Quantum chaos in clean many-body systems - Tomaž Prosen

Quantum chaos in clean many-body systems
Tomaž Prosen
Faculty of mathematics and physics, University of Ljubljana, SLOVENIA
Let’s face Complexity, Como, 4-8 September 2017
Tomaž Prosen Quantum chaos in clean many-body systems

What is quantum chaos....

What is quantum chaos....
.. and how does it ﬁt to many-body physics and statistical mechanics?

Outline
Quantum chaos and Quantum chaology
The quantum chaos conjecture: semiclassical versus many-body
Quantum ergodicity, decay of correlations and Loschmidt echo
Integrability breaking ergodicity/non-ergodicity transition in spin chains
Quantum chaos in non-equilibrium steady states of open many body
systems
Out-of-time order correlation functions and Weak quantum chaos

Chaos and random matrix theory
[Berry 1977, Casati, Guarneri and Val-Gris 1982, Bohigas, Giannoni and
Schmit 1984, ...]
August 2, 2016 Advances in Physics Review
Figure 1. Examples of trajectories of a particle bouncing in a cavity: (a) non-chaotic circular and (b) chaotic
Bunimovich stadium. The images were taken from scholarpedia [60].
of freedom N:
{Ij, H} = 0, {Ij, Ik} = 0, where {f, g} =
X
j=1,N
@f
@qj
@g
@pj
@f
@pj
@g
@qj
. (1)
From Liouville’s integrability theorem [59], it follows that there is a canonical trans-
formation (p, q) ! (I, ⇥) (where I, ⇥ are called action-angle variables) such that
H(p, q) = H(I) [58]. As a result, the solutions of the equations of motion for the action-
angle variables are trivial: Ij(t) = I0
j = constant, and ⇥j(t) = ⌦jt + ⇥j(0). For obvious
reasons, the motion is referred to as taking place on an N-dimensional torus, and it is
not chaotic.
To get a feeling for the di↵erences between integrable and chaotic systems, in Fig. 1,
Physics Review
uantum Chaos in Physical Systems
Examples of Wigner-Dyson and Poisson Statistics
m matrix statistics has found many applications since its introduction by Wigner.
xtend far beyond the framework of the original motivation, and have been inten-
xplored in many fields (for a recent comprehensive review, see Ref. [93]). Examples
ntum systems whose spectra exhibit Wigner-Dyson statistics are: (i) heavy nuclei
) Sinai billiards (square or rectangular cavities with circular potential barriers in
nter) [85], which are classically chaotic as the Bunimovich stadium in Fig. 1, (iii)
excited levels5 of the hydrogen atom in a strong magnetic field [95], (iv) Spin-1/2
s and spin-polarized fermions in one-dimensional lattices [69, 70]. Interestingly, the
r-Dyson statistics is also the distribution of spacings between zeros of the Riemann
nction, which is directly related to prime numbers. In turn, these zeros can be
eted as Fisher zeros of the partition function of a particular system of free bosons
ppendix B). In this section, we discuss in more detail some examples originating
ver 30 years of research.
nuclei - Perhaps the most famous example demonstrating the Wigner-Dyson
cs is shown in Fig. 2. That figure depicts the cumulative data of the level spacing
ution obtained from slow neutron resonance data and proton resonance data of
30 di↵erent heavy nuclei [71, 96]. All spacings are normalized by the mean level
g. The data are shown as a histogram and the two solid lines depict the (GOE)
r-Dyson distribution and the Poisson distribution. One can see that the Wigner-
distribution works very well, confirming Wigner’s original idea.
Nearest neighbor spacing distribution for the “Nuclear Data Ensemble” comprising 1726 spacings
m) versus normalized (to the mean) level spacing. The two lines represent predictions of the random
OE ensemble and the Poisson distribution. Taken from Ref. [96]. See also Ref. [71].
August 2, 2016 Advances in Physics Review
0.7r
0.4r
⌦
(1, 1)
(0, 0)
Figure 1. One of the regions proven by Sinai to
be classically chaotic is this region Ω
constructed from line segments and circular
arcs.
Traditionally, analysis of the spectrum recovers
information such as the total area of the billiard,
from the asymptotics of the counting function
N(λ) = #{λn ≤ λ}: As λ → ∞, N(λ) ∼ area
4π
λ
(Weyl’s law). Quantum chaos provides completely
different information: The claim is that we should
be able to recover the coarse nature of the dynam-
ics of the classical system, such as whether they
are very regular (“integrable”) or “chaotic”. The
term integrable can mean a variety of things, the
least of which is that, in two degrees of freedom,
there is another conserved quantity besides ener-
gy, and ideally that the equations of motion can be
explicitly solved by quadratures. Examples are the
rectangular billiard, where the magnitudes of the
momenta along the rectangle’s axes are conserved,
or billiards in an ellipse, where the product of an-
gular momenta about the two foci is conserved,
and each billiard trajectory repeatedly touches a
conic confocal with the ellipse. The term chaotic
indicates an exponential sensitivity to changes
of initial condition, as well as ergodicity of the
motion. One example is Sinai’s billiard, a square
billiard with a central disk removed; another class
of shapes investigated by Sinai, and proved by him
Figure 2. This figure gives some idea of how
classical ergodicity arises in Ω.
s = 0 1 2 3 4
y = e s
Figure 3. It is conjectured that the distribution
of eigenvalues π2
(m2
/a2
+ n2
/b2
) of a
rectangle with sufficiently incommensurable
sides a, b is that of a Poisson process. The
mean is 4π/ab by simple geometric reasoning,
s = 0 1 2 3 4
GOE distribution
Figure 4. Plotted here are the normalized gaps
between roughly 50,000 sorted eigenvalues
for the domain Ω, computed by Alex Barnett,
compared to the distribution of the
ties in the
mass at th
chaotic ca
empirical
the “gener
3 and 4 sh
Some p
the case of
by Sarnak
Marklof. H
even ther
conjecture
gaps in t
rectangles
The beh
tics of the
whose loc
length L
between le
offer infor
are often
is the var
believed to
(in “generi
amples). A
0.7r
0.4r
⌦
(1, 1)
(0, 0)
Figure 1. One of the regions proven by Sinai to
be classically chaotic is this region Ω
constructed from line segments and circular
arcs.
Traditionally, analysis of the spectrum recovers
information such as the total area of the billiard,
from the asymptotics of the counting function
N(λ) = #{λn ≤ λ}: As λ → ∞, N(λ) ∼ area
4π
λ
(Weyl’s law). Quantum chaos provides completely
different information: The claim is that we should
be able to recover the coarse nature of the dynam-
ics of the classical system, such as whether they
are very regular (“integrable”) or “chaotic”. The
term integrable can mean a variety of things, the
least of which is that, in two degrees of freedom,
there is another conserved quantity besides ener-
gy, and ideally that the equations of motion can be
explicitly solved by quadratures. Examples are the
rectangular billiard, where the magnitudes of the
momenta along the rectangle’s axes are conserved,
or billiards in an ellipse, where the product of an-
gular momenta about the two foci is conserved,
and each billiard trajectory repeatedly touches a
conic confocal with the ellipse. The term chaotic
indicates an exponential sensitivity to changes
of initial condition, as well as ergodicity of the
motion. One example is Sinai’s billiard, a square
billiard with a central disk removed; another class
of shapes investigated by Sinai, and proved by him
Figure 2. This figure gives some idea of how
classical ergodicity arises in Ω.
s = 0 1 2 3 4
y = e s
Figure 3. It is conjectured that the distribution
of eigenvalues π2
(m2
/a2
+ n2
/b2
) of a
rectangle with sufficiently incommensurable
sides a, b is that of a Poisson process. The
mean is 4π/ab by simple geometric reasoning,
Figure 3. (Left panel) Distribution of 250,000 single-particle energy level spacings in a rectangular two-
dimensional box with sides a and b such that a/b = 4
p
5 and ab = 4⇡. (Right panel) Distribution of 50,000
single-particle energy level spacings in a chaotic cavity consisting of two arcs and two line segments (see inset).Tomaž Prosen Quantum chaos in clean many-body systems

