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arXiv:1302.2953 · Physical Review B

Correlation functions in the prethermalized regime after a quantum quench of a spin chain

Aditi Mitra

Published May 9, 2013• 68 citations
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Abstract

Results are presented for a two-point correlation function of a spin chain after a quantum quench for an intermediate time regime where inelastic effects are weak. A Callan-Symanzik-like equation for the correlation function is explicitly constructed which is used to show the appearance of three distinct scaling regimes. One is for spatial separations within a light cone, the second is for spatial separations on the light cone, and the third is for spatial separations outside the light cone. In these three regimes, the correlation function is found to decay with power laws with nonequilibrium exponents that differ from those in equilibrium, as well as from those obtained from quenches in a quadratic Luttinger liquid theory. A detailed discussion is presented on how the existence of scaling depends on the properties of the initial state before the quench.

DOI
10.1103/physrevb.87.205109

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Results are presented for a two-point correlation function of a spin chain after a quantum quench for an intermediate time regime where inelastic effects are weak. A Callan-Symanzik-like equation for the correlation function is explicitly constructed which is used to show the appearance of three distinct scaling regimes. One is for spatial separations within a light cone, the second is for spatial separations on the light cone, and the third is for spatial separations outside the light cone. In these three regimes, the correlation function is found to decay with power laws with nonequilibrium exponents that differ from those in equilibrium, as well as from those obtained from quenches in a quadratic Luttinger liquid theory. A detailed discussion is presented on how the existence of scaling depends on the properties of the initial state before the quench.
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.
We employ equation-of-motion techniques to study the nonequilibrium dynamics in a lattice model of weakly interacting spinless fermions. Our model provides a simple setting for analyzing the effects of weak integrability-breaking perturbations on the time evolution after a quantum quench. We establish the accuracy of the method by comparing results at short and intermediate times to time-dependent density matrix renormalization group computations. For sufficiently weak integrability-breaking interactions we always observe prethermalization plateaus, where local observables relax to nonthermal values at intermediate time scales. At later times a crossover towards thermal behavior sets in. We determine the associated time scale, which depends on the initial state, the band structure of the noninteracting theory, and the strength of the integrability-breaking perturbation. Our method allows us to analyze in some detail the spreading of correlations and in particular the structure of the associated light cones in our model. We find that the interior and exterior of the light cone are separated by an intermediate region, the temporal width of which appears to scale with a universal power law t1/3{t}^{1/3}t1/3.
We study the short-time Loschmidt dynamics after a small sudden quench near a quantum critical point. We show that the initial quadratic growth of the Loschmidt rate function is governed by the variance of the quench operator per system size. For a local quench operator, this variance density is exactly equal to the spatial sum of the equal-time connected two-point correlation function in the initial ground state. This relation connects the early-time Loschmidt response to static critical correlations. Assuming power-law correlations at criticality, we classify the finite-size scaling of the short-time coefficient by the scaling dimension of the quench operator. In one dimension, the coefficient is finite, logarithmically enhanced, or algebraically enhanced with system size. We illustrate this operator dependence in the transverse-field Ising chain, where transverse-field and longitudinal-field quenches couple to different critical operators. We also discuss the first correction beyond the quadratic regime using the fourth cumulant of the post-quench Hamiltonian.
We investigate the effects of fluctuations on the dynamics of an isolated quantum system represented by a ${\left({\mathbf{\ensuremath{\phi}}}^{2}\right)}^{2}$ field theory with O(N)O(N)O(N) symmetry after a quench in d>2d>2d>2 spatial dimensions. A perturbative renormalization-group approach involving a dimensional expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ is employed in order to study the evolution within a prethermal regime controlled by elastic dephasing. In particular, we focus on a quench from a disordered initial state to the critical point, which introduces an effective temporal boundary in the evolution. At this boundary, the relevant fields acquire an anomalous scaling dimension, while the evolution of both the order parameter and its correlation and response functions display universal aging. Since the relevant excitations propagate ballistically, a light cone in real space emerges. At longer times, the onset of inelastic scattering appears as secularly growing self-generated dissipation in the effective Keldysh field theory, with the strength of the dissipative vertices providing an estimate for the time needed to leave the prethermal regime.
In non-relativistic quantum theories the Lieb-Robinson bound defines an effective light cone with exponentially small tails outside of it. In this work we use it to derive a bound for the correlation function of two local disjoint observables at different times if the initial state has a power-law decay. We show that the exponent of the power-law of the bound is identical to the initial (equilibrium) decay. We explicitly verify this result by studying the full dynamics of the susceptibilities and correlations in the exactly solvable Luttinger model after a sudden quench from the non-interacting to the interacting model.
We study the relaxation dynamics of interacting one-dimensional fermions with band curvature after a weak quench in the interaction parameter at zero temperature. Our model lies within the class of interacting Luttinger liquids, where the harmonic Luttinger theory is extended by a weak-integrability-breaking phonon scattering term. In order to solve for the nonequilibrium time evolution, we use quantum kinetic equations exploiting the resonant but subleading character of the phonon interaction term. The interplay between phonon scattering and the quadratic Luttinger theory leads to the emergence of three distinct spatiotemporal regimes for the fermionic real-space correlation function. It features the crossover from a prequench to a prethermal state, finally evolving towards a thermal state on increasing length and time scales. The characteristic algebraically decaying real-space correlations in the prethermalized regime become modulated by an amplitude that is decaying in time according to a stretched exponential as an effect of the interactions. The asymptotic thermalization dynamics is governed by energy transport over large distances from the thermalized to the nonthermalized regions via macroscopic, dynamical slow modes. This is revealed in an algebraic decay of the system's effective temperature. The numerical value of the associated exponent agrees with the dynamical critical exponent of the Kardar-Parisi-Zhang universality class. We also discuss a criterion for the applicability of this theory away from the integrable limit of noninteracting fermions.