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arXiv:2308.05182 · Physical review. B./Physical review. B

Effect of long-range hopping on dynamic quantum phase transitions of an exactly solvable free-fermion model: Nonanalyticities at almost all times

J. C. Xavier, José A. Hoyos

Published Dec 5, 2023• 2 citations
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Abstract

In this work, we investigate quenches in a free-fermion chain with long-range hopping, which decay with the distance with an exponent $\ensuremath{\nu}$ and has range $D$. By exploring the exact solution of the model, we found that the dynamic free energy is nonanalytical in the thermodynamic limit, whenever the sudden quench crosses the equilibrium quantum critical point. We were able to determine the nonanalyticities of dynamic free energy $f(t)$ at some critical times ${t}^{c}$ by solving nonlinear equations. We also show that the Yang-Lee-Fisher (YLF) zeros cross the real-time axis at those critical times. We found that the number of nontrivial critical times, ${N}_{s},$ depends on $\ensuremath{\nu}$ and $D$. In particular, we show that for small $\ensuremath{\nu}$ and large $D$ the dynamic free energy presents nonanalyticities in any time interval $\mathrm{\ensuremath{\Delta}}t\ensuremath{\sim}1/D\ensuremath{\ll}1$, i.e., there are nonanalyticities at almost all times. For the spacial case $\ensuremath{\nu}=0$, we obtain the critical times in terms of a simple expression of the model parameters and also show that $f(t)$ is nonanalytical even for finite system under antiperiodic boundary condition, when we consider some special values of quench parameters. We also show that, generically, the first derivative of the dynamic free energy is discontinuous at the critical time instant when the YLF zeros are nondegenerate. On the other hand, when they become degenerate, all derivatives of $f(t)$ exist at the associated critical instant.

DOI
10.1103/physrevb.108.214303

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In this work, we investigate quantum quenches in the two-leg Creutz model with long-range hopping, where the hopping amplitudes decay with distance as a power law characterized by an exponent ννν and have a finite range DDD. We first obtain the exact solution of a generic two-band model in momentum space. This allows us to compute the Loschmidt amplitude and, consequently, the dynamical free energy f(t)f(\texttt{t})f(t) of the two-band model. We also show how to determine the Yang-Lee Fisher (YLF) zeros by solving a nonlinear equation. We demonstrate that the two-leg Creutz model in momentum space is a special case of the generic two-band model. Using these results, we identify the non-analyticities in the dynamical free energy f(t)f(\texttt{t})f(t) at critical times tc\texttt{t}_{c}tc​. We find that the number of nontrivial critical times NsN_{s}Ns​ depends on both ννν and DDD. In particular, we show that for small ννν and large DDD the critical times become increasingly dense, leading, in the appropriate regime, to non-analyticities at an increasingly dense set of times---similar to what was observed by Xavier and Hoyos [Phys. Rev. B, 108\textbf{108}108, 214303 (2023)] in the Su-Schrieffer-Heeger (SSH) model with long-range hopping terms.
We investigate the dynamics following sudden quenches across quantum critical points belonging to different universality classes. Specifically, we use matrix product state methods to study the quantum Ising chain in the presence of two additional terms which break integrability. We find that in all models the rate function for the return probability to the initial state becomes a nonanalytic function of time in the thermodynamic limit. This so-called ``dynamical phase transition'' was first observed in a recent work by Heyl, Polkovnikov, and Kehrein [Phys. Rev. Lett. 110, 135704 (2013)] for the exactly-solvable quantum Ising chain, which can be mapped to free fermions. Our results for ``interacting theories'' indicate that nonanalytic dynamics is a generic feature of sudden quenches across quantum critical points. We discuss potential connections to the dynamics of the order parameter.
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We investigate the nonequilibrium dynamics induced by a finite-time linear quench in the XYXYXY chain. Initially, we examine the dynamical quantum phase transition (DQPT-LE), characterized by the nonanalytic behavior of the Loschmidt amplitude. We find distinct behaviors of DQPT-LEs during and following the ramp. Following the ramp, the ramp crossing the critical point hc{h}_{c}hc​ is the sufficient condition for the occurrence of DQPT-LE, but it is not during the ramp. Through adiabatic-impulse-adiabatic (AIA) approximation analysis, we establish that adequate distancing from the critical point is crucial for DQPT-LE manifestation during the ramp, elucidating the absence of DQPT-LE as the ramp gets faster. Additionally, we explore another type of dynamical phase transition (DQPT-OP), describing the long-term relaxation behavior of the order parameter. Our finding indicates that the asymptotic behavior of the time-dependent part induced by the linear quench is equivalent to that following a sudden quench, i.e., the time-dependent part exhibits power-law decays of $\ensuremath{\sim}{t}^{\ensuremath{-}3/2}$ and $\ensuremath{\sim}{t}^{\ensuremath{-}1/2}$ for the ramp to the commensurate and incommensurate phases, respectively. Moreover, we also delve into the steady part, which showcases nonanalytic singularities at the critical point.
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