1,394 research outputs found
Finite-size corrections vs. relaxation after a sudden quench
We consider the time evolution after sudden quenches of global parameters in
translational invariant Hamiltonians and study the time average expectation
values and entanglement entropies in finite chains. We show that in
noninteracting models the time average of spin correlation functions is
asymptotically equal to the infinite time limit in the infinite chain, which is
known to be described by a generalized Gibbs ensemble. The equivalence breaks
down considering nonlocal operators, and we establish that this can be traced
back to the existence of conservation laws common to the Hamiltonian before and
after the quench. We develop a method to compute the leading finite-size
correction for time average correlation functions and entanglement entropies.
We find that large corrections are generally associated to observables with
slow relaxation dynamics.Comment: 12 pages, 2 figures; V2: minor changes and reference adde
New insights into the entanglement of disjoint blocks
We study the entanglement of two disjoint blocks in spin-1/2 chains obtained
by merging solvable models, such as XX and quantum Ising models. We focus on
the universal quantities that can be extracted from the R\'enyi entropies
S_\alpha. The most important information is encoded in some functions denoted
by F_\alpha. We compute F_2 and we show that F_\alpha-1 and F_{v.N.},
corresponding to the von Neumann entropy, can be negative, in contrast to what
observed in all models examined so far. An exact relation between the
entanglement of disjoint subsystems in the XX model and that in a chain
embodying two quantum Ising models is a by-product of our investigations.Comment: 6 pages, 4 figures, revised version accepted for publication in EP
Local conservation laws in spin-1/2 XY chains with open boundary conditions
We revisit the conserved quantities of the spin-1/2 XY model with open
boundary conditions. In the absence of a transverse field, we find new families
of local charges and show that half of the seeming conservation laws are
conserved only if the number of sites is odd. In even chains the set of
noninteracting charges is abelian, like in the periodic case when the number of
sites is odd. In odd chains the set is doubled and becomes non-abelian, like in
even periodic chains. The dependence of the charges on the parity of the
chain's size undermines the common belief that the thermodynamic limit of
diagonal ensembles exists. We consider also the transverse-field Ising chain,
where the situation is more ordinary. The generalization to the XY model in a
transverse field is not straightforward and we propose a general framework to
carry out similar calculations. We conjecture the form of the bulk part of the
local charges and discuss the emergence of quasilocal conserved quantities. We
provide evidence that in a region of the parameter space there is a reduction
of the number of quasilocal conservation laws invariant under chain inversion.
As a by-product, we study a class of block-Toeplitz-plus-Hankel operators and
identify the conditions that their symbols satisfy in order to commute with a
given block-Toeplitz.Comment: 49 pages, 5 figures, 3 tables; published versio
Conservation laws for a class of generic Hamiltonians
Within a strong coupling expansion, we construct local quasi-conserved
operators for a class of Hamiltonians that includes both integrable and
non-integrable models. We explicitly show that at the lowest orders of
perturbation theory the structure of the operators is independent of the system
details. Higher order contributions are investigated numerically by means of an
ab initio method for computing the time evolution of local operators in the
Heisenberg picture. The numerical analysis suggests that the quasi-conserved
operators could be approximations of a quasi-local conservation law, even if
the model is non-integrable.Comment: 5+2 pages, 2+1 figure
Higher-Order Hydrodynamics in 1D: a Promising Direction and a Null Result
We derive a Moyal dynamical equation that describes exact time evolution in
generic (inhomogeneous) noninteracting spin-chain models. Assuming
quasistationarity, we develop a hydrodynamic theory. The question at hand is
whether some large-time corrections are captured by higher-order hydrodynamics.
We consider in particular the dynamics after that two chains, prepared in
different conditions, are joined together. In these situations a light cone,
separating regions with macroscopically different properties, emerges from the
junction. In free fermionic systems some observables close to the light cone
follow a universal behavior, known as Tracy-Widom scaling. Universality means
weak dependence on the system's details, so this is the perfect setting where
hydrodynamics could emerge. For the transverse-field Ising chain and the XX
model, we show that hydrodynamics captures the scaling behavior close to the
light cone. On the other hand, our numerical analysis suggests that
hydrodynamics fails in more general models, whenever a condition is not
satisfied.Comment: 7+2 pages, 1+2 figure
Dynamical Phase Transitions as Properties of the Stationary State: Analytic Results after Quantum Quenches in the Spin-1/2 XXZ Chain
The (Loschmidt) overlap between the state at different times after a quantum
quench is attracting increasing interest, as it was recently shown that in the
thermodynamic limit its logarithm per unit of length has a non-analytic
behavior if a Hamiltonian parameter is quenched across a critical point. This
phenomenon was called a "dynamical phase transition" in analogy with the
behavior of the canonical partition function at an equilibrium phase
transition. We distinguish between local and nonlocal contributions to the
aforementioned quantity and derive an analytic expression for the time
evolution of the local part after quantum quenches in the XXZ spin-1/2 chain.
The state that describes the stationary properties of (local) observables can
be represented by a Gibbs ensemble of a generalized Hamiltonian; we reveal a
deep connection between the appearance of singularities and the excitation
energies of the generalized Hamiltonian.Comment: 5+2 pages, 1+1 figures. In Version 2 an error in the discussion of
noninteracting models is fixed and the quantity under investigation is
revisite
On Conservation Laws, Relaxation and Pre-relaxation after a Quantum Quench
We consider the time evolution following a quantum quench in spin-1/2 chains.
It is well known that local conservation laws constrain the dynamics and,
eventually, the stationary behavior of local observables. We show that some
widely studied models, like the quantum XY model, possess extra families of
local conservation laws in addition to the translation invariant ones. As a
consequence, the additional charges must be included in the generalized Gibbs
ensemble that describes the stationary properties. The effects go well beyond a
simple redefinition of the stationary state. The time evolution of a
non-translation invariant state under a (translation invariant) Hamiltonian
with a perturbation that weakly breaks the hidden symmetries underlying the
extra conservation laws exhibits pre-relaxation. In addition, in the limit of
small perturbation, the time evolution following pre-relaxation can be
described by means of a time-dependent generalized Gibbs ensemble.Comment: 28 pages, 5 figures; v2: minor changes, a few references adde
Prethermalization at Low Temperature: the Scent of Long-Range Order
Non-equilibrium time evolution in isolated many-body quantum systems
generally results in thermalization. However, the relaxation process can be
very slow, and quasi-stationary non-thermal plateaux are often observed at
intermediate times. The paradigmatic example is a quantum quench in an
integrable model with weak integrability breaking; for a long time, the state
can not escape the constraints imposed by the approximate integrability. We
unveil a new mechanism of prethermalization, based on the presence of a
symmetry of the pre-quench Hamiltonian, which is spontaneously broken at zero
temperature and is explicitly broken by the post-quench Hamiltonian. The
typical time scale of the phenomenon is proportional to the thermal correlation
length of the initial state, which diverges as the temperature is lowered. We
show that the prethermal quasi-stationary state can be approximated by a mixed
state that violates cluster decomposition property. We consider two examples:
the transverse-field Ising chain, where the full time evolution is computed
analytically, and the (non integrable) ANNNI model, which is investigated
numerically.Comment: 6 pages, 2 figures, accepted for publication in PR
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
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