tion of such spacings below a given bound x is
x
−∞ P(s)ds. A dramatic insight of quantum chaos
is given by the universality conjectures for P(s):
• If the classical dynamics is integrable, then
P(s) coincides with the corresponding quantity for
a sequence of uncorrelated levels (the Poisson en-
semble) with the same mean spacing: P(s) = ce−cs
,
c = area/4π (Berry and Tabor, 1977).
• If the classical dynamics is chaotic, then P(s)
coincides with the corresponding quantity for the
eigenvalues of a suitable ensemble of random
matrices (Bohigas, Giannoni, and Schmit, 1984).
Remarkably, a related distribution is observed for
the zeros of Riemann’s zeta function.
Not a single instance of these conjectures is
known, in fact there are counterexamples, but
the conjectures are expected to hold “generically”,
that is unless we have a good reason to think oth-
erwise. A counterexample in the integrable case
is the square billiard, where due to multiplici-
syst
Eme
MN
New
[3] J
S. Z
fun
Vol
Tsu
34 Notices of the AMS
lower energies) and chaotic (occurring at higher energies) motion [98]. Results of numeri-
cal simulations (see Fig. 4) show a clear interpolation between Poisson and Wigner-Dyson
level statistics as the dimensionless energy (denoted by Ê) increases [95]. Note that at
intermediate energies the statistics is neither Poissonian nor Wigner-Dyson, suggesting
Figure 4. The level spacing distribution of a hydrogen atom in a magnetic field. Di↵erent plots correspond to
di↵erent mean dimensionless energies Ê, measured in natural energy units proportional to B2/3, where B is the
magnetic field. As the energy increases one observes a crossover between Poisson and Wigner-Dyson statistics.
The numerical results are fitted to a Brody distribution (solid lines) [87], and to a semi-classical formula due to
Berry and Robnik (dashed lines) [97]. From Ref. [95].
Hydrogen atom in uniform magnetic field [Wintgen and Friedrich 1987].

The proof of Quantum chaos conjecture: Semiclassics
[Berry 1985, Sieber and Richter 2001, Müller, ...and Haake 2004,2005]
A simple key object: Spectral correlation function
R( ) =
1
¯ρ2
ρ(E +
2π¯ρ
)ρ(E −
2π¯ρ
− 1
or the spectral form factor
K(τ) =
1
π
∞
−∞
d R( )e2i τ
∼
n
ei nτ
2

The proof of Quantum chaos conjecture: Semiclassics
[Berry 1985, Sieber and Richter 2001, Müller, ...and Haake 2004,2005]
A simple key object: Spectral correlation function
R( ) =
1
¯ρ2
ρ(E +
2π¯ρ
)ρ(E −
2π¯ρ
− 1
or the spectral form factor
K(τ) =
1
π
∞
−∞
d R( )e2i τ
∼
n
ei nτ
2
In non-integrable systems with a chaotic classical lomit, form factor has two
regimes:
universal described by RMT,
non-universal described by short classical periodic orbits.

For chaotic (hyperbolic) systems, K(τ), to all orders in τn
, agrees with RMT!
(based on small asymptotics!)

, agrees with RMT!
To ﬁrst order, this is captured by the diagonal approximation (Berry)

, agrees with RMT!
To second order, the RMT term is reproduced by considering so-called
Sieber-Richter pairs of orbits
4
Figure 1. Bunch of 72 (pseudo-)orbits differing in two three-encounters and one
two-encounter (a pseudo-orbit is a set of several disjoined orbits; see below).
Different orbits not resolved except in blowups of encounters.
Figure 2. Sieber–Richter pair.
where two stretches of an orbit are close (see figure 2). Full agreement with all coefficients cn
from RMT was established by the present authors in [13, 14].
In none of these works, oscillatory contributions could be obtained (note however
courageous forays by Keating [16] and Bogomolny and Keating [17]), due to the fact that
Gutzwiller’s formula for the level density is divergent. To enforce convergence, one needs to
allow for complex energies with imaginary parts large compared to the mean level spacing,
Im ✏ 1. Oscillatory terms proportional to e2i✏
then become exponentially small and cannot be
resolved within the conventional semiclassical approach.
In [18], we proposed a way around this difficulty that we here want to elaborate in detail.Tomaž Prosen Quantum chaos in clean many-body systems

