# Noncommutative effective LQC: inclusion of potential term.

Luis Rey Díaz-Barrón<EMAIL_ADDRESS>Unidad Profesional Interdisciplinaria de
Ingeniería Campus Guanajuato del Instituto Politécnico Nacional.
Av. Mineral de Valenciana No. 200, Col. Fraccionamiento Industrial Puerto
Interior, C.P. 36275, Silao de la Victoria, Guanajuaṫo, México.  Abraham
Espinoza-García<EMAIL_ADDRESS>Unidad Profesional Interdisciplinaria de
Ingeniería Campus Guanajuato del Instituto Politécnico Nacional.
Av. Mineral de Valenciana No. 200, Col. Fraccionamiento Industrial Puerto
Interior, C.P. 36275, Silao de la Victoria, Guanajuaṫo, México.  S. Pérez-
Payán<EMAIL_ADDRESS>Unidad Profesional Interdisciplinaria de Ingeniería
Campus Guanajuato del Instituto Politécnico Nacional.
Av. Mineral de Valenciana No. 200, Col. Fraccionamiento Industrial Puerto
Interior, C.P. 36275, Silao de la Victoria, Guanajuaṫo, México.  J. Socorro
<EMAIL_ADDRESS>Departamento de Física, DCeI, Universidad de
Guanajuato-Campus León, C.P. 37150, León, Guanajuato, México.

###### Abstract

We construct and study a simple noncommutative scheme (theta-deformation) for
the effective Loop Quantum Cosmology of the flat Friedmann-Lemaître-Robertson-
Walker model in the presence of a homogeneous scalar field $\phi$ with a
potential $\mathcal{V}(\phi)=\frac{1}{2}m^{2}\phi^{2}$. We first conduct a
simple analysis from the corresponding Hamilton equations of motion
considering a generic term $\mathcal{V}(\phi)$. It is observed that the
characteristic Big Bounce of Loop Quantum Cosmology is preserved under such
noncommutative extension. When specializing to the quadratic case, numerical
solutions to the corresponding Hamilton equations exhibiting an early
inflationary epoch with a sufficiently large number of e-foldings are found.
It is concluded that, in this noncommutative setup, solutions exist which are
in the overall compatible with the early universe predicted by standard
(effective) Loop Quantum Cosmology (i.e. a bouncing and inflationary early
universe). The issue of the genericness of a sufficiently long inflationary
period on the space of solutions in this noncommutative construct remains to
be addressed.

##  I Introduction

The somewhat recent investigations connes-douglas , seiberg-witten  resulted
in the resurrection of the noncommutativity paradigm, which had remained
burried for roughly five decades. The original idea can be traced back to the
work of Snyder snyder , where noncommutativity in fixed Minkowski spacetime is
proposed as a mechanism to regularize quantum field theory. On the other hand,
the theoretical evidence provided by leading approaches to quantum gravity
(e.g. string theory, loop quantum gravity (LQG) —a standard monograph is
thiemann ) that the fundamental structure of spacetime is expected to
incorporate some sort of discreteness, has motivated the use of the
noncommutativity paradigm to approximately model such a granular picture, and
study its consequences (see the reviews nekrasov ,szabo  for a string theory
community perspective of noncommutative field theory). Although most of the
noncommutative constructs have relied upon a fixed flat spacetime background,
some attempts to incorporate noncommutativity directly into the full
gravitational field have been performed (see, for instance, mofat -kober ).
The inherent complexity of these proposals have led theoreticians to consider
toy models in which the gravitational field can be described by a finite
number of degrees of freedom in order to extract, in a simple way, some of the
physical consequences of noncommutativity. In the seminal investigation
compean3  the authors implemented a star product on the Wheeler-DeWitt
equation of the Kantowski-Sachs anisotropic model in order to study the
effects of an assumed noncommutativity (theta deformation) in the
corresponding quantum minisuperspace. On the other hand, in barbosa , the
authors considered the effects of noncommutativity already at the classical
configuration manifold of the Kantowski-Sachs spacetime. These two
investigations spurred noncommutative extensions of relevant cosmological and
astrophysical scenarios (e.g. wally-1 -Sinuhe_2 ), at both classical and
quantum levels.

The present research work is a follow up to hindawi  and ijmpd , in which a
canonical noncommutativity at the effective scheme of the loop quantum
cosmology (LQC) of the flat Friedmann-Lemaître-Robertson-Walker (FLRW) model
in the presence of a free homogeneous scalar field was considered. One of the
main motivations of the work hindawi  was to study to what extend a simple
theta deformation at the so called effective scheme could prompt radical
modifications of key features of the LQC paradigm (e.g. singularity resolution
through a Big Bounce). It was concluded that a triad-scalar field momentum
theta-deformation does not modify the scalar field energy density,
furthermore, such noncommutativity allows to maintain the single Big Bounce
(associated to the maximum value $\rho_{c}$ of the energy density) of the
standard LQC paradigm. Additionally, in ijmpd  it was observed that as a
result of a configuration sector theta-deformation the a priori free scalar
field $\phi$ acquires an effective potential, and the question of whether this
could prompt an inflationary phase with enough e-foldings was addressed. The
answer was in the negative. On the other hand, it is well known that the
inclusion of a homogeneous scalar field with quadratic potential in the
effective LQC of the flat FLRW results in a period of inflation with enough
e-foldings, and that this is a rather generic feature of the solutions: the a
priori probability of picking a solution which generates an inflationary
period with at least 68 e-foldings is very close to one sloan . Therefore, the
(effective) LQC paradigm not only resolves the classical initial singularity,
but it is also capable of reproducing the inflation mechanism rather
generically. The present investigation aims at establishing whether
inflationary solutions exist which do not depart radically from the early
universe depicted by standard effective LQC (for the case of a quadratic
potential term), when implementing a theta-deformation in the momentum sector
of phase space.

The manuscript is organized as follows. Section II is devoted to the
construction and study of a noncommutative extension of the standard case
(which is briefly reviewed in this Introduction), but considering a generic
potential term $\mathcal{V}(\phi)$. We undertake a simple analysis via the
Hamiltonian constraint and the equations of motion, which enables us to single
out a particular representation of the considered noncommutativity. Next, in
section III, we specialize the discussion of section II to a quadratic
potential. Here, numerical solutions which exhibit a sufficiently long early
inflationary epoch are presented. Section IV is devoted to make some remarks
and state some conclusions.

In the following we briefly review the effective LQC paradigm. In the LQC
literature, it is well known (see, for instance, the reviews bojowald-lrr ,
lqc-status-report , and references therein) that in the case of the flat FLRW
cosmology in the presence of a scalar field $\phi$ with potential term

$\mathcal{V}(\phi)=\frac{1}{2}m^{2}\phi^{2},$ (1)

the effective Hamiltonian takes the form

$\mathcal{H}_{eff}=-\frac{3}{8\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta)V+\frac{(p_{\phi})^{2}}{2V}+\frac{1}{2}m^{2}\phi^{2}V.$
(2)

The associated Hamilton equations are

$\displaystyle\dot{\beta}$ $\displaystyle=$
$\displaystyle-\frac{3}{\gamma\lambda^{2}}\sin^{2}(\lambda\beta)+4\pi G\gamma
m^{2}\phi^{2},$ (3) $\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{p_{\phi}}{V},$ (4) $\displaystyle\dot{V}$ $\displaystyle=$
$\displaystyle\frac{3}{\gamma\lambda}V\sin(\lambda\beta)\cos(\lambda\beta),$
(5) $\displaystyle\dot{p}_{\phi}$ $\displaystyle=$ $\displaystyle-m^{2}V\phi.$
(6)

This so called effective scheme is based on a geometrical formulation of
quantum mechanics (QM) ashtekar-gfqm . That this semiclassical approximation
reproduces remarkably well the quantum behavior given by full LQC has been
established via numerical simulations in diener1 .

With the help of the Hamiltonian constraint, we readily note:

  * •

The equation of motion for $\beta$ indicates that this dynamical variable is a
decreasing function of time. In exactly soluble LQC, it is shown that $\beta$
takes values in the interval $\left(0,\frac{\pi}{\lambda}\right)$robustness
(the interval
$\left[\left.-\frac{\pi}{2\lambda},\frac{\pi}{2\lambda}\right)\right.$ is also
employed in this effective scheme – see for instance corichi-tatiana ).

  * •

Equation for $V$ implies that a bounce takes place at $t=t_{c}$ with
$\beta(t_{c})=\frac{\pi}{2\lambda}$, and that this bounce corresponds to the
maximum of the energy density:
$\rho(t_{c})=\frac{1}{2}\dot{\phi}^{2}(t_{c})+\frac{m^{2}}{2}\phi^{2}(t_{c})=\frac{3}{8\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta(t_{c}))=\frac{3}{8\pi
G\gamma^{2}\lambda^{2}}$ (where in arriving at the second equality the
equation of motion for $\phi$ and the Hamiltonian constraint were used). In
the following, we denote by $\rho_{c}$ the maximum value attained by the
energy density function. For definiteness, the bounce is naturally set at
$t_{c}=0$.

