# The sinh-Gordon model beyond the self dual point
and the freezing transition in disordered systems

Denis<EMAIL_ADDRESS>and André
<EMAIL_ADDRESS>Laboratoire de Physique de l’Ecole Normale
Supérieure, CNRS, ENS & Université PSL, Sorbonne Université, Université de
Paris, 75005 Paris, France  Cornell University, Physics Department, Ithaca, NY
14850, USA

###### Abstract

The S-matrix of the well-studied sinh-Gordon model possesses a remarkable
strong/weak coupling duality $b\to 1/b$. Since there is no understanding nor
evidence for such a duality based on the quantum action of the model, it
should be questioned whether the properties of the model for $b>1$ are simply
obtained by analytic continuation of the weak coupling regime $0<b<1$. In this
article we assert that the answer is no, and we develop a concrete and
specific proposal for the properties when $b>1$. Namely, we propose that in
this region one needs to introduce a background charge $Q_{\infty}=b+1/b-2$
which differs from the Liouville background charge by the shift of $-2$. We
propose that in this regime the model has non-trivial massless renormalization
group flows between two different conformal field theories. This is in
contrast to the weak coupling regime which is a theory of a single massive
particle. Evidence for our proposal comes from higher order beta functions. We
show how our proposal correctly reproduces the freezing transitions in the
multi-fractal exponents of a Dirac fermion in $2+1$ dimensions in a random
magnetic field, which provides a strong check since such transitions have
several detailed features. We also point out a connection between a semi-
classical version of this transition and the so-called Manning condensation
phenomena in polyelectrolyte physics.

###### Contents

  1. I Introduction
  2. II Sinh-Gordon conventions
  3. III Sinh-Gordon theory with a background charge
    1. III.1 Generalities
    2. III.2 Liouville case
    3. III.3 Freezing transition at the self-dual point
  4. IV Massless Renormalization group flows in the sine- and sinh-Gordon models
    1. IV.1 Higher order beta functions
    2. IV.2 Identification of sinh and sine Gordon phases
    3. IV.3 Massless flows in the “imaginary” sine-Gordon model
    4. IV.4 RG flows in the sinh-Gordon model: $b<1$ verses $b>1$.
    5. IV.5 Remarks on real verses imaginary $\mu$
  5. V Application to the freezing transition in disordered systems
    1. V.1 Dirac fermions in a random U(1) gauge field
    2. V.2 Map to the sinh-Gordon model
    3. V.3 Multi-fractal density of states exponents
    4. V.4 Multi-fractal spectrum
  6. VI Summary and discussion
  7. VII Acknowledgements
  8. A Semi-classical freezing and the Manning condensation

##  I Introduction

The sinh-Gordon model is the simplest relativistic model in $1+1$ dimensions
that is integrable. It can be defined by the action

${\cal S}=\int
d^{2}x\left(\dfrac{1}{8\pi}\partial_{\mu}\phi\partial_{\mu}\phi+2\mu\,\cosh(\sqrt{2}b\phi)\right),$
(1)

with $b$ is a real parameter and $\phi$ a real scalar field. The current
understanding is that the spectrum consists of a single massive particle, with
an S-matrix that is factorizable in terms of the two particle S-matrix sinhG1
, eqn. (3) below. Based on this, a great deal is known about the model. As a
partial list of references, let us mention the following. The form factors
have been computed MussardoSinh1 ; MussardoSinh2 , which can be used to
calculate correlation functions. Even finite temperature 1-point correlations
are computable LeClairMussardo . The thermodynamic Bethe ansatz (TBA) has also
been investigated ZamoTBAsinhG ; we will say more about this later.

In spite of this vast amount of known results concerning the sinh-Gordon
model, one important aspect has essentially remained unanswered. Remarkably,
the S-matrix satisfies the strong/weak coupling duality in that it is
invariant under $b\to 1/b$. The most common viewpoint is that one first
defines the theory for $0<b<1$ where one can trust perturbation theory around
$b=0$, and then one defines the theory for $1<b<\infty$ using the duality.
Since the form factors and TBA are invariant under $b\to 1/b$, from this
perspective the theory for $b>1$ is the same as the dual theory at $0<b<1$.
However an important question arises. Since there is no indication of a $b\to
1/b$ duality based on the action (1) itself, the analytic continuation $b\to
1/b$ may actually not be valid. Relatively recently this issue was studied in
much detail using a truncated Hilbert space approach KonikMussardo  and indeed
indications were found that this approach breaks down as $b$ approaches the
self-dual point $b=1$. It was suggested there that for $b>1$ the theory may
actually be massless, however definite properties of such a theory remained
unspecified and are still unknown. A similar breakdown of analyticity is known
to occur in freezing phenomena, such as in disordered systems Derrida , and
was recently shown to be present in Coulomb gas systems BouchaudFyodorov
which bear analogies with the sinh-Gordon model. The freezing transition in
the sinh-Gordon, in connection with disordered fermions, was also considered
in Doussal ; Doussal2 . We will say more about this connection later.

The purpose of the present article is to make a definite proposal for the
behavior of the sinh-Gordon model for $b>1$ that is not a simple analytic
continuation $b\to 1/b$ of the $0<b<1$ regime. Our specific proposal is easily
described. For $0<b<1$ the theory can be defined as a perturbation of a free
massless boson in the ultraviolet (UV), and the standard properties based on a
single massive particle with the known S-matrix all apply. However at $b=1$
and above, a background charge $Q_{\infty}$ is spontaneously generated. This
background charge is not the same as one would obtain if one views the sinh-
Gordon theory as a perturbation of the Liouville conformal field theory (CFT),
but is rather given by

$Q_{\infty}=b+1/b-2$ (2)

which is a shift of the Liouville value by $-2$. Furthermore, the $b>1$ regime
is a massless phase, but is not conformally invariant. Rather the theory can
be described by a massless renormalization group (RG) flow between two
conformal field theories.

In Section III we propose our main result (2) based on some rather simple
criteria. However these simple arguments by themselves are not enough to
understand the true nature of the theory beyond the self-dual point.
Ultimately the properties of this theory should be tied to properties of the
RG, since the non-zero background charge $Q_{\infty}$ affects anomalous
dimensions, etc. To this aim we study the RG for the sinh-Gordon model based
on the beta functions proposed in Moriconi  and understood in more detail in
BL . These beta functions are well suited to our purposes since they are
ultimately based on the action (1) of the sinh-Gordon theory itself. More
importantly, the physics of interest here concerns massless flows between
conformal field theories, and it was shown in BL  that these proposed beta
functions predict RG flows that agree precisely with exact results for
massless flows in the so-called “imaginary” sine Gordon model FSZ1 ; FSZ2 . We
will review this below. These beta functions also correctly predicted cyclic
RG flows. This gives us some confidence in at least attempting to use these
beta functions to explore the physics we are trying to understand. As we will
show in Section IV.1 there is a clear difference between the RG flows for
$b<1$ verses $b>1$ since these beta functions do not have the symmetry $b\to
1/b$. Furthermore, we can argue based on these beta functions that $b>1$ is a
massless phase and provide support for our proposed $Q_{\infty}$.

The beta functions in Moriconi  are based on anisotropic current-current
perturbations of a Wess-Zumino-Witten model at a level $k$, and the map to
sine and sinh Gordon theory was made in BL , which only involves level $k=1$.
It needs to be mentioned that the beta functions in Moriconi  are still
conjectural. It was argued in Wiese  that there are corrections at 4-loops.
One of us has also pointed out that there could be $1/k$ corrections to these
beta functions for a different class of “flavor” anisotropic models Chiral .
Similar kinds of “all-orders” beta functions were considered in Greeks1 ;
Greeks2 ; Greeks3 ; Tseytlin  using rather different gravitational methods.
There it was also argued that there are higher $1/k$ corrections. On the other
hand, as already stated above, the renormalization scheme used in Moriconi  to
obtain an all-orders beta function has already been shown to provide exact
results for the kind of physics being explored here. In light of these
statements, in this paper we will simply assume the beta functions in Moriconi
; BL  to be correct enough to capture the physics we are trying to understand
and leave aside the issue of possible corrections and whether they affect our
conclusions. Our analysis of these beta functions at this stage should be
viewed as supportive, but not indisputable, evidence for our main proposal
described above. In any case, irrespective of the present work, it is not at
all understood how these proposed $1/k$ corrections can be reconciled with the
correct exact predictions on massless flows in BL ; FSZ1 ; FSZ2 .

In Section V we apply our proposal to the freezing transition of a Dirac
fermion in $2+1$ dimensions in a random $U(1)$ gauge field, namely a magnetic
field. We first map the problem to the sinh-Gordon theory. Then using our
proposed $Q_{\infty}$, we compute in detail the multi-fractal exponents and
their transitions, which can all be traced to the transition at $b=1$ of the
sinh-Gordon model. Our results agree with known results based on Derrida’s
random energy model or other random fermion models Derrida ; Castillo ;
Doussal ; Doussal2 . This provides rather strong evidence for our proposals
concerning the transition in the sinh-Gordon model at the self-dual point.

