\lambda\prod_{j=1}^{N}\mathrm{e}^{-V(\lambda_{j})},$
(11.87)

including the cases $Z_{0,\ell}(V,w)=1$ and
$Z_{1,\ell}(V,w)=\ell\int_{-\infty}^{w}\mathrm{d}\lambda_{1}\exp(-V(\lambda_{1}))$.
The factor $\ell^{N}$ results from the $\phi$-integrations. By introducing the
chemical potential $\mu$, dual to the particle number, we switch to the grand
canonical ensemble as

$Z_{\ell}(V,w)=\sum_{N=0}^{\infty}\mathrm{e}^{\mu N}Z_{N,\ell}(V,w)$ (11.88)

and note that

$Z_{\ell}^{\prime}(V,w)=\ell\mathrm{e}^{-V(w)+\mu}Z_{\ell}(V,w-(2\pi/\ell))$
(11.89)

with $Z_{\ell}^{\prime}=(\mathrm{d}/\mathrm{d}w)Z_{\ell}$. The constrained
free energy per unit length is

$F_{\ell}(V,w)=\frac{1}{\ell}\log
Z_{\ell}(V,w),\qquad\lim_{\ell\to\infty}\frac{1}{\ell}\log
Z_{\ell}(V,w)=F(V,w).$ (11.90)

Commonly the free energy would be $-F(V,w)$. With our definition,
$F^{\prime}(V,w)>0$ that will be more convenient. Inserting in (11.89), one
concludes that in the infinite volume limit

$F^{\prime}(V,w)=\mathrm{e}^{-V(w)+\mu}\mathrm{e}^{-2\pi F^{\prime}(V,w)}.$
(11.91)

The connection with the Toda fluid can now established by introducing

$F^{\prime}(V)=\mathrm{e}^{-\varepsilon}=\rho_{\mathsf{n}}.$ (11.92)

Then (11.90) turns into

$\varepsilon=V-\mu+2\pi\mathrm{e}^{-\varepsilon},$ (11.93)

which one recognizes as TBA equation. The $T$ operator has the kernel

$T(w,w^{\prime})=-2\pi\delta(w-w^{\prime}),$ (11.94)

in accordance with zero scattering shift. Observe that for hard rods the
corresponding operator $T$ is the projection onto the constant function, while
here $-T/2\pi$ is the identity matrix. The $\ell\to\infty$ limit of the free
energy per unit volume is given by

$F(V,\infty)=\int_{\mathbb{R}}\mathrm{d}wF^{\prime}(V,w).$ (11.95)

To complete the analogy, we have to still determine the density of states of
the Lax matrix per unit volume, which amounts to the average

$\ell^{-1}\langle\sum_{j=1}^{N}f(\lambda_{j})\rangle_{V,\ell},$ (11.96)

average with respect to the grand canonical density as in the partition
function (11.87) at $w=\infty$. This average can be accomplished by perturbing
the confining potential as $V+\delta f$. Then to first order in $\delta$

$-\int_{\mathbb{R}}\mathrm{d}wf(w)\partial_{\delta}F^{\prime}(V,w)|_{\delta=0}=\int_{\mathbb{R}}\mathrm{d}wf(w)\rho_{\mathsf{p}}(w).$
(11.97)

Using (11.91) to work out the derivative with respect to $\delta$ at
$\delta=0$ yields

$\rho_{\mathsf{p}}(w)=\frac{F^{\prime}(V,w)}{1+2\pi
F^{\prime}(V,w)},\qquad\rho_{\mathsf{s}}(w)=\frac{1}{1+2\pi F^{\prime}(V,w)}.$
(11.98)

Then indeed $\rho_{\mathsf{p}}=\rho_{\mathsf{n}}\rho_{\mathsf{s}}$ and
$\rho_{\mathsf{s}}=1+T\rho_{\mathsf{p}}$. As a final step, the effective
velocity can be determined through (6.20) with the result

$v_{\mathrm{eff}}(w)=w,$ (11.99)

implying the completely decoupled set of hydrodynamic equation

$\partial_{t}\rho_{\mathsf{p}}(x,t;v)+v\partial_{x}\rho_{\mathsf{p}}(x,t;v)=0.$
(11.100)

Notes and references

Section 11.0

The famous hierarchical model was introduced by Dyson [100] to study
ferromagnetic Ising models in one dimension with long range interactions.
Johansson [166] established the liquid-gas transition for a one-dimensional
fluid with a potential decaying as $-r^{-\alpha}$, $1\leq\alpha<2$. The
functional relations for $\alpha,\beta,\gamma,V_{\mathrm{mec}}$ are studied in
Calogero [46], Olshanetsky and Perelomov [224]. Interestingly enough, the
construction of the Calogero soliton uses the Lax matrix $L^{[3]}$ Gon and
Kulkarni [135].

Section 11.1

Calogero [44] and Sutherland [274] independently discovered the integrability
of the quantum many-body system with $1/x^{2}$ interaction potential. A few
years later the classical version was investigated by Calogero [45] as part of
a wider class of classical systems having a Lax matrix pair. This class also
includes the two models discussed in this chapter. Independently Moser [215]
discovered the Lax matrix for the rational and trigonometric model Moser. In
the rational case he proves that the $N$-particle system has zero scattering
shift. For the trigonometric model he found concise expressions for the time-
dependent solution. In honor of these early contributions, the names Calogero-
Moser-Sutherland are attached to the models already listed and their vast
generalizations discovered in the mean time, as documented in the book edited
by van Diejen and Vinet [79]. We cover only two specific cases, namely the
Calogero fluid, interaction potential $1/\sinh^{2}$, and the Calogero-Moser
model, interaction potential $1/x^{2}$ and its periodized version
$1/\sin^{2}$. The 2001 monograph by Calogero [46] is a rich source on
integrable classical particle models. The hyperbolic model is discussed in
Calogero [46], Section 2.1.5. The connection to Lie algebras is reviewed by
Olshanetsky and Perelomov [224]. When adding a fine-tuned external potential
integrability is preserved, see the contributions by Inozemtsev [164],
Polychronakos [235], and Kulkarni and Polychronakos [181]. The strength can be
arbitrary, but the decay has to match the one of the $\sinh^{-2}$ potential.
The term $-\tfrac{1}{12}$ appearing in (11.22) is discussed by Ruijsenaars
[250].

Section 11.2

The scattering coordinates for the hyperbolic Calogero model have been
constructed by Ruijsenaars [247]. Our exposition follows the very readable
contribution of Bogomolny et al. [24], who provide the required algebra. The
definition of the scattering shifts through (11.41) is based on analogy with
earlier contributions by Airault et al. [9] and Adler [8]. The difficult part
is to establish agreement with the dynamical approach in (2.43). Only then the
required properties of $\Phi$ can be proved, as accomplished by Ruijsenaars
[247]. Ruijsenaars[250] provides an instructive summary of his own
contributions.

Section 11.3

While there is some discussion in the literature, for example Polychronakos
[236], the variational principle for the free energy is a recent result. The
two-particle scattering shift has been computed by Calogero [46], Section
2.1.5. The Calogero fluid with external potential $g_{1}\cosh x+g_{2}\cosh
2x$, compare with (11.46), is still integrable, see van Diejen [78] for a
proof. But the respective Lax matrix is less accessible.

Section 11.5

Classical Bethe equations seem to be novel. To study such a problem, a
convenient starting point would be to assume a toy model defined by
$\phi_{\mathrm{ca}}>0$. Then the Jacobian matrix is positive and Bethe
equations have a unique solution.

Section 11.6

The construction of action-angle variables can be found in the article
Ruijsenaars [249] under model $III_{nr}$. Since also other models are
discussed in parallel, the results of interest in the context of our
discussions are scattered throughout the article. A concise account of the
algebraic steps are provided by Bogomolny et al. [24]. Our discussion is based
on the clarifying contribution by Choquard [57], who covers also the quantized
model. In the recent contribution by Bulchandani et al. [41] both the quantum
and classical rational Calogero-Moser model is studied, emphasizing the aspect
of quasiparticles. Presented are plots of the DOS and also time-dependent
solutions.

##  12 Discretized nonlinear Schrödinger equation

###  12.1 Continuum wave equations

A famous integrable classical field theory in $1+1$ dimensions is the
nonlinear Schrödinger equation (NLS). In the defocusing case the wave field,
$\psi(x,t)\in\mathbb{C}$, is governed by

$\mathrm{i}\partial_{t}\psi=-\partial_{x}^{2}\psi+2|\psi|^{2}\psi.$ (12.1)

Our interest will be random initial conditions linked to the hydrodynamic
scale. For NLS the densities of the locally conserved fields are known. The
starting entries of the list read

$Q^{[0]}(x)=|\psi(x)|^{2},\qquad
Q^{[1]}(x)=-\mathrm{i}\bar{\psi}(x)\partial_{x}\psi(x),\qquad
Q^{[2]}(x)=|\partial_{x}\psi(x)|^{2}+|\psi(x)|^{4}.$ (12.2)

For a given bounded interval $\Lambda\subset\mathbb{R}$, with periodic
boundary conditions, the total conserved quantities then become

$Q_{\Lambda}^{[n]}=\int_{\Lambda}\mathrm{d}xQ^{[n]}(x),$ (12.3)

$n=0,1,\ldots.$ . Formally, the time-stationary generalized Gibbs ensembles
are of the form

$\exp\Big{[}-\sum_{n=0}^{\infty}\mu_{n}Q_{\Lambda}^{[n]}\Big{]}\prod_{x\in\Lambda}\mathrm{d}^{2}\psi(x),$
(12.4)

where the $\mu_{n}$’s are suitable chemical potentials. For the thermal case,
$n=0,1,2,$ much work has been invested to construct a proper probability
measure. The basic idea of the construction is easily explained. Obviously,
one can lattice discretize the volume $\Lambda$. Then the product Lebesgue
measure makes sense. However, the limit of zero lattice spacing is ill-defined
within standard measure theory. On the other hand the weighted measure

$(Z_{\mathrm{nls},N})^{-1}\prod_{j=1}^{N}\mathrm{d}x_{j}\exp\Big{(}-\mu_{0}\sum_{j=1}^{N}|\psi(j)|^{2}-\mu_{2}\sum_{j=1}^{N}|\psi(j+1)-\psi(j)|^{2}\Big{)},$
(12.5)

with boundary condition $\psi(N+1)=\psi(1)$, has a well defined continuum
limit, which is Gaussian and well-known as stationary $\mathbb{R}^{2}$-valued
Ornstein-Uhlenbeck process. The remaining GGE piece reads

$\exp\Big{(}-\mu_{2}\int_{\Lambda}\mathrm{d}x|\psi(x)|^{4}\Big{)}.$ (12.6)

If $\mu_{2}>0$, it is integrable with respect to the Ornstein-Uhlenbeck
process. As a separate issue, it has to be shown that the so constructed
measure is invariant under the NLS dynamics. Recently such method has been
extended to a much larger class of generalized Gibbs measures in case the sum
in (12.4) is restricted to some highest even $n$. To provide a very rough
idea, the Gaussian a priori measure becomes now a generalized Ornstein-
Uhlenbeck process with action

$\int_{\Lambda}\mathrm{d}x\big{(}|\partial_{x}^{n/2}\psi(x)|^{2}+|\psi{(x)}|^{2}\big{)}.$
(12.7)

As for $n=2$, the technical part is to establish that, for an appropriate
choice of chemical potentials, the exponential of all remaining terms can be
integrated with respect to this Gaussian measure. In essence, the thereby
constructed measure is concentrated on $(n-2)$ times differentiable paths.
Time-stationarity is established separately. To go beyond such an existence
result seems to be a challenging problem.

Given the difficulties in even obtaining the basic building blocks, a
different strategy might be pursued. In numerical simulations of NLS one
discretizes the equation. While generically this would break integrability,
surprisingly enough, in many cases there is one very specific discretization
for which integrability is maintained. For NLS such a discretization was
discovered in 1975 by Ablowitz and Ladik and is usually referred to as IDNLS,
i.e. integrable discrete NLS. For convenience we will use here AL as acronym.

Another example in the same spirit is the classical sinh-Gordon equation,

$\partial_{t}^{2}\phi-\partial_{x}^{2}\phi+\sinh\phi=0,$ (12.8)

with $\phi(x,t)$ a real-valued wave-field. This dynamics is integrable, while
its naive discretization is not. But there is a more subtle lattice version
which is still integrable. Its construction is based on an analytic
continuation of the discretized sine-Gordon equation.

A further much studied classical system is the Landau-Lifshitz model of a one-
dimensional magnet. Here the spin field is a three-vector, $\vec{S}(x,t)$,
with $|\vec{S}|=1$. In the simplest case of an isotropic interaction, no
external magnetic field, the continuum equations of motion are

$\partial_{t}\vec{S}=\vec{S}\wedge\partial_{x}^{2}\vec{S},$ (12.9)

which is integrable. In the naive discretization the hamiltonian would be

$H_{\mathrm{ll}}=\sum_{j\in\mathbb{Z}}\vec{S}_{j}\cdot\vec{S}_{j+1},$ (12.10)

which is not integrable, while the integrable discrete model turns out to be
governed by the hamiltonian

$\tilde{H}_{\mathrm{ll}}=\sum_{j\in\mathbb{Z}}\log(1+\vec{S}_{j}\cdot\vec{S}_{j+1}).$
(12.11)

The nonlinear Schrödinger equation and its Ablowitz-Ladik discretization are
solvable through IST, compare with Section 10.1. IST is a powerful and
systematic method to obtain $n$-soliton solutions. Generically the one-soliton
solution is still explicit, while the asymptotics of the two-soliton solution
yields the scattering shift. Thus the basic input for a soliton-based
hydrodynamic scale is available. For NLS the spectral parameter turns out to
be complex, which foreshadows the possibility of spectral parameters different
from $\mathbb{R}$. Here we will pursue the in spirit particle-based scheme for
the Ablowitz-Ladik lattice.

