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arXiv:2608.24514

Dynamic quantum phase transitions in the two-leg Creutz ladder with long-range hopping

J. A. da Silva, J. C. Xavier

Published Aug 25, 2026• 0 citations
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Abstract

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 $ν$ and have a finite range $D$. 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 $f(\texttt{t})$ 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 $f(\texttt{t})$ at critical times $\texttt{t}_{c}$. We find that the number of nontrivial critical times $N_{s}$ depends on both $ν$ and $D$. In particular, we show that for small $ν$ and large $D$ 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, $\textbf{108}$, 214303 (2023)] in the Su-Schrieffer-Heeger (SSH) model with long-range hopping terms.

DOI
10.48550/arxiv.2608.24514

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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.
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 DDD. 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)f(t)f(t) at some critical times tc{t}^{c}tc 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, Ns,{N}_{s},Ns​, depends on $\ensuremath{\nu}$ and DDD. In particular, we show that for small $\ensuremath{\nu}$ and large DDD 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)f(t)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)f(t)f(t) exist at the associated critical instant.
Abstract The dynamics of quantum phase transitions of topological systems has attracted interest. Here we consider the effect of long-range hopping terms on the dynamic behavior. The long-range hoppings preserve chiral symmetry and decay as a function of distance, with an exponent β. We also consider a situation where the hoppings are modulated by oscillations. We compare the results with those obtained for short-range models such as the Su–Schrieffer–Heeger model or its extension to second neighboring cells. While often the lowest critical times coincide with the first peak of the Loschmidt rate, several anomalous cases are identified. Also, the number of critical times often exceeds the absolute value of the change of the winding number in a quench. Various correlation functions are considered, and their time-dependence may often be used to identify the early-time dynamical phase transitions.
In most lattice models, the closing of a band gap typically occurs at high-symmetry points in the Brillouin zone. Differently, in the Creutz model---describing a system of spinless fermions hopping on a two-leg ladder pierced by a magnetic field---the gap closing at the quantum phase transition between the two topologically nontrivial phases of the model can be moved by tuning the hopping amplitudes. We take advantage of this property to examine the nonequilibrium dynamics of the model after a sudden quench of the magnetic flux through the plaquettes of the ladder. For a quench to one of the equilibrium quantum critical points, we find that the revival period of the Loschmidt echo---measuring the overlap between initial and time-evolved states---is controlled by the gap closing zero-energy modes. In particular, and contrary to expectations, the revival period of the Loschmidt echo for a finite ladder does not scale linearly with size but exhibits jumps determined by the presence or absence of zero-energy modes. We further investigate the conditions for the appearance of dynamical quantum phase transitions in the model and find that, for a quench to an equilibrium critical point, such transitions occur only for ladders of sizes which host zero-energy modes. Exploiting concepts from quantum thermodynamics, we show that the average work and the irreversible work per lattice site exhibit a weak dependence on the size of the system after a quench across an equilibrium critical point, suggesting that quenching into a different phase induces effective correlations among the particles.
Abstract Disorders and long-range hoppings can induce exotic phenomena in condensed matter and artificial systems. We study the topological and dynamical properties of the quasiperiodic Su–Schrier–Heeger model with long-range hoppings. It is found that the interplay of quasiperiodic disorder and long-range hopping can induce topological Anderson insulator phases with non-zero winding numbers ω = 1 , 2 , and the phase boundaries can be consistently revealed by the divergence of zero-energy mode localization length. We also investigate the nonequilibrium dynamics by ramping the long-range hopping along two different paths. The critical exponents extracted from the dynamical behavior agree with the Kibble–Zurek mechanic prediction for the path with W = 0.90 . In particular, the dynamical exponent of the path crossing the multicritical point is numerical obtained as 1 / 6 ∼ 0.167 , which agrees with the unconventional finding in the previously studied XY spin model. Besides, we discuss the anomalous and non-universal scaling of the defect density dynamics of topological edge states in this disordered system under open boundary condictions.
We investigate dynamical quantum phase transitions (DQPTs) in the extended Agassi model following sudden quenches of same level pairing (ΣΣΣ), monopole-monopole (χχχ) and cross level pairing (ΛΛΛ) strengths. Using exact diagonalization, we characterize the non-equilibrium dynamics through the Loschmidt echo and intensive return-rate function. Quenches across equilibrium phase boundaries produce cusp-like structures in the return rate that shows non-analyticities with increasing system size. Equilibrium finite size to large scaling identifies the χχχ-driven transition as a second order phase transition similar to mean-field, Dicke, and Lipkin-Meshkov-Glick (LMG) models critical behaviour with the exponent of the order parameter β≃1/2β\simeq 1/2β≃1/2, while the ΣΣΣ- and ΛΛΛ-driven transitions are first order. We further characterize the associated dynamical singularities through a local scaling exponent ααα, with αχ≃2/3α_χ\simeq 2/3αχ​≃2/3, αΛ≃1/3α_Λ\simeq 1/3αΛ​≃1/3, and αΣ≃3/2α_Σ\simeq 3/2αΣ​≃3/2 of all three parameters. The distinct scaling behaviours across the parameter channels show that dynamical criticality is not determined solely by the equilibrium order of the transition, revealing multiple dynamical scaling regimes. These non-analyticities are associated with the Fisher zeros of the Loschmidt amplitude, which cross the real time axis at critical times and the time derivatives of the return-rate function to characterize the order of DQPTs within the extended Agassi model.
Research on topological models unveils fascinating physics, especially in the realm of dynamical quantum phase transitions (DQPTs). However, the understanding of entanglement structures and properties near DQPT in models with longer-range hoppings is far from complete. In this work, we study DQPTs in the quenched extended Su-Schrieffer-Heeger model. Anomalous DQPTs, where the number of critical momenta exceeds the winding number differences between the prequench and postquench phases, are observed. We find that the entanglement exhibits local maximum (minimum) around the anomalous DQPTs, in line with the level crossings (separations) around the middle of the correlation matrix spectrum. We further categorize the phases in the equilibrium model into two classes and distinctive features in the time evolution of the entanglement involving quenches within and across the two classes are identified. The findings pave the way to a better understanding of topological models with longer-range hoppings in the out-of-equilibrium regime.