  * •

Since $\beta$ presents a monotonic behavior, such bounce is always featured
and is achieved exactly once. Therefore, in this effective scheme, the Big
Bounce is generic. This singularity resolution via a Big Bounce is a key
feature of (effective) LQC.

For completeness, we obtained some numerical solutions to the equations of
motion. A particular set of which is shown in Figs. 1a-1e. We also reproduce,
in Fig. 2, the familiar phase portrait (compare with, e.g. mukhanov , agullo-
corichi ) for the scalar field $\phi$ by explicitly evaluating some of the
numerical solutions obtained.

(a)

(b)

(c)

(d)

(e)

Figure 1: Numerical solutions to (3)-(6). (a) $\beta(t)$. (b) volume function,
$V(t)$. (c) scalar field, $\phi(t)$. (d) momentum $p_{\phi}(t)$ canonically
conjugated to the scalar field; (e) energy density,
$\rho(t)=\dot{\phi}^{2}/2+m^{2}\phi^{2}/2$. The bounce was set to take place
at $t_{c}=0$. The following initial conditions and values were employed:
$v(0)=\pi/2\lambda$, $\beta(0)=\pi/2\lambda$, $\phi(0)=0.3$,
$p_{\phi}(0)=0.3$, an the constants $b=1$, $a=-1+4\pi$, $m=.2$ Figure 2: Phase
portrait of the scalar field $\phi$. The attractor is located at $\pm
m/\sqrt{12\pi}$

Now, the corresponding Friedmann equation and Klein-Gordon equation are given
by ($H=\frac{\dot{a}}{a}=\frac{\dot{V}}{3V}$ is the Hubble parameter):

$H^{2}=\frac{8\pi
G}{3}\rho\left(1-\frac{\rho}{\rho_{c}}\right),\qquad\ddot{\phi}+3H\dot{\phi}+m^{2}\phi=0.$
(7)

Note that the singularity resolution is incorporated in this modified
Friedmann equation in a rather simple way: $\dot{a}=0$ at $\rho=\rho_{c}$.

##  II Noncommutative effective LQC of the flat FLRW with scalar field $\phi$
with generic potential term $\mathcal{V}(\phi)$

We first review the ideas and methods related to the noncommutativity paradigm
as implemented in the minisuperspace approximation.

Recall (see, e.g. choquet ) that the phase space $\Gamma=T^{\star}\mathcal{Q}$
(the cotangent bundle of $\mathcal{Q}$), where $\mathcal{Q}$ is the
configuration space of a mechanical system, is a symplectic manifold: a
$2n$-dimensional manifold endowed with a closed ($d\omega=0$) and invertible
($\mathrm{rank}(\omega)=2n$) two-form
$\omega=\omega_{\alpha\beta}dx^{\alpha}\wedge dx^{\beta}$, called the
symplectic structure (in this particular case of the cotangent bundle,
$\omega$ is in fact exact: $\omega=d\vartheta$, $\vartheta$ being the so
called symplectic potential).

The symplectic structure provides an isomorphism
$\omega^{\flat}:T_{p}\mathcal{Q}\rightarrow T_{p}^{\star}\mathcal{Q}$, for
each $p\in\mathcal{Q}$, by
$\omega^{\flat}(\mathsf{X})=\omega(\mathsf{X},\cdot)$; with inverse
$\omega^{\sharp}:T_{p}^{\star}{\mathcal{Q}}\rightarrow T_{p}\mathcal{Q}$, for
each $p\in\mathcal{Q}$, given by
$\omega^{\sharp}(\eta)=\mathsf{P}(\eta,\cdot)$, where
$\mathsf{P}=\omega^{\alpha\beta}\partial_{\alpha}\wedge\partial_{\beta}$ is
the Poisson structure (an antisymmetric bivector field on
$T^{\star}\mathcal{Q}$ satisfying
$\omega^{\alpha\beta}\omega_{\beta\gamma}=\delta^{\alpha}_{\gamma}$). As a
consequence, one can write the equations of motion

$\dot{q}^{a}=\partial_{p_{a}}H,\quad\dot{p}_{a}=-\partial_{q^{a}}H$ (8)

of a mechanical system with Hamiltonian function $H$ in a geometric and
coordinate-independent way:

$\dot{\mathsf{x}}=-\omega^{\sharp}(dH),$ (9)

where $\mathsf{x}=(x^{1},...,x^{2n})$ are the phase space dynamical variables,
$dH=\partial_{\alpha}dx^{\alpha}$, and $\omega^{\sharp}(dH)=-\mathsf{X}_{H}$
the associated vector field to $dH$:
$-\mathsf{X}_{H}=\mathsf{P}(dH,\cdot)\leftrightarrow\omega(-\mathsf{X}_{H},\cdot)=dH$.

In canonical coordinates
$(x^{1},...,x^{2n})=(q^{1},q^{2},...,q^{n},p_{1},p_{2},...,p_{n})$ (i.e. in a
Darboux chart), the symplectic structure acquires the usual form ($a=1,...,n$)

$\omega=J_{\alpha\beta}dx^{\alpha}\wedge dx^{\beta}=dx^{n+a}\wedge
dx^{a}=dp_{a}\wedge dq^{a},$ (10)

that is,

$J_{\alpha\beta}=\begin{pmatrix}0_{n}&-I_{n}\\\ I_{n}&0_{n}\end{pmatrix}.$
(11)

We therefore have (in canonical coordinates)

$\mathsf{P}=J^{\alpha\beta}\partial_{\alpha}\wedge\partial_{\beta}=\partial_{a}\wedge\partial_{n+a}=\partial_{q^{a}}\wedge\partial_{p_{a}},$
(12)

and so,

$J^{\alpha\beta}=\begin{pmatrix}0_{n}&I_{n}\\\ -I_{n}&0_{n}\end{pmatrix}.$
(13)

(We reserve the notation $J_{\alpha\beta}$, $J^{\alpha\beta}$ for the
components of the symplectic and Poisson structures when expressed in a
Darboux charts —i.e. in canonical coordinates.)

Recall also that Poisson brackets can be defined by the Poisson structure or
the symplectic structure:

$-\omega(\mathsf{X}_{f},\mathsf{X}_{g})=:\\{f,g\\}:=\mathsf{P}(df,dg)$ (14)

hence, in canonical coordinates, we arrive at the familiar form (sum over
$a$):

$\\{f,g\\}=\partial_{q^{a}}f\partial_{p_{a}}g-\partial_{q^{a}}g\partial_{p_{a}}f.$
(15)

In particular, for the canonical coordinates $(x^{1},x^{2},...,x^{2n})$, we
have

$\\{q^{a},p_{b}\\}=\delta^{a}_{b},\qquad\\{q^{a},q^{b}\\}=\\{p_{a},p_{b}\\}=0.$
(16)

The so called correspondence principle leads to the canonical commutation
relations

$[\hat{q}^{i},\hat{p}_{j}]=i\hbar\delta^{i}_{j},\quad[\hat{q}^{i},\hat{q}^{j}]=[\hat{p}_{i},\hat{p}_{j}]=0$
(17)

Now, leading approaches to quantum gravity (e.g. Loop Quantum Gravity) point
to a discrete picture of spacetime at distances comparable to the Planck
length $\ell_{p}\sim 10^{-35}\mathrm{cm}$: i.e. in a full quantum theory of
the gravitational field, a quantization of spacetime itself could be in order.
A possible way to account for such discrete picture is provided by an
uncertainty relation such as

$[q^{i},q^{j}]=i\theta^{ij}.$ (18)

The noncommutativity paradigm which we consider here is closely related to
Quantum Mechanics in Phase Space (QMPS), which, as its name suggests, provides
an alternate way to consistently formulate Quantum Mechanics in classical
phase space (an introductory account of QMPS can be found in curtright ). In
essence, in QMPS, the canonical commutation relations (17) are realized by
means of a deformation of the usual pointwise multiplication of phase space
functions: $f\cdot g\mapsto f\star g$. This star-product ($\star$-product) can
be thought of as a “power series” of $\frac{i\hbar}{2}\mathsf{P}(df,dg)$
(where $\mathsf{P}$ is the Poisson structure of classical phase space) in
which the “zeroth power” is the usual commutative product $f\cdot g$. We can
therefore obtain standard Classical Mechanics by simply taking $\hbar\to 0$.
By considering the (classical) commutator $[f,g]_{\star}=f\star g-g\star f$,
the usual canonical commutation relations (17) are realized in classical phase
space $\Gamma$. Central to the scheme of QMPS is the so called Wigner
function, a quasi-probability distribution in phase space which was originally
introduced in the context of Classical Statistical Mechanics in order to
account for quantum corrections to the classical Liouville measure. In QMPS,
probability amplitudes are calculated with the help of the Wigner function.