###  12.2 Ablowitz-Ladik discretization

Upon discretization, the wave field is over the one-dimensional lattice
$\mathbb{Z}$, $\psi_{j}(t)\in\mathbb{C}$, and is governed by

$\mathrm{i}\frac{d}{dt}\psi_{j}=-\psi_{j-1}+2\psi_{j}-\psi_{j+1}+|\psi_{j}|^{2}(\psi_{j-1}+\psi_{j+1}),$
(12.12)

hence

$\mathrm{i}\frac{d}{dt}\psi_{j}=-(1-|\psi_{j}|^{2})(\psi_{j-1}+\psi_{j+1})+2\psi_{j}.$
(12.13)

Setting $\alpha_{j}(t)=\mathrm{e}^{2\mathrm{i}t}\psi_{j}(t)$, one arrives at
the standard version

$\frac{d}{dt}\alpha_{j}=\mathrm{i}\rho_{j}^{2}(\alpha_{j-1}+\alpha_{j+1}),\quad\rho_{j}^{2}=1-|\alpha_{j}|^{2}.$
(12.14)

Clearly the natural phase space is $\alpha_{j}\in\mathbb{D}$ with the unit
disk $\mathbb{D}=\\{z||z|\leq 1\\}$. In principle, whenever $\alpha_{j}(t)$
hits the boundary of $\mathbb{D}$, it freezes and thereby decouples the
system. As we will discuss, a conservation law ensures that, if initially away
from the boundary, the solution will stay so forever.

For determining the generalized free energy, the method of scattering
coordinates does not seem to be available. An alternative set-up would be the
closed Ablowitz-Ladik model, which consists of a finite ring of $N$ sites,
labelled as $j=0,\ldots,N-1$, with periodic boundary conditions,
$\alpha_{j+N}=\alpha_{j}$. While the system with periodic boundary conditions
is integrable, there seems to be no method for obtaining its generalized free
energy in the limit $N\to\infty$. However, there is an analogue of
$T=L^{\diamond}$ from the Toda chain, in the sense that the interactions for
bonds $(-1,0)$ and $(N-2,N-1)$ are modified. In analogy this is called the
open Ablowitz-Ladik lattice. As for the Toda chain, to compute the free energy
one has to impose a linear pressure ramp of slope $1/N$. Closed (periodic) and
open chain will be discussed separately.

Conserved fields. We consider a ring of $N$ sites. The evolution equations are
of hamiltonian form by regarding $\alpha_{j}$ and its complex conjugate
$\bar{\alpha}_{j}$ as canonically conjugate variables and by introducing the
weighted Poisson bracket

$\\{f,g\\}_{\mathrm{AL}}=\mathrm{i}\sum_{j=0}^{N-1}\rho_{j}^{2}\big{(}\partial_{\bar{\alpha}_{j}}f\partial_{\alpha_{j}}g-\partial_{\alpha_{j}}f\partial_{\bar{\alpha}_{j}}g\big{)}.$
(12.15)

The hamiltonian of the Ablowitz-Ladik system then reads

$H_{\mathrm{al},N}=-\sum_{j=0}^{N-1}\big{(}\alpha_{j-1}\bar{\alpha}_{j}+\bar{\alpha}_{j-1}\alpha_{j}\big{)}.$
(12.16)

One readily checks that indeed

$\frac{d}{dt}\alpha_{j}=\\{\alpha_{j},H_{\mathrm{al},N}\\}_{\mathrm{AL}}=\mathrm{i}\rho_{j}^{2}(\alpha_{j-1}+\alpha_{j+1}).$
(12.17)

The next step is to find out the locally conserved fields. As discovered by
Irena Nenciu, the Ablowitz-Ladik model has also a Lax matrix, this time in the
form of a Cantero-Moral-Velázquez (CMV) matrix. The basic building blocks are
$2\times 2$ matrices, which requires $N$ to be even because of periodic
boundary conditions. One defines

$\Xi_{j}=\begin{pmatrix}\bar{\alpha}_{j}&\rho_{j}\\\ \rho_{j}&-\alpha_{j}\\\
\end{pmatrix}$ (12.18)

and forms the $N\times N$ matrices

$L_{N}=\mathrm{diag}(\Xi_{0},\Xi_{2},\dots,\Xi_{N-2}),\quad
M_{N}=\mathrm{diag}(\Xi_{1},\Xi_{3},\dots,\Xi_{N-3}),$ (12.19)

where $\Xi_{N-1}$ respects the periodic boundary conditions. More pictorially
$L_{N}$ corresponds to the $2\times 2$ blocks $(0,1),\ldots,(N-2,N-1)$, while
$M_{N}$ uses $2\times 2$ blocks shifted by one, $(1,2),\ldots,(N-1,0)$. The
CMV matrix associated to the coefficients $\alpha_{0},\dots,\alpha_{N-1}$ is
then given by

$C_{N}=L_{N}M_{N}.$ (12.20)

Obviously, $L_{N},M_{N}$ are unitary and so is $C_{N}$. The eigenvalues of
$C_{N}$ are denoted by $\mathrm{e}^{\mathrm{i}\vartheta_{j}}$,
$\vartheta_{j}\in[0,2\pi]$, $j=1,\dots,N$. Of course, the eigenvalues depend
on $N$, which is suppressed in our notation.

Next we define for a general matrix, $A$, the $+$ operation as

$(A_{+})_{i,j}=\begin{cases}A_{i,j}&\mathrm{if}\,\,i<j,\\\
\tfrac{1}{2}A_{j,j}&\mathrm{if}\,\,i=j,\\\ 0&\mathrm{if}\,\,i>j.\end{cases}$
(12.21)

Then one version of the Lax pair reads

$\\{C_{N},\mathrm{tr}(C_{N})\\}_{\mathrm{AL}}=\mathrm{i}[C_{N},(C_{N})_{+}],\qquad\\{C_{N},\mathrm{tr}(C_{N}^{*})\\}_{\mathrm{AL}}=\mathrm{i}[C_{N},(C_{N+})^{*}].$
(12.22)

Since the Poisson bracket acts as a derivative, one deduces

$\\{(C_{N})^{n},\mathrm{tr}(C_{N})\\}_{\mathrm{AL}}=\sum_{m=0}^{n-1}(C_{N})^{m}\mathrm{i}[C_{N},C_{N+}](C_{N})^{n-1-m}=\mathrm{i}[(C_{N})^{n},C_{N+}],$
(12.23)

and similarly

$\\{(C_{N})^{n},\mathrm{tr}(C_{N}^{*})\\}_{\mathrm{AL}}=\mathrm{i}[(C_{N})^{n},(C_{N+})^{*}].$
(12.24)

Therefore locally conserved fields are given by

$Q^{[n],N}=\mathrm{tr}\big{[}(C_{N})^{n}\big{]}.$ (12.25)

Their mutual Poisson brackets vanish,

$\\{Q^{[n],N},Q^{[m],N}\\}_{\mathrm{AL}}=0\vskip 12.0pt plus 4.0pt minus
4.0pt.$ (12.26)

The fields can be turned real-valued by considering real and imaginary parts,

$\displaystyle
Q^{[n,+],N}=\tfrac{1}{2}\mathrm{tr}\big{[}(C_{N})^{n}+(C_{N}^{*})^{n}\big{]}=\mathrm{tr}\big{[}\cos((C_{N})^{n})\big{]},$
$\displaystyle
Q^{[n,-],N}=-\tfrac{1}{2}\mathrm{i}\,\mathrm{tr}\big{[}(C_{N})^{n}-(C_{N}^{*})^{n}\big{]}=\mathrm{tr}\big{[}\sin((C_{N})^{n})\big{]},$
(12.27)

with $n=1,\dots,N/2$. In particular
$H_{\mathrm{al},N}=2Q^{[1,+]}=\mathrm{tr}[C_{N}+C_{N}^{*}]$. These fields have
a density, respectively given by

$Q_{j}^{[n],N}=Q_{j}^{[n,+],N}+\mathrm{i}Q_{j}^{[n,-],N}=((C_{N})^{n})_{j,j}.$
(12.28)

Although the matrices $L_{N},M_{N}$ have a basic $2\times 2$ structure, the
densities of the conserved fields are shift covariant by $1$ in the sense

$Q^{[n,\sigma],N}_{j+1}(\alpha)=Q^{[n,\sigma],N}_{j}(\tau\alpha),\qquad\sigma=\pm
1,$ (12.29)

with the shift operator $(\tau\alpha)_{j}=\alpha_{j+1}$ $\mathrm{mod}(N)$. To
confirm, one introduces the unitary shift matrix $S$ through
$(S^{*}AS)_{i,j}=A_{i+1,j+1}$ $\mathrm{mod}(N)$. Then

$S^{*}C_{N}({\alpha,\bar{\alpha}})S=S^{*}L_{N}(\alpha,\bar{\alpha})SS^{*}M_{N}(\alpha,\bar{\alpha})S=M_{N}(\tau\alpha,\tau\bar{\alpha})L_{N}(\tau\alpha,\tau\bar{\alpha})=C_{N}(\tau\alpha,\tau\bar{\alpha})^{\mathrm{T}},$
(12.30)

which implies (12.29).

Figure 12: An admissable walk from $(0,1)$ to $(4,-1)$. According to the rules
its weight is given by $\rho_{0}(-\alpha_{-1})\bar{\alpha}_{0}\rho_{-1}$.

Later on, we will consider the infinite volume limit, $N\to\infty$. This will
always be understood as a two-sided limit. For example the infinite volume
limit of $L_{N}$, denoted by $L$, is
$L=\mathrm{diag}(\dots,\Xi_{-2},\Xi_{0},\Xi_{2},\dots)$ and correspondingly
$M=\mathrm{diag}(\dots,\Xi_{-1},\Xi_{1},\Xi_{3},\dots)$. $L,M$ are unitary
operators on the Hilbert space $\ell_{2}(\mathbb{Z})$, and so is $C=LM$. The
traces in (12.25) have no limit, but densities do. The matrix elements of
$C^{n}$ can be expanded as the sum

$(C^{n})_{i,j}=\sum_{j_{1}\in\mathbb{Z}}\dots\sum_{j_{2n}\in\mathbb{Z}}L_{i,j_{1}}M_{j_{1},j_{2}}\dots
L_{j_{2n-1},j_{2n}}M_{j_{2n},j},$ (12.31)

which consists of a finite number of terms, only. For the infinite system the
index $n$ runs over all positive integers. The infinite volume densities,
$Q^{[n,\sigma]}_{j}$, are strictly local functions of $\alpha$ with support of
at most $2n$ sites. The sum in (12.31) can be viewed as resulting from a
nearest neighbor $2n$ step random walk from left to right. For this purpose
one considers a checkerboard on $[0,2n]\times\mathbb{R}$. The unit square with
corners $(0,0),(1,0),(1,1),(0,1)$ is white. Single steps of the walk are
either horizontal, $j\leadsto j$, or up-down, $j\leadsto j\pm 1$. Such
diagonal steps are permitted only on white squares, compare with Figure 12.
The matrix element $(C^{n})_{i,j}$ is then the sum over all walks with $2n$
steps starting at $i$ and ending at $j$. Each walk represents a particular
polynomial obtained by taking the product of local weights along the walk. The
weights are

$\rho_{j}$ | for the diagonal steps $j\leadsto j+1$ and $j+1\leadsto j$,
---|---
$\bar{\alpha}_{j}$ | for the horizontal step $j\leadsto j$ in case its lower square is black,
$-\alpha_{j-1}$ | for the horizontal step $j\leadsto j$ in case its upper square is black.

As examples, $C_{j,j}=-\alpha_{j-1}\bar{\alpha}_{j}$ and
$(C^{2})_{j,j}=\alpha_{j-1}^{2}\bar{\alpha}_{j}^{2}-\rho_{j-1}^{2}\alpha_{j-2}\bar{\alpha}_{j}-\rho_{j}^{2}\alpha_{j-1}\bar{\alpha}_{j+1}$.
Note that densities are not unique in general, while the total conserved
fields, $Q^{[n],N}$, are unique. To illustrate, in the previous formula, an
equivalent density would be
$\alpha_{j-1}^{2}\bar{\alpha}_{j}^{2}-2\rho_{j}^{2}\alpha_{j-1}\bar{\alpha}_{j+1}$.

The CMV matrix misses one physically very important field namely

$Q^{[0],N}=-\sum_{j=0}^{N-1}\log(\rho_{j}^{2}).$ (12.32)

To simplify notation we set $[0]=[0,\sigma]$ and $0=0\sigma$. The time-
derivative of $Q^{[0],N}$ yields a telescoping sum, which vanishes on a ring.
In lack of a common name we call $Q^{[0],N}$ the log intensity. The log
intensity vanishes for small amplitudes $|\alpha_{j}|^{2}$ and diverges at the
maximal value, $|\alpha_{j}|^{2}=1$. Note also that

$\exp\big{(}-Q^{[0],N}\big{)}=\prod_{j=0}^{N-1}\rho_{j}^{2}$ (12.33)

is conserved. Thus if initially $\exp\big{(}-Q^{[0],N}\big{)}>0$, it stays so
for all times, guaranteeing that the phase space boundary is never reached.