By studying a simple semi-classical evaluation of one-point functions in the
sinh-Gordon model, one can understand how the premises of a transition can be
found in such an approximation, and how it is related to the well-known
counter-ion Manning’s condensation Manning  in polyelectrolyte solutions. This
semi-classical computation actually points towards a freezing transition in
the spectrum of possible exponential operators, as a function of their weights
for fixed value of the sinh-Gordon parameter $b$. We relegated these results
to an Appendix since the calculation is semi-classical and detailed properties
are beyond its scope. However we found it instructive to include this analysis
since it provides a simple intuitive picture for the transition.

##  II Sinh-Gordon conventions

Since there are several conventions in the literature, and factors of $1/4\pi$
and $\sqrt{2}$ are important here, we clearly define our conventions. In the
standard understanding, almost certainly valid in the weak coupling regime
$0<b<1$, the spectrum consists of a single particle of mass $m_{\rm phys}$
with two particle S-matrix

$S(\theta)=\dfrac{\sinh\theta-i\sin\pi\gamma}{\sinh\theta+i\sin\pi\gamma},~{}~{}~{}~{}~{}~{}~{}\gamma\equiv
b^{2}/(1+b^{2}).$ (3)

Here $\theta$ is the difference of the usual rapidity parameterization of
energy/momentum: $(E,p)=m_{\rm phys}\,(\cosh\theta,\sinh\theta)$. This
S-matrix is invariant under the duality $b\to 1/b$ which corresponds to
$\gamma\to 1-\gamma$.

The free gaussian field when $\mu=0$ can be decomposed as
$\phi(z,{\overline{z}})=\varphi(z)+\overline{\varphi}({\overline{z}})$, where
$z=x+iy$ and ${\overline{z}}=x-iy$. With the $1/8\pi$ in the action, the above
fields have the canonical two point functions,
$\langle\varphi(z)\varphi(w)\rangle=-\log(z-w)$, and similarly for
$\overline{\varphi}$, and its Virasoro central charge is $c=1$. It is most
natural to view the $\cosh$ potential as a perturbation of the free gaussian
field. Let $[[*]]$ denote the total scaling dimension of $*$ in mass units
(for fields this is the sum of left and right conformal dimensions
$\Delta+\overline{\Delta}$). One has

$[[\cosh(\sqrt{2}b\phi)]]=-2b^{2},$ (4)

which is always relevant for real $b$. Thus $\mu=[{\rm mass}]^{2+2b^{2}}$ for
some mass parameter. One can take the latter as the physical mass of the
single sinh-Gordon particle, such that

$m_{\rm phys}=F(b)\,\,\mu^{1/(2+2b^{2})}.$ (5)

The non-trivial function $F(b)$ was computed by Al. Zamolodchikov by comparing
conformal perturbation theory with the thermodynamic Bethe ansatz (TBA) since
the latter is expressed in terms of $m_{\rm phys}$ ZamoMassScale . We will not
need the explicit form of $F(b)$ which is somewhat complicated but only its
value in the limit $b\to 1$.

We can now clearly address the issue we are proposing to resolve in this paper
that was referred to in the Introduction. The S-matrix (3) is invariant under
the strong/weak duality $b\to 1/b$. The self-dual point is $b=1$. This,
combined with the $Z_{2}$ symmetry $b\to-b$ naively suggests one need only
solve the theory the region $0\leq b\leq 1$, and then analytically continue
the result to all $b$ on the real line. First of all, there is no guarantee
this analytic continuation is valid since it is not a symmetry of the
lagrangian whatsoever. Moreover there is concrete evidence that some
phenomenon is going on at $b=1$ which as yet is not understood. One indication
is that from the exact form of $F(b)$, one finds

$\lim_{b\to 1}\,m_{\rm phys}=\lim_{b\to
1}\,\,\dfrac{4\sqrt{\pi}}{\Gamma\left(\tfrac{1}{4}\right)^{2}}\left(\dfrac{\pi\mu}{\Gamma(1-b^{2})}\right)^{1/4}=0,$
(6)

which formally implies $m_{\rm phys}=0$ at $b=1$. For $b>1$ it would appear
$m_{\rm phys}$ is complex. If the physical mass is zero at $b=1$, then the
S-matrix (3) does not make much sense since rapidity is not defined if $m_{\rm
phys}=0$. More recently the sinh-Gordon model was studied starting from the
action in a truncated Hilbert space approach and clear deviations from the TBA
predictions were observed as $b\to 1$ KonikMussardo . On the other hand, for
$b$ not too large but still in the region $0\leq b\leq 1$, results based on a
truncated space of the free gaussian field work very well. Based on this one
could conclude that the theory defined by the lagrangian for $b>1$ has
different properties than one would expect from an analytic continuation of
$0\leq b<1$. Another piece of evidence for this transition comes from the
analysis of random fermions done in Doussal ; Doussal2 . There, the existence
of a freezing transition in the sinh-Gordon was conjectured in connection with
glassy behavior of random Dirac fermions. Their analysis was based on using
one-loop RG equations, in the form of the so-called KPP equations, and their
traveling wave solutions. Up to now these works, though very interesting, have
not provided a concrete indication of the properties of the theory for $b>1$.

##  III Sinh-Gordon theory with a background charge

###  III.1 Generalities

In this section we consider the sinh-Gordon model with a background charge
$Q_{\infty}$ at infinity. Formally one can deform the action as follows:

${\cal S}={\cal S}_{\rm shG}\,+\,Q_{\infty}\,\phi_{\infty}.$ (7)

Alternatively one can couple the field $\phi$ to the curvature $R$, adding a
term proportional to $Q_{\infty}\,R\,\phi$ to the lagrangian. Either way, in
the unperturbed $\mu=0$ conformal field theory, the effect is to modify the
conformal stress tensor

$T(z)=-\tfrac{1}{2}\partial_{z}\varphi\partial_{z}\varphi+\tfrac{Q_{\infty}}{\sqrt{2}}\,\,\partial_{z}^{2}\varphi,$
(8)

and similarly for $\overline{T}({\overline{z}})$. The Virasoro central charge
is now

$c=1+6\,Q_{\infty}^{2}.$ (9)

The main effect of non-zero $Q_{\infty}$ is to change the scaling dimensions
of operators in the free boson CFT :

$[[e^{\sqrt{2}a\phi}]]=2a(Q_{\infty}-a).$ (10)

The two exponentials in the $\cosh$ now have different scaling dimensions,
thus one should write

$2\mu\,\cosh(\sqrt{2}b\phi)~{}~{}~{}\to~{}~{}\mu_{+}e^{\sqrt{2}b\phi}+\mu_{-}e^{-\sqrt{2}b\phi}.$
(11)

Although the dimensions of $e^{\pm\sqrt{2}b\phi}$ differ, their sum adds up to
$-4b^{2}$ for any $Q_{\infty}$.

For a weight “$a$” in $e^{\sqrt{2}a\phi}$, let us define its dual
$\widetilde{a}$ :

$\widetilde{a}\equiv Q_{\infty}-a,$ (12)

and similarly for the sinh-Gordon coupling $\widetilde{b}\equiv Q_{\infty}-b$.
Note that for zero $Q_{\infty}$, $\widetilde{a}=-a$, including of course
$\widetilde{b}=-b$. Thus with no background charge, the duality
$b\to\widetilde{b}$ simply corresponds to the $Z_{2}$ symmetry of the action
$b\to-b$.

For the CFT with non-zero $Q_{\infty}$, one has the duality that the dimension
of $e^{\sqrt{2}a\phi}$ is invariant under $a\to\widetilde{a}$. Coulomb gas
techniques indicate the equivalence

$e^{\sqrt{2}a\phi}~{}\simeq~{}e^{\sqrt{2}(Q_{\infty}-a)\phi},$ (13)

in the CFT correlation functions. In the TBA this equivalence can be expressed
in terms of so-called reflection amplitudes $R(a)$

$e^{\sqrt{2}a\phi}~{}=R(a)\,e^{\sqrt{2}(Q_{\infty}-a)\phi}$ (14)

which are known for the Liouville case LiouvilleShG2 . This reflection
symmetry is known to be valid in the Liouville theory but only conjectural in
the sinh-Gordon theory (see the Appendix for a discussion of this point).

For any background charge $Q_{\infty}$, the effective central charge $c_{\rm
eff}$ of the TBA is the same if the particle is considered massive. The TBA
equations based on the S-matrix (3) do not depend explicitly on $Q_{\infty}$.
However the effective UV central charge is $c_{\rm eff}=c_{\rm vir}-12d_{0}$
where $c_{\rm vir}$ is the Virasoro central charge and $d_{0}$ is the ground
state energy. Now $d(a)=2a(Q_{\infty}-a)$ which has a maximum at
$a=Q_{\infty}/2$ which corresponds to $d_{0}=Q_{\infty}^{2}/2$. Thus $c_{\rm
eff}=1+6Q_{\infty}^{2}-12(Q_{\infty}^{2}/2)=1$, independently of $Q_{\infty}$.

