arXiv:2112.12092 · Journal of Physics A Mathematical and Theoretical

Exact time evolution formulae in the XXZ spin chain with domain wall initial state

Jean-Marie Stéphan

Published Mar 22, 2022• 5 citations

Abstract

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.

DOI: 10.1088/1751-8121/ac5fe8

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arXiv:2112.12092 · Journal of Physics A Mathematical and Theoretical

Exact time evolution formulae in the XXZ spin chain with domain wall initial state

Jean-Marie Stéphan

Published Mar 22, 2022• 5 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1751-8121/ac5fe8

Related papers

Domain Walls in the Heisenberg-Ising Spin-1/2 Chain

Axel Saenz, Craig A. Tracy, Harold Widom

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 $Δ= 0$ .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 $Δ= 0$.

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Stefano Scopa, Dragi Karevski

Abstract We revisit the out-of-equilibrium physics arising during the unitary evolution of a one-dimensional XXZ spin chain initially prepared in a domain wall state $\vert \psi _0\rangle =\vert \dots \uparrow \uparrow \downarrow \downarrow \dots \rangle$ | ψ 0 ⟩ = | ⋯ ↑ ↑ ↓ ↓ ⋯ ⟩ . In absence of interactions, we review the exact lattice calculation of several conserved quantities, including e.g. the magnetization and the spin current profiles. At large distances x and times t, we show how these quantities allow for a ballistic scaling behavior in terms of the scaling variable $\zeta = x/t$ ζ = x / t , with exactly computable scaling functions. In such a limit of large space-time scales, we show that the asymptotic behavior of the system is suitably captured by the local occupation function of spinless fermionic modes, whose semi-classical evolution in phase space is given by a Euler hydrodynamic equation. Similarly, analytical results for the asymptotic fronts dynamics are obtained for the interacting chain via Generalized Hydrodynamics. In the last part of the work, we include large-scale quantum fluctuations on top of the semi-classical hydrodynamic background in the form of a conformal field theory that lives along the evolving Fermi contour. With this procedure, dubbed quantum generalized hydrodynamics, it is possible to obtain exact asymptotic results for the entanglement spreading during the melting dynamics.Abstract We revisit the out-of-equilibrium physics arising during the unitary evolution of a one-dimensional XXZ spin chain initially prepared in a domain wall state $$\vert \psi _0\rangle =\vert \dots \uparrow \uparrow \downarrow \downarrow \dots \rangle$$ | ψ 0 ⟩ = | ⋯ ↑ ↑ ↓ ↓ ⋯ ⟩ . In absence of interactions, we review the exact lattice calculation of several conserved quantities, including e.g. the magnetization and the spin current profiles. At large distances x and times t, we show how these quantities allow for a ballistic scaling behavior in terms of the scaling variable $$\zeta = x/t$$ ζ = x / t , with exactly computable scaling functions. In such a limit of large space-time scales, we show that the asymptotic behavior of the system is suitably captured by the local occupation function of spinless fermionic modes, whose semi-classical evolution in phase space is given by a Euler hydrodynamic equation. Similarly, analytical results for the asymptotic fronts dynamics are obtained for the interacting chain via Generalized Hydrodynamics. In the last part of the work, we include large-scale quantum fluctuations on top of the semi-classical hydrodynamic background in the form of a conformal field theory that lives along the evolving Fermi contour. With this procedure, dubbed quantum generalized hydrodynamics, it is possible to obtain exact asymptotic results for the entanglement spreading during the melting dynamics.

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V. E. Korepin, Ovidiu I. Pâţu

arXiv:cond-mat/0701491Jan 20, 20075 citations

XXX spin chain: From Bethe solution to open problems

V. E. Korepin, Ovidiu I. Pâţu

Talk given atJan 1, 20071 citation

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

We study the dynamics of an isotropic spin-

\frac{1}{2}

Heisenberg chain starting in a domain-wall initial condition where the spins are initially up on the left half-line and down on the right half-line. We focus on the long-time behavior of the magnetization profile. We perform extensive time-dependent density-matrix renormalization-group simulations (up to

t=350

) and find that the data are compatible with a diffusive behavior. Subleading corrections decay slowly blurring the emergence of the diffusive behavior. We also compare our results with two alternative scenarios: superdiffusive behavior and enhanced diffusion with a logarithmic correction. We finally discuss the evolution of the entanglement entropy.

Abstract We revisit the out-of-equilibrium physics arising during the unitary evolution of a one-dimensional XXZ spin chain initially prepared in a domain wall state

\vert \psi _0\rangle =\vert \dots \uparrow \uparrow \downarrow \downarrow \dots \rangle

| ψ 0 ⟩ = | ⋯ ↑ ↑ ↓ ↓ ⋯ ⟩ . In absence of interactions, we review the exact lattice calculation of several conserved quantities, including e.g. the magnetization and the spin current profiles. At large distances x and times t, we show how these quantities allow for a ballistic scaling behavior in terms of the scaling variable

\zeta = x/t

ζ = x / t , with exactly computable scaling functions. In such a limit of large space-time scales, we show that the asymptotic behavior of the system is suitably captured by the local occupation function of spinless fermionic modes, whose semi-classical evolution in phase space is given by a Euler hydrodynamic equation. Similarly, analytical results for the asymptotic fronts dynamics are obtained for the interacting chain via Generalized Hydrodynamics. In the last part of the work, we include large-scale quantum fluctuations on top of the semi-classical hydrodynamic background in the form of a conformal field theory that lives along the evolving Fermi contour. With this procedure, dubbed quantum generalized hydrodynamics, it is possible to obtain exact asymptotic results for the entanglement spreading during the melting dynamics.