Generalized Gibbs ensemble. The natural a priori measure of the Ablowitz-Ladik
model is the product measure

$\prod_{j=0}^{N-1}\mathrm{d}^{2}\alpha_{j}(\rho_{j}^{2})^{P-1}=\prod_{j=0}^{N-1}\mathrm{d}^{2}\alpha_{j}(\rho_{j}^{2})^{-1}\exp\big{(}-PQ^{[0],N}\big{)}$
(12.34)

on $\mathbb{D}^{N}$. To normalize the measure, $P>0$ is required. The log
intensity is controlled by the parameter $P$ which, in analogy to the Toda
lattice, is called pressure. Small $P$ corresponds to maximal log intensity,
i.e. $|\alpha_{j}|^{2}\to 1$, and large $P$ to low log intensity, i.e.
$|\alpha_{j}|^{2}\to 0$. In the grand-canonical ensemble, the Boltzmann weight
is constructed from a linear combination of the conserved fields, which is
written as

$\sum_{n\in\mathbb{Z}}\mu_{n}\mathrm{tr}\big{[}(C_{N})^{n}\big{]}=\mathrm{tr}\big{[}\mathsfit{V}(C_{N})\big{]},\qquad\mathsfit{V}(z)=\sum_{n\in\mathbb{Z}}\mu_{n}z^{n}.$
(12.35)

The chemical potentials, $\mu_{n}$, are complex and assumed to be independent
of $N$. To have the trace real-valued one imposes $\mu_{n}=\bar{\mu}_{-n}$. In
fact is suffices to assume that $\mathsfit{V}$ is real-valued and continuous
on the circle $|z|=1$. Combining (12.34) and (12.35) yields the GGE as

$Z_{\mathrm{al},N}(P,V)^{-1}\prod_{j=0}^{N-1}\mathrm{d}^{2}\alpha_{j}(\rho_{j}^{2})^{P-1}\exp\big{(}-\mathrm{tr}[\hat{V}(C_{N})]\big{)}.$
(12.36)

$Z_{\mathrm{al},N}(P,V)$ is the normalizing partition function. As label, the
more natural object turns out to be the Fourier transform of the sequence
$\\{\mu_{n},n\in\mathbb{Z}\\}$,

$V(w)=\mathsfit{V}(\mathrm{e}^{\mathrm{i}w}).$ (12.37)

$V$ is a real-valued continuous function on $[0,2\pi]$. The corresponding
object for the Toda lattice lives on $\mathbb{R}$ and was called confining
potential, since it confines the eigenvalues of the Lax matrix. While for the
CMV matrix the eigenvalues are on the unit circle, hence there is nothing to
confine, for convenience we still stick to “confining” so to distinguish from
other potentials. If $\mathsfit{V}$ is given by a finite sum, then the
interaction of the Gibbs measure in (12.36) is of finite range. In this case
the infinite volume limit can be controlled through transfer matrix methods.
In particular, the limit measure is expected to have a finite correlation
length. As before, finite volume expectations with respect to the measure in
(12.36) are denoted by $\langle\cdot\rangle_{P,V,N}$ and their infinite volume
limit by $\langle\cdot\rangle_{P,V}$.

The generalized free energy, $F_{\mathrm{al}}$, is defined through

$\lim_{N\to\infty}-\frac{1}{N}\log
Z_{\mathrm{al},N}(P,V)=F_{\mathrm{al}}(P,V).$ (12.38)

In the hydrodynamic context of particular interest is the empirical density of
states,

$\rho_{\mathrm{Q},N}(w)=\frac{1}{N}\sum_{j=1}^{N}\delta(w-\vartheta_{j}),$
(12.39)

where $\\{\mathrm{e}^{\mathrm{i}\vartheta_{j}}\\}$ are the eigenvalues of
$C_{N}$. $\rho_{\mathrm{Q},N}$ is a probability measure on $[0,2\pi]$ and has
an almost sure limit as

$\lim_{N\to\infty}\rho_{\mathrm{Q},N}(w)=\rho_{\mathrm{Q}}(w).$ (12.40)

To grasp the significance of the DOS, we first introduce the trigonometric
functions $\varsigma_{0}(w)=1$, $\varsigma_{n-}(w)=\sin(nw)$, and
$\varsigma_{n+}(w)=\cos(nw)$, $n=1,2,\dots$ . They span the Hilbert space
$L^{2}([0,2\pi],\mathrm{d}w)$. Then, for the trigonometric moments of
$\rho_{\mathrm{Q},N}(w)$,

$\langle\rho_{\mathrm{Q},N}\varsigma_{n\sigma}\rangle=N^{-1}\langle
Q^{[n,\sigma],N}\rangle_{P,V,N}$ (12.41)

and

$\lim_{N\to\infty}N^{-1}\langle Q^{[n,\sigma],N}\rangle_{P,V,N}=\langle
Q_{0}^{[n,\sigma]}\rangle_{P,V}=\langle\rho_{\mathrm{Q}}\varsigma_{n\sigma}\rangle.$
(12.42)

Consistent with our previous notation, $\langle\cdot\rangle$ is simply a short
hand for the integration over $[0,2\pi]$. The limit value can also be
expressed as variational derivative of the generalized free energy per site,

$\frac{\mathrm{d}}{\mathrm{d}\kappa}F_{\mathrm{al}}(P,V+\kappa\varsigma_{n\sigma})\big{|}_{\kappa=0}=\langle
Q_{0}^{[n,\sigma]}\rangle_{P,V}.$ (12.43)

In addition, one introduces the average log intensity, denoted by $\nu$, for
which

$\nu=\langle Q_{0}^{[0]}\rangle_{P,V}=\partial_{P}F_{\mathrm{al}}(P,V).$
(12.44)

Note that by definition $\nu>0$. For $V=0$, one readily confirms
$F_{\mathrm{al}}(P,0)=\log(P/\pi)$ with log intensity $\nu(P)=P^{-1}>0$. Hence
there is no high pressure phase as encountered for the Toda lattice, see end
of Section 9.1.

###  12.3 Circular random matrices with pressure ramp

We modify the CMV matrix at its two boundaries. As before, the number $N$ of
sites is even. $L_{N}$ remains unchanged, $M_{N}$ is modified to
$M_{N}^{\diamond}$, where $(M_{N}^{\diamond})_{0,0}=1$,
$(M_{N}^{\diamond})_{0,N-1}=0$, $(M_{N}^{\diamond})_{N-1,0}=0$, and
$(M_{N}^{\diamond})_{N-1,N-1}=\mathrm{e}^{\mathrm{i}\phi}$, $\phi\in[0,2\pi]$.
This leads to the particular CMV matrix

$C_{N}^{\diamond}=L_{N}M_{N}^{\diamond}.$ (12.45)

For the a priori measure (12.34) the pressure $P$ is constant. To arrive at a
tractable expression, the constant pressure is replaced by a linearly changing
pressure with a yet arbitrary slope $-\tfrac{1}{2}\beta$, $\beta>0$, according
to

$\mathrm{d}\phi\prod_{j=0}^{N-2}(\rho_{j}^{2})^{-1}(\rho_{j}^{2})^{\beta(N-1-j)/2}\mathrm{d}^{2}\alpha_{j}.$
(12.46)

Surprisingly, Killip and Nenciu succeeded to compute the joint distribution of
eigenvalues of $C_{N}^{\diamond}$ under this measure. We define the
Vandermonde determinant

$\Delta(z_{1},\ldots,z_{N})=\prod_{1\leq i<j\leq N}(z_{j}-z_{i}).$ (12.47)

Denoting the eigenvalues of $C_{N}^{\diamond}$ by
$\mathrm{e}^{\mathrm{i}\vartheta_{1}},\dots,\mathrm{e}^{\mathrm{i}\vartheta_{N}}$,
their joint (not normalized) distribution under the measure in (12.46) is
given by

$\tilde{\zeta}_{N}(\beta)|\Delta(\mathrm{e}^{\mathrm{i}\vartheta_{1}},\ldots,\mathrm{e}^{\mathrm{i}\vartheta_{N}})|^{\beta}\prod_{j=1}^{N}\mathrm{d}\vartheta_{j},$
(12.48)

where the prefactor turns out to equal

$\tilde{\zeta}_{N}(\beta)=2^{(1-N)}\frac{1}{N!}\frac{\Gamma(\beta/2)^{N}}{\Gamma(N\beta/2)}.$
(12.49)

Since $\beta$ is a free parameter, one can choose specifically

$\beta=\frac{2P}{N}.$ (12.50)

Now the ramp has slope $-P/N$ and, in the limit $N\to\infty$, close to the
lattice point $(1-u)N$, $0<u<1$, the measure of (12.46) will converge to the
product measure of (12.34) with pressure $uP$. Since

$\mathrm{tr}\big{[}\mathsfit{V}(C_{N}^{\diamond})]=\sum_{j=1}^{N}V(\vartheta_{j}),$
(12.51)

the Boltzmann weight can be naturally included in (12.46). Hence the partition
function of the system with boundary conditions turns into

$\displaystyle
Z_{\mathrm{kn},N}(P,V)=\int_{[0,2\pi]^{N-1}}\prod_{j=0}^{N-2}\mathrm{d}^{2}\alpha_{j}\int_{0}^{2\pi}\mathrm{d}\phi\prod_{j=0}^{N-2}(\rho_{j}^{2})^{-1}(\rho_{j}^{2})^{P(N-1-j)/N}\exp\big{(}-\mathrm{tr}[V(C_{N}^{\diamond})]\big{)}$
$\displaystyle=\zeta_{N}(P)\int_{0}^{2\pi}\\!\\!\mathrm{d}\vartheta_{1}\dots\int_{0}^{2\pi}\\!\\!\mathrm{d}\vartheta_{N}\exp\Big{(}-\sum_{j=1}^{N}V(\vartheta_{j})+P\frac{1}{N}\sum_{i,j=1,i\neq
j}^{N}\log|\mathrm{e}^{\mathrm{i}\vartheta_{i}}-\mathrm{e}^{\mathrm{i}\vartheta_{j}}|\Big{)}$
(12.52)

with $\zeta_{N}(P)=\tilde{\zeta}_{N}(2P/N)$. In statistical mechanics the
probability distribution

$\frac{1}{Z_{\log,N}}\exp\Big{(}-\sum_{j=1}^{N}V(\vartheta_{j})+P\frac{1}{N}\sum_{i,j=1,i\neq
j}^{N}\log|\mathrm{e}^{\mathrm{i}\vartheta_{i}}-\mathrm{e}^{\mathrm{i}\vartheta_{j}}|\Big{)}$
(12.53)

is known as circular log-gas. Since the coupling strength is proportional to
$1/N$, it is the mean-field version of the log-gas.

The Ablowitz-Ladik model with linearly varying pressure has a free energy per
site defined through

$F_{\mathrm{kn}}(P,V)=\lim_{N\to\infty}-\frac{1}{N}\log
Z_{\mathrm{kn},N}(P,V).$ (12.54)

Because the pressure ramp has slope $1/N$, in the limit free energies merely
add up as

$F_{\mathrm{kn}}(P,V)=\int_{0}^{1}\mathrm{d}uF_{\mathrm{al}}(uP,V).$ (12.55)

Before studying the infinite volume free energy, we remark that the CMV matrix
$C_{N}^{\diamond}$ is still linked to a suitably modified Ablowitz-Ladik
dynamics governed by the hamiltonian

$H_{N}^{\diamond}=\mathrm{tr}\big{[}C_{N}^{\diamond}+{C_{N}^{\diamond}}^{*}\big{]}.$
(12.56)

Working out the Poisson brackets leads to the evolution equation

$\frac{d}{dt}\alpha_{j}=\mathrm{i}\rho_{j}^{2}(\alpha_{j-1}+\alpha_{j+1}),$
(12.57)

$j=0,\dots,N-2$, with the boundary conditions $\alpha_{-1}=-1$ and
$\alpha_{N-1}=\mathrm{e}^{\mathrm{i}\phi}$. As before
$\mathrm{tr}\big{[}{(C_{N}^{\diamond}})^{n}\big{]}$ is preserved under the
dynamics. However, the a priori measure (12.46) is no longer stationary. The
long time dynamics on a ring differs qualitatively from the one with open
boundary conditions.