, agrees with RMT!
To second order, the RMT term is reproduced by considering so-called
Sieber-Richter pairs of orbits
two-encounter (a pseudo-orbit is a set of several disjoined orbits; see below).
Different orbits not resolved except in blowups of encounters.
Figure 2. Sieber–Richter pair.
where two stretches of an orbit are close (see figure 2). Full agreement with all coefficients cn
from RMT was established by the present authors in [13, 14].
In none of these works, oscillatory contributions could be obtained (note however
courageous forays by Keating [16] and Bogomolny and Keating [17]), due to the fact that
Gutzwiller’s formula for the level density is divergent. To enforce convergence, one needs to
allow for complex energies with imaginary parts large compared to the mean level spacing,
Im ✏ 1. Oscillatory terms proportional to e2i✏
then become exponentially small and cannot be
resolved within the conventional semiclassical approach.
In [18], we proposed a way around this difficulty that we here want to elaborate in detail.
The key idea is to represent the correlation function through derivatives of a generating function
involving spectral determinants. Two such representations are available and entail different
semiclassical periodic-orbit expansions. One of them recovers the non-oscillatory part of the
asymptotic expansion (3), essentially in equivalence to [12]–[14]; the other representation
breaks new ground by giving the oscillatory part of (3).
The full random-matrix result also, and in fact most naturally, arises within an alternative
semiclassical approximation scheme proposed by Berry and Keating in [19]. That scheme
To all orders, RMT terms is reproduced by considering full combinatorics of
self-encountering orbits (Müller et al, 2004)4
two-encounter (a pseudo-orbit is a set of several disjoined orbits; see below).Tomaž Prosen Quantum chaos in clean many-body systems

The many-body quantum chaos problem at “ = 1”
A lot of numerics accumulated to day, showing that integrable many body
systems, such as an interacting chain of spinless fermions:
H =
L−1
j=0
(−Jc†
j cj+1 − J c†
j cj+2 + h.c. + Vnj nj+1 + V nj nj+2), nj = c†
j cj .016 Advances in Physics Review
0
0.5
1
P
0
0.5
1
P
0 2 4
ω
0 2 4
ω
0
0.5
1
P
0 2 4
ω
0.1 1
J’=V’
0
0.5
1
ωmax
L=18
L=21
L=24
J’=V’=0.00 J’=V’=0.02 J’=V’=0.04 J’=V’=0.08
J’=V’=0.16 J’=V’=0.32 J’=V’=0.64
(a) (b) (c) (d)
(e) (f) (g) (h)
Figure 5. (a)–(g) Level spacing distribution of spinless fermions in a one-dimensional lattice with Hamiltonian
(40). They are the average over the level spacing distributions of all k-sectors (see text) with no additional
symmetries (see Ref. [69] for details). Results are reported for L = 24, N = L/3, J = V = 1 (unit of energy), and
J0 = V 0 (shown in the panels) vs the normalized level spacing !. The smooth continuous lines are the Poisson
and Wigner-Dyson (GOE) distributions. (h) Position of the maximum of P(!), denoted as !max, vs J0 = V 0, for
three lattice sizes. The horizontal dashed line is the GOE prediction. Adapted from Ref. [69].
[from: Rigol and Santos, 2010]
More data: Montamboux et al.1993, Prosen 1999, 2002, 2005, 2007, 2014,
Kollath et al. 2010, ...

My favourite toy model of many-body quantum chaos: Kicked Ising Chain
Prosen PTPS 2000, Prosen PRE 2002
H(t) =
L−1
j=0
Jσz
j σz
j+1 + (hxσx
j + hzσz
j )
m∈Z
δ(t − m)
UFloquet = T exp −i
1+
0+
dt H(t ) =
j
exp −i(hxσx
j + hzσz
j ) exp −iJσz
j σz
j+1
where [σα
j , σβ
k ] = 2iεαβγσγ
j δjk .

H(t) =
L−1
j=0
Jσz
j σz
j+1 + (hxσx
j + hzσz
j )
m∈Z
δ(t − m)
1+
0+
dt H(t ) =
j
exp −i(hxσx
j + hzσz
j ) exp −iJσz
j σz
j+1
where [σα
j , σβ
j δjk .
The model is completely integrable in terms of Jordan-Wigner transformation if
hx = 0 (longitudinal ﬁeld)
hz = 0 (transverse ﬁeld)

H(t) =
L−1
j=0
Jσz
j σz
j+1 + (hxσx
j + hzσz
j )
m∈Z
δ(t − m)
1+
0+
dt H(t ) =
j
exp −i(hxσx
j + hzσz
j ) exp −iJσz
j σz
j+1
where [σα
j , σβ
j δjk .
The model is completely integrable in terms of Jordan-Wigner transformation if
hx = 0 (longitudinal ﬁeld)
hz = 0 (transverse ﬁeld)
Time-evolution of local observables is quasi-exact, e.g. for computing
U−t
Floquetσα
j Ut
Floquet
only 2t + 1 sites in the range [j − t, j + t] are needed!.
Quantum cellular automaton in the sense of Schumacher and Werner (2004).

Quasi-energy level statistics of KI
[C. Pineda, TP, PRE 2007]
Fix J = 0.7, hx = 0.9, hz = 0.9, s.t. KI is (strongly) non-integrable.
Diagonalize
UFloquet|n = exp(−iϕn)|n .
For each conserved total momentum K quantum number, we ﬁnd N ∼ 2L
/L
levels, normalized to mean level spacing as en = N
2π
ϕn, sn = en+1 − en.

Short-range statistics: Nearest neighbor level spacings
We plot cumulative level spacing distribution
W (s) =
s
0
dsP(s) = Prob{sn+1 − sn < s}.
0 1 2 3 4
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0 1 2 3
0
0.01
0 1 2 3 4
-0.004
-0.002
0
0.002
0.004
0.006
0.008
ss
W−WWignerW−WWigner
The noisy curve shows the difference between the numerical data for 18 qubits,
averaged over the different momentum sectors, and the Wigner RMT surmise.
The smooth (red) curve is the difference between infinitely dimensional COE
solution and the Wigner surmise. In the inset we present a similar figure with
the results for each of quasi-moemtnum sector K.

