477 research outputs found
Pre-relaxation in weakly interacting models
We consider time evolution in models close to integrable points with hidden
symmetries that generate infinitely many local conservation laws that do not
commute with one another. The system is expected to (locally) relax to a
thermal ensemble if integrability is broken, or to a so-called generalised
Gibbs ensemble if unbroken. In some circumstances expectation values exhibit
quasi-stationary behaviour long before their typical relaxation time. For
integrability-breaking perturbations, these are also called pre-thermalisation
plateaux, and emerge e.g. in the strong coupling limit of the Bose-Hubbard
model. As a result of the hidden symmetries, quasi-stationarity appears also in
integrable models, for example in the Ising limit of the XXZ model. We
investigate a weak coupling limit, identify a time window in which the effects
of the perturbations become significant and solve the time evolution through a
mean-field mapping. As an explicit example we study the XYZ spin-
chain with additional perturbations that break integrability. One of the most
intriguing results of the analysis is the appearance of persistent oscillatory
behaviour. To unravel its origin, we study in detail a toy model: the
transverse-field Ising chain with an additional nonlocal interaction
proportional to the square of the transverse spin per unit length [Phys. Rev.
Lett. 111, 197203 (2013)]. Despite being nonlocal, this belongs to a class of
models that emerge as intermediate steps of the mean-field mapping and shares
many dynamical properties with the weakly interacting models under
consideration.Comment: 69 pages, 17 figures, improved exposition, figures 1 and 13 added,
some typos correcte
Entanglement spreading in a minimal model of maximal many-body quantum chaos
The spreading of entanglement in out-of-equilibrium quantum systems is
currently at the centre of intense interdisciplinary research efforts involving
communities with interests ranging from holography to quantum information. Here
we provide a constructive and mathematically rigorous method to compute the
entanglement dynamics in a class of "maximally chaotic", periodically driven,
quantum spin chains. Specifically, we consider the so called "self-dual" kicked
Ising chains initialised in a class of separable states and devise a method to
compute exactly the time evolution of the entanglement entropies of finite
blocks of spins in the thermodynamic limit. Remarkably, these exact results are
obtained despite the models considered are maximally chaotic: their spectral
correlations are described by the circular orthogonal ensemble of random
matrices on all scales. Our results saturate the so called "minimal cut" bound
and are in agreement with those found in the contexts of random unitary
circuits with infinite-dimensional local Hilbert space and conformal field
theory. In particular, they agree with the expectations from both the
quasiparticle picture, which accounts for the entanglement spreading in
integrable models, and the minimal membrane picture, recently proposed to
describe the entanglement growth in generic systems. Based on a novel
"duality-based" numerical method, we argue that our results describe the
entanglement spreading from any product state at the leading order in time when
the model is non-integrable.Comment: 25 pages, 10 figures; v2 improved presentation; v3: 28 pages 11
figures, presentation improved, Section 7 rewritte
Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos
The most general and versatile defining feature of quantum chaotic systems is
that they possess an energy spectrum with correlations universally described by
random matrix theory (RMT). This feature can be exhibited by systems with a
well defined classical limit as well as by systems with no classical
correspondence, such as locally interacting spins or fermions. Despite great
phenomenological success, a general mechanism explaining the emergence of RMT
without reference to semiclassical concepts is still missing. Here we provide
the example of a quantum many-body system with no semiclassical limit (no large
parameter) where the emergence of RMT spectral correlations is proven exactly.
Specifically, we consider a periodically driven Ising model and write the
Fourier transform of spectral density's two-point function, the spectral form
factor, in terms of a partition function of a two-dimensional classical Ising
model featuring a space-time duality. We show that the self-dual cases provide
a minimal model of many-body quantum chaos, where the spectral form factor is
demonstrated to match RMT for all values of the integer time variable in
the thermodynamic limit. In particular, we rigorously prove RMT form factor for
odd , while we formulate a precise conjecture for even . The results
imply ergodicity for any finite amount of disorder in the longitudinal field,
rigorously excluding the possibility of many-body localization. Our method
provides a novel route for obtaining exact nonperturbative results in
non-integrable systems.Comment: 6 + 22 pages, 3 figures; v2: improved presentation of the proofs in
the appendices; v3: as appears in Physical Review Letter
Exact Correlation Functions for Dual-Unitary Lattice Models in 1+1 Dimensions
We consider a class of quantum lattice models in dimensions represented
as local quantum circuits that enjoy a particular "dual-unitarity" property. In
essence, this property ensures that both the evolution "in time" and that "in
space" are given in terms of unitary transfer matrices. We show that for this
class of circuits, generically non-integrable, one can compute explicitly all
dynamical correlations of local observables. Our result is exact,
non-pertubative, and holds for any dimension of the local Hilbert space. In
the minimal case of qubits () we also present a classification of all
dual-unitary circuits which allows us to single out a number of distinct
classes for the behaviour of the dynamical correlations. We find
"non-interacting" classes, where all correlations are preserved, the ergodic
and mixing one, where all correlations decay, and, interestingly, also classes
that are are both interacting and non-ergodic.Comment: 6+5 pages, no figures; v2 minor changes; v3 as appears in Phys. Rev.