The prefactor in (12.3) can be easily handled with the result

$\lim_{N\to\infty}-\frac{1}{N}\log\zeta_{N}(P)=\log(2P)-1.$ (12.58)

The sum in the exponential of (12.53) has a term involving the confining
potential $V$, which is linear in the empirical measure, $\varrho_{N}$, of the
$\\{\vartheta_{j}\\}$’s. The double sum of the interaction is quadratic in
$\varrho_{N}$. Thus the limiting free energy is determined by a variational
principle. We first define the free energy functional

$\mathcal{F}_{\mathrm{kn}}^{\circ}(\varrho)=\int_{0}^{2\pi}\mathrm{d}w\varrho(w)\Big{(}V(w)+\log\varrho(w)+\log
P-P\int_{0}^{2\pi}\mathrm{d}w^{\prime}\log|\mathrm{e}^{\mathrm{i}w}-\mathrm{e}^{\mathrm{i}w^{\prime}}|\varrho(w^{\prime})\Big{)}.$
(12.59)

This functional has to be minimized over all densities $\varrho$, with the
constraints $\varrho(w)\geq 0$ and $\langle\varrho\rangle=1$. Denoting by
$\varrho^{\star}$ the unique minimizer, one arrives at

$F_{\mathrm{kn}}(P,V)=\log
2-1+\mathcal{F}_{\mathrm{kn}}^{\circ}(\varrho^{\star})$ (12.60)

and thus, using (12.55),

$F_{\mathrm{al}}(P,V)=\partial_{P}(PF_{\mathrm{kn}}(P,V)).$ (12.61)

As for the Toda lattice, it turns out to be more convenient to absorb $P$ into
$\varrho$ by setting $\rho=P\varrho$. Then
$P\mathcal{F}_{\mathrm{kn}}^{\circ}(P^{-1}\rho)=\mathcal{F}_{\mathrm{kn}}(\rho)-P\log
P$ with the transformed free energy functional

$\mathcal{F}_{\mathrm{kn}}(\rho)=\int_{0}^{2\pi}\mathrm{d}w\rho(w)\Big{(}V(w)+\log\rho(w)-\int_{0}^{2\pi}\mathrm{d}w^{\prime}\log|\mathrm{e}^{\mathrm{i}w}-\mathrm{e}^{\mathrm{i}w^{\prime}}|\rho(w^{\prime})\Big{)}.$
(12.62)

$\mathcal{F}_{\mathrm{kn}}$ has to be minimized under the constraint

$\rho(w)\geq 0,\quad\int_{0}^{2\pi}\mathrm{d}w\rho(w)=P$ (12.63)

with minimizer denoted by $\rho^{\star}$. Then

$F_{\mathrm{al}}(P,V)=\partial_{P}\mathcal{F}_{\mathrm{kn}}(\rho^{\star})-1+\log
2.$ (12.64)

The constraint (12.63) is removed by introducing the Lagrange multiplier $\mu$
as

$\mathcal{F}_{\mathrm{kn}}^{\bullet}(\rho)=\mathcal{F}_{\mathrm{kn}}(\rho)-\mu\int_{0}^{2\pi}\mathrm{d}w\rho(w).$
(12.65)

The unique minimizer of $\mathcal{F}_{\mathrm{kn}}^{\bullet}(\rho)$ is denoted
by $\rho_{\mathsf{n},\mu}$ and determined as solution of the TBA equation

$V(w)-\mu-2\int_{0}^{2\pi}\mathrm{d}w^{\prime}\log|\mathrm{e}^{\mathrm{i}w}-\mathrm{e}^{\mathrm{i}w^{\prime}}|\rho_{\mathsf{n},\mu}(w^{\prime})+\log\rho_{\mathsf{n},\mu}(w)=0.$
(12.66)

The parameter $\mu$ has to be adjusted such that

$P=\int_{0}^{2\pi}\mathrm{d}w\rho_{\mathsf{n},\mu}(w).$ (12.67)

To obtain the Ablowitz-Ladik free energy, we differentiate as

$\partial_{P}\mathcal{F}_{\mathrm{kn}}(\rho^{\star})=\int_{0}^{2\pi}\mathrm{d}w\partial_{P}\rho^{\star}(w)\Big{(}V(w)+\log\rho^{\star}(w)-2\int_{0}^{2\pi}\mathrm{d}w^{\prime}\rho^{\star}(w^{\prime})\log|\mathrm{e}^{\mathrm{i}w}-\mathrm{e}^{\mathrm{i}w^{\prime}}|\Big{)}+1.$
(12.68)

Integrating (12.66) against $\partial_{P}\rho^{\star}$ one arrives at

$\partial_{P}\mathcal{F}_{\mathrm{kn}}(\rho_{\mathsf{n},\mu})=\mu+1$ (12.69)

and thus

$F_{\mathrm{al}}(P,V)=\mu(P,V)+\log 2.$ (12.70)

The Ablowitz-Ladik lattice has the property that its free energy is determined
through an explicit variational problem. At this stage, a structure comparable
to the Toda lattice has been reached. The modifications are minimal. The
integration is over the interval $[0,2\pi]$ corresponding to the eigenvalues
$\mathrm{e}^{\mathrm{i}w}$. The basis functions are $\varsigma_{0}(w)=1$,
$\varsigma_{n-}(w)=\sin(nw)$, $\varsigma_{n+}(w)=\cos(nw)$ and the kernel of
the operator $T$ is adjusted to

$Tf(w)=2\int_{0}^{2\pi}\mathrm{d}w^{\prime}\log|2\sin((w-w^{\prime})/2)|f(w^{\prime}),\quad
w\in[0,2\pi].$ (12.71)

With such modifications the results in Sections 3.3 and 3.4 hold ad verbatim.
In particular, under the GGE in (12.36) one obtains the $N\to\infty$ limit of
the DOS as

$\rho_{\mathrm{Q}}=\partial_{P}\rho_{\mathsf{n}}.\vskip 12.0pt plus 4.0pt
minus 4.0pt$ (12.72)

We arrived at a somewhat perplexing conclusion. Clearly the structure
discussed very strongly resembles the one obtained for the Toda lattice. Our
scheme is particle-based. Nevertheless, so far we did not identify an
underlying particle structure. Their potential two-particle scattering shift
arrives only indirectly through the generalized free energy. For the currents
we will use the availability of a conserved currents and thus bypass the
collision rate ansatz.

###  12.4 Average currents

As a fairly general fact, the average currents are related to fluctuations in
the density of states. The same strategy will be pursued for the Ablowitz-
Ladik lattice.

For the infinite lattice, the conserved fields satisfy a continuity equation
of the form

$\frac{\mathrm{d}}{\mathrm{d}t}Q_{j}^{[n,\sigma],N}=\\{Q_{j}^{[n,\sigma],N},H_{\mathrm{al},N}\\}_{\mathrm{AL}}=J_{j}^{[n,\sigma],N}-J_{j+1}^{[n,\sigma],N}.$
(12.73)

It will be convenient to work with the complex currents

$J_{j}^{[n],N}=J_{j}^{[n,+],N}+\mathrm{i}J_{j}^{[n,-],N}.$ (12.74)

Using Eq. (12.22) together with the fact that the matrix $C_{+}$ has
nonvanishing matrix elements only for $(j,j+\ell)$ with $\ell=0,1,2$, the time
derivative is obtained to

$\displaystyle\\{(C^{n})_{j,j},H_{\mathrm{al}}\\}_{\mathrm{AL}}$
$\displaystyle=\sum_{\ell=1,2}\mathrm{i}\Big{(}C_{j-\ell,j}(C^{n})_{j,j-\ell}-C_{j,j+\ell}(C^{n})_{j+\ell,j}+\bar{C}_{j,j+\ell}(C^{n})_{j,j+\ell}-\bar{C}_{j-\ell,j}(C^{n})_{j-\ell,j}\Big{)}.$
(12.75)

The term with $\ell=0$ does not contribute. The remaining terms look like a
shift difference, but it is not, since the off-diagonal matrix elements are
only two-periodic.

For $n=1$, one finds

$\\{C_{j,j},\mathrm{tr}(C)\\}_{\mathrm{AL}}=\mathrm{i}\big{(}-\rho_{j-1}^{2}\alpha_{j-2}\bar{\alpha}_{j}+\rho_{j}^{2}\alpha_{j-1}\bar{\alpha}_{j+1}\big{)},$
(12.76)

while

$\\{C_{j,j},\mathrm{tr}(C^{*})\\}_{\mathrm{AL}}=\mathrm{i}\big{(}\rho_{j}^{2}\alpha_{j+1}\bar{\alpha}_{j+1}-\rho_{j-1}^{2}\alpha_{j-2}\bar{\alpha}_{j-2}+\rho_{j}^{2}\rho_{j+1}^{2}-\rho_{j-2}^{2}\rho_{j-1}^{2}\big{)}=\mathrm{i}\big{(}\alpha_{j-1}\bar{\alpha}_{j-1}-\alpha_{j}\bar{\alpha}_{j}\big{)}$
(12.77)

for even $j$ and

$\\{C_{j,j},\mathrm{tr}(C^{*})\\}_{\mathrm{AL}}=\mathrm{i}\big{(}\rho_{j}^{2}\alpha_{j-1}\bar{\alpha}_{j-1}-\rho_{j-1}^{2}\alpha_{j}\bar{\alpha}_{j}\big{)}=\mathrm{i}\big{(}\alpha_{j-1}\bar{\alpha}_{j-1}-\alpha_{j}\bar{\alpha}_{j}\big{)}$
(12.78)

for odd $j$. Adding the two terms, the first current reads

$J_{j}^{[1]}=\mathrm{i}\big{(}-\rho_{j-1}^{2}\alpha_{j-2}\bar{\alpha}_{j}+\alpha_{j-1}\bar{\alpha}_{j-1}\big{)}.$
(12.79)

For higher $n$, it seems to be difficult to guess the cancellations leading to
an explicit difference form. But for the moment we only need the property of
currents being local, which can be ensured by an abstract argument.

$\blackdiamond\blackdiamond$ Existence of currents. We want to establish that
there is a local current, $J^{[n]}_{j}$, such that

$\\{(C^{n})_{j,j},H_{\mathrm{al}}\\}_{\mathrm{AL}}=J^{[n]}_{j}-J^{[n]}_{j+1}.$
(12.80)

For this purpose, we consider a ring of size $N$ and fix $n$ such that $N>4n$.
For clarity the index $N$ is suppressed in our notation.
$\\{Q^{[n]}_{0},H_{\mathrm{al}}\\}_{\mathrm{AL}}$ is a polynomial of degree
$2n+2$. This polynomial is decomposed into patterns. A pattern consists of a
specific monomial together with a collection of some of its translates. By
construction patterns are distinct. The monomial defining a pattern is denoted
by $\omega_{1}$ and its translates by $\omega_{j}$. Then, understood modulo
$N$, $\\{Q^{[n]}_{0},H_{\mathrm{al}}\\}_{\mathrm{AL}}$ is a sum of terms of
the form

$\sum_{|\ell|\leq 2n+2}a^{(\ell)}\omega_{\ell}$ (12.81)

with some complex coefficients $a^{(\ell)}$, which may vanish. Since
$\\{Q^{[n]},H_{\mathrm{al}}\\}_{\mathrm{AL}}=0$, shift invariance implies that
for every pattern

$0=\sum_{j=0}^{N-1}\sum_{|\ell|\leq
2n+2}a^{(\ell)}\omega_{\ell+j}=\sum_{|\ell|\leq
2n+2}a^{(\ell)}\Big{(}\sum_{j=0}^{N-1}\omega_{\ell+j}\Big{)}.$ (12.82)

and as a consequence

$\sum_{|\ell|\leq 2n+2}a^{(\ell)}=0.$ (12.83)

Eliminating the nonzero $a^{(\ell)}$ with largest index, one obtains unique
coefficients $b^{(\ell)}$ such that

$\sum_{|\ell|\leq
2n+2}b^{(\ell)}(\omega_{\ell}-\omega_{\ell+1})=J^{\omega}_{0}-J^{\omega}_{1}.$
(12.84)

For the actual local current one still has to sum over all patterns.

The total fields $Q^{[n],N}$ are unique, but the one-shift covariant local
densities $Q^{[n],N}_{j}$ are not. Once the density is chosen, the
corresponding one-shift covariant current density is determined up to a
constant. $\blackdiamond\blackdiamond$

To compute the GGE averaged currents, one notes that

$J^{0}_{j}=2Q_{j}^{[1,-]}.$ (12.85)

Hence the path outlined in Section 6.1 can be followed. Omitting all
intermediate steps, the result is

$\partial_{P}\big{(}\langle
J^{[n,\sigma]}_{0}\rangle_{P,V}+2P\langle\varsigma_{1-},C^{\sharp}\varsigma_{n\sigma}\rangle\big{)}=0,$
(12.86)

implying that the round bracket has to be independent of $P$, in particular
equal to its value at $P=0$. Since
$\langle\varsigma_{1-},C^{\sharp}\varsigma_{n\sigma}\rangle$ is bounded in
$P$, the second summand vanishes at $P=0$. For the first summand, one notes
that in the limit $P\to 0$, for each $j$ the a priori measure (12.34) becomes
uniform on the unit circle. Hence, $\langle\rho_{j}^{2}\rangle_{P}\to 0$ as
$P\to 0$ and the CMV matrix turns diagonal. Denoting
$\alpha_{j}=\mathrm{e}^{\mathrm{i}\phi_{j}}$, $\phi_{j}\in[0,2\pi]$, in the
limit $P\to 0$ the GGE (12.36) converges to

$\frac{1}{Z_{\mathrm{al},N}(0,V)}\prod_{j=0}^{N-1}\mathrm{d}\phi_{j}\exp\Big{(}-\sum_{j=0}^{N-1}V(\phi_{j+1}-\phi_{j})\Big{)}$
(12.87)

with boundary conditions $\phi_{N}=\phi_{0}$. According to (12.79), one
observes that $\langle J_{0}^{[1]}\rangle_{0,V}=\mathrm{i}$. By a somewhat
lengthy computation $\langle J_{0}^{[2]}\rangle_{0,V}=0$. To extend the
average to general $n$ seems to be difficult, since a sufficiently explicit
formula for $J_{0}^{[n]}$ is missing. Let us then assume that $\langle
J_{0}^{[n]}\rangle_{0,V}=d_{n}$ with some constant $d_{n}$ independent of $V$.
Next we substitute $P\varrho^{*}=\rho_{\mathsf{n}}$ with the result