In principle, the sinh-Gordon model can be considered with any $Q_{\infty}$.
In this paper we will only consider two choices, the Liouville case and the
choice described in the subsequent subsection. Let us consider the first.

###  III.2 Liouville case

This is the most natural choice besides the perturbation of the $Q_{\infty}=0$
free massless boson. Many works indicate that the sinh-Gordon model may be
viewed as a perturbation of the Liouville theory, in particular LiouvilleShG1
; LiouvilleShG2 ; LiouvilleShG3 . In this choice,

$Q_{\infty}=b+1/b,~{}~{}~{}~{}\Longrightarrow~{}~{}[[e^{\sqrt{2}b\phi}]]=2.$
(15)

Namely the positive exponential is an exactly marginal operator,
$[[\mu_{+}]]=2$, and the additional $e^{-\sqrt{2}b\phi}$ is viewed as a
relevant perturbation of the Liouville CFT. Although this may seem like an
unnecessary complication, surprisingly it has been shown that the conformal
perturbation theory of this model with non-zero $Q_{\infty}$ can reproduce the
perturbation theory with $Q_{\infty}=0$ when $0<b<1$ KonikMussardo .

For $Q_{\infty}=b+1/b$, the duality (12) is $b\to\widetilde{b}=1/b$. This
indicates that this Liouville formulation of the sinh-Gordon model is unable
to address the problem posed in this paper since this dual is the usual one
that maps the region $0\leq b\leq 1$ to $1\leq b\leq\infty$. Thus it has
nothing to say about any novel behavior for $b>1$.

###  III.3 Freezing transition at the self-dual point

Our aim is to find a different choice of background charge $Q_{\infty}$ that
can define the sinh-Gordon model for $b>1$, which is expected to have
different properties. As discussed above, the model with $|b|\leq 1$ appears
to be well-defined as a perturbation of the free massless boson with zero
$Q_{\infty}$, or as a perturbed Liouville theory. Taking the simpler option,
we assume there is no background charge in this region, i.e.

$Q_{\infty}=0,~{}~{}~{}~{}{\rm{for}~{}~{}0<b<1}.$ (16)

At $b=1$ we introduce a non-zero $Q_{\infty}$. It would be very interesting to
understand what precise mechanism spontaneously generates this non-zero
$Q_{\infty}$, however we leave aside that question in this work.

The conditions we impose on $Q_{\infty}$ for $|b|>1$ are quite natural and are
the following :

  * •

Based on the symmetry of the S-matrix, we require $Q_{\infty}$ to be self-
dual, as for the Liouville case. This implies we can expand $Q_{\infty}$ as a
series in $b+1/b$ :
$Q_{\infty}=\sum_{n=0}^{\infty}\alpha_{n}\left(b+1/b\right)^{n}$. Since there
are not enough constraints to fix all $\alpha_{n}$, we assume only
$\alpha_{0}$ and $\alpha_{1}$ are non-zero, as in the Liouville theory.

  * •

For continuity with $|b|<1$, we require the background charge $Q_{\infty}=0$
at $b=1$ : $Q_{\infty}=\alpha_{0}+\alpha_{1}(b+1/b)=0$ at $b=1$, so that
$\alpha_{0}=-2\alpha_{1}$. This in turn implies that at the self-dual point
$b=1$, in the UV one has $c=1$, as for $0<b<1$. Thus in the UV, the central
charge $c$ is continuous and only changes at the self-dual point $b=1$.

  * •

To fix $\alpha_{1}$ we need a condition at $b=\infty$. We require that under
the duality $b\to\widetilde{b}=Q_{\infty}-b$, the dual coupling constant
$\widetilde{b}$ remains in the non-zero $Q_{\infty}$ region
$|\widetilde{b}|\geq 1$. For a fixed $b$, this amounts to
$\widetilde{b}\leq-1$, which implies $\alpha_{1}\leq b/(b-1)$. Requiring the
above for all $b$, in particular $b=\infty$, leads to the minimal choice
$\alpha_{1}=1$. Although at this stage this appears somewhat ad hoc, as we
will see it leads to correct predictions for the random energy model.

In summary, we thus propose

$Q_{\infty}=b+1/b-2,~{}~{}~{}~{}{\rm{for}~{}~{}b>1},$ (17)

and zero otherwise. This is just a shift of the Liouville background charge by
the integer $-2$. Note that as $b\to\infty$, $Q_{\infty}$ is the same as for
the Liouville case. Notice also the dichotomy: For $Q_{\infty}=0$ and the
Liouville choice $Q_{\infty}=b+1/b$, $\widetilde{b}=-b$ and $1/b$
respectively, whereas for the above choice $\widetilde{b}=-2+1/b$, which
equals $-b$ for $b=1$.

With this choice of $Q_{\infty}$ one has

$[[e^{\sqrt{2}b\phi}]]=2-4b,~{}~{}~{}~{}~{}~{}[[e^{-\sqrt{2}b\phi}]]=4b-2-4b^{2}.$
(18)

Thus the dimensions of the parameters $\mu_{\pm}$ are

$[[\mu_{+}]]=4b,~{}~{}~{}~{}~{}~{}[[\mu_{-}]]=4(b^{2}-b+1).$ (19)

For $b\geq 1$, both operators $e^{\pm\sqrt{2}b\phi}$ are then relevant. The
ultra-violet limit is controlled by the highest dimension operator, namely the
least relevant. This is the operator $e^{+\sqrt{2}b\phi}$. We thus propose
that in this frozen phase $b>1$ one effectively has
$[[\cosh(\sqrt{2}b\phi)]]\sim 2-4b$. The term “frozen” is borrowed from the
theory of disordered systems; see below.

In summary, we have proposed that

$[[\cosh(\sqrt{2}b\phi)]]=\begin{cases}-2b^{2},~{}~{}~{}~{}~{}~{}&{\rm
for}~{}~{}0<b<1,\\\ 2-4b,~{}~{}~{}~{}&{\rm for}~{}~{}b>1.\end{cases}$ (20)

This transition is induced by the generation of a background charge for $b>1$.
We conjecture that the sinh-Gordon model, which is well defined for $0<b<1$
with $Q_{\infty}=0$, is actually ill-defined for $b>1$ without background
charge but well-defined with the background charge $Q_{\infty}=b+1/b-2$ as in
eq.(17).

##  IV Massless Renormalization group flows in the sine- and sinh-Gordon
models

In the present context by “massless flows” we mean the following. Suppose an
RG flow originates as a perturbation of an UV fixed point CFT by a relevant
operator of dimension $\Gamma_{UV}<2$ and flows to a different non-trivial
fixed point in the infrared (IR), necessarily arriving there via an irrelevant
operator of dimension $\Gamma_{IR}>2$. Generally in the flow to the IR,
massive particles decouple, thus if the IR theory is non-trivial some massless
particles must survive the flow. In the deep IR, the theory is approximated by
the interactions of these massless degrees of freedom.

###  IV.1 Higher order beta functions

The sinh-Gordon model can be viewed as a current-current perturbation of an
SU(2) WZW model at level $1$ with action,

${\cal S}=S_{{\rm wzw}}+\dfrac{1}{2\pi}\int
d^{2}x\left(g_{1}\big{[}J^{+}\overline{J}^{-}+J^{-}\overline{J}^{+}\big{]}+g_{2}J_{3}\overline{J}_{3}\right),$
(21)

with $J^{\pm}=\tfrac{1}{\sqrt{2}}e^{\pm i\sqrt{2}\varphi}$ and
$J_{3}=\tfrac{i}{\sqrt{2}}\partial_{z}\varphi$, where $\varphi(z)$ is the
$z$-dependent part of a free massless scalar field
$\phi(z,{\overline{z}})=\varphi(z)+\overline{\varphi}({\overline{z}})$. The
advantage of doing this is that current algebra Ward identities allow an
easier approach to calculating higher order corrections to the beta functions
since both couplings $g_{1,2}$ are marginal. The bosonized form of the action
is now

${\cal S}=\dfrac{1}{4\pi}\int
d^{2}x\left(\tfrac{1}{2}(\partial\phi)^{2}+g_{1}\,\cosh(\sqrt{2}b\phi)\right).$
(22)

The coupling $b$ is a function of $g_{2}$ presented below, and can be real or
imaginary corresponding to either sinh or sine Gordon phases BL . For reasons
that will become clear, let us postpone this identification for now since such
an identification depends on $Q_{\infty}$, and first describe the general
properties of the flows based solely on the beta functions for $g_{1},g_{2}$.