Short and long range: Spectral form factor
Spectral form factor K2(τ) is for nonzero integer t deﬁned as
K2(t/N) =
1
N
tr Ut 2
=
1
N n
e−iϕnt
2
.
0.25 0.5 0.75 1 1.25 1.5 1.75 2
0.2
0.4
0.6
0.8
1
0.5 1 1.5
0.02
0.02
t/τH
K2
We show the behavior of the form factor for L = 18 qubits. We perform
averaging over short ranges of time (τH/25). The results for each of the
K-spaces are shown in colors. The average over the diﬀerent spaces as well as
the theoretical COE(N) curve is plotted as a black and red curve, respectively.

Surprise!? Deviaton from universality at short times
Similarly as for semi-classical systems, we ﬁnd notable statistically signiﬁcant
deviations from universal COE/GOE predictions for short times of few kicks.
10 12 14 16 18 20
-6
-5
-4
-3
-2
10 12 14 16 18 20
-6
-5
-4
-3
-2
log10K2(1/τH)log10K2(1/τH)
LL
10 12 14 16 18 20
-2
-1
0
1
2
3
1
4
2
nσ
L
But there is no underlying classical structure! Dynamical explanation of this
phenomenon needed!

OPEN PROBLEM: Quantum chaos conjecture for clean many body systems
Consider a quantum lattice system with local interaction
H =
j,k
h(j, k)
where the interaction term h is ﬁxed, i.e. has no "random parameters".
Prove that the spectral correlations of H, once one removes trivial symmetries
such as translation invariance, are described by RMT in the thermodynamic limit
(L → ∞), as soon as the model is non-integrable.

Eigenstate thermalization hypothesis
Ergodicity of individual (almost all!) many-body eigenstates
[Deutsch 1991, Srednicki 1994, TP 1999, Rigol et al 2008]
For a typical (say local) observable A
Ψn|A|Ψn ≈ a(En), a(E(β)) =
tr A exp(−βH)
tr exp(−βH)
, E(β) = −
∂
∂β
log tr exp(−βH).August 2, 2016 Advances in Physics Review
Figure 13. Eigenstate expectation values of the occupation of the zero momentum mode [(a)–(c)] and the kinetic
energy per site [(d)–(f)] of hard-core bosons as a function of the energy per site of each eigenstate in the entire
spectrum, that is, the results for all k-sectors are included. We report results for three system sizes (L = 18, 21,
and 24), a total number of particles N = L/3, and for two values of J0 = V 0 [J0 = V 0 = 0.16 in panels (b) and (e)
and J0 = V 0 = 0.64 in panels (c) and (f)] as one departs from the integrable point [J0 = V 0 = 0 in panels (a) and
(d)]. In all cases J = V = 1 (unit of energy). See also Ref. [157].
zero momentum mode occupation
[from: D’Alesio et al, Adv. Phys. 2016]
Top: A = leading Fourier mode occupation number (nonlocal),
Bottom: A = average kinetic energy (local).

From the spectrum to time evolution ...

Quantum ergodicity transition and the "order parameter"
Consider many-body Floquet dynamics for simplicity:
Temporal correlation of an extensive traceless observable A
(tr A = 0, tr A2
∝ L):
CA(t) = lim
L→∞
1
L2L
tr AU−t
AUt
Average correlator
DA = lim
T→∞
1
T
T−1
t=0
CA(t)
signals quantum ergodicity if DA = 0.
Quantum chaos regime in KI chain seems compatible with exponential
decay of correlations. For integrable, and weakly non-integrable cases,
though, we ﬁnd saturation of temporal correlations D = 0.