$P\langle\varsigma_{1-},C^{\sharp}\varsigma_{n\sigma}\rangle=\langle\varsigma_{1-},(1-\rho_{\mathsf{n}}T)^{-1}\rho_{\mathsf{n}}\varsigma_{n\sigma}\rangle-\nu\langle\varsigma_{1-},(1-\rho_{\mathsf{n}}T)^{-1}\rho_{\mathsf{n}}\rangle\langle\varsigma_{n\sigma},(1-\rho_{\mathsf{n}}T)^{-1}\rho_{\mathsf{n}}\rangle.$
(12.88)

Noting that $(1-\rho_{\mathsf{n}}T)^{-1}\rho_{\mathsf{n}}$ is a symmetric
operator, one finally arrives at

$\langle
J^{[n,\sigma]}_{0}\rangle_{P,V}-d_{n}=-2\big{(}\langle\rho_{\mathsf{n}}\varsigma_{1-}^{\mathrm{dr}}\varsigma_{n\sigma}\rangle-
q_{1-}\langle\rho_{\mathsf{p}}\varsigma_{n\sigma}\rangle\big{)}$ (12.89)

with

$q_{1-}=\langle Q^{1,-}_{0}\rangle_{P,V}.$ (12.90)

The spectral function for the energy is $E(w)=2\cos w$, hence
$E^{\prime}(w)=-2\sin w$. To emphasize the correspondence with the Toda
lattice, see Eqs. (6.1) and (6.18), we define the effective velocity through

$v^{\mathrm{eff}}=\frac{(-2\varsigma_{1-})^{\mathrm{dr}}}{\varsigma_{0}^{\mathrm{dr}}},\qquad\tilde{q}_{1-}=-2q_{1-}.$
(12.91)

Then

$\langle J^{[0]}_{0}\rangle_{P,V}=-\tilde{q}_{1-},\qquad\langle
J^{[n,\sigma]}_{0}\rangle_{P,V}-d_{n}=\langle(v^{\mathrm{eff}}-\tilde{q}_{1-})\rho_{\mathsf{p}}\varsigma_{n\sigma}\rangle.$
(12.92)

###  12.5 Hydrodynamic equations

On the hydrodynamic scale the local GGE is characterized by the log intensity
$\nu$ and the CMV density of states $\nu\rho_{\mathsf{p}}$, both of which now
become spacetime dependent. Merely inserting the average currents, and since
$d_{n}$ is assumed to be a constant, one arrives at the Euler type
hydrodynamic evolution equations,

$\displaystyle\partial_{t}\nu(x,t)+2\partial_{x}q_{1-}(x,t)=0,$
$\displaystyle\partial_{t}\big{(}\nu(x,t)\rho_{\mathsf{p}}(x,t;v)\big{)}+\partial_{x}\big{(}2(v^{\mathrm{eff}}(x,t;v)+q_{1-}(x,t))\rho_{\mathsf{p}}(x,t;v)\big{)}=0.$
(12.93)

Here we followed a standard convention, which amounts to the free dispersion
relation $E(w)=2\cos w$. Upon adopting $E(w)=\cos w$ the extra factors of 2 in
(12.5) would be removed. Equation (12.5) is based on the assumption of local
GGE. To actually establish such an equation from the underlying Ablowitz-Ladik
lattice seems to be a difficult task.

As a general feature of generalized hydrodynamics, the equations can be
transformed explicitly to a quasilinear form through

$\rho_{\mathsf{n}}=\rho_{\mathsf{p}}(1+(T\rho_{\mathsf{p}}))^{-1},$ (12.94)

compare with (6.24). Then Eq. (12.5) turns into the normal form

$\partial_{t}\rho_{\mathsf{n}}+2\nu^{-1}(v^{\mathrm{eff}}+q_{1-})\partial_{x}\rho_{\mathsf{n}}=0.$
(12.95)

Since $\nu>0$, no apparent singularities are encountered.

For the defocusing discrete NLS in one dimension, we established the form of
the hydrodynamic equations. As a novel feature, their structure is determined
by the mean-field version of the log gas corresponding to CUE random matrices.
Thus formally expanding the circle to a line, one recovers the hydrodynamic
equations of the Toda lattice. Our analysis is pretty much on the same level
as the one for the Toda lattice. Only the handling of average currents in the
limit $P\to 0$ is incomplete. We hope to return to this point in the future.

###  12.6 Modified Korteweg-de Vries equation

Instead of the real part of $Q^{[1],N}$ one can also consider its imaginary
part as hamiltonian,

$H_{\mathrm{kv},N}=-\mathrm{i}\sum_{j=0}^{N-1}\big{(}\alpha_{j-1}\bar{\alpha}_{j}-\bar{\alpha}_{j-1}\alpha_{j}\big{)}=-\mathrm{i}\,\mathrm{tr}\big{[}C_{N}-C_{N}^{*}\big{]}.$
(12.96)

Then

$\frac{d}{dt}\alpha_{j}=\\{\alpha_{j},H_{\mathrm{kv},N}\\}_{\mathrm{AL}}=\rho_{j}^{2}(\alpha_{j+1}-\alpha_{j-1}),$
(12.97)

which is known as Schur flow. Through a formal Taylor expansion, one argues
that the continuum limit of Eq. (12.97) yields the modified Korteweg-de Vries
equation,

$\partial_{t}u=\partial_{x}^{3}u-6u^{2}\partial_{x}u,$ (12.98)

which is a good reason to briefly touch upon (12.97).

As before $\alpha_{j}\in\mathbb{D}$ and the conservation laws remain
unchanged. However the currents have to be modified from $J^{[n]}$ to
$\breve{J}^{[n]}$. For the log intensity current one obtains

$\breve{J}^{[0],N}_{j}=2Q^{[1,+],N}_{j}=H_{\mathrm{kv},N,j}.$ (12.99)

Thus the role of $1-$ and $1+$ are interchanged. The arguments in Section 12.4
can be repeated without modifications. In the hydrodynamic equations (12.92),
$q_{1-}$ is replaced by $q_{1+}$ and the effective velocity becomes

$v^{\mathrm{eff}}=\frac{\varsigma_{1+}^{\mathrm{dr}}}{\varsigma_{0}^{\mathrm{dr}}}\,.$
(12.100)

These considerations badly miss that the wave field of the modified Korteweg-
de Vries equation cabn be taken as real-valued. If the initial $\alpha_{j}$
are real, then according to the evolution equation (12.97) this property is
preserved in time. From the perspective of GGE, such initial conditions amount
to a set of measure zero and one has to reconsider the analysis. Fortunately
the relevant transformation formula has been proved already by Killip and
Nenciu. To avoid duplication of symbols, in the remainder of this section
$\bar{\alpha}_{j}=\alpha_{j}$ everywhere and hence $\alpha_{j}\in[-1,1]$. The
equations of motion read

$\frac{d}{dt}\alpha_{j}=\rho_{j}^{2}(\alpha_{j+1}-\alpha_{j-1}),$ (12.101)

where $\rho_{j}^{2}=1-\alpha_{j}^{2}$ as before. While an obvious hamiltonian
structure is lost, one readily checks that the a priori measure

$\prod_{j=0}^{N-1}\mathrm{d}\alpha_{j}(\rho_{j}^{2})^{P-1}=\prod_{j=0}^{N-1}\mathrm{d}\alpha_{j}(\rho_{j}^{2})^{-1}\exp\big{(}-PQ^{[0],N}\big{)}$
(12.102)

is still stationary under the dynamics. The densities of the conserved fields
are

$Q_{j}^{[m]}=(C^{m})_{j,j},$ (12.103)

$m=1,2,\ldots\,$, as before. But the imaginary part is dropped and there is no
longer a distinction of $\pm$. In particular,

$Q^{[0]}_{j}=-\log\rho_{j}^{2},\quad Q^{[1]}_{j}=-\alpha_{j-1}\alpha_{j},\quad
Q^{[2]}_{j}=\alpha_{j-1}^{2}\alpha_{j}^{2}-\rho_{j-1}^{2}\alpha_{j-2}\alpha_{j}-\rho_{j}^{2}\alpha_{j-1}\alpha_{j+1}.$
(12.104)

For the log intensity current, $\breve{J}^{[0]}_{j}=2Q^{[1]}_{j}$. Thus
$\breve{J}^{[0]}$ is conserved and the Toda strategy is still applicable.

Since $C_{N}$ is now a real matrix, its eigenvalues come in pairs. If
$\mathrm{e}^{\mathrm{i}\vartheta}$ is an eigenvalue, so is
$\mathrm{e}^{-\mathrm{i}\vartheta}$. For a system of size $N$, there are only
$\bar{N}=N/2$ independent eigenvalues, denoted by
$\mathrm{e}^{\mathrm{i}\vartheta_{1}},\ldots,\mathrm{e}^{\mathrm{i}\vartheta_{\\!\bar{N}}}$.
Rather than using a DOS reflecting such symmetry, it is more convenient to
restrict the phase as $0\leq\vartheta_{j}\leq\pi$. The empirical DOS is given
by

$\rho_{\mathrm{Q},\bar{N}}(w)=\frac{1}{\bar{N}}\sum_{j=1}^{\bar{N}}\delta(w-\vartheta_{j}),\qquad
w\in[0,\pi],$ (12.105)

where we switched to $\bar{N}$ as size parameter. In the limit
$\bar{N}\to\infty$, $\rho_{\mathrm{Q},\bar{N}}(w)$ converges to the
deterministic limit $\rho_{\mathrm{Q}}(w)$. The GGE expectations are then

$\langle
Q_{0}^{[n]}\rangle_{P,V}=2\int_{0}^{\pi}\mathrm{d}w\rho_{\mathrm{Q}}(w)\varsigma_{n}(w),\quad\cos(nw)=\varsigma_{n}(w),$
(12.106)

and the confining potential becomes

$V(w)=\sum_{n=1}^{\infty}\mu_{n}\cos(nw)$ (12.107)

with real chemical potentials $\mu_{n}$. The confining potential is assumed to
be continuous.

Under the measure

$\prod_{j=0}^{N-2}(1-\alpha_{j}^{2})^{-1}(1-\alpha_{j}^{2})^{\beta(N-j-1)/4}(1-\alpha_{j})^{a+1-(\beta/4)}(1+(-1)^{j}\alpha_{j})^{b+1-(\beta/4)}\mathrm{d}\alpha_{j}$
(12.108)

on $[-1,1]^{N-1}$ with $\beta>0$ and $a,b>-1+(\beta/4)$, the joint (not
normalized) distribution of the eigenvalues of $C_{N}$, imposing
$\alpha_{N-1}=1$, is given by

$\tilde{\zeta}_{\bar{N}}(\beta)2^{\kappa}|\Delta(2\cos\vartheta_{1},\ldots,2\cos\vartheta_{\bar{N}})|^{\beta}\prod_{j=1}^{\bar{N}}\big{(}1-\cos\vartheta_{j}\big{)}^{a}\big{(}1+\cos\vartheta_{j})^{b}\sin\vartheta_{j}\mathrm{d}\vartheta_{j}$
(12.109)

on $[0,\pi]^{\bar{N}}$. The proportionality factor $\tilde{\zeta}_{\bar{N}}$
is defined in (12.48) and $\kappa=(\bar{N}-1)(-\tfrac{1}{2}\beta+a+b+2)$. To
achieve a pressure ramp of slope $-P/2\bar{N}$, one has to set

$\beta=\frac{2P}{\bar{N}},\qquad a=b=-1+\tfrac{1}{4}\beta.$ (12.110)

Then $\kappa=0$ and under the measure

$\prod_{j=0}^{N-2}(1-\alpha_{j}^{2})^{-1}(1-\alpha_{j}^{2})^{P(N-j-1)/N}\mathrm{d}\alpha_{j}$
(12.111)

the eigenvalues of $C_{N}$ have the distribution

$\zeta_{\bar{N}}(P)|\Delta(2\cos\vartheta_{1},\ldots,2\cos\vartheta_{n})|^{2P/\bar{N}}\prod_{j=1}^{\bar{N}}(\sin\vartheta_{j})^{((P/\bar{N})-1)}\mathrm{d}\vartheta_{j}.$
(12.112)

Now the strategy of Section 12.3 is in force. In (12.112) we add a confining
potential. The power of $\sin\vartheta_{j}$ converges to $-1$ and in the limit
$\bar{N}\to\infty$ the free energy functional becomes

$\displaystyle\mathcal{F}_{\mathrm{kv}\triangleright}^{\circ}(\varrho)=\int_{0}^{\pi}\mathrm{d}w\varrho(w)\Big{(}V(w)+\log\sin
w+\log\varrho(w)$
$\displaystyle-P\int_{0}^{\pi}\mathrm{d}w^{\prime}\varrho(w^{\prime})\log(2|\cos
w-\cos w^{\prime}|)\Big{)}.$ (12.113)

$\mathcal{F}_{\mathrm{kv}\triangleright}^{\circ}$ has to be minimized over all
$\varrho\geq 0$ with $\int_{0}^{\pi}\mathrm{d}w\varrho(w)=1$ and boundary
condition $\rho(-1)=\rho(1)$. Actually our mean-field limit is somewhat
singular, since at $a=b=-1+(P/N)$ the a priori measure is barely integrable.
The quadratic energy term is repulsive, but the linear $\log\sin$ term pushes
the eigenvalues towards the two end points. It is not so obvious, whether and
how the two terms balance. Fortunately, the particular case $V=0$ has been
studied in detail, thereby confirming (12.6) and a normalizable DOS.