In our original treatment BL , $g_{1}$ was taken to be real. For several
reasons in this section we present our conclusions for $g_{1}$ imaginary. One
reason is that for our purposes we are interested in massless RG flows, and
this case provides a point of comparison with the known exact results FSZ1 ;
FSZ2 . The second is that for the map to sinh-Gordon for disordered systems,
equation (52) below, $g_{1}$ is indeed imaginary. Thirdly, under the
continuation $g_{1}\to ig_{1}$, the poles in the beta functions in BL  no
longer exist and one does not have to deal with continuing the flow through
these poles. Fortunately as we will comment on at the end of this section, for
the sinh-Gordon flows we are interested in the distinction between real and
imaginary $g_{1}$ does not matter as far as the endpoints of the flows are
concerned, even though the details of the RG trajectories do differ.

For the reasons just described above, we extend the results in BL  to
$g_{1}\to ig_{1}$, corresponding to imaginary $\mu$ in (1). Taking $g_{1}\to
ig_{1}$, the beta functions in Moriconi  become the reasonably simple
functions

$\displaystyle\dfrac{dg_{1}}{d\log a}$
$\displaystyle\equiv\beta_{g_{1}}=\dfrac{g_{1}(g_{2}+g_{1}^{2}/4)}{(1+g_{1}^{2}/16)(1+g_{2}/4)}~{},$
(23a) $\displaystyle\dfrac{dg_{2}}{d\log a}$
$\displaystyle\equiv\beta_{g_{2}}=-\dfrac{g_{1}^{2}(1-g_{2}/4)^{2}}{(1+g_{1}^{2}/16)^{2}}~{},$
(23b)

where $a$ is a cut-off scale. With these conventions, the flow to the IR
corresponds to $a\to\infty$. Flows with $g_{1}<0$ are just a mirror image of
those with $g_{1}>0$ since the beta functions are invariant under $g_{1}\to-
g_{1}$, thus we will only discuss the case $g_{1}>0$.

The above beta functions have a remarkable strong/weak coupling duality. For
both $\beta_{g_{1},g_{2}}$ (recall that beta functions transform as vector
fields) :

$\beta(16/g_{1},16/g_{2})=-\beta(g_{1},g_{2}).$ (24)

Also, again rather remarkably, there exists an RG invariant which allows us to
map out basic features of the flows without explicitly solving the coupled
differential equations based on the beta functions. Such an invariant
$\mathcal{I}$ satisfies $\sum_{g}\,\beta_{g}\,\partial_{g}\mathcal{I}=0$. One
may check that the following $\mathcal{I}$ is an invariant BL

$\mathcal{I}(g_{1},g_{2})=\dfrac{g_{1}^{2}+g_{2}^{2}}{(g_{2}-4)^{2}(g_{1}^{2}+16)}.$
(25)

This invariant satisfies the strong-weak coupling duality of the beta
functions (24),

$\mathcal{I}(16/g_{1},16/g_{2})=\mathcal{I}(g_{1},g_{2}).$ (26)

The line $g_{1}=0$ is a line of fixed points where both beta functions are
zero. By computing the slope of the beta function near $g_{1}=0$, one can
determine the dimension $\Gamma_{0}$ of the perturbation
$\cosh(\sqrt{2}b\phi)$ there with the general formula

$\beta(g)=(2-\Gamma(g_{c}))(g-g_{c})+\cdots,$ (27)

near a critical point $g_{c}$. Since
$\beta_{g_{1}}=\left[4g_{2}/(4+g_{2})\right]g_{1}$ near $g_{1}=0$, equating
the slope with $2-\Gamma_{0}(g_{2})$ yields

$\Gamma_{0}(g_{2})\equiv\Gamma(g_{1}=0)=2\left(\dfrac{4-g_{2}}{4+g_{2}}\right).$
(28)

Thus, the $g_{2}$ axis (at $g_{1}$ =0) is divided into several regions, where
the perturbations are classified as relevant $(\Gamma_{0}<2)$ or irrelevant
$(\Gamma_{0}>2)$. We thus identify three distinction regions at $g_{1}=0$ :

– Relevant : $-\infty<g_{2}<-4$;
– Irrelevant : $-4<g_{2}<0$;
– Relevant : $\ 0<g_{2}<\infty$.

For the relevant regions the flows originate at $g_{1}=0$ and flow toward
increasing $g_{1}$. For the irrelevant region, the flows terminate at
$g_{1}=0$ arriving from positive $g_{1}$.

As we will see, based on the invariant $\mathcal{I}$, many of the flows that
originate at $g_{1}=0$ end up at $g_{1}=\infty$. Not all however, depending on
whether $g_{1}$ is real or imaginary, see below. But in the case of interest,
namely the sinh-Gordon model, flows indeed start at $g_{1}=0$ in the UV and
flow to $g_{1}=\infty$ in the IR. Whereas the dimensions of the perturbations
around $g_{1}=0$ are unambiguous as a function of $g_{2}$ since $g_{1}$ is an
obvious line of fixed points (see $\Gamma_{0}(g_{2})$ in (28)), the scaling
dimension of the perturbation at $g_{1}=\infty$ is less obvious. We propose
the following identification. Based on the duality of the beta functions (24),
flows at $g_{1}=0$ can be mapped into flows at $g_{1}=\infty$, where formally
if the beta functions are zero at $g_{1}=0$ they are also zero at
$g_{1}=\infty$ if one uses $(16/g_{1},16/g_{2})$ as coordinates. However the
minus sign in (24) implies the UV and IR are exchanged, since they are related
by $a\to 1/a$. We propose that along the flow the dimension of the
perturbation is given by the same functional form as $\Gamma_{0}(g_{2})$ for
$g_{2}$ as a function of the RG scale. To be more precise, we are assuming
that the dimension of the perturbation $\Gamma(t)$, where $t=\log a$ is the RG
time, satisfies $\Gamma(t)=\Gamma_{0}(g_{2}(t))$. Thus at $t=0$,
$\Gamma=\Gamma_{0}(g_{2}(0))=\Gamma_{0}({g_{2}^{\rm UV}})=\Gamma_{\rm UV}$. On
the other hand at $t=\infty$,

$\Gamma=\Gamma_{0}(g_{2}(\infty))\equiv\Gamma_{\rm IR}.$ (29)

The flow of $g_{2}(t)$ thus implies a relation between $\Gamma_{\rm UV}$ and
$\Gamma_{\rm IR}$.

For instance if a flow originates at $g_{1}=0$ from a relevant perturbation
with $\Gamma_{\rm UV}=\Gamma_{0}({g_{2}^{\rm UV}})$, and ${g_{2}^{\rm UV}}$
flows to ${g_{2}^{\rm IR}}$, then in the IR at $g_{1}=\infty$ we identify the
dimension as $\Gamma_{\rm IR}$ where

$\Gamma_{\rm IR}=\Gamma_{0}({g_{2}^{\rm IR}})=2\left(\dfrac{4-{g_{2}^{\rm
IR}}}{4+{g_{2}^{\rm IR}}}\right),~{}~{}~{}\mathrm{at}\ g_{1}=\infty.$ (30)

Let us make several remarks supporting our rather natural proposal (29) since
it will be essential in the following :

  * •

The equation (29) correctly predicts the exact relation between $\Gamma_{\rm
UV}$ and $\Gamma_{\rm IR}$ for massless flows that both begin and end at
$g_{1}=0$. These are the flows in FSZ1 ; FSZ2 , see below.

  * •

At $g_{1}=\infty$, the coupling $g_{2}$ stops flowing, i.e. remains constant,
as it does at $g_{1}=0$, which implies the dimensions $\Gamma_{\rm UV}$ and
$\Gamma_{\rm IR}$ are constant there. One can see this as follows. One can
eliminate $g_{1}$ and write the beta function in terms of $g_{2}$ and
$\mathcal{I}$ only:

$\beta_{g_{2}}=\dfrac{16\left(g_{2}^{2}-16\mathcal{I}(g_{2}-4)^{2}\right)\left(1-\mathcal{I}(g_{2}-4)^{2}\right)}{(g_{2}+4)^{2}}$
(31)

Now one has

$\mathcal{I}_{0}\equiv\mathcal{I}(g_{1}=0,g_{2})=\dfrac{g_{2}^{2}}{16(g_{2}-4)^{2}},~{}~{}~{}~{}~{}~{}\mathcal{I}_{\infty}\equiv\mathcal{I}(g_{1}=\infty,g_{2})=\dfrac{1}{(g_{2}-4)^{2}}.$
(32)

One sees that for both $\mathcal{I}=\mathcal{I}_{0}$ and
$\mathcal{I}=\mathcal{I}_{\infty}$, the beta function $\beta_{g_{2}}=0$, thus
$g_{2}$ is a constant in RG time both at $g_{1}=0$ and $g_{1}=\infty$.
Alternatively, $\beta_{g_{2}}$ vanishes either for
$16\mathcal{I}(g_{2}-4)^{2}=g_{2}^{2}$ or for $\mathcal{I}(g_{2}-4)^{2}=1$.
The former corresponds to $g_{1}=0$, the latter to $g_{1}=\infty$.

