744 research outputs found
Excited state correlations of the finite Heisenberg chain
We consider short range correlations in excited states of the finite XXZ and
XXX Heisenberg spin chains. We conjecture that the known results for the
factorized ground state correlations can be applied to the excited states too,
if the so-called physical part of the construction is changed appropriately.
For the ground state we derive simple algebraic expressions for the physical
part; the formulas only use the ground state Bethe roots as an input. We
conjecture that the same formulas can be applied to the excited states as well,
if the exact Bethe roots of the excited states are used instead. In the XXZ
chain the results are expected to be valid for all states (except certain
singular cases where regularization is needed), whereas in the XXX case they
only apply to singlet states or group invariant operators. Our conjectures are
tested against numerical data from exact diagonalization and coordinate Bethe
Ansatz calculations, and perfect agreement is found in all cases. In the XXX
case we also derive a new result for the nearest-neighbour correlator
, which is valid for non-singlet states as well. Our
results build a bridge between the known theory of factorized correlations, and
the recently conjectured TBA-like description for the building blocks of the
construction.Comment: 27 pages, v2: minor modifications, a table is added (displaying the
numerical errors), v3: minor modification
Overlaps between eigenstates of the XXZ spin-1/2 chain and a class of simple product states
We consider a class of quantum quenches in the spin-1/2 XXZ chain, where the
initial state is of a simple product form. Specific examples are the N\'eel
state, the dimer state and the q-deformed dimer state. We compute determinant
formulas for finite volume overlaps between the initial state and arbitrary
eigenstates of the spin chain Hamiltonian. These results could serve as a basis
for calculating the time dependence of correlation functions following the
quantum quench.Comment: 17 pages, 1 figure, v2: one reference added, v3: minor modification,
v4: minor modification, v5: calculation of the determinant formula simplified
and a note added, v6: references update
Form factor approach to diagonal finite volume matrix elements in Integrable QFT
We derive an exact formula for finite volume excited state mean values of
local operators in 1+1 dimensional Integrable QFT with diagonal scattering. Our
result is a non-trivial generalization of the LeClair-Mussardo series, which is
a form factor expansion for finite size ground state mean values.Comment: 29 page
Failure of the Generalized Eigenstate Thermalization Hypothesis in integrable models with multiple particle species
It has been recently observed for a particular quantum quench in the XXZ spin
chain that local observables do not equilibrate to the predictions of the
Generalized Gibbs Ensemble (GGE). In this work we argue that the breakdown of
the GGE can be attributed to the failure of the Generalized Eigenstate
Thermalization Hypothesis (GETH), which has been the main candidate to explain
the validity of the GGE. We provide explicit counterexamples to the GETH and
argue that generally it does not hold in models with multiple particle species.
Therefore there is no reason to assume that the GGE should describe the long
time limit of observables in these integrable models.Comment: 16 pages, 2 figures, v2: minor modification
Mean values of local operators in highly excited Bethe states
We consider expectation values of local operators in (continuum) integrable
models in a situation when the mean value is calculated in a single Bethe state
with a large number of particles. We develop a form factor expansion for the
thermodynamic limit of the mean value, which applies whenever the distribution
of Bethe roots is given by smooth density functions. We present three
applications of our general result: i) In the framework of integrable Quantum
Field Theory (IQFT) we present a derivation of the LeClair-Mussardo formula for
finite temperature one-point functions. We also extend the results to boundary
operators in Boundary Field Theories. ii) We establish the LeClair-Mussardo
formula for the non-relativistic 1D Bose gas in the framework of Algebraic
Bethe Ansatz (ABA). This way we obtain an alternative derivation of the results
of Kormos et. al. for the (temperature dependent) local correlations using only
the concepts of ABA. iii) In IQFT we consider the long-time limit of one-point
functions after a certain type of global quench. It is shown that our general
results imply the integral series found by Fioretti and Mussardo. We also
discuss the generalized Eigenstate Thermalization hypothesis in the context of
quantum quenches in integrable models. It is shown that a single mean value
always takes the form of a thermodynamic average in a Generalized Gibbs
Ensemble, although the relation to the conserved charges is rather indirect.Comment: 32 pages, v2: minor changes, v3: minor correction
Generalized Gibbs Ensemble for Heisenberg Spin Chains
We consider the Generalized Gibbs Ensemble (GGE) in the context of global
quantum quenches in XXZ Heisenberg spin chains. Embedding the GGE into the
Quantum Transfer Matrix formalism we develop an iterative procedure to fix the
Lagrange-multipliers and to calculate predictions for the long-time limit of
short-range correlators. The main idea is to consider truncated GGE's with only
a finite number of charges and to investigate the convergence of the numerical
results as the truncation level is increased. As an example we consider a
quantum quench situation where the system is initially prepared in the N\'eel
state and then evolves with an XXZ Hamiltonian with anisotropy Delta>1. We
provide predictions for short range correlators and gather numerical evidence
that the iterative procedure indeed converges. The results show that the system
retains memory of the initial condition, and there are clear differences
between the numerical values of the correlators as calculated from the purely
thermal and the Generalized Gibbs ensembles.Comment: 21 pages, 12 figures. v2: minor changes, v3: minor change
Overlaps with arbitrary two-site states in the XXZ spin chain
We present a conjectured exact formula for overlaps between the Bethe states
of the spin-1/2 XXZ chain and generic two-site states. The result takes the
same form as in the previously known cases: it involves the same ratio of two
Gaudin-like determinants, and a product of single-particle overlap functions,
which can be fixed using a combination of the Quench Action and Quantum
Transfer Matrix methods. Our conjecture is confirmed by numerical data from
exact diagonalization. For one-site states the formula is found to be correct
even in chains with odd length, where existing methods can not be applied. It
is also pointed out, that the ratio of the Gaudin-like determinants plays a
crucial role in the overlap sum rule: it guarantees that in the thermodynamic
limit there remains no piece in the Quench Action.Comment: 22 pages, v2: references adde
Real-time dynamics in a strongly interacting bosonic hopping model: Global quenches and mapping to the XX chain
We study the time evolution of an integrable many-particle system, described
by the -boson Hamiltonian in the limit of strong interactions .
It is shown that, for a particular class of pure initial states, the analytical
calculation of certain observables simplifies considerably. Namely, we provide
exact formulas for the calculation of the Loschmidt-echo and the emptiness
formation probability, where the computational time scales polynomially with
the particle number. Moreover, we construct a non-local mapping of the
-boson model to the XX spin chain, and show how this can be utilized to
obtain the time evolution of various local bosonic observables for
translationally invariant initial states. The results obtained via the bosonic
and fermionic picture show perfect agreement. In the infinite volume and large
time limits, we rigorously verify the prediction of the Generalized Gibbs
Ensemble for homogeneous initial Fock states.Comment: 26 pages, 3 figures, v2: minor mistakes in Appendix 2 corrected, v3:
minor modification
Short distance correlators in the XXZ spin chain for arbitrary string distributions
In this letter we consider expectation values of local correlators in highly
excited states of the spin-1/2 XXZ chain. Assuming that the string hypothesis
holds we formulate the following conjecture: The correlation functions can be
computed using the known factorized formulas of the finite temperature
situation, if the building blocks are computed via certain linear integral
equations using the string densities only. We prove this statement for the
nearest neighbour z-z correlator for states with arbitrary string densities.
Also, we check the conjecture numerically for other correlators in the finite
temperature case. Our results pave the way towards the computation of the
stationary values of correlators in non-equilibrium situations using the
so-called quench-action approach.Comment: 15 pages, v2: minor modifications, v3: minor modification
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