Let
Transport in the sine-Gordon field theory: from generalized hydrodynamics to semiclassics
The semiclassical approach introduced by Sachdev and collaborators proved to
be extremely successful in the study of quantum quenches in massive field
theories, both in homogeneous and inhomogeneous settings. While conceptually
very simple, this method allows one to obtain analytic predictions for several
observables when the density of excitations produced by the quench is small. At
the same time, a novel generalized hydrodynamic (GHD) approach, which captures
exactly many asymptotic features of the integrable dynamics, has recently been
introduced. Interestingly, also this theory has a natural interpretation in
terms of semiclassical particles and it is then natural to compare the two
approaches. This is the objective of this work: we carry out a systematic
comparison between the two methods in the prototypical example of the
sine-Gordon field theory. In particular, we study the "bipartitioning protocol"
where the two halves of a system initially prepared at different temperatures
are joined together and then left to evolve unitarily with the same
Hamiltonian. We identify two different limits in which the semiclassical
predictions are analytically recovered from GHD: a particular non-relativistic
limit and the low temperature regime. Interestingly, the transport of
topological charge becomes sub-ballistic in these cases. Away from these limits
we find that the semiclassical predictions are only approximate and, in
contrast to the latter, the transport is always ballistic. This statement seems
to hold true even for the so-called "hybrid" semiclassical approach, where
finite time DMRG simulations are used to describe the evolution in the internal
space.Comment: 30 pages, 6 figure
Quantum quench in the sine-Gordon model
We consider the time evolution in the repulsive sine-Gordon quantum field
theory after the system is prepared in a particular class of initial states. We
focus on the time dependence of the one-point function of the semi-local
operator . By using two different methods
based on form-factor expansions, we show that this expectation value decays to
zero exponentially, and we determine the decay rate by analytical means. Our
methods generalise to other correlation functions and integrable models.Comment: 41 pages, 1 figure, some typos correcte
Entanglement dynamics in Rule 54: Exact results and quasiparticle picture
We study the entanglement dynamics generated by quantum quenches in the
quantum cellular automaton Rule . We consider the evolution from a recently
introduced class of solvable initial states. States in this class relax
(locally) to a one-parameter family of Gibbs states and the thermalisation
dynamics of local observables can be characterised exactly by means of an
evolution in space. Here we show that the latter approach also gives access to
the entanglement dynamics and derive exact formulas describing the asymptotic
linear growth of all R\'enyi entropies in the thermodynamic limit and their
eventual saturation for finite subsystems. While in the case of von Neumann
entropy we recover exactly the predictions of the quasiparticle picture, we
find no physically meaningful quasiparticle description for other R\'enyi
entropies. Our results apply to both homogeneous and inhomogeneous quenches.Comment: 33 pages, 5 figures; v2 accepted versio
Scrambling in Random Unitary Circuits: Exact Results
We study the scrambling of quantum information in local random unitary
circuits by focusing on the tripartite information proposed by Hosur et al. We
provide exact results for the averaged R\'enyi- tripartite information in
two cases: (i) the local gates are Haar random and (ii) the local gates are
dual-unitary and randomly sampled from a single-site Haar-invariant measure. We
show that the latter case defines a one-parameter family of circuits, and prove
that for a "maximally chaotic" subset of this family quantum information is
scrambled faster than in the Haar-random case. Our approach is based on a
standard mapping onto an averaged folded tensor network, that can be studied by
means of appropriate recurrence relations. By means of the same method, we also
revisit the computation of out-of-time-ordered correlation functions,
re-deriving known formulae for Haar-random unitary circuits, and presenting an
exact result for maximally chaotic random dual-unitary gates.Comment: 29 pages, 7 figure
Entanglement Barriers in Dual-Unitary Circuits
After quantum quenches in many-body systems, finite subsystems evolve
non-trivially in time, eventually approaching a stationary state. In typical
situations, the reduced density matrix of a given subsystem begins and ends
this endeavour as a low-entangled vector in the space of operators. This means
that if its operator space entanglement initially grows (which is generically
the case), it must eventually decrease, describing a barrier-shaped curve.
Understanding the shape of this "entanglement barrier" is interesting for three
main reasons: (i) it quantifies the dynamics of entanglement in the (open)
subsystem; (ii) it gives information on the approximability of the reduced
density matrix by means of matrix product operators; (iii) it shows qualitative
differences depending on the type of dynamics undergone by the system,
signalling quantum chaos. Here we compute exactly the shape of the entanglement
barriers described by different R\'enyi entropies after quantum quenches in
dual-unitary circuits initialised in a class of solvable matrix product states
(MPS)s. We show that, for free (SWAP-like) circuits, the entanglement entropy
behaves as in rational CFTs. On the other hand, for completely chaotic
dual-unitary circuits it behaves as in holographic CFTs, exhibiting a longer
entanglement barrier that drops rapidly when the subsystem thermalises.
Interestingly, the entanglement spectrum is non-trivial in the completely
chaotic case. Higher R\'enyi entropies behave in an increasingly similar way to
rational CFTs, such that the free and completely chaotic barriers are identical
in the limit of infinite replicas (i.e. for the so called min-entropy). We also
show that, upon increasing the bond dimension of the MPSs, the barrier
maintains the same shape. It simply shifts to the left to accommodate for the
larger initial entanglement.Comment: 27 pages, 4 figures; v2 as appears in the journa
Growth of entanglement of generic states under dual-unitary dynamics
Dual-unitary circuits are a class of locally-interacting quantum many-body
systems displaying unitary dynamics also when the roles of space and time are
exchanged. These systems have recently emerged as a remarkable framework where
certain features of many-body quantum chaos can be studied exactly. In
particular, they admit a class of ``solvable" initial states for which, in the
thermodynamic limit, one can access the full non-equilibrium dynamics. This
reveals a surprising property: when a dual-unitary circuit is prepared in a
solvable state the quantum entanglement between two complementary spatial
regions grows at the maximal speed allowed by the local structure of the
evolution. Here we investigate the fate of this property when the system is
prepared in a generic pair-product state. We show that in this case the
entanglement increment during a time step is sub-maximal for finite times,
however, it approaches the maximal value in the infinite-time limit. This
statement is proven rigorously for dual-unitary circuits generating high enough
entanglement, while it is argued to hold for the entire class.Comment: 17 pages, 3 figure
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