  * •

Since under the duality $g_{2}\to 16/g_{2}$ the IR and UV limits are exchanged
due to (24), one should expect that $\Gamma_{0}(16/g_{2})=-\Gamma_{0}(g_{2})$,
which is satisfied.

  * •

When $g_{2}=4$, $g_{2}$ does not flow at all since $\beta_{g_{2}}=0$ for all
$g_{1}$. Thus it must be that $\Gamma_{\rm UV}=\Gamma_{\rm
IR}=\Gamma_{0}(4)=0$. As we will see below, this corresponds to the $b=0$
point of the sinh-Gordon theory which is just a free massive boson.

Notice that at $g_{1}\to\infty$ in the IR, the theory might either be the
trivial massive theory, with all degrees of freedom frozen, or a non trivial
theory, potentially a conformally invariant theory.

###  IV.2 Identification of sinh and sine Gordon phases

It remains to identify where the above model with $g_{1},g_{2}$ corresponds to
the sinh-Gordon model. This identification clearly depends on whether we
assume the presence of a background charge or not, and this fact will be
important later. If we view the $\cosh$ potential as a perturbation of the
free gaussian field with no background charge, then $\Gamma_{0}=-2b^{2}$:

$b^{2}=\dfrac{(g_{2}-4)}{(g_{2}+4)}.$ (33)

Again – at the price of repeating ourselves – this identification relies on a
specifically chosen relation between the scaling dimension and the parameter
$b$ (which here assumes the absence of background charge). If this dimension
is positive we view the potential as being in a sine-Gordon regime
$g_{1}\cos(\sqrt{2}\beta\phi)$ with $\beta=ib$. There are now four distinct
regions:

\- Relevant : $-\infty<g_{2}<-4$, sinh-Gordon with $1<b^{2}<\infty$;
\- Irrelevant : $-4<g_{2}<0$, sine-Gordon with $1<\beta^{2}<\infty$;
\- Relevant : $0<g_{2}<4$, sine-Gordon with $0<\beta^{2}<1$;
\- Relevant : $4<g_{2}<\infty$, sinh-Gordon with $0<b^{2}<1$.

The above regions are the same as those already identified in BL . Note
already that the regions $b<1$ and $b>1$ are clearly distinguished.

For $g_{1}=0$ in the UV, the duality $g_{2}\to 16/g_{2}$ corresponds to $b\to
ib$, i.e. maps from the sinh-Gordon to sine-Gordon regimes. On the other hand,
the usual hypothetical sinh-Gordon duality $b\to 1/b$ corresponds to
$g_{2}\to-g_{2}$. However the latter is not a symmetry of the beta functions
and indicates that the RG properties of $0<b<1$ verses $b>1$ are indeed
different. This is one of the main points of this paper which we will
subsequently explore in more detail.

###  IV.3 Massless flows in the “imaginary” sine-Gordon model

Following the terminology in FSZ1 ; FSZ2  we refer to the sinh-Gordon action
(1) with $\mu\propto g_{1}$ imaginary as the “imaginary” sine-Gordon model. In
this case there are flows that both begin and end at $g_{1}=0$, indicating a
massless flow between two different CFT’s, both at $c=1$, which differ in
their radius of compactification $\beta$. Here since the flows both start and
end at $g_{1}=0$, there is no ambiguity in determining anomalous dimensions in
the UV nor the IR. This situation was already explained in BL  based on the
beta functions above, however we review it here since it represents a
prototype of the kinds of flows we will propose in the sinh-Gordon case.

In the sine-Gordon regime with small couplings $g_{1}$ and $g_{2}$,
$\mathcal{I}\propto(g_{1}^{2}+g_{2}^{2})$, thus the RG flows are approximately
circles. This implies that flows can both begin and end on the $g_{2}$ axis,
which is a massless flow as defined above. Such flows are straightforward to
analyze to all orders. For $g_{1}=0$, $\mathcal{I}=\mathcal{I}_{0}$ defined in
(32). Since $\mathcal{I}$ is preserved along the flow, one must have

$\dfrac{{g_{2}^{\rm UV}}}{{g_{2}^{\rm UV}}-4}=-\dfrac{{g_{2}^{\rm
IR}}}{{g_{2}^{\rm IR}}-4}~{}~{}\Longrightarrow~{}~{}{g_{2}^{\rm
IR}}=\dfrac{2{g_{2}^{\rm UV}}}{{g_{2}^{\rm UV}}-2}.$ (34)

In terms of $\beta$,

$\beta_{\rm IR}^{2}=\dfrac{\beta_{UV}^{2}}{2\beta_{UV}^{2}-1},$ (35)

which implies the dimensions of the perturbation in the UV verses IR are
related as follows

$\Gamma_{\rm IR}=\dfrac{\Gamma_{\rm UV}}{\Gamma_{\rm UV}-1}.$ (36)

For irrelevance in the IR, $\Gamma_{\rm IR}>2$, requires $0<g_{2}<4/3$ or
equivalently $1/2<\beta^{2}<1$, consistent with FSZ1 ; FSZ2 . A contour plot
of such a flow is shown in Figure 1. The existence of the flows and the
relation (36) have been conjectured long ago in FSZ1 ; FSZ2 . The fact that we
recover them and the correct relation (36) provides further support for the
effectiveness of the beta functions (23a) in understanding this kind of
physics.

Figure 1: Massless flow in the imaginary sine-Gordon model from ${g_{2}^{\rm
UV}}=1$ to ${g_{2}^{\rm IR}}=-2$. This corresponds to a flow from $\beta_{\rm
UV}=\sqrt{3/5}$ to $\beta_{\rm IR}=\sqrt{3}$. What is shown is a contour plot
of the RG invariant $\mathcal{I}$.

###  IV.4 RG flows in the sinh-Gordon model: $b<1$ verses $b>1$.

Here we consider flows in the different regimes $0<b<1$ and $1<b<\infty$ for
the sinh-Gordon with imaginary $\mu$. As explained at the end of this section,
the case of real $\mu$ is not very different. Recall these regimes correspond
to $g_{2}>4$ and $g_{2}<-4$ at $g_{1}=0$, respectively. For such large
coupling $g_{2}$, constant $\mathcal{I}$ is not at all approximated by a
circle as in the sine-Gordon case. All flows originating at $g_{1}=0$ end up
at $g_{1}=\infty$. As we now explain, there are two cases which have rather
different behavior, and correspond precisely to $0<b<1$ verses $b>1$. We need
to relate $g_{2}$ in the UV and IR. Since $\mathcal{I}$ is preserved along the
flow, one must have $\mathcal{I}_{0}({g_{2}^{\rm
UV}})=\mathcal{I}_{\infty}({g_{2}^{\rm IR}})$. A fortunate and promising
result that has not been pre-programmed into the above beta functions is that
the two cases correspond precisely to weak verses strong coupling:

$\bullet$ For $0<b<1$ : The flows originating at $(g_{1},g_{2})=(0,{g_{2}^{\rm
UV}}>4)$ end up at $(\infty,{g_{2}^{\rm IR}}))$ where

$\dfrac{{g_{2}^{\rm UV}}}{4({g_{2}^{\rm UV}}-4)}=\dfrac{1}{({g_{2}^{\rm
IR}}-4)}~{}~{}~{}~{}\Longrightarrow~{}~{}{g_{2}^{\rm IR}}=\dfrac{8({g_{2}^{\rm
UV}}-2)}{{g_{2}^{\rm UV}}}.$ (37)

Expressing this in terms of the scaling dimensions :

$\Gamma_{\rm IR}=\dfrac{\Gamma_{\rm UV}}{1-\Gamma_{\rm UV}}~{}.$ (38)

Identifying the parameter $b$ using the relation $\Gamma=-2b^{2}$ yields
$b_{\rm IR}^{2}={b_{\rm UV}^{2}}/({1+2b_{\rm UV}^{2}})$. Both $b_{\rm UV}$ and
$b_{\rm IR}$ then remain in the weak coupling region $0<b<1$. This should be a
massive flow since $\Gamma_{\rm IR}$ still signifies a relevant perturbation,
although with an imaginary coupling $\mu$. One instance of it is shown in
Figure 2.

Figure 2: A massive flow in the imaginary sine-Gordon model from ${g_{2}^{\rm
UV}}=5$ to ${g_{2}^{\rm IR}}=24/5$. This corresponds to a flow from $b_{\rm
UV}=1/3$ to $b_{\rm IR}=1/\sqrt{11}$. What is shown is a contour plot of the
RG invariant $\mathcal{I}$ where the $x,y$ axes corresponds to $g_{1},g_{2}$.