Surprisingly, deviating from the conventional script, the confining potential
$V$ is corrected by the $\log\sin$ potential, which is attractive and favors
accumulation of eigenvalues at $0,\pi$. The prior computation of the average
currents is carried out for fixed $V$, which has now to be replaced by
$\tilde{V}(w)=V(w)+\log(\sin w)$. Of course, $\rho_{\mathsf{n}}$ and
$\rho_{\mathsf{p}}$ depend on $\tilde{V}$. Also the operator $T$ has to be
adjusted to

$Tf(w)=2\int_{0}^{\pi}\mathrm{d}w^{\prime}\log|2(\cos w-\cos
w^{\prime})|f(w^{\prime}),\quad w\in[0,\pi].$ (12.114)

With these modifications, the TBA formalism can be taken over with no further
changes. In Eq. (6.19) the bare velocity $v$ is replaced by $2\cos v$ and the
kernel $2\log|v-w|$ is substituted by $2\log|2(\cos v-\cos w)|$. Then the
effective velocity is given by

$v^{\mathrm{eff}}=\frac{\varsigma_{1}^{\mathrm{dr}}}{\varsigma_{0}^{\mathrm{dr}}},$
(12.115)

upon adopting the dressing operator $T$ from (12.114) and more implicitly
$\tilde{V}$. The hydrodynamic equations are argued along the standard route.
The subtraction term reads
$q_{1}=2\nu\int_{0}^{\pi}\mathrm{d}w\rho_{\mathsf{p}}(w)\cos w$.

The discrete modified KdV equation is an integrable wave equation with no
obvious hamiltonian structure. Compared to the Ablowitz-Ladik lattice, CUE is
replaced by COE. Interestingly enough, in the standard coordinates, an
additional confining potential is generated. Such a feature has not yet been
encountered before.

Notes and references

Section 12.1

A most useful source of information are the monographs by Ablowitz and Segur
[6] and by Ablowitz et al. [7]. In the book by Grébert and Kappeler [140] the
integrable structure of continuum NLS is discussed in great detail.
Equilibrium measures were first studied by Lebowitz et al. [191], see also the
more recent contributions by Oh and Quastel [222] and Fröhlich et al. [123].
More general nonlinearities are covered in Bourgain [35]. GGEs with $V$ a
finite polynomial are studied by Zhidkov [301]. The discretization discussed
in this section is based on Ablowitz and Ladik [4, 5]. The GHD of the
continuum sinh-Gordon equation has been developed by Bastianello et al. [18],
see also De Luca and Mussardo [72]. The integrable lattice version is reported
by Orfanidis [227]. The integrable structure of continuum spin chain has been
studied first by Takhtajan [282] and Sklyanin [262]. Its integrable
discretization is covered by Faddeev and Takhtajan [108], see also the more
recent contribution by Krajnik et al. [178]. Numerical simulations of the spin
chain are reported by Das et al. [68]. The article by Spohn and Lebowitz [273]
is an early investigation of what is now called GGE for wave equations.

Section 12.2

Nenciu [219, 220] discovered the role played by the Cantero-Moral-Velázquez
matrices, which were originally introduced with distinct goals, see Cantero et
al. [47, 48]. Further accounts are Killip and Nenciu [171] and Simon [260].

Section 12.3

The change of volume elements for the linear pressure ramp has been
accomplished by Killip and Nenciu [170] with motivation from the corresponding
result by Dumitriu and Edelman [96] for the Toda lattice. The circular log-gas
is summarized in Forrester [118] and its mean-field version is studied by
Trinh and Trinh [291] and Hardy and Lambert [148]. The results reported in the
text are based on Spohn [272]. Independently, similar results are worked out
by Grava and Mazzuca [144]. In particular, for polynomial confining potentials
the free energy is proved to be the solution of the variational problem in
(12.59). Mazzuca and Memin [199] extend such results to all continuous
confining potentials $V$, conceptually is an important step. More generally
one would like to prove the following property of GHD. One starts from an
initial profile $x\mapsto\nu\rho_{\mathsf{p}}(x)$, such that every
$\nu\rho_{\mathsf{p}}(x)$ determines a unique GGE. To be shown is propagation
of such a property in time. Numerical simulations of the Ablowitz-Ladik chain
are reported by Mendl and Spohn [202] with results to some extent resembling
Fig. 8. A comparison with GHD is still missing. Based on Killip and Nenciu
[171] the $N$-particle scattering is elucidated by Brollo and Spohn [37].

Section 12.5

From a general perspective the expressions (12.91) and (12.115) are somewhat
puzzling. The numerator is the dressing of the derivative of energy, i.e.
$-\sin w$ for (12.91) and $\cos w$ for (12.115). However, according to the
rules, the denominator should be the derivative of momentum, thus respectively
$\cos w$ and $-\sin w$, while we obtained the dressing of the constant
function.

Section 12.6

The notion Schur flow for the dynamics generated by (12.97) goes back to
Golinskiǐ [134]. The transformation of measure is proved by Killip and Nenciu
[170]. In Trinh and Trinh [291] and Forrester and Mazzuca [119] explicit
formulas are obtained for the thermal case with $a,b>-1$. With the methods
Trinh and Trinh [291] also the diagonal limit, $a=b=-1+(P/N)$, is proved to
yield the stated result. Surprisingly, the two-fold degeneracy of the spectrum
of the CMV matrix results in asymptotically two-particle bound states, see
Brollo and Spohn [37].

##  13 Hydrodynamics for the Lieb-Liniger $\delta$-Bose gas

Much of the interest related to GHD is rooted in the study of integrable
quantum many-body systems. Over the past decade the number of contributions
staggered and it would be difficult to present a balanced view. Reviews and
journal special volumes are available. On the other hand, without mentioning
quantum models the GHD story would be utterly incomplete. Our compromise
consists of two elements. The first acquisition is the quantized version of
the Toda lattice, the topic of Chapter 14. Due to missing physical
realizations this model received less attention. Still the model is
instructive because a close theoretical comparison between classical and
quantum is feasible.

The by far most widely studied integrable model is the quantized nonlinear
Schrödinger equation, better known as the $\delta$-Bose gas because the
interaction between bosons is point-like. In their pioneering work from 1963
Elliott Lieb and Werner Liniger established that eigenvalues and
eigenfunctions of the hamiltonian can be obtained through the Bethe ansatz.
This discovery triggered the field of integrable quantum many-body systems.
For GHD the Lieb-Liniger model also plays a very special role. It serves as
paradigm for the study of more difficult models as the XXZ quantum spin chain
and the spin-$\tfrac{1}{2}$ Fermi-Hubbard model. Even more importantly,
Rubidium atoms enclosed in a narrow tube are well modelled by bosons
interacting through an extremely short range potential. Experimental
realizations are available through the amazing advances in cold atom physics,
see the Insert in Section 13.3. Thus as a second acquisition we discuss
aspects of GHD for the $\delta$-Bose gas. In particular, we will emphasize the
structural similarity with the DOS of a classical Lax matrix.

For all quantum mechanical models the hydrodynamic scale is particle-based.
The most convincing evidence is the two-particle scattering shift, which is
computed from the scattering of two quantum particles. Mathematically this is
very different from a two-soliton solution. The quantum particles are
described by a wave equation in $\mathbb{R}^{2}$ governing the motion of the
two-particle wave function $\psi(x_{1},x_{2},t)$, while a two-soliton wave
field refers to a wave equation over $\mathbb{R}$. The construction of
soliton-based hydrodynamics for quantum models remains as a task for the
future.

###  13.1 Bethe ansatz

To quantize the continuum nonlinear Schrödinger equation of the introduction
to Chapter 12, one starts from a scalar bosonic field $\Psi(x)$,
$x\in\mathbb{R}$, with canonical commutation relations

$[\Psi(x),\Psi(x^{\prime})^{*}]=\delta(x-x^{\prime}),\quad[\Psi(x),\Psi(x^{\prime})]=0,\quad[\Psi(x)^{*},\Psi(x^{\prime})^{*}]=0.$
(13.1)

In normal ordered form the Lieb-Liniger hamiltonian reads then

$H_{\mathrm{li}}=\tfrac{1}{2}\int_{\mathbb{R}}\mathrm{d}x\big{(}\partial_{x}\Psi(x)^{*}\partial_{x}\Psi(x)+c\Psi(x)^{*}\Psi(x)^{*}\Psi(x)\Psi(x)\big{)}.$
(13.2)

Here $c$ is the coupling constant with $c\geq 0$ imposed. The attractive case,
$c<0$, is considerably more complicated because of bound states. In the widely
accepted standard convention the factor $\tfrac{1}{2}$ in front of the
integral is omitted. We inserted here so have an energy-momentum relation of
the free theory identical to the one of the noninteracting Toda fluid. The
Lieb-Liniger hamiltonian is formal because of point-interactions. Fortunately,
there is the first quantized version, which in the $N$-particle sector reads

$H_{\mathrm{li},N}=-\sum_{j=1}^{N}\tfrac{1}{2}(\partial_{x_{j}})^{2}+c\sum_{1\leq
i<j\leq N}\delta(x_{i}-x_{j}).$ (13.3)

Bosonic means that the operator acts on wave functions symmetric under
particle exchange. The case of interest is a spatial interval $[0,\ell]$,
$\ell>0$, with periodic boundary conditions, which will be distinguished by
either a sub - or superscript $\ell$. Of course, one could also consider a
finite number of particles moving on the real line, which would be the natural
set-up for multi-particle scattering.

Let us first recall, how to handle the delta potential. The simplest example
is $N=2$, for which the relative motion, $y=x_{2}-x_{1}$, is governed by the
hamiltonian

$H_{\mathrm{rel}}=-\partial_{y}^{2}+c\,\delta(y).$ (13.4)

Away from $y=0$, $H_{\mathrm{rel}}\psi(y)=-\partial_{y}^{2}\psi(y)$ with
$\psi\in\mathcal{C}^{2}(\mathbb{R}\setminus\\{0\\})$, the twice continuously
differentiable functions on $\mathbb{R}\setminus\\{0\\}$ with bounded
derivatives. The $\delta$-potential translates to the boundary condition

$\partial_{y}\psi(0_{+})-\partial_{y}\psi(0_{-})=2c\psi(0).$ (13.5)

The same mechanism works for the $N$-particle hamiltonian
$H_{\mathrm{li},\ell,N}$ acting on symmetric wave functions. For them one can
restrict the construction to the Weyl chamber $\mathbb{W}_{N,\ell}=\\{0\leq
x_{1}\leq\ldots\leq x_{N}\leq\ell\\}$. By permutation symmetry the hamiltonian
can then be extended to all other sectors. Away from the boundary the
hamiltonian is the $N$-dimensional Laplacian,

$H_{\mathrm{li},\ell,N}=-\sum_{j=1}^{N}\tfrac{1}{2}(\partial_{x_{j}})^{2},$
(13.6)

which acts on
$\psi\in\mathcal{C}^{2}(\mathbb{W}_{N,\ell}\setminus\partial\mathbb{W}_{N,\ell})$.
Periodic boundary conditions mean

$\psi(0,x^{\bot})=\psi(x^{\bot},\ell),\quad\partial_{x_{1}}\psi(0,x^{\bot})=\partial_{x_{N}}\psi(x^{\bot},\ell),$
(13.7)

while the interaction is encoded by the boundary condition

$(\partial_{x_{j+1}}-\partial_{x_{j}})\psi(x)=c\psi(x)\big{|}_{x_{j}=x_{j+1}},\quad
j=1,\ldots,N-1,$ (13.8)

where the limit is taken from the interior of $\mathbb{W}_{N,\ell}$. It can be
shown that the Laplacian with these boundary conditions defines a unique self-
adjoint operator on $L^{2}(\mathbb{W}_{N,\ell})$.

On $\mathbb{W}_{\ell,N}$ an eigenfunction of $H_{\mathrm{li},\ell,N}$ has the
Bethe ansatz form

$\psi(x_{1},\ldots,x_{N})=\sum_{\sigma\in\mathcal{S}_{N}}\Big{[}\exp\Big{(}\mathrm{i}\sum_{j=1}^{N}k_{\sigma(j)}x_{j}\Big{)}\prod_{\begin{subarray}{c}i<j\\\
\sigma(i)>\sigma(j)\end{subarray}}A(k_{i},k_{j})\Big{]},$ (13.9)

where the sum is over all permutations $\sigma$ of $(1,\ldots,N)$. The
$k_{j}$’s are called rapidities, so to distinguish from the momenta of
noninteracting particles. $\psi$ has to satisfy the boundary conditions
(13.8), which leads to

$A(k_{1},k_{2})=\frac{\mathrm{i}(k_{1}-k_{2})-c}{\mathrm{i}(k_{1}-k_{2})+c}=\mathrm{e}^{-\mathrm{i}\theta_{\mathrm{li}}(k_{1}-k_{2})}.$
(13.10)

Here the two-particle phase shift equals

$\theta_{\mathrm{li}}(w)=2\arctan(w/c),$ (13.11)

implying the scattering shift

$\theta_{\mathrm{li}}^{\prime}(w)=\frac{2c}{w^{2}+c^{2}}=\phi_{\mathrm{li}}(w).$
(13.12)

Thus when two bosons scatter, their distance is reduced in comparison to the
free particle motion. Equivalently one can think of a delay time. When two
bosons scatter, they effectively stick for a short time span which depends on
the incoming momenta.

