153 research outputs found
Time-dependent Correlation Functions in Open Quadratic Fermionic Systems
We formulate and discuss explicit computation of dynamic correlation
functions in open quadradic fermionic systems which are driven and dissipated
by the Lindblad jump processes that are linear in canonical fermionic
operators. Dynamic correlators are interpreted in terms of local quantum quench
where the pre-quench state is the non-equilibrium steady state, i.e. a fixed
point of the Liouvillian. As an example we study the XY spin 1/2 chain and the
Kitaev Majorana chains with boundary Lindblad driving, whose dynamics exhibits
asymmetric (skewed) light cone behaviour. We also numerically treat the two
dimensional XY model and the XY spin chain with additional
Dzyaloshinskii-Moriya interactions. The latter exhibits a new non-equilibrium
phase transition which can be understood in terms of bifurcations of the
quasi-particle dispersion relation. Finally, considering in some detail the
periodic Kitaev chain (fermionic ring) with dissipation at a single (arbitrary)
site, we present analytical expressions for the first order corrections (in the
strength of dissipation) to the spectrum and the non-equilibrium steady state
(NESS) correlation functions.Comment: 25 pages, 10 figure
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
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
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
Many-body quantum chaos: Analytic connection to random matrix theory
A key goal of quantum chaos is to establish a relationship between widely
observed universal spectral fluctuations of clean quantum systems and random
matrix theory (RMT). For single particle systems with fully chaotic classical
counterparts, the problem has been partly solved by Berry (1985) within the
so-called diagonal approximation of semiclassical periodic-orbit sums.
Derivation of the full RMT spectral form factor from semiclassics has
been completed only much later in a tour de force by Mueller et al (2004). In
recent years, the questions of long-time dynamics at high energies, for which
the full many-body energy spectrum becomes relevant, are coming at the
forefront even for simple many-body quantum systems, such as locally
interacting spin chains. Such systems display two universal types of behaviour
which are termed as `many-body localized phase' and `ergodic phase'. In the
ergodic phase, the spectral fluctuations are excellently described by RMT, even
for very simple interactions and in the absence of any external source of
disorder. Here we provide the first theoretical explanation for these
observations. We compute explicitly in the leading two orders in and
show its agreement with RMT for non-integrable, time-reversal invariant
many-body systems without classical counterparts, a generic example of which
are Ising spin 1/2 models in a periodically kicking transverse field.Comment: 10 pages in RevTex with 4 figures and a few diagrams; v3: version
accepted by PR
Chapter Performance Analysis of Empirical Ionosphere Models by Comparison with CODE Vertical TEC Maps
Earth science
Performance Analysis of Empirical Ionosphere Models by Comparison with CODE Vertical TEC Maps
Earth science
Scrambling is Necessary but Not Sufficient for Chaos
We show that out-of-time-order correlators (OTOCs) constitute a probe for
Local-Operator Entanglement (LOE). There is strong evidence that a volumetric
growth of LOE is a faithful dynamical indicator of quantum chaos, while OTOC
decay corresponds to operator scrambling, often conflated with chaos. We show
that rapid OTOC decay is a necessary but not sufficient condition for linear
(chaotic) growth of the LOE entropy. We analytically support our results
through wide classes of local-circuit models of many-body dynamics, including
both integrable and non-integrable dual-unitary circuits. We show sufficient
conditions under which local dynamics leads to an equivalence of scrambling and
chaos.Comment: 6+16 Pages. Comments welcom
Correlations in Perturbed Dual-Unitary Circuits: Efficient Path-Integral Formula
Interacting many-body systems with explicitly accessible spatio-temporal
correlation functions are extremely rare, especially in the absence of
integrability. Recently, we identified a remarkable class of such systems and
termed them dual-unitary quantum circuits. These are brick-wall type local
quantum circuits whose dynamics are unitary in both time and space. For these
systems the spatio-temporal correlation functions are non-trivial only at the
edge of the causal light cone and can be computed in terms of one-dimensional
transfer matrices. Dual-unitarity, however, requires fine-tuning and the degree
of generality of the observed dynamical features remained unclear. Here we
address this question by introducing arbitrary perturbations of the local
gates. Considering fixed perturbations, we prove that for a particular class of
unperturbed elementary dual-unitary gates the correlation functions are still
expressed in terms of one-dimensional transfer matrices. These matrices,
however, are now contracted over generic paths connecting the origin to a fixed
endpoint inside the causal light cone. The correlation function is given as a
sum over all such paths. Our statement is rigorous in the "dilute limit", where
only a small fraction of the gates is perturbed, and in the presence of random
longitudinal fields, but we provide theoretical arguments and stringent
numerical checks supporting its validity even in the clean case and when all
gates are perturbed. As a byproduct, in the case of random longitudinal fields
-- which turns out to be equivalent to certain classical Markov chains -- we
find four types of non-dual-unitary(and non-integrable) interacting many-body
systems where the correlation functions are exactly given by the path-sum
formula.Comment: 28 pages, 9 figures, 2 tables; v3: 32 pages, 10 figures, 2 tables;
presentation improved, Section 3 rewritte
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