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arXiv:1707.06625 · Journal of Statistical Mechanics Theory and Experiment

Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain

Jean-Marie Stéphan

Published Oct 1, 2017• 73 citations

Abstract

Abstract We study the return probability and its imaginary ( τ ) time continuation after a quench from a domain wall initial state in the XXZ spin chain, focusing mainly on the region with anisotropy | Δ | < 1 . We establish exact Fredholm determinant formulas for those, by exploiting a connection to the six-vertex model with domain wall boundary conditions. In imaginary time, we find the expected scaling for a partition function of a statistical mechanical model of area proportional to τ 2 , which reflects the fact that the model exhibits the limit shape phenomenon. In real time, we observe that in the region | Δ | < 1 the decay for long time t is nowhere continuous as a function of anisotropy: it is Gaussian at roots of unity and exponential otherwise. We also determine that the front moves as x f ( t ) = t 1 − Δ 2 , by the analytic continuation of known arctic curves in the six-vertex model. Exactly at | Δ | = 1 , we find the return probability decays as e − ζ ( 3 / 2 ) t / π t 1 / 2 O ( 1 ) . It is argued that this result provides an upper bound on spin transport. In particular, it suggests that transport should be diffusive at the isotropic point for this quench.

DOI: 10.1088/1742-5468/aa8c19

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Abstract We study the return probability and its imaginary ( τ ) time continuation after a quench from a domain wall initial state in the XXZ spin chain, focusing mainly on the region with anisotropy | Δ | < 1 . We establish exact Fredholm determinant formulas for those, by exploiting a connection to the six-vertex model with domain wall boundary conditions. In imaginary time, we find the expected scaling for a partition function of a statistical mechanical model of area proportional to τ 2 , which reflects the fact that the model exhibits the limit shape phenomenon. In real time, we observe that in the region | Δ | < 1 the decay for long time t is nowhere continuous as a function of anisotropy: it is Gaussian at roots of unity and exponential otherwise. We also determine that the front moves as x f ( t ) = t 1 − Δ 2 , by the analytic continuation of known arctic curves in the six-vertex model. Exactly at | Δ | = 1 , we find the return probability decays as e − ζ ( 3 / 2 ) t / π t 1 / 2 O ( 1 ) . It is argued that this result provides an upper bound on spin transport. In particular, it suggests that transport should be diffusive at the isotropic point for this quench.

Abstract We study the time evolution of the spin-1/2 XXZ chain initialized in a domain wall state, where all spins to the left of the origin are up, all spins to its right are down. The focus is on exact formulae, which hold for arbitrary finite (real or imaginary) time. In particular, we compute the amplitudes corresponding to the process where all but k spins come back to their initial orientation, as a k -fold contour integral. These results are obtained using a correspondence with the six vertex model, and taking a somewhat complicated Hamiltonian/Trotter-type limit. Several simple applications are studied and also discussed in a broader context.

In this paper we obtain formulas for the distribution of the left-most up-spin in the Heisenberg-Ising spin-1/2 chain with anisotropy parameter

Δ

, also known as the XXZ spin-1/2 chain, on the one-dimensional lattice

\mathbb{Z}

with domain wall initial conditions. We use the Bethe Ansatz to solve the Schr

ö

dinger equation and a recent antisymmetrization identity of Cantini, Colomo, and Pronko (arXiv:1906.07636) to simplify the marginal distribution of the left-most up-spin. In the

Δ=0

case, the distribution

F_2

arises. In the

Δ\neq 0

case, we propose a conjectural series expansion type formula based on a saddle point analysis. The conjectural formula turns out to be a Fredholm series expansion in the

Δ\rightarrow 0

limit and recovers the result for

Δ= 0

.

Abstract The open XXZ spin chain with the anisotropy parameter Δ = − 1 2 and diagonal boundary magnetic fields that depend on a parameter x is studied. For real x > 0, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in x with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik–Zamolodchikov equations associated with the R -matrix and diagonal K -matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.

Results are presented for the nonequilibrium dynamics of a quantum XXZ -spin chain whose spins are initially arranged in a domain wall profile via the application of a magnetic field in the z direction, which is spatially varying along the chain. The system is driven out of equilibrium in two ways: a). by rapidly turning off the magnetic field, b). by rapidly quenching the interactions at the same time as the magnetic field is turned off. The time evolution of the domain wall profile as well as various two-point spin correlation functions are studied by the exact solution of the fermionic problem for the XX chain and via a bosonization approach and a mean-field approach for the XXZ chain. At long times the magnetization is found to equilibrate (reach the ground state value), while the two-point correlation functions in general do not. In particular, for quenches within the gapless XX phase, the spin correlation function transverse to the z direction acquires a spatially inhomogeneous structure at long times whose details depend on the initial domain wall profile. The spatial inhomogeneity is also recovered for the case of classical spins initially arranged in a domain wall profile and shows that the inhomogeneities arise due to the dephasing of transverse spin components as the domain wall broadens. A generalized Gibbs ensemble approach is found to be inadequate in capturing this spatially inhomogeneous state.

Abstract We study numerically the density profile in the six-vertex model with domain wall boundary conditions. Using a Monte Carlo algorithm originally proposed by Allison and Reshetikhin we numerically evaluate the inhomogeneous density profiles in the disordered and antiferromagnetic regimes where frozen corners appear. At the free fermion point we present an exact finite-size formula for the density on the horizontal edges that relies on the imaginary time transfer matrix approach. In all cases where exact analytic forms for the density and the arctic curves are known the numerical method shows perfect agreement with them. This also suggests the possibility of its use for accurate quantitative purposes.

The six-vertex model on an N × N square lattice with domain wall boundary conditions is considered. A Fredholm determinant representation for the partition function of the model is given. The kernel of the corresponding integral operator is of the so-called integrable type, and involves classical orthogonal polynomials. From this representation, a ‘reconstruction’ formula is proposed, which expresses the partition function as the trace of a suitably chosen quantum operator, in the spirit of corner transfer matrix and vertex operator approaches to integrable spin models.