The connection with canonical Quantum Mechanics is given by working in the
Weyl representation (which is unitary equivalent to the standard Schrödinger
representation), where the canonical commutation relations (17) acquires an
“exponentiated form”; this representation is encoded in a quantization
prescription, the so called Weyl map $\mathcal{W}$, which associates a
(symmetric ordered) quantum operator $\hat{f}$ to a phase space function $f$.
The Weyl map $\mathcal{W}$ is invertible ($\mathcal{W}^{-1}(\hat{f})$ is
called the Weyl symbol of $\hat{f}$, or the Wigner transform of $\hat{f}$).
The Wigner function maps (up to multiplicative factors of $\hbar$) to the
density operator $\hat{\rho}$ under the Weyl transform (i.e. the Wigner
function is the Wigner transform of the density operator). The $\star$-product
is the image of the operator product (in the Weyl representation) under the
Wigner transform: $f\star g=\mathcal{W}^{-1}(\mathcal{W}(f)\mathcal{W}(g))$.

For the case of the canonical algebra (17) we have

$\displaystyle f\star g$ $\displaystyle=$
$\displaystyle\exp\left(\frac{i\hbar}{2}\mathsf{P}(df,dg)\right)=f(x)\exp{\left(\frac{i\hbar}{2}\partial_{\mu}J^{\mu\nu}\partial_{\nu}\right)}g(x)$
(19) $\displaystyle=$ $\displaystyle
fg+\frac{1}{2}\\{f,g\\}+\mathcal{O}(\hbar^{2})$

where in the expansion, terms of the form
$\mathsf{P}^{r}(df,dg)=J^{\mu_{1}\nu_{1}}\cdot\cdot\cdot
J^{\mu_{r}\nu_{r}}\left(\partial_{\mu_{1}...\mu_{r}}f\right)\left(\partial_{\nu_{1}...\nu_{r}}g\right)$
(the “rth power” of the Poisson bracket of $f$ and $g$) are to be considered.
This is the Moyal $\star$-product moyal , it replaces the ordinary pointwise
multiplication in the algebra of functions defined in phase space. The Moyal
$\star$-bracket $[f,g]_{\star}:=f\star g-g\star f$ is thus responsible for the
realization of the canonical algebra (17). This product, together with the
Wigner function, is the cornerstone of QMPS.

The noncommutativity paradigm employs some of the ideas and methods of QMPS
(most notably, the $\star$-product), and can therefore be viewed from the
perspective of the QMPS scheme: given more general commutation relations, e.g.

$\left[\hat{q}^{a},\hat{p}_{b}\right]=i\hbar\delta^{a}_{b},\quad[\hat{q}^{a},\hat{q}^{b}]=i\theta^{ab},\quad[\hat{p}_{a},\hat{p}_{b}]=i\kappa_{ab},$
(20)

construct the corresponding $\star$-product by accordingly considering an
appropriate modification $\tilde{\mathsf{P}}$ to $\mathsf{P}$. This
$\tilde{\mathsf{P}}$ will be such that the classical Poisson relations
corresponding to the above commutation relations are fulfilled. Once the
$\star$-product is constructed, the advancement of a Noncommutative Quantum
Mechanics in Phase Space (NCQMPS) can be pursued. A related point of view is
to consider (20) as an extension to standard Quantum Mechanics, resulting in a
more noncommutative framework than the original standard one. The
implementation of such additional noncommutativity is achieved by implementing
a $\star$-product on quantum phase space. This point of view results in a
framework known as Noncommutative Quantum Mechanics (NCQM) mezin .

The introduction of the modification $\tilde{\mathsf{P}}$ can, at the
classical level, be interpreted as a manifestation of the phase space
coordinates not being canonical, which in turn would imply that a new phase
space $\tilde{\Gamma}$ is being considered (a “deformation” to the original
one, $\Gamma$). An interesting point of view is the assumption that this new
$\tilde{\Gamma}$ is the true phase space of the mechanical system under study,
and that it was initially overlooked due to the noncommutative scale being
unnoticed. This phase space $\tilde{\Gamma}$ would therefore carry
(noncommutative) quantum corrections in a purely classical setup. In this
interpretation, the classical equations of motion would not be in canonical
form in the original (now non-canonical) variables $y^{\alpha}$, but the
Hamiltonian function would remain unchanged. This deviation from canonicity of
the equations of motion (noncommutative equations of motion) would yield such
quantum corrections. Due to the Darboux theorem, one would be able to find a
new chart with local coordinates $x^{\alpha}$ ($y=T(x)$) in which
$\tilde{\mathsf{P}}$ acquires the canonical form (13) (so that the classical
equations of motion would be in canonical form), but the Hamiltonian function
would get modified ($H(y)\mapsto H(x)=H(T(x))$): the quantum corrections from
noncommutativity would in this case descend to the Hamiltonian function (this
point of view was adopted in Ref. barbosa  in order to present a
noncommutative classical cosmology of the Kantowski-Sachs model). In our
investigations, we follow this latter deformed Hamiltonian (or noncommutative
Hamiltonian) approach: we construct the “noncommutative Hamiltonian”
corresponding to a simple theta-deformation in order to consider
noncommutative corrections to the (formally classical) effective scheme of the
flat FLRW LQC (which already carries quantum corrections from LQG).

In view of the above discussion, consider the algebra

$\\{V^{nc},p^{nc}_{\phi}\\}=\theta,\ \\{\beta^{nc},V^{nc}\\}=4\pi G\gamma,\
\\{\phi^{nc},p^{nc}_{\phi}\\}=1,$ (21)

with the remaining brackets being zero.

The above relations can be implemented by working with the shifted variables

$V^{nc}=V+a\theta\phi,\ p_{\phi}^{nc}=p_{\phi}+b\theta\beta,\
\beta^{nc}=\beta,\ \phi^{nc}=\phi,$ (22)

where $a$ and $b$ satisfy the relation $a-4\pi G\gamma b=1$. Note that
$(\beta^{nc},\phi^{nc},V^{nc},p_{\phi}^{nc})$ denote the non-canonical
variables satsifying (21) in the deformed phase space $\tilde{\Gamma}$. If we
were to write the equations of motion employing these non-canonical variables
we would have to do so by considering $\tilde{\mathsf{P}}$ in such
coordinates. As announced, we prefer to write the equations of motion in
standard form by employing appropiate canonical coordinates in
$\tilde{\Gamma}$, which we denote by $(\beta,\phi,V,p_{\phi})$ (the same
notation for the canonical coordinates in the undeformed phase space
$\Gamma$).

Relations (22) enables to write the noncommutative Hamiltonian, it takes the
form

$\mathcal{H}^{nc}_{eff}=-\frac{3}{8\pi
G\gamma^{3}\lambda^{2}}\sin^{2}(\lambda\beta)V^{nc}+\frac{(p_{\phi}^{nc})^{2}}{2V^{nc}}+\mathcal{V(\phi)}V^{nc}.$
(23)

The respective noncommutative equations of motion are given by

$\displaystyle\dot{\beta}=4\pi G\gamma\left[-\frac{3}{4\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta)+2\mathcal{V(\phi)}\right],$ (24)
$\displaystyle\dot{V}=\frac{3}{\gamma\lambda}V^{nc}\sin(\lambda\beta)\cos(\lambda\beta)-\frac{4\pi
G\gamma b\theta p_{\phi}^{nc}}{V^{nc}},$ (25)
$\displaystyle\dot{\phi}=\frac{p_{\phi}^{nc}}{V^{nc}},$ (26)
$\displaystyle\dot{p}_{\phi}=\frac{3a\theta}{4\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta)-2a\theta\mathcal{V(\phi)}-V^{nc}\partial_{\phi}\mathcal{V(\phi)},$
(27)

in the limit $\theta\rightarrow 0$ we recover the commutative field equations.
From the equation of motion for $\dot{\phi}$ and $\mathcal{H}_{eff}^{nc}$, we
can see that the energy density takes the form
$\rho=\dot{\phi}^{2}/2+V(\phi)=\rho_{c}\sin^{2}(\lambda\beta)$, which is the
same functional form (in terms of $\beta$) as in the standard case, and
therefore attains the same maximum value $\rho_{c}$. However, the functional
form in terms of $t$ would not be in general the same as the standard case,
since the solution of $\beta(t)$ is likely to depend on the noncommutative
parameter $\theta$ (nonetheless, given the monotonic behavior of $\beta$, it
is expected that the behavior of the energy density function is in the overall
the same as in the standard case). It is also noted that several stationary
values of the volume function could in principle be attained, which in turn
could result in several bouncing scenarios (depending on the evolution of all
dynamical variables).