VOLUME 80, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 2 MARCH 1998
Time Evolution of a Quantum Many-Body System: Transition from Integrability
to Ergodicity in the Thermodynamic Limit
Tomaˇz Prosen
Physics Department, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia
(Received 17 July 1997)
Numerical evidence is given for nonergodic (nonmixing) behavior, exhibiting ideal transport, of a
simple nonintegrable many-body quantum system in the thermodynamic limit, namely, the kicked t-V
model of spinless fermions on a ring. However, for sufficiently large kick parameters t and V we
recover quantum ergodicity, and normal transport, which can be described by random matrix theory.
[S0031-9007(98)05420-9]
PACS numbers: 05.30.Fk, 05.45.+b, 72.10.Bg
A simple question is addressed here: “Do intermedi-
ate quantum many-body systems, which are neither in-
tegrable nor ergodic, exist in the thermodynamic limit”?
While it is clear that integrable systems are rather excep-
tional, it is an important open question whether a finite
generic perturbation of an integrable system becomes er-
godic or not in the thermodynamic limit (TL), size ! `
and fixed density. It is known that local statistical prop-
erties of quantum systems with few degrees of freedom
whose classical limit is completely chaotic/ergodic, are
universally described by random matrix theory (RMT);
while in the other extreme case of integrable systems,
Poissonian statistics may typically be applied [1,2] (with
some notable nongeneric exceptions such as finite dimen-
sional harmonic oscillator). This statement has also been
recently verified numerically for integrable and strongly
nonintegrable many-body systems of interacting fermions
[3] which do not have a classical limit.
Having lost the reference to classical dynamics, we re-
sort to the definition of quantum ergodicity (also termed
quantum mixing) [4] as the decay of time correlations
kAstdBs0dl 2 kAl kBl of any pair of quantum observables
A and B in TL, taking the time limit t ! ` in the end.
In [4] a many-body system of interacting bosons has been
studied, and it has been shown that quantum ergodic-
ity corresponds to strongly chaotic (ergodic) dynamics of
associated nonlinear mean-field equations. As a conse-
quence of linear response theory, quantum ergodicity also
implies normal transport and finite transport coefficients
(such as dc electrical conductivity). On the other hand,
integrable systems, which are solvable by Bethe ansatz
or quantum inverse scattering, are characterized by (infin-
itely many) conservation laws and are thus nonergodic. It
has been pointed out recently [5] that integrability implies
nonvanishing stiffness, i.e., ideal conductance with infi-
nite transport coefficients (or ideal insulating state). As
we argue below, any deviation from quantum ergodicity
generically implies nonvanishing long-time current auto-
correlation and therefore an infinite transport coefficient.
statistics smoothly interpolating from Poisson to RMT), it
is thus important to question if and when such nonergod-
icity can survive TL.
In this Letter we introduce a family of simple many-
body systems smoothly interpolating between integrable
and ergodic regimes, namely, kicked t-V model (KtV)
of spinless fermions with periodically switched nearest-
neighbor interaction on a 1D lattice of size L and peri-
odic boundary conditions L ; 0, with a time-dependent
Hamiltonian,
Hstd ≠
L21X
j≠0
f2
1
2 tsc
y
j cj11 1 H.c.d 1 dpstdVnjnj11g ,
(1)
and give numerical evidence for the existence of an inter-
mediate nonergodic regime in TL by direct simulation of
the time evolution. c
y
j , cj, nj are fermionic creation, anni-
hilation, and number operators, respectively, and dpstd ≠P`
m≠2` dst 2 md. Deviations from quantum ergodicity
(or mixing) are characterized by several different quanti-
ties as described below.
The KtV model (1) is a many-body analog of popular
1D nonintegrable kicked systems [2] such as, e.g., kicked
rotor: Its evolution (Floquet) operator over one period,
U ≠ ˆT expf2i
R11
01 dtHstdg s ¯h ≠ 1d, factorizes into the
product of a kinetic and potential part,
U ≠ exp
√
2iV
L21X
j≠0
njnj11
!
3 exp
√
it
L21X
k≠0
cosssk 1 fdñk
!
, (2)
where s ≠ 2pyL. The flux parameter f is used in order
to introduce a current operator J ≠ siytdUy≠fUjf≠0 ≠PL21
k≠0 sinsskdñk, elsewhere we put f :≠ 0. The tilde de-
notes the operators which refer to momentum variable
k, ˜ck ≠ L21y2
PL21
j≠0 expsisjkdcj, ñk ≠ ˜c
y
k ˜ck. The KtV
[S0031-9007(98)05420-9]
PACS numbers: 05.30.Fk, 05.45.+b, 72.10.Bg
A simple question is addressed here: “Do intermedi-
ate quantum many-body systems, which are neither in-
tegrable nor ergodic, exist in the thermodynamic limit”?
While it is clear that integrable systems are rather excep-
tional, it is an important open question whether a finite
generic perturbation of an integrable system becomes er-
godic or not in the thermodynamic limit (TL), size ! `
and fixed density. It is known that local statistical prop-
erties of quantum systems with few degrees of freedom
whose classical limit is completely chaotic/ergodic, are
universally described by random matrix theory (RMT);
while in the other extreme case of integrable systems,
Poissonian statistics may typically be applied [1,2] (with
some notable nongeneric exceptions such as finite dimen-
sional harmonic oscillator). This statement has also been
recently verified numerically for integrable and strongly
nonintegrable many-body systems of interacting fermions
[3] which do not have a classical limit.
Having lost the reference to classical dynamics, we re-
sort to the definition of quantum ergodicity (also termed
quantum mixing) [4] as the decay of time correlations
kAstdBs0dl 2 kAl kBl of any pair of quantum observables
A and B in TL, taking the time limit t ! ` in the end.
In [4] a many-body system of interacting bosons has been
studied, and it has been shown that quantum ergodic-
ity corresponds to strongly chaotic (ergodic) dynamics of
associated nonlinear mean-field equations. As a conse-
quence of linear response theory, quantum ergodicity also
implies normal transport and finite transport coefficients
(such as dc electrical conductivity). On the other hand,
integrable systems, which are solvable by Bethe ansatz
or quantum inverse scattering, are characterized by (infin-
itely many) conservation laws and are thus nonergodic. It
has been pointed out recently [5] that integrability implies
nonvanishing stiffness, i.e., ideal conductance with infi-