$\bullet$ For $1<b<\infty$ : The flows originating at
$(g_{1},g_{2})=(0,{g_{2}^{\rm UV}}<-4)$ end up at $(\infty,{g_{2}^{\rm IR}}))$
where there is just a change of sign in (37):

$\dfrac{{g_{2}^{\rm UV}}}{4({g_{2}^{\rm UV}}-4)}=-\dfrac{1}{({g_{2}^{\rm
IR}}-4)}~{}~{}~{}~{}\Longrightarrow~{}~{}{g_{2}^{\rm
IR}}=\dfrac{16}{{g_{2}^{\rm UV}}}.$ (39)

Expressing this in terms of the dimensions $\Gamma$:

$\Gamma_{\rm IR}=-\Gamma_{\rm UV}.$ (40)

Now in this case, $\Gamma_{\rm IR}>2$, i.e. irrelevant, and this is thus a
massless flow. This flow in the sinh-Gordon region is rather analogous to the
massless flows in the sine-Gordon model described above, since in the IR they
both end up in the irrelevant regime where $-4<{g_{2}^{\rm IR}}<0$. The
details of the flow are however more intricate compared to the previous case.
Rather the flows start at $(g_{1},g_{2})=(0,{g_{2}^{\rm UV}})$ and first flow
to $g_{2}=-\infty$. This occurs at $g_{1}=g_{1}^{*}$ such that
$\mathcal{I}(0,g_{2})=\mathcal{I}(g_{1}^{*},\pm\infty)$, that is
$g_{1}^{*}={8\sqrt{4-2g_{2}}}/{|g_{2}|}$. Using the cylindrical topology
proposed in BL  which identifies $g_{2}$ with $-g_{2}$ at $|g_{2}|=\infty$,
the flow then continues from $g_{2}=\infty$ to $g_{1}=\infty$ but with a
different $g_{2}={g_{2}^{\rm IR}}$, which is actually the dual of $g_{2}$.
This implies the flow

$(g_{1},g_{2})=(0,g_{2})~{}~{}~{}{\overset{UV\to
IR}{\longrightarrow}}~{}~{}~{}(\infty,16/g_{2}).$ (41)

The self-dual point $b_{\rm UV}=1$ flows to a marginally irrelevant
perturbation in the IR, i.e. $\Gamma_{\rm IR}=2^{+}$, which seems desirable if
it is indeed a massless flow. Such flows are sketched in Figure 3 and were
verified numerically.

Figure 3: Sketch of the massless flow in the imaginary sinh-Gordon model from
${g_{2}^{\rm UV}}<-4$ to ${g_{2}^{\rm IR}}=16/{g_{2}^{\rm UV}}$. The value
$g_{1}^{*}$ is given in the text: $g_{1}^{*}={8\sqrt{4-2g_{2}}}/{|g_{2}|}$.

We now explain why the above RG flows cannot be properly interpreted if we
stick to the relation $\Gamma=-2b^{2}$, and argue that they acquire a natural
interpretation if we introduce the background charge $Q_{\infty}=b+1/b-2$.
This is one of the main points of this paper.

Let us present supporting arguments for the introduction of a background
charge. If we continue to identify the scaling dimension $\Gamma_{0}$ with
$-2b^{2}$ then the above relation (40) implies the peculiarity of $b_{\rm
IR}=ib_{\rm UV}$, i.e. becoming imaginary. The flow to imaginary $b$ seems
unsatisfactory since it takes us out of the proper sinh-Gordon regime manifold
and into the sine-Gordon one. This would correspond, roughly speaking, from a
flow from a non-compact model to a compact one. We suggest that this problem
arose since we identified the coupling $b$ with the dimension
$\Gamma_{0}=-2b^{2}$ which assumed there was no background charge.
Introduction of the background charge $Q_{\infty}$ in (2) can resolve this
issue.

The perturbative calculations that led to the above beta functions (23a) did
not incorporate a background charge. However the flows do predict dimensions
of operators $\Gamma_{\rm UV,IR}$ regardless of the free gaussian
identification $\Gamma=-2b^{2}$. Incorporating a background charge should just
modify this identification, while preserving the flows in $(g_{1},g_{2})$. We
can indeed modify this identification, but still must preserve the relation
$\Gamma_{\rm IR}=-\Gamma_{\rm UV}$, since the latter is predicted by the beta
functions regardless of the identification relating $\Gamma$ and $b$.

We require that both the UV and IR are in the same regime of “$b$”. Let us
identify $\Gamma_{0}$ with the dimension proposed in Section III

$\Gamma_{0}=2-4b$ (42)

which was based on a background charge $Q_{\infty}=b+1/b-2$ in the region
$b>1$. The identification (42) modifies the relation between
$g_{2}^{\mathrm{UV}}$ and $b_{\mathrm{UV}}$, as
$4b_{\mathrm{UV}}=2-\Gamma_{0}(g_{2}^{\mathrm{UV}})$, which then reads
$b_{\mathrm{UV}}=g_{2}^{\mathrm{UV}}/(4+g_{2}^{\mathrm{UV}})$. For
$g_{2}^{\mathrm{UV}}<-4$, we still have $b_{\mathrm{UV}}$ in the strong
coupling regime $b_{\mathrm{UV}}>1$. Then $\Gamma_{\rm IR}=-\Gamma_{\rm UV}$
in (40) implies the simple relation

$b_{\rm IR}=1-b_{\rm UV}.$ (43)

This has the desired property that the whole region $b_{\rm UV}>1$ is mapped
to $b_{\rm IR}<0$ which excludes the usual sinh-Gordon region $0<b<1$.
Importantly, note that $b_{\rm UV}=1$ is mapped to $b_{\rm IR}=0$, where
$\Gamma_{\rm IR}=2$, thus the $b_{\rm UV}=1$ theory is marginally irrelevant
in the IR, consistent with a massless flow.

One can also argue that the background charge must be
$Q_{\infty}=\alpha_{1}(b+1/b-2)$ with $\alpha_{1}=1$ as follows. We identify
$\Gamma=2b(Q_{\infty}-b)$ based on above considerations. The RG flows predict
$\Gamma_{\rm IR}=-\Gamma_{\rm UV}$, which is a complicated relation between
$b_{\rm IR}$ and $b_{\rm UV}$ for generic value of $\alpha_{1}$. One can check
that unless $\alpha_{1}=1$, for $b_{UV}>1$, $b_{\rm IR}$ is generally complex.
Only for $\alpha_{1}=1$ does one have the simple relation $b_{\rm IR}=1-b_{\rm
UV}$.

###  IV.5 Remarks on real verses imaginary $\mu$

Let use make a few remarks concerning the $g_{1}$ real case originally
considered in BL . First of all, the massless flows for the imaginary sine-
Gordon theory no longer exist, since for small $g_{1,2}$ the RG trajectories
based on the RG invariant $\mathcal{I}$ are no longer approximately circles,
but rather hyperbolas. However the flows that begin at $g_{1}=0$ and end up at
$g_{1}=\infty$ have the same endpoints, and the relations between $\Gamma_{\rm
UV}$ and $\Gamma_{\rm IR}$ presented above remain the same. This can be seen
from the fact that equations (32) are the same. However the detailed
trajectories are different. One can easily see with contour plots of
$\mathcal{I}$ for real verses imaginary $g_{1}$ that the topologies of the
flows in Figures 2 and 3 are essentially interchanged.

##  V Application to the freezing transition in disordered systems

We now make contact with disordered systems and explain the relation between
the above sinh-Gordon model, with imaginary coupling, and Dirac fermions in
random gauge field. See also Doussal ; Doussal2 . This was actually our
motivation when we started looking at this problem fifteen years ago, and left
it aside for a short while.

###  V.1 Dirac fermions in a random U(1) gauge field

We consider two-component Dirac fermions in a random gauge field $A_{\mu}$ in
two spatial dimensions $(x,y)$ plus time. Defining complex spatial coordinates
$z=(x+iy)/\sqrt{2}$ and ${\overline{z}}=(x-iy)/\sqrt{2}$, the model is defined
by the random hermitian hamiltonian

$H=\left(\begin{matrix}0&-i\partial_{\overline{z}}+A_{\overline{z}}\\\
-i\partial_{z}+A_{z}&0\\\ \end{matrix}\right).$ (44)

The probability distribution will be specified below.

