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arXiv:1903.07682 · Journal of High Energy Physics

Quantum quench in non-relativistic fermionic field theory: harmonic traps and 2d string theory

Sumit R. Das, Shaun D. Hampton, Sinong Liu

Published Aug 1, 2019• 15 citations
arXivHTMLPDF

Abstract

Abstract We investigate a class of exactly solvable quantum quench protocols with a finite quench rate in systems of one dimensional non-relativistic fermions in external harmonic oscillator or inverted harmonic oscillator potentials, with time dependent masses and frequencies. These hamiltonians arise, respectively, in harmonic traps, and the c = 1 Matrix Model description of two dimensional string theory with time dependent string coupling. We show how the dynamics is determined by a single function of time which satisfies a generalized Ermakov-Pinney equation. The quench protocols we consider asymptote to constant masses and frequencies at early times, and cross or approach a gapless potential. In a right side up harmonic oscillator potential we determine the scaling behavior of the one point function and the entanglement entropy of a subregion by obtaining analytic approximations to the exact answers. The results are consistent with Kibble-Zurek scaling for slow quenches and with perturbation calculations for fast quenches. For cis-critical quench protocols the entanglement entropy oscillates at late times around its initial value. For end-critical protocols the entanglement entropy monotonically goes to zero inversely with time, reflecting the spread of fermions over the entire line. For the inverted harmonic oscillator potential, the dual collective field description is a scalar field in a time dependent metric and dilaton background.