The equation of motion for $V$ indicates that the simultaneous fulfillment of

$b=0,\ \ V^{nc}(t_{c})>0$ (28)

(where, as before, $t_{c}$ is such that $\rho(t_{c})=\rho_{c}$) is a
sufficient condition for a minimum volume bounce to be reached when the
maximum $\rho_{c}$ of the energy density $\rho$ is attained (this condition is
already present in the free case, but it was not spelled out explicitly in
Ref. hindawi ). That is to say, condition (28) implies that the standard LQC
Big Bounce is preserved. This will be the only bouncing scenario since $\beta$
is a decreasing function of time (as (24) indicates), taking values on
$\left(0,\frac{\pi}{\lambda}\right)$, and since $V^{nc}=0$ is not allowed
during evolution (in view of the equation of motion for $\phi$ and the energy
density being bounded).

On the other hand, the simultaneous fulfillment of

$\dot{\phi}(t_{c})=0,\ b\theta\ddot{\phi}(t_{c})<0$ (29)

which translates into

$\rho(\phi(t_{c}))=\mathcal{V}(\phi(t_{c})),\ b\theta\ddot{\phi}(t_{c})<0,$
(30)

is also a sufficient condition for the occurrence of a minimum volume bounce;
however, it does not precludes the existence of additional bounces (which
could lead to additional minima of volume, not necessarily all positive),
hence, singularity resolution is not guaranteed to always be achieved in this
latter scenario. Condition (29) would allow to fix the value of $\phi$ at this
minimum volume bounce. Furthermore, from the equation of motion for $\phi$ it
is observed that
$p_{\phi}(t_{c})=-b\theta\beta(t_{c})=-\frac{b\theta\pi}{2\lambda}$ must hold.
We are facing the following situation: by demanding that a minimum volume
bounce takes place when the energy density reaches its maximum value, but not
restricting to the particular noncommutativity representation given by $b=0$,
initial conditions for all dynamical variables (except for the volume) get
fixed at the bounce. This condition therefore leads to a dramatically reduced
number of solutions which feature a bounce somewhat resembling the (single)
Big Bounce of standard LQC. This state of affairs is obviously not
particularly appealing.

Of course, the conditions (28) and (29) do not exhaust all the possibilities
for the existence of a minimum volume bounce, but they do exhaust all the
possibilities for a minimum volume bounce taking place at $\rho=\rho_{c}$.
Additionally, these conditions exclude each other: they cannot both occur for
solutions featuring such a bouncing event.

For the scalar field we can construct the modified Klein-Gordon equation using
(24)-(27):

$\ddot{\phi}+\frac{1}{V^{nc}}\left[\dot{V}^{nc}+a\theta\phi\right]\dot{\phi}+\partial_{\phi}\mathcal{V}(\phi)=0,$
(31)

we can see that in the limit $\theta\to 0$ we recover the commutative Klein-
Gordon equation for the scalar field. Now we construct the corresponding
noncommutative modified Friedmann equation. Since
$H=\frac{\dot{a}}{a}=\frac{\dot{V}(t)}{3V(t)}$, taking into account the
effective Hamiltonian constraint $\mathcal{H}^{nc}_{eff}\approx 0$ and the
equation of motion for $V$, we have

$H^{2}=\frac{8\pi
G}{3}\rho\left[1-\frac{\rho+\rho_{\theta}}{\rho_{c}}\right],$ (32)

where $\rho_{\theta}$ is the contribution of the noncommutative parameter to
the generalization of the modified Friedmann equation. This $\rho_{\theta}$
can be written as linear order contributions and quadratic contributions in
$\theta$, $\rho_{\theta}=\rho_{1}(\theta)+\rho_{2}(\theta^{2})$, and these are
given as

$\displaystyle\rho_{1}(\theta)$ $\displaystyle=$
$\displaystyle\frac{2a\theta\phi}{V}(\rho_{c}-\rho)$ (33)
$\displaystyle+\frac{\sqrt{2}b\theta}{\gamma\lambda
V}\sqrt{\frac{\rho_{c}}{\rho}\left(1-\frac{\rho}{\rho_{c}}\right)(\rho-\mathcal{V}(\phi))}$
$\displaystyle\rho_{2}(\theta^{2})$ $\displaystyle=$
$\displaystyle\frac{a^{2}\theta^{2}\phi^{2}}{V^{2}}(\rho_{c}-\rho)$ (34)
$\displaystyle+\frac{\sqrt{2}ab\theta^{2}\phi}{\gamma\lambda
V}\sqrt{\frac{\rho_{c}}{\rho}\left(1-\frac{\rho}{\rho_{c}}\right)(\rho-\mathcal{V}(\phi))}$
$\displaystyle+\frac{4\pi
Gb^{2}\theta^{2}}{3V^{2}}\rho_{c}\left(1-\frac{\mathcal{V}(\phi)}{\rho}\right).$

From eq. (32) it is observed that if condition (28) is imposed, the
noncommutative contribution is $zero$ at $t=t_{c}$. On the other hand, for the
limited number of bouncing solutions given by imposing condition (29), the
contribution at $t=t_{c}$ of terms related to noncommutativity gets also
suppressed.

##  III Noncommutative effective LQC of flat FLRW with scalar field $\phi$
with potential term $\mathcal{V}(\phi)=\frac{1}{2}m^{2}\phi^{2}$

In the following we concentrate on finding particular solutions which fulfill
either (29) or (28) for the case of a quadratic potential term, and on
establishing if solutions exist which feature an inflationary scenario with a
sufficient number of e-foldings.

When considering the quadratic potential term (1), the Hamiltonian (23), and
the equations of motion emanating from it, are given by

$H^{nc}_{eff}=-\frac{3}{8\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta)V^{nc}+\frac{(p^{nc}_{\phi})^{2}}{2V^{nc}}+\frac{1}{2}m^{2}\phi^{2}V^{nc}$
(35) $\displaystyle\dot{\beta}$ $\displaystyle=$
$\displaystyle-\frac{3}{\gamma\lambda^{2}}\sin^{2}(\lambda\beta)+4\pi G\gamma
m^{2}\phi^{2},$ (36) $\displaystyle\dot{V}$ $\displaystyle=$
$\displaystyle\frac{3}{\gamma\lambda}V^{nc}\sin(\lambda\beta)\cos(\lambda\beta)-4\pi
G\gamma b\theta\frac{p_{\phi}^{nc}}{V^{nc}},$ (37) $\displaystyle\dot{\phi}$
$\displaystyle=$ $\displaystyle\frac{p^{nc}_{\phi}}{V^{nc}},$ (38)
$\displaystyle\dot{p}_{\phi}$ $\displaystyle=$
$\displaystyle\frac{3a\theta}{8\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta)+a\theta\frac{(p_{\phi}^{nc})^{2}}{2(V^{nc})^{2}}-m^{2}V^{nc}\phi$
(39) $\displaystyle-\frac{1}{2}a\theta m^{2}\phi^{2},$

Of course, these field equations reduce to the commutative ones when taking
$\theta\rightarrow 0$, and exhibit the features already noticed in the general
case presented in section II.

###  III.1 Particular inflationary solutions related to condition $b=0,\ \
V^{nc}(t_{c})>0$

As we saw earlier, condition (28) ensures that all solutions undergo a single
minimum volume bounce at $\rho_{c}$; in other words, a single standard LQC Big
Bounce event is always featured for each solution satisfying (28).
Trajectories with a sufficiently long inflationary phase which pertain to such
solutions sector are therefore the best candidates for reproducing an early
universe compatible with the (effective) LQC paradigm.

Analytical solutions to the equations of motion (24-27) could not be found.
However, some curves constructed via numerical solutions are shown in Figs.
3a-3e. We can explicitly observe that the bounce is preserved, as expected.
The energy density, Fig. 3e, has the same behavior that the one exhibited in
standard LQC. Trajectories in the $\phi-\dot{\phi}$ plane are presented in
Fig. 4.