nite transport coefficients (or ideal insulating state). As
we argue below, any deviation from quantum ergodicity
generically implies nonvanishing long-time current auto-
correlation and therefore an infinite transport coefficient.
Since generic nonintegrable systems of finite size (num-
ber of degrees of freedom) are nonergodic (obeying mixed
statistics smoothly interpolating from Poisson to RMT), it
is thus important to question if and when such nonergod-
icity can survive TL.
In this Letter we introduce a family of simple many-
body systems smoothly interpolating between integrable
and ergodic regimes, namely, kicked t-V model (KtV)
of spinless fermions with periodically switched nearest-
neighbor interaction on a 1D lattice of size L and peri-
odic boundary conditions L ; 0, with a time-dependent
Hamiltonian,
Hstd ≠
L21X
j≠0
f2
1
2 tsc
y
j cj11 1 H.c.d 1 dpstdVnjnj11g ,
(1)
and give numerical evidence for the existence of an inter-
mediate nonergodic regime in TL by direct simulation of
the time evolution. c
y
j , cj, nj are fermionic creation, anni-
hilation, and number operators, respectively, and dpstd ≠P`
m≠2` dst 2 md. Deviations from quantum ergodicity
(or mixing) are characterized by several different quanti-
ties as described below.
The KtV model (1) is a many-body analog of popular
1D nonintegrable kicked systems [2] such as, e.g., kicked
rotor: Its evolution (Floquet) operator over one period,
U ≠ ˆT expf2i
R11
01 dtHstdg s ¯h ≠ 1d, factorizes into the
product of a kinetic and potential part,
U ≠ exp
√
2iV
L21X
j≠0
njnj11
!
3 exp
√
it
L21X
k≠0
cosssk 1 fdñk
!
, (2)
where s ≠ 2pyL. The flux parameter f is used in order
to introduce a current operator J ≠ siytdUy≠fUjf≠0 ≠PL21
k≠0 sinsskdñk, elsewhere we put f :≠ 0. The tilde de-
notes the operators which refer to momentum variable
k, ˜ck ≠ L21y2
PL21
j≠0 expsisjkdcj, ñk ≠ ˜c
y
k ˜ck. The KtV
model is integrable if either t ≠ 0, or V ≠ 0 smod 2pd,
or tV ! 0 and tyV finite (continuous time t-V model
1808 0031-9007y98y80(9)y1808(4)$15.00 © 1998 The American Physical Society
matrices
reation
in all
particle
Fourier
t from
ces of
ore, the
mdl, us-
oughly
uperior
FPO],
um dy-
of the
n func-
ymJUm
ge.” J
kl, and
of time
,
(3)
For
mple of
n order
md for
og2 N
ne can
fferent
d total
eigen-
For sufficiently large control parameters the system is
quantum ergodic (case t ≠ V ≠ 4 of Fig. 1), DJ goes
to zero, and s remains finite as L ! ` sN ! `d and
r ≠ NyL fixed, whereas in the other case (t ≠ V ≠ 1
of Fig. 1), DJ remains well above zero as we approach
TL, whereas conductivity s diverges [8]. In Fig. 2 we
have analyzed 1yL scaling of DJ. Again, for large val-
ues of parameters, say t ≠ V ≠ 4, DJ is already practi-
cally zero for L ¯ 20, while for smaller (but not small)
control parameters DJ ¯ D`
J 1 byL, where D`
J . 0. In
the close-to-critical case t ≠ V ≠ 2, we find a larger cor-
relation time Mp
, 102
, and hence use a longer aver-
aging time M0 ≠ 200. In Fig. 3 we illustrate an ideal
transport for t , V , 1 by plotting a persistent cur-
rent J
p
$k0 ≠ limM!`s1yMd
PM
m≠1 k$k0jJsmdj$k0l vs the initial
FIG. 1. Current autocorrelation function CJ smd against dis-
crete time m for the quantum ergodic (t ≠ V ≠ 4, lower set of
curves for various sizes L) and intermediate regimes (t ≠ V ≠
1, upper set of curves) with density r ≠
1
4 . Averaging over
the entire Fock space is performed, N 0
≠ N , for L # 20,
whereas random samples of N 0
≠ 12 000 and N 0
≠ 160 ini-
tial states have been used for L ≠ 24 and L ≠ 32, respectively.
1809
dic, t ≠
≠ 2 and
e as in
regime
or t ,
nt cur-
≠ aJ$k0 .
Eq. (4)]
frs1 2
momen-
quan-
ion of
NK ¯
$kl with
m state
e m is
so [7])
entropy
es j$k0l,
pace as
smd
!
.
Rsmd as
e same
t value
conser-
have no
ey may
OE) of
e max-
ocaliza-
onds to
become
he cor-
kJl2
≠
btained
V ≠ 4,
aged over bins of size DJ ≠ 0.05) in the ergodic, t ≠ V ≠ 4,
(nearly) ergodic, t ≠ V ≠ 2, and intermediate, t ≠ V ≠ 1 and
t ≠ 1, V ≠ 2, regimes (L ≠ 24 and r ≠
1
4 .)
while for smaller values of parameters t, V, Rsmd satu-
rates to a smaller value indicating that there may exist
approximate conservation laws causing nontrivial local-
ization inside the Fock space. Scaling with 1yL suggests
that, even in TL, ¯R is smaller than ¯RCOE for the interme-
diate regime t , V , 1 (Fig. 5).
Finally, we discuss current fluctuations, or more gener-
ally, current distribution PcsId ≠ kcjdsI 2 Jdjcl giving
a probability density of having a current I in a state jcl.
We let the state c with a “good” known initial current I0
evolve for a long time from which we compute a steady-
state current distribution (SSCD),
PsI; I0d ≠ lim
M!`
1
M
MX
m≠1
kdsssI0 2 Js0dddddsssI 2 Jsmddddl .
Of course, delta functions should have a finite small width
providing averaging over several states j$kl with J$k ¯ I0.
In the quantum ergodic regime all states eventually be-
come populated, so SSCD PsI; I0d should be independent
of the initial current I0 and equal to the microcanoni-
cal current distribution PmcsId ≠ kdsI 2 Jdl. It has
been shown by elementary calculation that in TL
the latter becomes a Gaussian, PmcsId ! PGausssId ≠
s1y
p
2pkJ2ld exps2
1
2 I2
ykJ2
ld, while at any finite size L
FIG. 4. Relative localization dimension in Fock space Rsmd
for data of Fig. 1.
VOLUME 80, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R
FIG. 2. Stiffness DJ vs 1yL at constant density r ≠
1
4 and
for different values of control parameters in the ergodic, t ≠
V ≠ 4 and t ≠ V ≠ 2, and intermediate, t ≠ 1, V ≠ 2 and
t ≠ V ≠ 1, regimes. Other parameters are the same as in
Fig. 1.
current J$k0 . The normal transport in the ergodic regime
t ≠ V ≠ 4 is characterized by J
p
$k0 ≠ 0, while for t ,
V , 1 we find the ideal transport with the persistent cur-
rent being proportional to the initial current, J
p
$k0 ≠ aJ$k0 .
Proportionality constant a can be computed from [Eq. (4)]
DJ ≠ s1yLd kJ$k0 J
p
$k0 l ≠ sayLd kJ2l, so a ≠ 2DJyfrs1 2
rdg, where kJ2
l is given below [Eq. (5)].
Because of translational symmetry, the total momen-P
FIG. 3. Persistent c
aged over bins of siz
(nearly) ergodic, t ≠
t ≠ 1, V ≠ 2, regim
while for smaller
rates to a smaller
approximate conse
ization inside the F
that, even in TL, ¯R
diate regime t , V
Finally, we discu
ally, current distrib
a probability densi
VOLUME 80, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R
FIG. 5. Limiting relative localization dimension ¯R vs 1yL for
data of Fig. 2.
FIG. 6. Steady-sta
ian PsI, I dyPTomaž Prosen Quantum chaos in clean many-body systems