The Green functions, Fourier transformed in time to energy ${\cal E}$, are
given by functional integrals with respect to the action

${\cal S}=i\int\dfrac{d^{2}x}{2\pi}\Psi^{\dagger}\left(H-{\cal E}\right)\Psi.$
(45)

Introducing component fields as follows,
$\Psi=\left(\begin{smallmatrix}\psi_{+}\\\ \overline{\psi}_{+}\\\
\end{smallmatrix}\right)$ and
$\Psi^{\dagger}=\left(\overline{\psi}_{-},\psi_{-}\right)$, one finds

${\cal
S}(\Psi,A)=\int\dfrac{d^{2}x}{2\pi}\left[\,\overline{\psi}_{-}(\partial_{z}-iA_{z})\overline{\psi}_{+}+\psi_{-}(\partial_{\overline{z}}-iA_{\overline{z}})\psi_{+}+i{\cal
E}\,(\overline{\psi}_{-}\psi_{+}+\psi_{-}\overline{\psi}_{+})\right].$ (46)

Disorder averaged correlation functions $\langle{\cal O}\rangle$ are then
defined as functional integrals over $A$:

$\overline{\langle{\cal O}\rangle}=\int DA\,P[A]\,\langle{\cal O}\rangle_{A}$
(47)

where the probability distribution for $A$ is taken to be gaussian:

$P[A]=\exp\left(-\dfrac{1}{g}\int\dfrac{d^{2}x}{2\pi}\,A_{z}A_{\overline{z}}\right).$
(48)

The coupling constant $g$ is a measure of the strength of the disorder. In
(47), $\langle{\cal O}\rangle_{A}$ is the correlation function in a given
realization of the disorder :

$\langle{\cal O}\rangle_{A}=\dfrac{1}{Z(A,{\cal E})}\int
D\Psi\,e^{-S(\Psi,A)}\,{\cal O}$ (49)

where $Z(A,{\cal E})$ is the partition function.

###  V.2 Map to the sinh-Gordon model

It is convenient to parameterize the gauge field in terms of a scalar field
$\eta$ as follows333In $\mathbb{R}^{2}$, any gauge potential can be decomposed
as
$A_{\mu}=\partial_{\mu}\vartheta+\tfrac{1}{2}\epsilon_{\mu\nu}\partial_{\nu}\eta$.
But the pure gauge part $A_{\mu}=\partial_{\mu}\vartheta$ can be gauged away
in (46) and only the component
$A_{\mu}=\tfrac{1}{2}\epsilon_{\mu\nu}\partial_{\nu}\eta$ matters. :

$A_{\mu}=\tfrac{1}{2}\epsilon_{\mu\nu}\partial_{\nu}\eta,~{}~{}~{}~{}~{}\Longrightarrow~{}~{}~{}P[\eta]=\exp\left(-\dfrac{1}{4g}\int\dfrac{d^{2}x}{4\pi}\left(\partial_{\mu}\eta\right)^{2}\right).$
(50)

The coupling of the fermions to the gauge field can then be removed by the
chiral gauge transformation:

$\overline{\psi}^{\prime}_{+}=e^{\eta/2}\,\overline{\psi}_{+},~{}~{}~{}\overline{\psi}^{\prime}_{-}=e^{-\eta/2}\,\overline{\psi}_{-},~{}~{}~{}\psi^{\prime}_{-}=e^{\eta/2}\,\psi_{-},~{}~{}~{}\psi^{\prime}_{+}=e^{-\eta/2}\,\psi_{+}~{},$
(51)

and the action becomes

${\cal
S}=\int\dfrac{d^{2}x}{2\pi}\left[\overline{\psi}^{\prime}_{-}\partial_{z}\overline{\psi}^{\prime}_{+}+\psi^{\prime}_{-}\partial_{\overline{z}}\psi^{\prime}_{+}+i{\cal
E}\left(e^{\eta}\,\overline{\psi}^{\prime}_{-}\psi^{\prime}_{+}+e^{-\eta}\,\psi^{\prime}_{-}\overline{\psi}^{\prime}_{+}\right)\right].$
(52)

To make further progress, we first consider ${\cal E}$ to be very small, and
later restore it as a perturbation. When ${\cal E}=0$, the jacobian which
arises in passing from $\Psi$ to $\Psi^{\prime}$ in the functional integral
precisely cancels the $1/Z$ factor in (49). This is easily seen by bosonizing
the fermions $\Psi$ with a single boson $\phi$ so that the action (46) becomes
(when ${\cal E}=0$):

${\cal
S}=\int\dfrac{d^{2}x}{4\pi}\Big{(}\dfrac{1}{2}\left(\partial\phi\right)^{2}+i\partial\eta\partial\phi\Big{)}.$
(53)

The functional integrals over $\Psi$ and $\Psi^{\prime}$ are then simply
related by the shift $\phi\to\phi^{\prime}-i\eta$.

When ${\cal E}=0$ the functional integrals over $\Psi^{\prime}$ can be done
and do not introduce any new $\eta$ dependence. To restore the ${\cal E}$
perturbation, we make a mean field approximation and replace the
$\Psi^{\prime}$ fermion bilinears by their one-point functions in a finite
geometry of size $L$. Since the fermions have dimension $1/2$, we have :

$\langle\overline{\psi}^{\prime}_{-}\psi^{\prime}_{+}\rangle\sim\langle\psi^{\prime}_{-}\overline{\psi}^{\prime}_{+}\rangle\sim
1/L~{}.$ (54)

One is finally left with the functional integral over $\eta$. Rescaling
$\eta=\sqrt{2g}\,\phi$, one finds the sinh-Gordon action

${\cal S}[\phi]=\int
d^{2}x\left(\dfrac{1}{8\pi}\left(\partial\phi\right)^{2}+2\mu\cosh\big{(}\sqrt{2}\,b\,\phi\big{)}\right)~{},$
(55)

where

$b=\sqrt{g},~{}~{}~{}~{}\mu=i\,\dfrac{{\cal E}}{2\pi L}~{}.$ (56)

The density of states operator is the one that couples to ${\cal E}$, which we
chose to normalize as follows :

$\rho\equiv\dfrac{1}{L}\cosh\big{(}\sqrt{2}\,b\,\phi\big{)}~{}.$ (57)

By definition one has $\overline{\langle\rho\rangle}=\int D\phi
e^{-S[\phi]}\,\rho$.

###  V.3 Multi-fractal density of states exponents

We first review some general standard definitions of exponents characterizing
the density of states. Let $\rho(x)$ denote the density of states field
operator. The physical density of states is its vacuum expectation value, i.e.
the 1-point function denoted as $\langle\rho\rangle$, and depends on the
realization of the disorder. Let $\overline{\langle\rho\rangle}$ denote the
disorder averaged quantity. For a system of size $L$, one defines the
fundamental exponent $\Gamma_{1}$ as

$\overline{\langle\rho\rangle}\sim L^{-\Gamma_{1}}.$ (58)

In other words, the exponent $\Gamma_{1}$ is just the anomalous dimension of
the operator $\rho$ in the disorder averaged theory.

Also of interest are multi-fractal exponents $\Gamma_{q}$ defined as follows :
$\Gamma_{q}$ is defined as the anomalous dimension of the $q$-th moment of
$\rho$:

$\Gamma_{q}=[[\,\overline{\langle\rho\rangle^{q}}\,]]~{},$ (59)

where we use the same notation as above, where $[[X]]$ denotes the scaling
dimension of $X$ in inverse length units.

Because it is related to the multi-fractal spectrum of the density
$\langle\rho(x)\rangle$, or of the associated measure
$\langle\rho(x)\rangle\,dx$, a related quantity that is often studied is the
normalized ratio

$P^{(q)}=\dfrac{\int d^{2}x\,\overline{\langle\rho(x)\rangle^{q}}}{\left(\int
d^{2}x\,\overline{\langle\rho(x)\rangle}\right)^{q}}~{}.$ (60)

Simple scaling leads to

$P^{(q)}\sim L^{-\tau(q)}$ (61)

where

$\tau(q)=\Gamma_{q}-q\Gamma_{1}+2(q-1).$ (62)

Legendre transform of $\tau(q)$ gives access to the spectrum of multi-fractal
dimensions of the density $\langle\rho(x)\rangle$.

###  V.4 Multi-fractal spectrum

Returning to our model of interest, using the mapping to the sinh-Gordon model
and (57), we have

$\Gamma_{1}(g)=1+\gamma(g)~{},$ (63)

where the “$1$” comes from the $1/L$ in (57), and $\gamma(g)$ the scaling
dimension of $\cosh(\sqrt{2g}\,\phi)$,

$\gamma(g)=[[\cosh(\sqrt{2g}\,\phi)]]~{}.$ (64)

For higher $q$, since the leading term in $\rho^{q}$ is
$\cosh(q\sqrt{2g}\,\phi)/L^{q}$, one has

$\Gamma_{q}(g)=q+[[\cosh(q\sqrt{2g}\,\phi)]]~{}.$ (65)

Since in the above equation the $\cosh$-operator is related to $\rho$ by $g\to
q^{2}g$, this immediately leads to the fundamental equation

$\Gamma_{q}(g)=\Gamma_{1}(q^{2}g)+q-1.$ (66)

The latter implies

$\tau(q)=\Gamma_{1}(q^{2}g)-q\Gamma_{1}(g)+3(q-1).$ (67)

Given (63), one sees that everything boils down to the dimension of the
$\cosh(\sqrt{2}\,b\phi)$ operator in the sinh-Gordon theory. According to our
proposal for the freezing transition in sinh-Gordon, we have
$\Gamma_{1}(g)=1-2g$ for $g<1$ and $\Gamma_{1}(g)=3-4\sqrt{g}$ for $g>1$.