(a)

(b)

(c)

(d)

(e)

Figure 3: Numerical solutions to (36)-(39) taking into account (28). The
dotted line corresponds to $\theta=0$, green line to $\theta=0.1$ and red line
to $\theta=0.05$. (a) $\beta(t)$. (b) volume function, $V(t)$. (c) scalar
field, $\phi(t)$. (d) the momentum $p_{\phi}(t)$ canonically conjugated to the
scalar field. (e) energy density, $\rho(t)=\dot{\phi}^{2}/2+m^{2}\phi^{2}/2$.
The following initial conditions and values were employed:
$v(0)=\pi/2\lambda$, $\beta(0)=\pi/2\lambda$, $\phi(0)=1$, $p_{\phi}(0)=1$,
and the constants , $a=-1$, $b=0$, $m=2\times 10^{-6}$. In order to better
appreciate the differences, we used $\lambda=G=\gamma=1$. Figure 4: Parametric
Plot $(\phi,\dot{\phi})$ constructed from numerical solutions to (36)-(39)
taking into account (28). The solid line corresponds to $\theta=0$, the tiny
dashed line to $\theta=0.5$ and the large dashed line to $\theta=0.1$. The
following initial conditions and values were employed: : $v(0)=\pi/2\lambda$,
$\beta(0)=\pi/2\lambda$, $\phi(0)=1$, $p_{\phi}(0)=1$, and the constants ,
$a=-1$, $b=0$, $m=2\times 10^{-6}$ Figure 5: The number $N$ of e-foldings as a
function of $t$ constructed from numerical solutions to (36)-(39) taking into
account (28). The dotted line corresponds to $\theta=0$, green line to
$\theta=0.1$ and red line to $\theta=0.05$. The following initial conditions
and values were employed: $v(0)=\pi/2\lambda$, $\beta(0)=\pi/2\lambda$,
$\phi(0)=1$, $p_{\phi}(0)=1$, and the constants , $a=-1$, $b=0$, $m=2\times
10^{-6}$ We observe that a sufficiently large number of e-foldings is
achieved.

We can compute the duration of the inflationary period by means of the number
$N$ of e-foldings (this epoch had to last for a long enough period, in order
to make the cosmological observations sufficiently flat within the
observational limits)

$N(t)=\int^{t}_{t_{i}}H(t)dt$ (40)

A period of early inflationary expansion consistent with observations demands
$N\geq 60$ wmap . Fig. 5 shows the behavior of $N(t)$, where initial
conditions associated to (28) have been considered. We observe that the model
presents an early inflationary period with $N\geq 60$.

###  III.2 Particular inflationary solutions related to condition
$\dot{\phi}(t_{c})=0,\ b\theta\ddot{\phi}(t_{c})<0$

In this case, condition (29) takes the following form

$p_{\phi}(0)=-\frac{b\theta\pi}{2\lambda},\ b\theta\ddot{\phi}(t_{c})<0,\
\phi(0)=\left\\{\begin{matrix}\sqrt{\frac{2\rho_{c}}{m^{2}}},&b\theta>0\\\
-\sqrt{\frac{2\rho_{c}}{m^{2}}},&b\theta<0\end{matrix}\right..$ (41)

In other words, under this assumption, the standard bounce takes place only
for the solutions related to the above initial conditions which correspond to
only one trajectory on the $\phi-\dot{\phi}$ plane for each case $b\theta>0$,
$b\theta<0$.

Condition (41) implies that, in the case $b\theta>0$, a local maximum of the
scalar field $\phi(t)$ is reached at $t_{c}$, for any value of the $\theta$
parameter. This can be seen represented in the numerical solution shown in
Fig. 6.

Figure 6: Numerical solution for (38) taking into account (41). The dotted
line corresponds to $\theta=0$, green line to $\theta=0.1$ and red line to
$\theta=0.05$. The following initial conditions and values were employed:
$\beta(0)=\pi/2\lambda$, $V(0)=\pi/2\lambda$,
$\phi(0)=\sqrt{2\rho_{c}/m^{2}}$, $p_{\phi}(0)=-(b\theta\pi)/2\lambda$;
$\lambda=G=\gamma=1$, $b=1$, $m=2\times 10^{-6}$.

From Fig. 7 we observe that the inclusion of noncommutativity results in a
greater volume expansion rate.

(a)  Figure 7: Numerical solution to (37) taking into account 41. The dotted
line corresponds to $\theta=0$, green line to $\theta=0.1$ and red line to
$\theta=0.05$. The following initial conditions and values were employed:
$\beta(0)=\pi/2\lambda$, $V(0)=\pi/2\lambda$,
$\phi(0)=\sqrt{2\rho_{c}/m^{2}}$, $p_{\phi}(0)=-(b\theta\pi)/2\lambda$;
$\lambda=G=\gamma=1$, $b=1$, $m=2\times 10^{-6}$.

The energy density $\rho=\frac{\dot{\phi}^{2}}{2}+\frac{1}{2}m^{2}\phi^{2}$ is
given by

$\rho=\frac{1}{2}\left(\frac{p_{\phi}^{nc}}{V^{nc}}\right)^{2}+\frac{1}{2}m^{2}\phi^{2}=\frac{3}{8\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta),$ (42)

In Fig. 8b the energy density is shown, it is observed that it has a
dependence on the noncommutative parameter, in contrast to the free case ijmpd
. As already pointed out, this is because the energy density (42) depends
explicitly on $\beta(t)$, but $\beta$ is $\theta$-dependent, as shown in
Figure (8a). It is important to emphasize that although the energy density
depends on the noncommutative parameter, the maximum $\rho_{c}$ is conserved
for any value of $\theta$ (this was also already pointed out when considering
the general case in the preceding section).

(a)

(b)

Figure 8: Numerical solutions to (36)-(39) taking into account 41. The dotted
line corresponds to $\theta=0$, green line to $\theta=0.1$ and red line to
$\theta=0.05$. (a): $\beta(t)$; (b): energy density function (42). The
following initial conditions and values were employed:
$\beta(0)=\pi/2\lambda$, $V(0)=\pi/2\lambda$,
$\phi(0)=\sqrt{2\rho_{c}/m^{2}}$, $p_{\phi}(0)=-(b\theta\pi)/2\lambda$;
$\lambda=G=\gamma=1$, $b=1$, $m=2\times 10^{-6}$.

As in the previous subsection, we can establish if an inflationary period with
enough e-foldings is achieved. 9a shows $N$ as a function of $t$, taking
initial conditions consistent with (41). The model presents a sufficiently
long period of inflation. On the other hand Fig. 9b returns the number $N$ of
e-foldings in terms of the scalar field $\phi$. The value employed for the
field is $\phi\approx 10^{6}$, which is determined from
$\mathcal{V}(\phi(t_{c}))=\rho_{c}$.

(a)

(b)

Figure 9: The number $N$ of e-foldings as a function of $t$ constructed from
numerical solutions to (36)-(39) taking into account (41). The dotted line
corresponds to $\theta=0$, green line to $\theta=0.1$ and red line to
$\theta=0.05$. The following initial conditions and values were employed:
$\beta(0)=\pi/2\lambda$, $V(0)=\pi/2\lambda$,
$\phi(0)=\sqrt{2\rho_{c}/m^{2}}$, $p_{\phi}(0)=-(b\theta\pi)/2\lambda$;
$\lambda=G=\gamma=1$, $b=1$, $m=2\times 10^{-6}$.

Finally, we spell out the noncommutative modified Friedmann equation (32)
specialized to the quadratic potential term given in (1):

$H^{2}=\frac{8\pi
G}{3}\rho\left[1-\frac{\rho+\rho_{\theta}}{\rho_{c}}\right],$ (43)

where $\rho^{\theta}=\rho_{1}(\theta)+\rho_{2}(\theta^{2})$ and

$\displaystyle\rho_{1}(\theta)$ $\displaystyle=$
$\displaystyle\frac{2a\theta\phi}{V}(\rho_{c}-\rho)$ (44)
$\displaystyle+\frac{\sqrt{2}b\theta}{\gamma\lambda
V}\sqrt{\frac{\rho_{c}}{\rho}\left(1-\frac{\rho}{\rho_{c}}\right)(\rho-\frac{1}{2}m^{2}\phi^{2})}$
$\displaystyle\rho_{2}(\theta^{2})$ $\displaystyle=$
$\displaystyle\frac{a^{2}\theta^{2}\phi^{2}}{V^{2}}(\rho_{c}-\rho)$ (45)
$\displaystyle+\frac{\sqrt{2}ab\theta^{2}\phi}{\gamma\lambda
V}\sqrt{\frac{\rho_{c}}{\rho}\left(1-\frac{\rho}{\rho_{c}}\right)(\rho-\frac{1}{2}m^{2}\phi^{2})}$
$\displaystyle+\frac{4\pi
Gb^{2}\theta^{2}}{3V^{2}}\rho_{c}\left(1-\frac{m^{2}\phi^{2}}{2\rho}\right).$

This equation gives us the dynamical evolution of the model. If we take the
limit $\theta\to 0$, we recover the usual modified Friedmann equation of
standard LQC.