DOI
10.1007/jhep08(2019)176

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Abstract We investigate a class of exactly solvable quantum quench protocols with a finite quench rate in systems of one dimensional non-relativistic fermions in external harmonic oscillator or inverted harmonic oscillator potentials, with time dependent masses and frequencies. These hamiltonians arise, respectively, in harmonic traps, and the c = 1 Matrix Model description of two dimensional string theory with time dependent string coupling. We show how the dynamics is determined by a single function of time which satisfies a generalized Ermakov-Pinney equation. The quench protocols we consider asymptote to constant masses and frequencies at early times, and cross or approach a gapless potential. In a right side up harmonic oscillator potential we determine the scaling behavior of the one point function and the entanglement entropy of a subregion by obtaining analytic approximations to the exact answers. The results are consistent with Kibble-Zurek scaling for slow quenches and with perturbation calculations for fast quenches. For cis-critical quench protocols the entanglement entropy oscillates at late times around its initial value. For end-critical protocols the entanglement entropy monotonically goes to zero inversely with time, reflecting the spread of fermions over the entire line. For the inverted harmonic oscillator potential, the dual collective field description is a scalar field in a time dependent metric and dilaton background.
Global quantum quench with a finite quench rate which crosses critical points is known to lead to universal scaling of correlation functions as functions of the quench rate. In this work, we explore scaling properties of the entanglement entropy of a subsystem in a harmonic chain during a mass quench which asymptotes to finite constant values at early and late times and for which the dynamics is exactly solvable. When the initial state is the ground state, we find that for large enough subsystem sizes the entanglement entropy becomes independent of size. This is consistent with Kibble–Zurek scaling for slow quenches, and with recently discussed “fast quench scaling” for quenches fast compared to physical scales, but slow compared to UV cutoff scales.
Abstract We consider the scaling behavior of circuit complexity under quantum quench in an a relativistic fermion field theory on a one dimensional spatial lattice. This is done by finding an exactly solvable quench protocol which asymptotes to massive phases at early and late times and crosses a critical point in between. We find a variety of scaling behavior as a function of the quench rate, starting with a saturation for quenches at the lattice scale, a “fast quench scaling” at intermediate rate and a Kibble Zurek scaling at slow rates.
The dissertation includes two parts. In Part I, we study non-equilibrium phenomena in various models associated with global quantum quench. It is known that local quantities, when subjected to global quantum quench across or approaching critical points, exhibit a variety of universal scaling behaviors at various quench rates. To investigate if similar scaling holds for non-local quantities, we consider the scaling behavior of circuit complexity under quantum quench across the critical massless point in Majorana fermion field theory of the one-dimensional integrable transverse field Ising model and find it obeys such scaling. To investigate if similar scaling holds for non-relativistic theories, we test various solvable critical quantum quench protocols in a theory of fermions in a harmonic oscillator potential and find local quantities as well as entanglement entropy obeys different scaling behaviors at different quench rates. We study quantum quench in the c=1 matrix model which is holographically dual to two-dimensional string theory. Unlike higher dimensional holographic setups where quantum quench leads to black holes, the emergent spacetime in this model generically develops cosmological singularities at late times. In Part II, we expand the proposal that target space entanglement provides a precise notion of entanglement in the bulk gravitational duals of Dp brane theories, which was shown in a gauge fixed formalism. We developed a gauge invariant description of target space entanglement in these theories and derived path integral expressions for the entanglement entropy which can be used in numerical calculations.
Quantum quenches in continuum field theory across critical points are known to display different scaling behaviours in different regimes of the quench rate. We extend these results to integrable lattice models such as the transverse field Ising model on a one-dimensional chain and the Kitaev model on a two-dimensional honeycomb lattice using a nonlinear quench protocol which allows for exact analytical solutions of the dynamics. Our quench protocol starts with a finite mass gap at early times and crosses a critical point or a critical region, and we study the behaviour of one point functions of the quenched operator at the critical point or in the critical region as a function of the quench rate. For quench rates slow compared to the initial mass gap, we find the expected Kibble-Zurek scaling. In contrast, for rates fast compared to the mass gap, but slow compared to the inverse lattice spacing, we find scaling behaviour similar to smooth fast continuum quenches. For quench rates of the same order of the lattice scale, the one point function saturates as a function of the rate, approaching the results of an abrupt quench. The presence of an extended critical surface in the Kitaev model leads to a variety of scaling exponents depending on the starting point and on the time where the operator is measured. We discuss the role of the amplitude of the quench in determining the extent of the slow (Kibble-Zurek) and fast quench regimes, and the onset of the saturation.
Abstract Quantum quenches in continuum field theory across critical points are known to display different scaling behaviours in different regimes of the quench rate. We extend these results to integrable lattice models such as the transverse field Ising model on a one-dimensional chain and the Kitaev model on a two-dimensional honeycomb lattice using a nonlinear quench protocol which allows for exact analytical solutions of the dynamics. Our quench protocol starts with a finite mass gap at early times and crosses a critical point or a critical region, and we study the behaviour of one point functions of the quenched operator at the critical point or in the critical region as a function of the quench rate. For quench rates slow compared to the initial mass gap, we find the expected Kibble-Zurek scaling. In contrast, for rates fast compared to the mass gap, but slow compared to the inverse lattice spacing, we find scaling behaviour similar to smooth fast continuum quenches. For quench rates of the same order of the lattice scale, the one point function saturates as a function of the rate, approaching the results of an abrupt quench. The presence of an extended critical surface in the Kitaev model leads to a variety of scaling exponents depending on the starting point and on the time where the operator is measured. We discuss the role of the amplitude of the quench in determining the extent of the slow (Kibble-Zurek) and fast quench regimes, and the onset of the saturation.
Abstract We study dynamics of quantum entanglement in smooth global quenches with a finite rate, by computing the time evolution of entanglement entropy in 1 + 1 dimensional free scalar theory with time-dependent masses which start from a nonzero value at early time and either crosses or approaches zero. The time-dependence is chosen so that the quantum dynamics is exactly solvable. If the quenches asymptotically approach a critical point at late time, the early-time and late-time entropies are proportional to the time and subsystem size respectively. Their proportionality coefficients are determined by scales: in a fast limit, an initial correlation length; in a slow limit, an effective scale defined when adiabaticity breaks down. If the quenches cross a critical point, the time evolution of entropy is characterized by the scales: the initial correlation length in the fast limit and the effective correlation length in the slow limit. The entropy oscillates, and the entanglement oscillation comes from a coherence between right-moving and left-moving waves if we measure the entropy after time characterized by the quench rate. The periodicity of the late-time oscillation is consistent with the periodicity of the oscillation of zero modes which are zero-momentum value of two point functions of a fundamental field and its conjugate momentum.