Transitions in the variable $q$ are thereby related to transitions in
$b=\sqrt{g}$. Using our proposal for a freezing transition in the sinh-Gordon
model (20), the two transition points are then $b=g=1$ and
$q^{2}g=q^{2}b^{2}=1$. There are thus $4$ distinct regimes. In terms of $g$
and $q$, they are:

$\displaystyle~{}g<1,~{}~{}q<1/\sqrt{g}:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau(q)=2(q-1)(1-qg)$
(68a)
$\displaystyle~{}g<1,~{}~{}q>1/\sqrt{g}:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau(q)=2q(1-\sqrt{g})^{2}$
(68b)
$\displaystyle~{}g>1,~{}~{}q<1/\sqrt{g}:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau(q)=-2(1-q\sqrt{g})^{2}$
(68c)
$\displaystyle~{}g>1,~{}~{}q>1/\sqrt{g}:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau(q)=0.$
(68d)

This agrees with known results Doussal ; Doussal2 ; Castillo .

##  VI Summary and discussion

We have presented a specific proposal for the behavior of the sinh-Gordon
model above the self-dual point $b>1$ that is quite different from the
analytic continuation $b\to 1/b$ of the well-understood properties of the
massive theory for $0<b<1$. The main properties of this theory is that unlike
the $0<b<1$ region it has a non-zero background charge $Q_{\infty}$ given in
(2). The theory is massless but not conformally invariant, but rather is a
relevant perturbation in the UV that flows to another CFT in the IR, arriving
there via an irrelevant operator. We provided two supporting arguments. The
first was based on the beta functions in Moriconi ; BL , which are ultimately
based on perturbation theory for the sinh-Gordon action, and do not show a
$b\to 1/b$ symmetry, and clearly predict different RG flows for $b<1$ verses
$b>1$. The second is that our proposal correctly reproduces known exact
results for a Dirac fermion in a random magnetic field, in particular all the
transitions in the multi-fractal exponents.

Since we are not able to provide a complete and rigorous derivation of our
proposal at this stage, it would be interesting to investigate the problem by
other means, for instance from a lattice formulation of the sinh-Gordon model,
or using continuous network tensor techniques adapted to field theory
CiracVerstraete ; Tilloy , or even a rigorous probabilistic construction as in
LiouvilleChaos . We should say that the validity of the beta functions we used
in Section IV.1 and our interpretation of scaling dimensions at
$g_{1}=\infty$, namely based on (29), could benefit from closer scrutiny, even
though we showed how these beta functions can reproduce known exact results on
massless flows in the sine-Gordon model FSZ1 ; FSZ2 . We refer to the
Introduction for further remarks about this.

If our proposal is indeed correct, it remains to determine the S-matrices for
the massless flow when $b>1$ along the lines formulated in ZZMassless . This
is beyond the original scope of this paper, however there are some natural
guesses. Letting $L$ and $R$ signify left verses right movers as in ZZMassless
, it is likely that the LL and RR S-matrices are $S_{\rm LL}=S_{\rm RR}=S_{\rm
shG}$ where $S_{\rm shG}$ is the function of rapidity in (3). This would
guarantee that in the IR, $c=1$. It remains to specify left-right scattering
$S_{\rm LR}$ which controls the UV. It is natural to consider $S_{\rm
LR}=S_{\rm shG}$ here also, however there are clearly other possibilities to
be explored, such as the very simplest possibility $S_{\rm
LR}(\theta)=-\tanh\left(\tfrac{\theta}{2}-\tfrac{i\pi}{4}\right)$. Clearly
more work needs to be done in this direction.

There are other possible applications of the freezing transition that our work
may shed some light on. An obvious one is to more complicated disordered
systems such as the quantum Hall transition. We also mention that it has been
applied to extreme values of the Riemann zeta function Keating1 ; Keating2 .

##  VII Acknowledgements

We would like to thank Pierre Le Doussal, Giuseppe Mussardo, Kostas Sfetos,
Kostas Siampos, Germán Sierra and Alyosha Zamolodchikov for discussions.
Alyosha first suggested to us long ago the check of our beta functions based
on comparison with massless flows in the imaginary sine-Gordon model. AL is
thankful for the support of the Ecole Normale Supérieure in Paris and the
organizers of the celebration of DB’s 60-th birthday which led to a re-
examination of this work we started and abandoned 15 years ago; he also wishes
to thank the Scuola Internazionale Superiore di Studi Avanzati (SISSA) in
Trieste, Italy, for support while this work was completed.

##  Appendix A Semi-classical freezing and the Manning condensation

Let us imagine computing semi-classically the one-point function of an
exponential operator in the sinh-Gordon theory. Restoring $\hbar$ so that the
action becomes ${\cal S}\to{\hbar}^{-1}{\cal S}$, the one-point function of
the operator $\exp({\sqrt{2}a}{\hbar}^{-1}\phi)$, $a>0$, located at the
position $x_{0}$, is represented by the functional integral

$\int D\phi\,e^{-{\hbar}^{-1}\big{(}{\cal S}-\sqrt{2}a\phi(x_{0})\big{)}}~{}.$
(69)

In the semi-classical limit $\hbar\to 0$, the integral is dominated by the
saddle point field configuration $\phi_{\mathrm{cl}}$, solutions of

$-\dfrac{1}{4\pi}\Delta_{x}\phi_{\mathrm{cl}}(x)+2\widehat{\mu}\,\sinh(\sqrt{2}b\phi_{\mathrm{cl}}(x))=\sqrt{2}a\,\delta^{(2)}(x-x_{0})~{},$
(70)

where $\Delta_{x}$ is the Laplacian in 2D and $\delta^{(2)}(x-x_{0})$ the
Dirac measure at $x_{0}$ and $\widehat{\mu}=\sqrt{2}b\mu$. To take care of the
$\delta$-function source, we should have
$\phi_{\mathrm{cl}}(x)\simeq-\sqrt{2}a\log|x-x_{0}|^{2}$ as $|x|\to x_{0}$.
Thus we set $\phi_{\mathrm{cl}}(x)=-\sqrt{2}a\log|x-x_{0}|^{2}+\varphi(x)$,
with $\varphi(x)$ vanishing near $x_{0}$. We take $\varphi$ decreasing as a
power law, so that

$\phi_{\mathrm{cl}}(x)=-\sqrt{2}a\log|x-x_{0}|^{2}+\mathrm{const.}\,|x-x_{0}|^{\sigma}+\cdots~{},$
(71)

with $\sigma>0$ (so that $\varphi$ vanishes at $x_{0}$) and where the dots
refer to subleading terms as $x$ approaches $x_{0}$. The exponent $\sigma$ is
found by matching the leading terms in $\Delta_{x}\phi_{\mathrm{cl}}$ and in
$\sinh(\sqrt{2}b\phi_{\mathrm{cl}})$. This yields

$|x-x_{0}|^{\sigma-2}\sim
e^{-2ab\log|x-x_{0}|^{2}}=|x-x_{0}|^{-4ab}\quad\Longrightarrow~{}\sigma=2(1-2ab)~{}.$
(72)

Since we should have $\sigma>0$, this is possible only for $a<a_{c}=1/2b$. For
$a>a_{c}$, the operator $\exp({\sqrt{2}a}{\hbar}^{-1}\phi)$ is actually
screened such that its effective weight $a_{\mathrm{eff}}$ at large scale is
$a_{c}$.

This semi-classical computation indicates the possibility of a freezing
transition. For any fixed sinh-Gordon parameter $b$, the exponential operators
$\exp({\sqrt{2}a}\,\phi)$ are well-defined for $a<a_{c}$ only, for some
critical value $a_{c}$, but they get frozen for $a>a_{c}$ to the critical
exponential operator $\exp({\sqrt{2}a_{c}}\,\phi)$ with critical weight
$a_{c}$. In view of the symmetry relation (14), valid in Liouville theory, it
is tempting to propose that $a_{c}=Q_{\infty}/2$. This is compatible with the
semi-classical limit $a_{c}\simeq 1/2b$ for $b\to 0$.

This phenomena is known in the physics of polyelectrolyte solutions as the
Manning condensation Manning . Imagine considering a positively charged
polymer, say an ADN, immersed in a polyelectrolyte made of positive and
negative charged ions, and ask what is the electrostatic potential for this
system. If we imagine the polymer to be straight along the $z$-axis, then (70)
is the Poisson-Boltzmann equation for this electrostatic problem in the 2D
transverse directions. If the charge density of the polymer is too high,
larger than a critical value $a_{c}$, it is screened by oppositely charged
ions which occupy a cylindrical volume around the polymer of diameter $r_{c}$,
so that the system formed by the polymer and these counter-ions behaves at a
distance higher than $r_{c}$ like a polymer of critical charge density
$a_{c}$. This is the Manning’s screening effect.

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