##  IV Discussion

The present investigation shows that the inclusion of a generic potential term
in the momentum sector noncommutativity encoded in the algebra (21), within
the LQC effective scheme of the flat FLRW, maintains the occurrence of the
single Big Bounce scenario characteristic of standard LQC (considering the
class of noncommutative models associated to condition (28)). The alternative
relations (29) were established as also a sufficient condition for retaining a
Big Bounce, however, such condition does not precludes additional stationary
points in the volume function (in contrast to condition (28)) which could lead
to several bouncing scenarios. A solution featuring a minimum volume bounce at
$\rho=\rho_{c}$ cannot fulfill both (28) and (29). As a first approach to
studying the inflationary scenario, we specialized to a quadratic potential
term (1). In this particular case, condition (29) reduces to (41) which, for
$b\theta>0$ ($b\theta<0$), is equivalent to demanding a maximum (minimum) of
the scalar field at the bounce; this in turn fixes initial conditions at the
bounce for the scalar field and its conjugate momentum: fine tuning reduces to
volume-tuning (since $\beta$ is already fixed at the bounce). Figures 8, 9
show the behavior of a particular set of solutions satisfying initial
conditions consistent with (41), it is observed that a sufficiently large
period of inflation is achieved. On the other hand, a satisfactory
inflationary period is also achieved for particular sets of solutions
consistent with condition (28). One of such sets is considered in Figs. 3a-3e;
5. We therefore conclude that solutions to the noncommutative paradigm encoded
in (21) exist, which depict an early universe whose behavior is essentially
the one emanating from the standard LQC paradigm. There remains to be
addressed the question of wether the sufficiently long inflationary period
exhibited in the specific solutions found is a rather generic feature (in the
sense of Ref. sloan ). Also, it is important to note that since conditions
(28) and (29) were established by considering a generic potential term
$\mathcal{V}$, the study of the occurrence of a more sensible inflationary
scenario by considering other meaningful potential terms is therefore a
natural pursuit. These and other related matters are currently being tackled
by the authors and will be reported elsewhere.

## Acknowledgments

J.S. was partially funded by PRODEP grant UGTO-CA-3. A.E.G. was partially
supported by a CONACyT PhD fellowship.

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# Universal spectral correlations in interacting chaotic few-body quantum
systems

Felix Fritzsch  Physics Department, Faculty of Mathematics and Physics,
University of Ljubljana, Ljubljana, Slovenia  Maximilian F. I. Kieler
Technische Universität Dresden, Institut für Theoretische Physik and Center
for Dynamics, Dresden, Germany

###### Abstract

The emergence of random matrix spectral correlations in interacting quantum
systems is a defining feature of quantum chaos. We study such correlations in
terms of the spectral form factor in interacting chaotic few- and many-body
systems, modeled by suitable random-matrix ensembles, and obtain exact results
for large Hilbert space dimensions. The transition of the spectral form factor
from the non-interacting to the strongly interacting case can be described as
a simple combination of these two limiting cases, which we confirm by
extensive numerical studies in few-body systems. This transition is
universally governed by a single scaling parameter. Moreover, our approach
accurately captures spectral correlations in actual physical system, which we
demonstrate for coupled kicked rotors.

The quantum chaos conjecture [1, 2, 3] predicts statistical properties of
energy levels in quantum systems whose classical limit is chaotic to follow
random matrix theory [4, 5, 6]. Using semiclassical periodic orbit theory this
connection has been shown to follow from only a few basic properties of the
chaotic classical dynamics [7, 8, 9, 10]. Subsequently random-matrix like
spectral statistics has become one of the most widely used definitions of
quantum chaos even in the absence of a classical limit. A distinguished
feature of the spectrum of such chaotic quantum systems and the corresponding
random matrix ensembles is the presence of correlations between energy levels
in contrast to the uncorrelated Poissonian spectrum of integrable [11] or
(many-body [12]) localized systems [13, 14]. These correlations are
conveniently detected by the spectral form factor (SFF) [15] which has
received growing attention in recent years in, e.g., high energy physics [16,
17, 18, 19] as well as condensed matter and many-body systems [20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. Recently, the SFF has
been shown to follow random matrix theory in various solvable instances of
chaotic many-body systems involving both homogeneous [20, 21, 22] and random
quantum circuit models [25, 26, 27, 28, 29]. The latter constitute random
matrix ensembles which incorporate the (spatial) locality of typical many-body
systems.

We study the SFF in a similar random matrix model, which is built from large
independent chaotic subsystems subject to an all-to-all, and hence spatially
non-local, interaction of tunable strength. For the bipartite case of just two
subsystems our setting reduces to the so-called random matrix transition
ensembles (RMTE) introduced in Ref. [38]. We therefore refer to our setting as
the extended RMTE henceforth. The bipartite RMTE models a universal transition
from an uncorrelated Poissonian spectrum with exponentially distributed level
spacings in the non-interacting case to a correlated spectrum whose spacings
follow Wigner Dyson statistics at strong interaction [38]. This universal
transition has been observed subsequently also in the average eigenstate
entanglement [39, 40, 41] and in the entanglement generation after a quench
[42, 43].

In the extended RMTE we describe the full transition of the SFF from a simple
product structure in the non-interacting case towards the full random matrix
result at strong interaction as a simple convex combination of these two
extreme cases. This transition is universally governed by a single scaling
parameter, which combines the dependence of the SFF on all parameters of the
system, namely the interaction strength as well as the size and the number of
subsystems, into a single number. The SFF signals an intricate interplay
between different time (and associated energy) scales, such as the Heisenberg
time of the subsystems and the full system, and most notably a non-trivial
Thouless time. We confirm our prediction by extensive numerical studies in
few-body systems but expect our results to hold even in the many-body case.
For the minimal setting of a bipartite system, we obtain a similar description
also for the moments of the spectral form factor, which characterize its
distribution and indicate correlations between multiple levels. Moreover we go
beyond random matrix models and demonstrate that the above results equally
well apply in quantized dynamical systems, i.e., a pair of coupled kicked
rotors. Ultimately, we complement our results by a perturbative treatment of
the interaction.

_Extended random matrix transition ensemble._ —To model interacting few- or
many-body systems, we generalize the RMTE introduced in Ref. [38] by allowing
for an arbitrary number $L$ of subsystems. We consider Floquet systems which
evolve in discrete time with unitary time evolution operator given by

$\displaystyle\mathcal{U}=\mathcal{U}_{\text{c}}(\epsilon)\left(\mathcal{U}_{1}\otimes\mathcal{U}_{2}\otimes\cdots\otimes\mathcal{U}_{L}\right).$
(1)

Each of the $\mathcal{U}_{i}$ is an independent, $N$-dimensional Haar random
unitary drawn from the circular unitary ensemble, CUE$(N)$, and models a
chaotic subsystem. By considering the CUE we restrict ourselves to systems
without anti-unitary symmetries, e.g., time-reversal invariance. The
interaction is introduced by the $N^{L}$-dimensional diagonal unitary matrix
$\mathcal{U}_{\text{c}}(\epsilon)$, where $\epsilon$ controls the strength of
the interaction with $\epsilon=0$ corresponding to the non-interacting
situation, $\mathcal{U}_{\text{c}}(0)=\mathds{1}$. In the canonical product
basis the interaction reads

$\displaystyle\left[\mathcal{U}_{\text{c}}(\epsilon)\right]_{j_{1}\cdots
j_{L}}^{i_{1}\cdots
i_{L}}=\delta^{i_{1}}_{j_{1}}\cdots\delta^{i_{L}}_{j_{L}}\exp\left(\text{i}\epsilon\xi_{i_{1}\cdots
i_{L}}\right).$ (2)

Here the phases $\xi_{i_{1}\cdots i_{L}}$ are i.i.d. random variables with
zero mean and variance $\sigma^{2}$, which gives rise to an effective
interaction strength $\sigma\epsilon$. For numerical simulations we use phases
uniformly distributed in $\left[-\pi,\pi\right]$. Note, that imposing a
spatial locality structure on $\mathcal{U}_{\text{c}}$ recovers the random-
phase circuit of Ref. [26].

_Spectral form factor._ —The SFF indicates correlations between the
eigenphases (or quasi-energies) $\phi_{i}$ defined by the eigenvalue equation
$\mathcal{U}\ket{i}=\exp\left(\text{i}\phi_{i}\right)\ket{i}$. For chaotic
Floquet systems the spectral density is a constant, but its two-point
correlation function $r_{2}(\omega)$ yields the probability of finding two
eigenphases with a distance $\omega$ and hence encodes spectral correlations.
The SFF $K(t)$ is then given by the Fourier transform of the connected part of
$r_{2}(\omega)$ and depends on a time variable $t$ conjugate to the quasi-
energy difference $\omega$. The SFF has a simple representation in terms of
the time evolution operator $\mathcal{U}$ as

$\displaystyle
K(t)=\big{\langle}|\text{tr}\left(\mathcal{U}^{t}\right)|^{2}\big{\rangle}-N^{2L}\delta_{t}^{0}.$
(3)

Here, the brackets denote an ensemble average over the subsystems, i.e., over
the $L$ independent CUE$(N)$, as well as an average over the random phases
$\xi_{i_{1}\cdots i_{L}}$. For numerical simulation we average over at least
1000 realizations. This averaging procedure is necessary as the SFF is not
self averaging [44] and fluctuates wildly for an individual realization.

