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Physical review. B./Physical review. B

Exploring dynamical quantum phase transitions in a spin model with deconfined critical point via the quantum steering ellipsoid

Changxiao Li, Sheng Yang, Jing‐Bo Xu, Hai‐Qing Lin

Published Feb 21, 2023• 11 citations
Publisher

Abstract

The dynamical quantum phase transition (DQPT), which is featured by the nonanalytic behavior of the Loschmidt rate function in the real time evolution, has attracted substantial attention as a valuable theoretical concept for characterizing nonequilibrium states of quantum matter. Although the link between the DQPT and many physical concepts has been established, a thorough understanding of this transition still calls for more studies. In this paper, from the perspective of the quantum steering ellipsoid (QSE), we investigate the DQPTs supported in a one-dimensional spin chain with a deconfined quantum critical point by using the global subspace expansion time-dependent variational principle algorithm. For the quench from the valence-bond-solid phase to the ferromagnetic phase, we find a clear correspondence between the vanishing of the QSE volume and the occurrence of the DQPT. For the quench in the opposite direction, however, the QSE exhibits a rather complicated behavior during the time evolution. We also calculate the quantum entanglement and quantum coherence in the quench processes to unveil the change of the quantum correlations encoded in the QSE picture. These findings could offer a fascinating possibility of revealing DQPTs in a geometrically discernible manner.

DOI
10.1103/physrevb.107.085130

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The dynamical quantum phase transition (DQPT), which is featured by the nonanalytic behavior of the Loschmidt rate function in the real time evolution, has attracted substantial attention as a valuable theoretical concept for characterizing nonequilibrium states of quantum matter. Although the link between the DQPT and many physical concepts has been established, a thorough understanding of this transition still calls for more studies. In this paper, from the perspective of the quantum steering ellipsoid (QSE), we investigate the DQPTs supported in a one-dimensional spin chain with a deconfined quantum critical point by using the global subspace expansion time-dependent variational principle algorithm. For the quench from the valence-bond-solid phase to the ferromagnetic phase, we find a clear correspondence between the vanishing of the QSE volume and the occurrence of the DQPT. For the quench in the opposite direction, however, the QSE exhibits a rather complicated behavior during the time evolution. We also calculate the quantum entanglement and quantum coherence in the quench processes to unveil the change of the quantum correlations encoded in the QSE picture. These findings could offer a fascinating possibility of revealing DQPTs in a geometrically discernible manner.
The past two decades have witnessed a surge of interest in borrowing tools from quantum information theory to investigate quantum phase transitions (QPTs). The best-known examples are entanglement measures whose nonanalyticities at critical points were tied to QPTs in a plethora of physical models. Here, focusing on the XXZXXZXXZ model, we show how QPTs can be revealed through the quantum steering ellipsoid (QSE), which is a geometric tool capable of characterizing both the strength and type of quantum correlations between two subsystems of a compound system. We find that the QSE associated with the XXZXXZXXZ model changes in shape with the QPTs; that is, it is a needle in the ferromagnetic phase, an oblate spheroid in the gapless phase, and a prolate spheroid in the antiferromagnetic phase. This finding offers an example demonstrating the intriguing possibility of unveiling QPTs in a geometrically visible fashion. Some connections between our results and previous ones are discussed.
Abstract Quantum steering ellipsoids (QSEs) can serve as a useful geometric tool for describing both the strength and type of quantum correlations between two subsystems of a compound system. By employing the quantum renormalization-group method, we focus on investigating the relation between QSEs and the quantum phase transition (QPT) in the anisotropic spin XY model. The results indicate that the QPT is well visualized in terms of the shape of the QSE, i.e. it is an oblate spheroid in the spin-fluid phase and a needle in the Néel phase. Meanwhile, after several iterations of renormalization, the QSE volume V undergoes a contraction mutation, and can develop two saturated values at the critical points associated with the QPT, which correspond to two different phases: the spin-fluid phase and the Néel phase. We also find that the QSE is closely associated with quantum entanglement in the model, i.e. the volume of the QSE between blocks is more than 4π/81 when the system is in the spin-fluid phase, which indicates that the system must be entangled. Furthermore, the nonanalytic and scaling behaviors of the volume of the QSE have been analyzed in detail, and the results convince us that the quantum critical properties are connected with the behavior of the QSE.
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.
The notion of a dynamical quantum phase transition (DQPT) was recently introduced [Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)] as the nonanalytic behavior of the Loschmidt echo at critical times in the thermodynamic limit. In this work the quench dynamics in the ground state sector of the two-dimensional Kitaev honeycomb model is studied regarding the occurrence of DQPTs. For general two-dimensional systems of BCS type it is demonstrated how the zeros of the Loschmidt echo coalesce to areas in the thermodynamic limit, implying that DQPTs occur as discontinuities in the second derivative. In the Kitaev honeycomb model DQPTs appear after quenches across a phase boundary or within the massless phase. In the 1d limit of the Kitaev honeycomb model it becomes clear that the discontinuity in the higher derivative is intimately related to the higher dimensionality of the nondegenerate model. Moreover, there is a strong connection between the stationary value of the rate function of the Loschmidt echo after long times and the occurrence of DQPTs in this model.
We analytically and numerically study the Loschmidt echo and the dynamical order parameters in a spin chain with a deconfined phase transition between a dimerized state and a ferromagnetic phase. For quenches from a dimerized state to a ferromagnetic phase, we find that the model can exhibit a dynamical quantum phase transition characterized by an associating dimerized order parameter. In particular, when quenching the system from the Majumdar-Ghosh state to the ferromagnetic Ising state, we find an exact mapping into the classical Ising chain for a quench from the paramagnetic phase to the classical Ising phase by analytically calculating the Loschmidt echo and the dynamical order parameters. By contrast, for quenches from a ferromagnetic state to a dimerized state, the system relaxes very fast so that the dynamical quantum transition may exist on only a short timescale. We reveal that the dynamical quantum phase transition can occur in systems with two broken symmetry phases and the quench dynamics may be independent of equilibrium phase transitions.
Dynamical quantum phase transitions (DQPTs), which refer to the criticality in time of a quantum many-body system, have attracted much theoretical and experimental research interest recently. Despite that DQPTs are defined and signaled by the nonanalyticities in the Loschmidt rate, its interrelation with various correlation measures such as the equilibrium order parameters of the system remains unclear. In this work, by considering the quench dynamics in an interacting topological model, we find that the equilibrium order parameters of the model in general exhibit signatures around the DQPT, in the short time regime. The first extrema of the equilibrium order parameters are connected to the first Loschmidt rate peak. By studying the unequal-time two-point correlation, we also find that the correlation between the nearest neighbors decays while that with neighbors further away builds up as time grows in the noninteracting case, and upon the addition of repulsive intracell interactions. On the other hand, the intercell interaction tends to suppress the two-site correlations. These findings could provide us insights into the characteristic of the system around DQPTs, and pave the way to a better understanding of the dynamics in nonequilibrium quantum many-body systems.