Decay of time correlatons in KI chain
Three typical cases of parameters:
(a) J = 1, hx = 1.4, hz = 0.0
(completely integrable).
(b) J = 1, hx = 1.4, hz = 0.4
(intermediate).
(c) J = 1, hx = 1.4, hz = 1.4
("quantum chaotic").
0.1
0 5 10 15 20 25 30 35 40 45 50
|<M(t)M>|/L
t
10
-2
10-3
(c)
L=20
L=16
L=12
0.25exp(-t/6)
0
0.2
0.4
0.6
0.8
<M(t)M>/L
(b)
DM/L=0.293
0
0.2
0.4
0.6
0.8
<M(t)M>/L
(a)
DM/L=0.485
|<M(t)M>-DM|/L
10-1
10-2
10-3
10 20 30 t

Loschmidt echo and decay of ﬁdelity
Decay of correlations is closely related to
ﬁdelity decay F(t) = U−t
Ut
δ(t) due to
perturbed evolution Uδ = U exp(−iδA)
(Prosen PRE 2002) e.g. in a linear re-
sponse approximation:
F(t) = 1 −
δ2
2
t
t ,t =1
C(t − t )
(a) J = 1, hx = 1.4, hz = 0.0
(completely integrable).
(b) J = 1, hx = 1.4, hz = 0.4
(intermediate).
(c) J = 1, hx = 1.4, hz = 1.4
("quantum chaotic").
0.1
0 50 100 150 200 250 300 350 400
|F(t)|
t
10
-2
10
-3
10
-4
δ’=0.04
δ’=0.02 δ’=0.01
(c)
L=20
L=16
L=12
theory
0.1
|F(t)|
10-2
10
-3
δ’=0.01
δ’=0.005
δ’=0.0025
(b)
0.1
|F(t)|
10
-2
10
-3
δ’=0.01 δ’=0.005
δ’=0.0025
(a)

Quantum Ruelle-Policot-like resonances
TP, J. Phys. A 35, L737 (2002)
Transfer matrix approach to exponential decay of correlation:
Truncated quantum Perron-Frobenius map and Ruelle-like resonances.

Quantum Ruelle-Policot-like resonances
TP, J. Phys. A 35, L737 (2002)
Transfer matrix approach to exponential decay of correlation:
Truncated quantum Perron-Frobenius map and Ruelle-like resonances.
We construct a matrix representation of the following dynamical Heisenberg
map
ˆTA = [U†
AU]r
truncated with respect to the following basis of translationally invariant
extensive observables
Z(s0s1...sr−1) =
∞
j=−∞
σs0
j σs1
j+1 · · · σ
sr−1
j+r−1
and inner product
(A|B) = lim
L→∞
1
L2L
tr A†
B, A =
s
as Zs ⇒ (A|A) =
s
|as |2
< ∞.

Quantum Ruelle resonances
J = 0.7, hx = 1.1
r=5
r=6
r=7
hz=0.5 hz=0.0
0.01
0.1
1
0 10 20 30 40 50 60 70
|C(t)|
t
UL: L=24
L=12
Tr: r=12
r=6
w1exp(-q1t)

Nonequilibrium quantum transport problem in one-dimension
Canonical markovian master equation for the many-body density matrix:
The Lindblad (L-GKS) equation:
dρ
dt
= ˆLρ := −i[H, ρ] +
µ
2LµρL†
µ − {L†
µLµ, ρ} .

Nonequilibrium quantum transport problem in one-dimension
Canonical markovian master equation for the many-body density matrix:
The Lindblad (L-GKS) equation:
dρ
dt
= ˆLρ := −i[H, ρ] +
µ
2LµρL†
µ − {L†
µLµ, ρ} .
Bulk: Fully coherent, local interactions,e.g. H = n−1
x=1 hx,x+1.
Boundaries: Fully incoherent, ultra-local dissipation,
jump operators Lµ supported near boundaries x = 1 or x = n.

Nonequilibrium steady state (NESS) density matrix
ˆLρNESS = 0.
Integrability vs. non-integrability of NESS [TP, J. Phys. A48, 373001(2015)]

ˆLρNESS = 0.
Spectral correlations of Heﬀ = − log ρNESS signal (non)integrability of NESS!

ˆLρNESS = 0.
Spectral correlations of Heff = − log ρNESS signal (non)integrability of NESS!
a current, as is the case in all NESSs studied here, we
expect the LSD to display GUE statistics and not the one
for the Gaussian orthogonal ensemble (GOE) irrespective
of the symmetry class to which the Hamiltonian belongs
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
p(s)
s=λj+1-λj
(b) staggered XXZ chain
0
0.2
0.4
0.6
0.8
1
p(s)
(a) XXZ chain, ∆=0.5
FIG. 2 (color online). LSD for NESSs of nonsolvable systems.
(a) XXZ chain with Á ¼ 0:5 (À ¼ 1, ¼ 0:2, ¼ 0:3).
(b) XXZ chain with Á ¼ 0:5 in a staggered field ( ¼ 0:1,
¼ 0, À ¼ 1). Both cases are for n ¼ 14 in the sector with
Z ¼ 7. Full black curve is the Wigner surmise for the GUE; the
dotted blue curve is for the Gaussian orthogonal ensemble.
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3 3.5 4
p(s)
s=λj+1-λj
(c)XXX chain, µ=1
0
0.2
0.4
0.6
0.8
1
p(s)
(b)XX chain with dephasing
0
0.2
0.4
0.6
0.8
1
p(s)
(a)XX chain
FIG. 1 (color online). LSD for NESSs of solvable nonequilib-
rium systems. (a) XX chain (n ¼ 16, Z ¼ 10). (b) XX chain with
dephasing of strength
¼ 1 (n ¼ 14, Z ¼ 7). (c) XXX chain
with maximal driving ¼ 1 (n ¼ 20, Z ¼ 5, Á ¼ 1). Cases (a)
and (b) are for À ¼ 1, ¼ 0:2, ¼ 0:3, while (c) is for À ¼
0:1, ¼ 1, ¼ 0.
PRL 111, 124101 (2013) P H Y S I C A L R E V I E W L E T T E R S
week ending
20 SEPTEMBER 2013
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
p(s)
s=λj+1-λj
(b) staggered XXZ chain
0
0.2
0.4
0.6
0.8
1
p(s)
(a) XXZ chain, ∆=0.5
FIG. 2 (color online). LSD for NESSs of nonsolvable systems.
(a) XXZ chain with Á ¼ 0:5 (À ¼ 1, ¼ 0:2, ¼ 0:3).
(b) XXZ chain with Á ¼ 0:5 in a staggered field ( ¼ 0:1,
¼ 0, À ¼ 1). Both cases are for n ¼ 14 in the sector with
Z ¼ 7. Full black curve is the Wigner surmise for the GUE; the
dotted blue curve is for the Gaussian orthogonal ensemble.
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3 3.5 4
p(s)
s=λj+1-λj
(c)XXX chain, µ=1
0
0.2
0.4
0.6
0.8
1
p(s)
(b)XX chain with dephasing
0
0.2
0.4
0.6
0.8
1
p(s)
(a)XX chain
1 (color online). LSD for NESSs of solvable nonequilib-
systems. (a) XX chain (n ¼ 16, Z ¼ 10). (b) XX chain with
L 111, 124101 (2013) P H Y S I C A L R E V I E W L E T T E R S
week ending
20 SEPTEMBER 2013
Exactly solvable (left) Non-exactly solvable (right)
[Prosen and Žnidarič, PRL 111, 124101 (2013)]

Redeﬁning quantum chaos? Out-of-time-order correlations.
CAB (t) = [A(t), B]2
− [A(t), B] 2
.