It is instructive to begin with the SFF for a single CUE of dimension $M$, for
which the SFF takes the simple form $K_{M}(t)=\min\\{t,M\\}$. The initial
linear ramp $\sim t$ indicates correlations in the spectrum and substantially
differs from the constant SFF of an uncorrelated Poissonian spectrum,
characteristic for, e.g., integrable systems [11]. Hence a linear ramp of the
SFF indicates quantum chaos and ergodicity. In interacting physical models the
linear ramp is usually approached after a non-universal time scale known as
Thouless time $t_{\text{Th}}$. It sets the energy scale $\sim 1/t_{\text{Th}}$
below which the system exhibits random matrix like spectral correlations and
hence indicates the onset of universal dynamics. In contrast for non-
interacting systems modeled by the tensor product of independent CUE$(N)$
matrices, e.g., the extended RMTE at $\epsilon=0$, the SFF factorizes into a
product $K(t)=\left[K_{N}(t)\right]^{L}$.

In the extended RMTE we expect a transition from this factorized SFF to the
full CUE$\left(N^{L}\right)$ SFF for increasing interaction strength. In the
following we fully characterize this transition and demonstrate that it
depends on a single scaling parameter only. To this end we adapt the large $N$
expansion of the SFF for the random-phase circuit of Ref. [26] based on the
Weingarten calculus for integration over unitary groups [45, 46] to the
extended RMTE. The average over the subsytems proceeds in the same fashion
whereas the average over the phases simplifies; see Ref. [47] for a detailed
derivation. Ultimately, as our first main result, we represent the SFF in the
simple form of a time-dependent convex combination of the two extreme cases
discussed above. It is given by

$\displaystyle
K(t)=|\chi(\epsilon)|^{2t}K_{N}(t)^{L}+\left(1-|\chi(\epsilon)|^{2t}\right)K_{N^{L}}(t),$
(4)

where
$\chi(\epsilon)=\langle\exp\left(\text{i}\epsilon\xi\right)\rangle_{\xi}$ is
the characteristic function of the distribution of the phases
$\xi_{i_{1}\cdots i_{L}}$. This result is exact in the limit $N\to\infty$. For
finite $N$ it provides the leading contribution (in $1/N$) for times
$t<t_{\text{SH}}=N$, i.e., smaller than the subsystems’ Heisenberg time
$t_{\text{SH}}$ set by the mean level spacing $2\pi/N$ of the subsystems. For
this times $K_{N}(t)=K_{N^{L}}(t)=t$. It has a natural extension to larger
times by including the plateaus of $K_{N}(t)=N$ for $t>t_{\text{SH}}$ and
$K_{N^{L}}(t)=N^{L}$ for times $t>t_{\text{H}}=N^{L}$, i.e., larger than the
full systems Heisenberg time $t_{\text{H}}$. This extension is an
approximation, which is in excellent agreement with numerical data as depicted
in Fig 1 with possible deviations occurring around Heisenberg time and for
small coupling. We emphasize, that requiring large $N$ limits numerical
studies to few-body systems, i.e., small $L$, while our arguments do not
depend on $L$ being small. We therefore expect our results to hold also in the
many-body setting. However, before discussing the qualitative features of the
SFF in the RMTE in more detail, we first point out its universal dependence on
a single scaling parameter.

Figure 1: SFF $\kappa(\tau)$ for the extended RMTE for different $N$, $L$ and
$\Gamma$. Black lines correspond to Eq. (4). Dashed gray lines correspond to
$\tau_{\text{SH}}$ and $\tau_{\text{H}}$.

_Universality._ —To compare the SFF for different systems it is appropriate to
measure both $K(t)$ and time $t$ in units of $t_{\text{H}}$ and to introduce
the rescaled SFF $\kappa(\tau)$ and the rescaled time $\tau$ via

$\displaystyle\kappa(\tau)=K(t)/N^{L}\quad\text{and }\,\tau=t/N^{L}.$ (5)

This results in a rescaled Heisenberg time $\tau_{\text{H}}=1$ of the full
system and $\tau_{\text{SH}}=N^{-L+1}$ of the subsystems. Apart from the
latter, the only $N$ dependence is implicitly contained in
$|\chi(\epsilon)|^{2t}$ via $t=N^{L}\tau$. By applying the central limit
theorem to the characteristic function the $N$ dependence together with the
dependence on the effective coupling strength $\sigma\epsilon$ can be
converted into the dependence on a single scaling parameter $\Gamma$ via

$\displaystyle|\chi(\epsilon)|^{2t}=\exp\left(-\Gamma^{2}\tau\right)\quad\text{with
}\,\Gamma=\sigma\epsilon N^{L/2}.$ (6)

Here we use the characteristic function $\exp\left(-x^{2}/2\right)$ of the
standard normal distribution. Consequently, the SFF becomes independent from
the concrete choice of the distribution of the phases $\xi_{i_{1}\cdots
i_{L}}$ entering $\mathcal{U}_{\text{c}}$. Moreover, it depends only on
$\Gamma$ for times $\tau>\tau_{\text{SH}}$. This universal dependence on a
single scaling parameter constitutes our second main result. It is well
confirmed in Fig. 1, where we depict the SFF for different combinations of
$N$, $L$, and $\epsilon$ all leading to the same $\Gamma$ and coinciding SFF
for $\tau>\tau_{\text{SH}}$. In the non-interacting case $\Gamma=0$ and hence
$\exp\left(-\Gamma^{2}\tau\right)=1$ the SFF initially grows as
$\kappa(\tau)=\tau^{L}$ up to $\tau=\tau_{\text{SH}}$ and subsequently is
constant, $\kappa(\tau)=1$ (not shown). For small $\Gamma$ we still observe an
initial growth of the SFF as $\kappa(\tau)\sim\tau^{L}$, but after times
larger than $\tau_{\text{SH}}$ the SFF drops down to the linear ramp
$\kappa(\tau)\sim\tau$ because all other terms are exponentially suppressed as
$\exp\left(-\Gamma^{2}\tau\right)$. This indicates the Thouless time
$\tau_{\text{Th}}$ as the smallest time for which $\kappa(\tau)\sim\tau$. For
intermediate $\Gamma$ one has $\tau_{\text{SH}}<\tau_{\text{Th}}<1$ and we
obtain [47]

$\displaystyle
t_{\text{Th}}=N^{L}\tau_{\text{Th}}=\frac{L\ln(N)}{2|\ln|\chi(\epsilon)||},$
(7)

which scales linear with the number of subsystems. This is in contrast with,
e.g. logarithmic scaling [20, 26, 28] for local interactions or even
$t_{\text{Th}}=0$ in local dual-unitary quantum circuits [21, 22]. For large
$\Gamma$ the linear ramp is approached earlier than $\tau_{\text{SH}}$, as
shown for $\Gamma=27.21$ for $N=80$ and $L=2$. Ultimately for very large
$\Gamma$ all terms involving the characteristic function are almost
immediately suppressed and the SFF reduces to the CUE$\left(N^{L}\right)$
result (not shown).

Figure 2: Second ($m=2$ ,left) and third ($m=3$, right) moment of the SFF
$\kappa_{m}(\tau)$ for the extended RMTE for different $N$, $L$ and $\Gamma$.
Black lines are obtained from Eq. (9). Dashed gray lines correspond to
$\tau=\tau_{\text{SH}}$ and $\tau=\tau_{\text{H}}$.

