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arXiv:1507.03492 · Physical Review B

Quantum quench within the gapless phase of the spin−12 Heisenberg XXZ spin chain

Mario Collura, Pasquale Calabrese, Fabian H. L. Eßler

Published Sep 17, 2015• 69 citations
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Abstract

We consider an interaction quench in the critical $\text{spin}\ensuremath{-}\frac{1}{2}$ Heisenberg XXZ chain. We numerically compute the time evolution of the two-point correlation functions of spin operators in the thermodynamic limit and compare the results to predictions obtained in the framework of the Luttinger liquid approximation. We find that the transverse correlation function $\ensuremath{\langle}{S}_{j}^{x}{S}_{j+\ensuremath{\ell}}^{x}\ensuremath{\rangle}$ agrees with the Luttinger model prediction to a surprising level of accuracy. The agreement for the longitudinal two-point function $\ensuremath{\langle}{S}_{j}^{z}{S}_{j+\ensuremath{\ell}}^{z}\ensuremath{\rangle}$ is found to be much poorer. We speculate that this difference between transverse and longitudinal correlations has its origin in the locality properties of the respective spin operator with respect to the underlying fermionic modes.

DOI
10.1103/physrevb.92.125131

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We consider an interaction quench in the critical spin\ensuremath−12\text{spin}\ensuremath{-}\frac{1}{2}spin\ensuremath−21​ Heisenberg XXZ chain. We numerically compute the time evolution of the two-point correlation functions of spin operators in the thermodynamic limit and compare the results to predictions obtained in the framework of the Luttinger liquid approximation. We find that the transverse correlation function \ensuremath⟨SjxSj+\ensuremathℓx\ensuremath⟩\ensuremath{\langle}{S}_{j}^{x}{S}_{j+\ensuremath{\ell}}^{x}\ensuremath{\rangle}\ensuremath⟨Sjx​Sj+\ensuremathℓx​\ensuremath⟩ agrees with the Luttinger model prediction to a surprising level of accuracy. The agreement for the longitudinal two-point function \ensuremath⟨SjzSj+\ensuremathℓz\ensuremath⟩\ensuremath{\langle}{S}_{j}^{z}{S}_{j+\ensuremath{\ell}}^{z}\ensuremath{\rangle}\ensuremath⟨Sjz​Sj+\ensuremathℓz​\ensuremath⟩ is found to be much poorer. We speculate that this difference between transverse and longitudinal correlations has its origin in the locality properties of the respective spin operator with respect to the underlying fermionic modes.
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We study the time evolution of the two-point correlation functions in the XXZ Heisenberg chain after a finite-time quantum quench in the anisotropy. We compare results from numerical simulations to ones obtained in the Luttinger model and find good agreement. We analyze the spreading of the correlations and the associated light-cone features. We observe a delay in the appearance of the light cone as compared to the sudden-quench setup, and link this delay to the properties of the quench protocol.
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We have studied the zero- and low-temperature behaviors of anisotropic alternating antiferromagnetic-ferromagnetic Heisenberg spin-12\frac{1}{2}21​ chains in a transverse magnetic field. Using the analytical spinless fermion approach, the thermodynamic behavior of the model has been studied. We have introduced a different order parameter to distinguish the gapless Luttinger liquid phase from the other gapped phases. The exact diagonalization Lanczos results are also used to compare with the spinless fermion ones. We have found a double peak structure in the specific-heat curves for the region between the two quantum critical points. Using the numerical full diagonalization results, the existence of the double peak structure is confirmed.
We consider the time evolution after quantum quenches in the spin-12\frac{1}{2}21​ Heisenberg XXZ quantum spin chain with Ising-type anisotropy. The time evolution of short-distance spin-spin correlation functions is studied by numerical tensor network techniques for a variety of initial states, including N\'eel and Majumdar-Ghosh states and the ground state of the XXZ chain at large values of the anisotropy. The various correlators appear to approach stationary values, which are found to be in good agreement with the results of exact calculations of stationary expectation values in appropriate generalized Gibbs ensembles. In particular, our analysis shows how symmetries of the post-quench Hamiltonian that are broken by particular initial states are restored at late times.
Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.