Imposing periodic boundary conditions as in (13.7), the rapidities have to
satisfy

$\mathrm{e}^{\mathrm{i}k_{j}\ell}=(-1)^{N-1}\prod_{i=1}^{N}\mathrm{e}^{-\mathrm{i}\theta_{\mathrm{li}}(k_{j}-k_{i})}.$
(13.13)

Equivalently $k_{1},\ldots,k_{N}$ solve the Bethe equations

$2\pi I_{j}=\ell k_{j}+\sum_{i=1}^{N}\theta_{\mathrm{li}}(k_{j}-k_{i}),\quad
j=1,\ldots,N,$ (13.14)

and hence are also referred to as Bethe roots. The quantum states are labelled
by vectors $I=(I_{1},\ldots,I_{N})$, constrained as $I_{1}<\ldots<I_{N}$ and
with entries from $\mathbb{Z}$ in case of odd and from
$\mathbb{Z}+\tfrac{1}{2}$ in case of even $N$. The set of quantum numbers is
denoted by $\mathbb{I}_{N}$. For each such $I\in\mathbb{I}_{N}$ the Bethe
equations have a unique solution denoted by $k(I)=k=(k_{1},\ldots,k_{N})$
ordered as $k_{1}<\ldots<k_{N}$. It is known that these eigenfunctions
constitute a complete orthonormal basis in $L^{2}(\mathbb{W}_{N,\ell})$.

$\blackdiamond\blackdiamond$ Volume factors. If the quantum numbers above
would be redefined to take values in $2\pi\mathbb{Z}$, resp.
$2\pi\big{(}\mathbb{Z}+\tfrac{1}{2}\big{)}$, then the factors of $2\pi$ would
disappear from our formulas. Thereby an optically even closer similarity to
the Toda fluid would be achieved. For the ease of comparison the conventional
notation is adopted, however. $\blackdiamond\blackdiamond$

Symmetric eigenfunctions on $L^{2}([0,\ell]^{N})$ can be written as

$\psi_{k}(x_{1},\ldots,x_{N})=\sum_{\sigma\in\mathcal{S}_{N}}(-1)^{\mathrm{sgn}(\sigma)}\exp\Big{(}\mathrm{i}\sum_{j=1}^{N}k_{\sigma(j)}x_{j}\Big{)}\prod_{1\leq
i<j\leq
N}\Big{(}k_{\sigma(i)}-k_{\sigma(j)}+\mathrm{i}c\,\mathrm{sgn}(x_{i}-x_{j})\Big{)}$
(13.15)

with $\mathrm{sgn}(\sigma)$ the sign of the permutation $\sigma$. $\psi_{k}$
is not normalized, the normalized state vector being denoted by
$|k\rangle=\langle\psi_{k},\psi_{k}\rangle^{-\frac{1}{2}}\psi_{k}$. The
normalization constants are known and the wave functions $|k\rangle$ span the
bosonic subspace $L^{2}([0,\ell]^{N})_{\mathrm{sym}}$. In particular

$H_{\mathrm{li},N}=\sum_{I\in\mathbb{W}_{N}}\Big{(}\tfrac{1}{2}\sum_{j=1}^{N}(k_{j})^{2}\Big{)}|k\rangle\langle
k|$ (13.16)

with $k=k(I)$. Up to the prefactor $\tfrac{1}{2}$, if the power $2$ is
substituted by $0$, one arrives at the number operator. Similarly power $1$
results in the total momentum. This suggests to introduce the $n$-th conserved
field (= charge) by

$Q^{[n],N}=\sum_{I\in\mathbb{I}_{N}}\Big{(}\sum_{j=1}^{N}(k_{j})^{n}\Big{)}|k\rangle\langle
k|.$ (13.17)

###  13.2 Bethe root densities, free energy, TBA equations

The Boltzmann weight involves a linear combination of charges. Therefore,
again following the Toda lattice blueprint, we introduce the confining
potential $V$ and set

$Q^{[V],N}=\sum_{I\in\mathbb{I}_{N}}\Big{(}\sum_{j=1}^{N}V(k_{j})\Big{)}|k\rangle\langle
k|.$ (13.18)

The precise class of confining potentials remains yet to be studied. A natural
choice seems to be $V$ continuous and $V(w)\geq c_{0}+c_{1}|w|$ with
$c_{1}>0$. The unnormalized GGE density matrix is defined by

$\exp\big{[}-Q^{[V],N}\big{]}$ (13.19)

as operator on $L^{2}([0,\ell]^{N})_{\mathrm{sym}}$. Hence the normalizing
partition function is given by

$Z_{\mathrm{li},N}(\ell,V)=\mathrm{tr}\big{[}\mathrm{e}^{-Q^{[V],N}}\big{]}=\sum_{I\in\mathbb{I}_{N}}\exp\Big{[}-\sum_{j=1}^{N}V(k_{j})\Big{]},$
(13.20)

trace over $L^{2}([0,\ell]^{N})_{\mathrm{sym}}$. Correspondingly the GGE
average of some operator $\mathcal{O}$ is defined by

$\langle\mathcal{O}\rangle_{\ell,V}=Z_{\mathrm{li},N}(\ell,V)^{-1}\mathrm{tr}\big{[}\mathrm{e}^{-Q^{[V],N}}\mathcal{O}\big{]}.$
(13.21)

The Bethe roots depend on $\ell$ and, as before, this dependence is indicated
in the partition function.

Our goal is the infinite volume

$\ell\to\infty,\qquad N=\nu\ell$ (13.22)

with $\nu>0$ and $\rho_{\mathsf{f}}=1/\nu$ the Bose gas density. As for
previously studied models, the free energy of the $\delta$-Bose gas will be
determined through a variational problem. In fact, the free energy functional
will be surprisingly similar to the one of the Toda and Calogero fluid,
compare with Eqs. (9.47) and (11.59). Hence it is of interest to have a closer
look at the derivation. The second item are GGE averaged charges
$\ell^{-1}\langle Q^{[n],N}\rangle_{\ell,V}$. These observables are diagonal
in the energy basis, which reduces our task to an unconventional problem of
classical statistical mechanics. As a simplification, the scattering shift
appears already explicitly through the Bethe equations.

We first introduce the empirical root density

$\varrho_{N}(w)=\frac{1}{N}\sum_{j=1}^{N}\delta(w-k_{j}),$ (13.23)

compare with (3.25). Then

$\sum_{j=1}^{N}V(k_{j})=N\int_{\mathbb{R}}\mathrm{d}w\varrho_{N}(w)V(w)=N\mathcal{E}(\varrho_{N}),$
(13.24)

an energy-like term, and our task is to express the sum over quantum states by
a sum over root densities, at least approximately for large $N$. For given
$I$, thus Bethe roots $k(I)$, we consider the equation

$y(\lambda)=\nu\lambda+\theta_{\mathrm{li}}*\varrho_{N}(\lambda)$ (13.25)

with the convolution

$\theta_{\mathrm{li}}*\varrho_{N}(\lambda)=\int_{\mathbb{R}}\mathrm{d}w\theta_{\mathrm{li}}(\lambda-w)\varrho_{N}(w).$
(13.26)

Since the function $y(\lambda)$ is strictly increasing, the relation

$y(k_{m}^{\mathrm{vac}})=\frac{2\pi}{N}m,\quad
m\in\mathbb{Z},\,\,\,\mathrm{resp.}\,\,m\in\mathbb{Z}+\tfrac{1}{2},$ (13.27)

defines the increasingly ordered collection of points
$\\{k_{m}^{\mathrm{vac}}\\}$. This set is viewed as a collection of
placeholders. Their empirical density is the empirical space density

$\varrho_{\mathsf{s},N}(w)=\frac{1}{N}\sum_{m\in\mathbb{Z}}\delta(w-k_{m}^{\mathrm{vac}}).$
(13.28)

Since $y(k_{j})=2\pi I_{j}/N$, the Bethe roots are a subset of
$\\{k_{m}^{\mathrm{vac}}\\}$. These are particle locations in distorted
momentum space. Particles partially fill the placeholder set
$\\{k_{m}^{\mathrm{vac}}\\}$. The complementary set, denoted by
$\\{k_{m}^{\mathrm{h}}\\}$, are hole locations in distorted momentum space.
The empirical hole density is then

$\varrho_{\mathsf{h},N}(w)=\frac{1}{N}\sum_{m\in\mathbb{Z}}\delta(w-k_{m}^{\mathsf{h}}).$
(13.29)

Obviously

$\varrho_{N}+\varrho_{\mathsf{h},N}=\varrho_{\mathsf{s},N}.$ (13.30)

We assume that, relative to the given GGE, with probability one the empirical
densities have a deterministic limit as $N\to\infty$. More precisely one picks
a well localized test function, $f$, and asserts that

$\lim_{N\to\infty}\frac{1}{N}\sum_{m\in\mathbb{Z}}f(k_{m}^{\mathrm{vac}})=\lim_{N\to\infty}\int_{\mathbb{R}}\mathrm{d}wf(w)\varrho_{\mathsf{s},N}(w)=\int_{\mathbb{R}}\mathrm{d}wf(w)\varrho_{\mathsf{s}}(w),$
(13.31)

where the expression on left side is random, while its limit is non-random.
Eq. (13.31) defines the infinite volume space density $\varrho_{\mathsf{s}}$,
correspondingly for $\varrho$ and $\varrho_{\mathsf{h}}$. According to
(13.30),

$\varrho+\varrho_{\mathsf{h}}=\varrho_{\mathsf{s}}.$ (13.32)

Next we study the limit related to (13.25). Setting
$\lambda=k_{j}^{\mathrm{vac}}$ and summing over $f(k_{j}^{\mathrm{vac}})$
yields

$\frac{1}{N}\sum_{j\in\mathbb{Z}}f(k_{j}^{\mathrm{vac}})\frac{2\pi}{N}j=\frac{1}{N}\sum_{j\in\mathbb{Z}}f(k_{j}^{\mathrm{vac}})\Big{(}\nu
k_{j}^{\mathrm{vac}}+\theta_{\mathrm{li}}*\varrho_{N}(k_{j}^{\mathrm{vac}})\Big{)}.$
(13.33)

By construction the right hand side converges to

$\int_{\mathbb{R}}\mathrm{d}wf(w)\varrho_{\mathsf{s}}(w)\big{(}\nu
w+\theta_{\mathrm{li}}*\varrho(w)\big{)}.$ (13.34)

For the left hand side, setting $i<j$, we start from

$\int_{k_{i}^{\mathrm{vac}}}^{k_{j}^{\mathrm{vac}}}\mathrm{d}w\varrho_{\mathsf{s},N}(w)=\frac{1}{N}(j-i).$
(13.35)

For $i\to-\infty$ the nonlinearity becomes negligible and
$k_{i}^{\mathrm{vac}}=2\pi i/N$. Therefore, by restricting the support of the
test function to $[-a,\infty)$ and setting $k_{i}^{\mathrm{vac}}=-a$ with $a$
sufficiently large,

$\frac{1}{N}\sum_{j\in\mathbb{Z}}f(k_{j}^{\mathrm{vac}})\Big{(}\int_{-a}^{k_{j}^{\mathrm{vac}}}2\pi\varrho_{\mathsf{s},N}(w^{\prime})\mathrm{d}w^{\prime}-a\Big{)}\to\int_{\mathbb{R}}\mathrm{d}wf(w)\varrho_{\mathsf{s}}(w)\Big{(}\int_{-a}^{w}2\pi\varrho_{\mathsf{s}}(w^{\prime})\mathrm{d}w^{\prime}-a\Big{)}$
(13.36)

as $N\to\infty$. Inserting (13.34) and (13.36) in Eq. (13.33), one arrives at
the pointwise identity

$\int_{-a}^{w}2\pi\varrho_{\mathsf{s}}(w^{\prime})\mathrm{d}w^{\prime}-a=\nu
w+\theta_{\mathrm{li}}*\varrho(w),$ (13.37)

which upon differentiating yields

$2\pi\varrho_{\mathsf{s}}(w)=\nu+\phi_{\mathrm{li}}*\varrho(w).$ (13.38)

Through the root density the scattering shift rules the space density.

The final step would require more details. The energy term appeared already in
(13.24). Hence one still has to figure out the entropy, namely the number of
$I$’s subject to our constraints. For this purpose considered is a small
volume element, $\mathrm{d}w$, which still contains a huge number of
placeholders given by $N\varrho_{\mathsf{s}}(w)\mathrm{d}w$. They split into
Bethe roots, $N\varrho(w)\mathrm{d}w$, and holes,
$N\varrho_{\mathsf{h}}(w)\mathrm{d}w$. Merely counting the corresponding
quantum numbers, the corresponding local entropy equals

$\displaystyle\log\frac{(N\varrho_{\mathsf{s}}(w)\mathrm{d}w)!}{(N\varrho(w)\mathrm{d}w)!(N\varrho_{\mathsf{h}}(w)\mathrm{d}w)!}$
$\displaystyle\approx
N\mathrm{d}w\big{(}\varrho_{\mathsf{s}}(w)\log\varrho_{\mathsf{s}}(w)-\varrho(w)\log\varrho(w)-\varrho_{\mathsf{h}}(w)\log\varrho_{\mathsf{h}}(w)\big{)},$
(13.39)

where the linear term vanishes because of (13.32). With $\varrho_{\mathsf{s}}$
defined in (13.38) and $\varrho_{\mathsf{h}}$ in (13.32), the resulting
entropy per particle is thus given by

$\mathcal{S}(\varrho)=\int_{\mathbb{R}}\mathrm{d}w\big{(}\varrho_{\mathsf{s}}\log\varrho_{\mathsf{s}}-\varrho\log\varrho-\varrho_{\mathsf{h}}\log\varrho_{\mathsf{h}}\big{)}.$
(13.40)

In approximation one arrives at

$Z_{\mathrm{li},N}(\ell,V)\approx\int\mathcal{D}(\varrho)\mathrm{e}^{-N\big{(}\mathcal{E}(\varrho)-\mathcal{S}(\varrho)\big{)}}$
(13.41)

involving a somewhat vague sum over all root densities.