CAB (t) = [A(t), B]2
− [A(t), B] 2
.
Key OTOC term: A(t)BA(t)B .

CAB (t) = [A(t), B]2
− [A(t), B] 2
.
Studied by Larkin and Ovchinikov (1969) in the context of semiclassical theory
of superconductivity. In the semiclassical limit → 0,
i
[A(t), B] → {Acl(t), Bcl}PB,
hence
CAB (t) ∼ eλLyapunovt
, t tEhrenfest ∼
log(1/ )
λLyapunov
.

CAB (t) = [A(t), B]2
− [A(t), B] 2
.
Studied by Larkin and Ovchinikov (1969) in the context of semiclassical theory
of superconductivity. In the semiclassical limit → 0,
i
[A(t), B] → {Acl(t), Bcl}PB,
hence
CAB (t) ∼ eλLyapunovt
, t tEhrenfest ∼
log(1/ )
λLyapunov
.
OTOC has been picked up as a deﬁnition of Quantum chaos in many-body
systems and QFT by the community working on AdS/CFT holography
[Kitaev 2014, Maldacena, Shenker and Stanford 2016 + cca.100 papers...]

Sachdev-Ye-Kitaev model and bound on chaos
H =
N
i,j,k,l=1
ξijkl γi γj γk γl , {γj , γk } = δj,k .
CAB (t) ∼
1
N
eλLt
where
λL ≤
2π
β
.

Sachdev-Ye-Kitaev model and bound on chaos
H =
N
i,j,k,l=1
ξijkl γi γj γk γl , {γj , γk } = δj,k .
CAB (t) ∼
1
N
eλLt
where
λL ≤
2π
β
.
However, for systems with local interactions H = x hx,x+1 and bounded local
operators A = u(x), B = v(y)
CAB (t) ≤ 4 u 2
v 2
min(1, eλ(t−|x−y|/v)
)
which is just a slightly nontrivial restatement of well-known Lieb-Robinson
bound (causality).
Information propagation in isolated quantum systems
David J. Luitz1, 2
and Yevgeny Bar Lev3
1
Department of Physics and Institute for Condensed Matter Theory,
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
2
Department of Physics, T42, Technische Universit¨at M¨unchen,
James-Franck-Straße 1, D-85748 Garching, Germany⇤
3
Department of Chemistry, Columbia University,
3000 Broadway, New York, New York 10027, USA†
nglement growth and out-of-time-order correlators (OTOC) are used to assess the propaga-
nformation in isolated quantum systems. In this work, using large scale exact time-evolution
w that for weakly disordered nonintegrable systems information propagates behind a ballisti-
oving front, and the entanglement entropy growths linearly in time. For stronger disorder the
of the information front is algebraic and sub-ballistic and is characterized by an exponent
epends on the strength of the disorder, similarly to the sublinear growth of the entanglement
. We show that the dynamical exponent associated with the information front coincides with
onent of the growth of the entanglement entropy for both weak and strong disorder. We also
trate that the temporal dependence of the OTOC is characterized by a fast nonexponential
followed by a slow saturation after the passage of the information front. Finally,we discuss
lications of this behavioral change on the growth of the entanglement entropy.
While the speed of light is the ab-
of information propagation in both
um relativistic systems, surprisingly a
s a similar role exists also for short-
onrelativistic quantum systems. This
the Lieb-Robinson velocity, bounds
rrelations in the system and implies
bout local initial excitations propa-
al “light-cone”, similarly to the light-
n the theory of special relativity [1].
ight-cone can be obtained from the
15 9 3 3 9
i
0
2
4
6
8
10
t
W = 0.3
15 9 3 3 9
i
W = 1.8 0.00
0.03
0.06
0.09

Density of OTOC and Weak Quantum Chaos
[Kukuljan, Grozdanov and TP, PRB 2017, arXiv:1701.09147]
C(t) =
1
N
( [U(t), V ]2
− [U(t), V ] 2
), U =
N
x=1
u(x), V =
N
x=1
v(x).
dOTOC satisﬁes a general polynomial bound (C(t) ct3d
in d dim.).
It grows linearly C(t) ∝ t for non-integrable kicked Ising dynamics
(Weak quantum chaos)
It saturates for integrable transverse ﬁeld Ising dynamics.

Density of OTOC and Weak Quantum Chaos
[Kukuljan, Grozdanov and TP, PRB 2017, arXiv:1701.09147]
C(t) =
1
N
( [U(t), V ]2
− [U(t), V ] 2
), U =
N
x=1
u(x), V =
N
x=1
v(x).
dOTOC satisﬁes a general polynomial bound (C(t) ct3d
in d dim.).
It grows linearly C(t) ∝ t for non-integrable kicked Ising dynamics
(Weak quantum chaos)
It saturates for integrable transverse ﬁeld Ising dynamics.
4
FIG. 1. Density of the OTOC of extensive observables for one-dimensional KI model (7) with periodic boundary conditions:
⇡

Conclusions
Urgent open questions:
The mechanism for random-matrix-behaviour of simple interacting
quantum many-body systems without disorder?
The nature of ergodicity-breaking transitions of quantum many-body
systems is not understood. No corresponding KAM-like theory?
Is there a measure of dynamical complexity (like K-S entropy) for quantum
chaotic dynamics, which could discriminate the information ‘pumped’ from
the initial condition (many-body initial state) from dynamically generated
information (entropy)?
This work has been continuously supported by the Slovenian Research Agency
(ARRS), and, since 2016 by
Thanks to many collaborators on this and related endevours, especially to:
Marko Žnidarič, Giulio Casati, Carlos Pineda, Thomas Seligman, Enej Ilievski,
. . .

Quantum chaos in clean many-body systems - Tomaž Prosen

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