_Higher moments._ —As the SFF is defined via an average over the RMTE one
might study its distribution via its moments of order $m$ defined by

$\displaystyle
K_{m}(t)=\big{\langle}\big{|}\text{tr}\left(\mathcal{U}^{t}\right)\big{|}^{2m}\big{\rangle}-N^{2Lm}\delta_{t}^{0}.$
(8)

For the CUE$(M)$ the SFF follows an exponential distribution, i.e.,
$K_{M,m}(t)=m!K_{M}(t)^{m}$ [48]. To compute the moments in the extended RMTE
for $t<t_{\text{SH}}$ we follow Ref. [29] to perform the average over the
independent CUE$(N)$. The remaining average over the phases $\xi_{i_{1}\cdots
i_{L}}$ yields [47]

$\displaystyle
K_{m}(t)=m!t^{m}\sum_{k=0}^{m}A_{k}(t)|\chi(\epsilon)|^{2t(m-k)}$ (9)

for initial times $t<t_{\text{SH}}$. Here the combinatorical factors
$A_{k}(t)$ are polynomials of degree $m(L-1)$ in $t$ which can be obtained
exactly only for the bipartite case $L=2$. Computing the latter for $L=2$ and
fixed $m$ allows for expressing the SFF as a time dependent convex combination
between the full random matrix result $K_{N^{2},m}(t)$ and the non-interacting
result $\left[K_{N,m}(t)\right]^{2}$ as well as additional terms involving
products of lower moments. For instance for the second moment, $m=2$, we find
[47]

$\displaystyle K_{2}(t)=$
$\displaystyle\,K_{N^{2},2}(t)\left(1-|\chi(\epsilon)|\right)^{2}+\left[K_{N,2}(t)\right]^{2}|\chi(\epsilon)|^{4t}$
$\displaystyle+4K_{N^{2}}(t)K_{N}(t)^{2}|\chi(\epsilon)|^{2t}\left(1-|\chi(\epsilon)|^{2t}\right)$
(10)

and similar for $m>2$. By explicitly including the plateaus for the moments of
the CUE spectral form factors the above results again extends also to times
$t>N$. Moreover, it reproduces the correct result for the non-interacting case
$\epsilon=0$ for all $m$ and for the interacting case implies $K(t)\sim
K_{N^{2},m}(t)$, i.e., an exponential distribution, for $t>t_{\text{Th}}$ as
all the terms involving $|\chi(\epsilon)|^{2t}$ have decayed. Given this
exponential distribution we define the rescaled moments via

$\displaystyle\kappa_{m}(\tau)=\frac{1}{N^{L}}\left(\frac{K_{m}(t)}{m!}\right)^{1/m}.$
(11)

Repeating the argument invoking the central limit theorem, we again find that
the rescaled moments of the SFF depend only on $\Gamma$ for times
$\tau>\tau_{\text{SH}}$. Both Eq. (10) and its variants for $m>2$, see [47],
as well as the universal dependence on $\Gamma$ for fixed $L=2$ is confirmed
in Fig. 2 for the second and third moment. Due to the rescaling (11) higher
moments exhibit the same phenomenology as the SFF $\kappa(\tau)$. They depend
on both $L$ and $\Gamma$ even for times
$\tau_{\text{SH}}<\tau\lesssim\tau_{\text{Th}}$ while coinciding with the
CUE$(N^{L})$ result afterwards.

_Coupled kicked rotors._ —To demonstrate, that the RMTE describes actual
physical systems, we apply our results to a quantized dynamical system given
by two coupled kicked rotors [49]. While individual kicked rotors [50] are a
paradigmatic model for both classical and single particle quantum chaos,
coupling two rotors provides an example for the corresponding two-body setting
[51, 52, 53, 54, 38, 39, 40]. We consider coupled kicked rotors with periodic
boundary conditions, whose classical phase space is the four torus with
canonical conjugate coordinates $(q_{1},q_{2},p_{1},p_{2})$. After
quantization the effective Planck’s constant $h$ is constraint to integer
values $1/h=N$. The time evolution operator is a $N^{2}$-dimensional unitary
of the form (1) with [55, 56, 57, 58, 59]

$\displaystyle\mathcal{U}_{i}=\text{e}^{-\pi\text{i}Np_{i}^{2}}\text{e}^{-\frac{\text{i}\gamma
N}{2\pi}\cos(2\pi q_{i})}.$ (12)

Here $k_{1}=9.7$ and $k_{2}=10.5$ governs the strength of the kicks end
ensures chaotic classical dynamics. The coupling is introduced by

$\displaystyle\mathcal{U}_{\text{c}}=\text{e}^{-\frac{\text{i}\gamma
N}{2\pi}\cos(2\pi[q_{1}+q_{2}])},$ (13)

with coupling strength $\gamma$ and effective $\epsilon=\gamma N/(2\pi)$. We
choose boundary conditions for the quantum states which break time-reversal
invariance and average over such boundary conditions in order to perform the
average in the definition of the SFF and its moments. The resulting SFF and
its second moment is depicted in Fig. 3 and shows qualitatively similar
behavior as in the RMTE. However, initial fluctuations are more pronounced,
which we attribute to short periodic orbits in the classical dynamics. In
order to model the coupled kicked rotors with the bipartite RMTE we choose
$\xi_{ij}=\cos(\eta_{ij})$ with i.i.d. and uniformly distributed $\eta_{ij}$.
This yields $\chi(\epsilon)=J_{0}(\gamma N/(2\pi))$. The corresponding RMTE
result is in good agreement with numerical data and again implies universal
dependence on $\Gamma$ for $\tau>\tau_{\text{SH}}$; see Fig. 3.

For scaling parameters $\Gamma$ for which the Thouless time is given by Eq.
(7) we note, that $t_{\text{Th}}$ does not coincide with the Ehrenfest time
$t_{\text{E}}$. The latter is the time it takes for an initially localized
wave packet to spread over the system and hence indicates the time for which
quantum follows classical dynamics. It is determined by the classical system’s
Lyuapunov exponents and for the coupled kicked rotors approximately reads
$t_{\text{E}}\approx\ln(N)/(2\ln(k_{\text{A}}k_{\text{B}}/4))$ [50]. For
chaotic subsystems $t_{\text{E}}$ is necessarily smaller than the subsystem’s
Heisenberg time $t=N$ and is also much smaller than $t_{\text{Th}}$ even
though both times scale logarithmic with $N$.

Figure 3: SFF ($m=1$, left) and its second moment ($m=2$, right)
$\kappa_{m}(\tau)$ for the coupled kick rotors for different $N$ and $\Gamma$.
Black lines depict the RTME results. Dashed gray lines correspond to
$\tau=\tau_{\text{SH}}$ and $\tau=\tau_{\text{H}}$.

_Perturbative regime._ —For very small scaling parameter extrapolating the
exact result from $t<N$ to larger times gives a less accurate description of
the SFF. This is visible already for $\Gamma=2.27$ in Fig. 1 around Heisenberg
time $\tau\approx\tau_{\text{H}}$. A natural approach for $\Gamma\ll 1$ is to
extend the regularized Raleigh-Schrödinger perturbation theory introduced in
Ref. [38] from the bipartite to the extended case of arbitrary $L$. Viewing
$\mathcal{U}_{\text{c}}(\epsilon)$ as a perturbation to the non-interacting
system the eigenphases $\phi_{i}=\phi_{i}(\epsilon)$ can be expanded in a
perturbative series in $\epsilon$ which allows for computing
$K(t)=\langle\exp\left(\text{i}\sum_{ij}\phi_{i}-\phi_{j}\right)\rangle$.
While Eq. (9) still holds for $\tau<\tau_{\text{SH}}$ the perturbative
approach yields [47]

$\displaystyle\kappa(\tau)=1-\Gamma^{2}\tau\text{e}^{-(\Gamma\tau)^{2}}$ (14)

for $\tau>\tau_{\text{SH}}$ up to arbitrary large times. Again, this
universally depends on the scaling parameter $\Gamma$ only. The validity of
the perturbative approach for very small $\Gamma$ is depicted in Fig. 4.

Figure 4: SFF $\kappa(\tau)$ for small $\Gamma$ for $\tau>\tau_{\text{SH}}$
and different $N,L$. We smooth the SFF by an additional moving time average.
The perturbative result (14) is depicted as black lines. Deviations from
universality at $\Gamma=0.199$ are of subleading order $1/N$.

_Summary and Outlook._ —We have given a simple description of the SFF (and its
moments for the bipartite case) for interacting chaotic subsystems as a convex
combination of the results for the non-interacting and the strongly
interacting case. We confirm this numerically for few-body systems and expect
it to hold also for many-body systems at large $N$. Interestingly relatively
small subsystem sizes, $N\approx 10$, seem to be large enough for our
description to apply. Our description additionally implies the universal
dependence of the SFF on a single scaling parameter $\Gamma$ and is
insensitive to the detailed statistics of the phases $\xi_{i_{1}\cdots
i_{L}}$. However, using i.i.d. phases we ignore all correlations in the phases
as they would be present for instance due to spatial locality of typical many-
body systems. It therefore is an interesting open question, whether such a
simple picture applies also for these situations. Moreover, our results for
the RMTE are exact only for small times $t<N$ whereas a derivation for larger
times might be possible using field theoretical methods [60, 61]. For systems
originating from the quantization of classically chaotic systems, e.g., the
coupled kicked rotors, semiclassical periodic-orbit based techniques might
shed further light on spectral correlations. The latter approaches, however,
are left for future research.

_Acknowledgements._ — We thank A. Bäcker for insightful discussions. FF
further acknowledges fruitful discussion with P. Kos, F. G. Montoya and T.
Prosen. The work has been supported by Deutsche Forschungsgemeinschaft (DFG),
Project No. 453812159 (FF) and Project No. 497038782 (MK).

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