The exponent defines the Yang-Yang free energy functional

$\mathcal{F}^{\circ}_{\mathrm{yy}}(\varrho)=\nu^{-1}\int_{\mathbb{R}}\mathrm{d}w\big{(}\varrho
V+\varrho\log\varrho+\varrho_{\mathsf{h}}\log\varrho_{\mathsf{h}}-\varrho_{\mathsf{s}}\log\varrho_{\mathsf{s}}\big{)}.$
(13.42)

The free energy is per unit length, compare with (9.47) for the Toda fluid. It
is convenient to switch to $\rho=\nu^{-1}\varrho$ and correspondingly
$\rho_{\mathsf{s}}=\nu^{-1}\varrho_{\mathsf{s}}$,
$\rho_{\mathsf{h}}=\nu^{-1}\varrho_{\mathsf{h}}$. We also introduce the $T$
operator

$Tf(w)=\frac{1}{2\pi}\int_{\mathbb{R}}\mathrm{d}w^{\prime}\phi_{\mathrm{li}}(w-w^{\prime})f(w^{\prime}).$
(13.43)

The kernel of $2\pi T$ equals the $\delta$-Bose scattering shift. Also
$T\varsigma_{0}=\varsigma_{0}$ and $T\to 1$ for $c\to 0$, while $T\to 0$ for
$c\to\infty$. The latter case corresponds to free fermions and the former to
free bosons. Eq. (13.32) becomes

$\rho+\rho_{\mathsf{h}}=\rho_{\mathsf{s}},$ (13.44)

while (13.38) transforms to

$2\pi\rho_{\mathsf{s}}=1+2\pi T\rho.$ (13.45)

Then, using (13.44), the free energy per unit length is written as

$\mathcal{F}_{\mathrm{yy}}(\rho)=\int_{\mathbb{R}}\mathrm{d}w\big{(}\rho
V+\rho\log\rho+\rho_{\mathsf{h}}\log\rho_{\mathsf{h}}-(\rho+\rho_{\mathsf{h}})\log(\rho+\rho_{\mathsf{h}})\big{)}.$
(13.46)

The variation is over all $\rho$ with $\rho\geq 0$ and
$\nu\langle\rho\rangle=1$. Note that, deviating from classical models, the
factor $2\pi$ appears as a result of momentum quantization.

$\blackdiamond\blackdiamond$ Convexity of the Yang-Yang free energy
functional. Following the original argument we establish that the functional
$\mathcal{F}_{\mathrm{yy}}$ is convex. We start from two arbitrary spectral
densities $\rho_{0},\rho_{1}$ of well-defined Yang-Yang free energy and
consider the linear interpolation $\rho_{u}=\rho_{0}(1-u)+u\rho_{1}$, $0\leq
u\leq 1$. The aim is to show that
$\mathcal{F}_{\mathrm{yy}}(\rho_{u})^{\prime\prime}>0$, the prime denoting
derivative with respect to $u$. Note that
$\rho_{u}^{\prime}=\rho_{1}-\rho_{0}$ and, in general,
$\rho^{\prime}+\rho_{\mathsf{h}}^{\prime}=T\rho^{\prime}$. With this input,
differentiating $\mathcal{F}_{\mathrm{yy}}(\rho(u))$ is straightforward with
the result

$\mathcal{F}_{\mathrm{yy}}(\rho_{u})^{\prime\prime}=\int_{\mathbb{R}}\mathrm{d}w\Big{(}\frac{1}{\rho_{u}}+\frac{1}{\rho_{\mathsf{h},u}}\Big{)}\Big{(}(\rho_{1}-\rho_{0})-\frac{\rho_{u}}{\rho_{u}+\rho_{\mathsf{h},u}}T(\rho_{1}-\rho_{0})\Big{)}^{2}\geq
0.$ (13.47)

While likely to be correct, strict convexity would require an extra argument.

$\blackdiamond\blackdiamond$

We remove the constraint $\nu\langle\rho\rangle=1$ by the Lagrange multiplier
$\mu$. Then the free energy becomes

$\mathcal{F}^{\bullet}_{\mathrm{yy}}(\rho)=\int_{\mathbb{R}}\mathrm{d}w\big{(}\rho(V-\mu)+\rho\log\rho+\rho_{\mathsf{h}}\log\rho_{\mathsf{h}}-(\rho+\rho_{\mathsf{h}})\log(\rho+\rho_{\mathsf{h}})\big{)}.$
(13.48)

The minimizing root density is the particle density $\rho_{\mathsf{p}}$, which
is solution of the saddle-point equation

$V-\mu-\log\frac{\rho_{\mathsf{h}}}{\rho_{\mathsf{p}}}-T\log\Big{(}1+\frac{\rho_{\mathsf{p}}}{\rho_{\mathsf{h}}}\Big{)}=0.$
(13.49)

Here it is understood that space and hole density associated with
$\rho_{\mathsf{p}}$ are again denoted by
$\rho_{\mathsf{s}},\rho_{\mathsf{h}}$. Hence
$\rho_{\mathsf{s}}=\rho_{\mathsf{p}}+\rho_{\mathsf{h}}$ and, in addition,

$\rho_{\mathsf{s}}=\frac{1}{2\pi}+T\rho_{\mathsf{p}},$ (13.50)

which confirms the usage of “space density”. For $c=0$, i.e. $T=0$, the Bethe
roots are merely the eigenvalues of the quantized momentum, which in the
infinite volume limit converge to the Lebesgue measure
$\frac{1}{2\pi}\mathrm{d}w$. Through the interaction this density is modified,
in the same spirit as the bare velocity is modified to $v^{\mathrm{eff}}$.
Particle and hole density are then taken relative to such modified space
density.

To solve (13.49) and (13.50), we introduce the number density

$\rho_{\mathsf{n}}=\frac{\rho_{\mathsf{p}}}{\rho_{\mathsf{p}}+\rho_{\mathsf{h}}}=\frac{\rho_{\mathsf{p}}}{\rho_{\mathsf{s}}},$
(13.51)

thereby yielding the quasi-energy $\varepsilon$ through

$\rho_{\mathsf{n}}=\frac{1}{1+\mathrm{e}^{\varepsilon}},\qquad\mathrm{e}^{\varepsilon}=\frac{\rho_{\mathsf{h}}}{\rho_{\mathsf{p}}}.$
(13.52)

Then, using the dressing transformation

$f^{\mathrm{dr}}=f+T\rho_{\mathsf{n}}f^{\mathrm{dr}},\quad
f^{\mathrm{dr}}=\big{(}1-T\rho_{\mathsf{n}}\big{)}^{-1}f,$ (13.53)

one obtains

$2\pi\rho_{\mathsf{s}}=\varsigma_{0}^{\mathrm{dr}}.$ (13.54)

With these conventions Eq. (13.49) turns into the quantum TBA equation

$\varepsilon=V-\mu-T\log(1+\mathrm{e}^{-\varepsilon}),$ (13.55)

compare with (9.50) for the Toda fluid.

To determine the free energy of the $\delta$-Bose gas one has to evaluate the
free energy functional $\mathcal{F}^{\bullet}_{\mathrm{yy}}$ at its minimizer
$\rho_{\mathsf{p}}$, which leads to

$\lim_{\ell\to\infty}-\frac{1}{\ell}\log
Z_{\mathrm{li},N}(\ell,\mu,V)=F_{\mathrm{yy}}(\mu,V)=\mathcal{F}^{\bullet}_{\mathrm{yy}}(\rho_{\mathsf{p}})=-\frac{1}{2\pi}\int\mathrm{d}w\log(1+\mathrm{e}^{-\varepsilon}),$
(13.56)

a form familiar from noninteracting fermions. Adding the small perturbation
$V+\kappa\varsigma_{n}$ and computing the linear response of the free energy
in $\kappa$ one obtains

$\lim_{\ell\to\infty}\ell^{-1}\langle
Q^{[n],\ell}\rangle_{V,\ell}=\langle\rho_{\mathsf{p}}\varsigma_{n}\rangle,\quad
n=0,1,\ldots\,.$ (13.57)

This identity suggests the proper interpretation of the minimizing particle
density $\rho_{\mathsf{p}}$. Under the GGE the Bethe roots
$k_{1},\ldots,k_{N}$ with variable $N$ are random and combine into a empirical
density as

$\rho_{\mathsf{p},\ell}(w)=\ell^{-1}\sum_{j=1}^{N}\delta(w-k_{j}),$ (13.58)

compare with Eq. (13.23). Obviously, $\rho_{\mathsf{p},\ell}(w)\geq 0$ with
normalization $\langle\rho_{\mathsf{p},\ell}\rangle=\nu^{-1}$. Eq. (13.57)
states that on average the $n$-th moment of $\rho_{\mathsf{p},\ell}$ converges
to the one of $\rho_{\mathsf{p}}$ as $\ell\to\infty$. In Chapter 4, we
explained that typical fluctuations of the empirical density of Lax matrix
eigenvalues are of order $1/\sqrt{N}$. For the Lieb-Liniger model one expects
the same behavior and

$\lim_{\ell\to\infty}\rho_{\mathsf{p},\ell}(w)=\rho_{\mathsf{p}}(w)$ (13.59)

with probability one, upon integrating against a localized smooth test
function.

###  13.3 Charge currents, hydrodynamic equations

To obtain the total currents, the usual strategy is to first figure out the
charge densities. They satisfy a local conservation law from which one deduces
the charge current density, and thereby also the total charge currents. The
total charges have been defined in (13.17). Then, to lowest order and on the
entire real line,

$\displaystyle Q^{[0]}(x)=\Psi(x)^{*}\Psi(x),\quad
Q^{[1]}(x)=-\mathrm{i}\Psi(x)^{*}\partial_{x}\Psi(x),$ $\displaystyle
Q^{[2]}(x)=\partial_{x}\Psi(x)^{*}\partial_{x}\Psi(x)+c\Psi(x)^{*}\Psi(x)^{*}\Psi(x)\Psi(x),$
(13.60)

which implies the current densities

$J^{[0]}(x)=Q^{[1]}(x),\qquad
J^{[1]}(x)=\partial_{x}\Psi(x)^{*}\partial_{x}\Psi(x)+\tfrac{1}{2}c\Psi(x)^{*}\Psi(x)^{*}\Psi(x)\Psi(x).$
(13.61)

Thus a natural strategy would be to start from the classical nonlinear
Schrödinger equation and its known sequence of charge densities involving
higher order field derivatives. As naive guess, imposing normal order will
yield the charges of the Lieb-Liniger model in second quantized form, which
would then be manifestly local. But this scheme works only up to $n=3$.
Already for $Q^{[4]}$ more complicated subtraction terms appear. For the total
charge currents similar difficulties will appear. We still freely use
$Q^{[n]}(x)$ and $J^{[n]}(x)$, assuming that at some point these difficulties
can be resolved. For infinite volume, formally, the charges satisfy a local
conservation law of the form

$\partial_{t}Q^{[n]}(x,t)+\partial_{x}J^{[n]}(x,t)=0.$ (13.62)

Just to be clear, here $t$ refers to the unitary time evolution. For the
operators at time zero, we omit the argument $t$, e.g.
$J^{[n]}(x)=J^{[n]}(x,t=0)$.

To compute the average current, one starts from a GGE at infinite volume,
$\ell=\infty$, average being denoted as $\langle\cdot\rangle_{V}$ and
truncation by $\langle fg\rangle_{V}^{\mathrm{c}}=\langle f\rangle_{V}\langle
g\rangle_{V}$. Then the charge-current correlator is given by

$B_{m,n}=\int_{\mathbb{R}}\mathrm{d}x\langle
J^{[m]}(x)Q^{[n]}(0)\rangle_{V}^{\mathrm{c}}.$ (13.63)

While the definition looks asymmetric, in fact,

$B_{m,n}=B_{n,m},$ (13.64)

as has been explained for other models before, compare with (6.1). Now,
setting $\partial_{0}=\partial_{\mu_{0}}$ and
$\partial_{1}=\partial_{\mu_{1}}$, one arrives at

$\partial_{0}\langle
J^{[n]}(0)\rangle_{V}=-B_{n,0}=-B_{0,n}=-\int_{\mathbb{R}}\mathrm{d}x\langle
Q^{[1]}(x)Q^{[n]}(0)\rangle_{V}=\partial_{1}\langle
Q^{[n]}(0)\rangle_{V}=\partial_{1}\langle\rho_{\mathsf{p}}\varsigma_{n}\rangle.$
(13.65)

The $\partial_{0}$ derivative of the average current is related to a
susceptibility, i.e. a particular second derivative of the generalized free
energy. This is a doable problem and we explain a short path towards its
solution.

Using the TBA equation (13.55) to differentiate $\rho_{\mathsf{n}}$ with
respect to $\partial_{0}$ and $\partial_{1}$, one obtains the relations

$\partial_{0}\rho_{\mathsf{n}}=-\rho_{\mathsf{n}}(1-\rho_{\mathsf{n}})\varsigma_{0}^{\mathrm{dr}},\quad\partial_{1}\rho_{\mathsf{n}}=-\rho_{\mathsf{n}}(1-\rho_{\mathsf{n}})\varsigma_{1}^{\mathrm{dr}}.$
(13.66)

Hence