Using local quantum fidelity distances, we study the dynamical quantum phase transition in integrable and nonintegrable one-dimensional Ising chains. Unlike the Loschmidt amplitude, the standard measure for distinguishing between two quantum states to describe the dynamical quantum phase transition, the local fidelity requires only a part of the system to characterize it. The nonanalyticities in the quantum distance between two subsystem density matrices identify the critical time and the corresponding critical exponent reasonably well in a finite-size system. Moreover, we propose a distance measure from the upper bound of the local quantum fidelity for certain quench protocols where the entanglement entropy features oscillatory growth in time. This local distance encodes the difference between the eigenvalue distribution of the initial and quenched subsystem density matrices and quantifies the critical properties. The alternative distance measure could be employed to examine the dynamical quantum phase transitions in a broader range of models, with implications for gaining insights into the transition from the entanglement perspective.
Using local quantum fidelity distances, we study the dynamical quantum phase transition in integrable and nonintegrable one-dimensional Ising chains. Unlike the Loschmidt amplitude, the standard measure for distinguishing between two quantum states to describe the dynamical quantum phase transition, the local fidelity requires only a part of the system to characterize it. The nonanalyticities in the quantum distance between two subsystem density matrices identify the critical time and the corresponding critical exponent reasonably well in a finite-size system. Moreover, we propose a distance measure from the upper bound of the local quantum fidelity for certain quench protocols where the entanglement entropy features oscillatory growth in time. This local distance encodes the difference between the eigenvalue distribution of the initial and quenched subsystem density matrices and quantifies the critical properties. The alternative distance measure could be employed to examine the dynamical quantum phase transitions in a broader range of models, with implications for gaining insights into the transition from the entanglement perspective.
In this dissertation, we address several problems in condensed matter physics, statistical mechanics, and quantum chaos from the perspective of the quantum information-theoretic notion of state distinguishability, as measured though evaluation of the appropriate fidelity measure. ❧ We find that the fidelity susceptibility, a measure for the response of the fidelity to infinitesimal changes of the Hamiltonian parameters, serves as a useful tool for studying the effects of disorder on quantum phase transitions. We study in detail the phase diagram of the well-known one-dimensional random quantum XY chain. Through an analysis of the fidelity susceptibility’s finite-size scaling and statistics for this model, we find that the fidelity approach reflects large variation between average and typical behavior, may detect the presence of Griffiths phases, and indicates the modifications to the phase diagram due to disorder. ❧ For this same model, though without disorder, we evaluate the dynamics following a sudden Hamiltonian quench using the Loschmidt echo, a form of the fidelity that compares the initial state with the time-evolved state. Through a central limit theorem-type argument we find that the long-time statistics of the Loschmidt echo takes one of two universal forms, even at finite temperature, depending on the system’s proximity to the quantum critical lines. ❧ Finally, for a canonical model exhibiting quantum chaos, the hydrogen atom in a uniform external magnetic field, we evaluate the operator fidelity susceptibility (OFS) between unitary operators generating the dynamics. We find that the part of the OFS which corresponds to the variation of the Hamiltonian eigenvectors serves to indicate the transition from the regular perturbative and n-mixing regimes to the quantum chaotic regime.
This dissertation puts forward the concept of fidelity as a way to understand and characterize quantum phase transitions involving topological phases. It lies at the interphase between Condensed Matter and Quantum Phase Transitions. ❧ We find that the fidelity metric, which characterizes the degree of distinghuishability between two quantum states, can serve as a useful tool in the understanding of transitions involving topological phases. ❧ The methodology does not require a complete knowledge of the physical mechanism involved in the occurrence of the critical point, such as the identification of a local order parameter and a symmetry breaking mechanism, the fundamental pillars of critical phenomena. Topological phases are unique in that they cannot be classified under these methodologies, due to their lack of local order parameters. It is one of the purposes of this thesis to understand how well we can characterize the phase boundaries between these phases using fidelity concepts. ❧ In the first part, we introduce two quantum models that exhibit a phase transition between an ordered and a topological phase, as quantified by the topological entanglement entropy. In the thermodynamic limit, the different components of the fidelity metric exhibit a divergence, signalling the occurrence of the critical point. ❧ In the second part, we introduce models that have phase boundaries between two distinct topological phases, and we show that the fidelity is successful at identifying the occurrence of a critical region. Moreover, a finite size scaling analysis is carried out, from which we determine the value of critical exponents.
Non-local orders, entanglement entropy, and quantum fidelity are investigated in an infinite-size bond-alternating Ising chain with the Dzyaloshinskii-Moriya interaction by employing the infinite matrix product state representation with the infinite time evolving block decimation method. Directly computing two distinct types of finite string correlations for very large lattice distances, in contrast to an extrapolated extreme value for finite size chains, reveals two topologically ordered phases. As the bond alternation varies, a topological quantum phase transition with continuously variable critical exponents along the phase boundary occurs between the two Haldane phases for a Dzyaloshinskii-Moriya interaction stronger than the Ising interaction while a topological quantum crossover between them happens through an intermediate antiferromagentic phase demonstrated with the quantum fidelity for a Dzyaloshinskii-Moriya interaction weaker than the Ising interaction. The critical exponents of the order parameters and the central charges from the entanglement entropy quantify the universality classes of the phase transition points. Anisotropic Heisenberg types of spin chains with bond alternations are finally discussed to share the same criticality.
We investigate the nonequilibrium quench dynamics of the one-dimensional transverse-field Ising model in both integrable and nonintegrable regimes. In particular, we report on a novel type of dynamical quantum phase transition (DQPT) that is characterized by a singular multipartite entanglement signature occurring at critical times in the post-quench dynamics. We show that this behavior is fundamentally distinct from previously studied DQPTs characterized by a nonanalytic rate function. We quantify the multipartite entanglement of the state by the quantum Fisher information and demonstrate that the DQPT belongs to a different universality class than the ground-state phase transition. Furthermore, we perform a spectral analysis of the DQPT and demonstrate that it is a genuine nonequilibrium transition arising from the constructive interference of excited states of the system during the many-body dynamics. Finally, we discuss potential experimental realizations in Rydberg platforms as well as applications in the context of quantum metrology.
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.
The study of quantum phase transitions requires the preparation of a many-body system near its ground state, a challenging task for many experimental systems. The measurement of quench dynamics, on the other hand, is now a routine practice in most cold atom platforms. Here we show that quintessential ingredients of quantum phase transitions can be probed directly with quench dynamics in integrable and nearly integrable systems. As a paradigmatic example, we study global quench dynamics in a transverse-field Ising model with either short-range or long-range interactions. When the model is integrable, we discover a new dynamical critical point with a nonanalytic signature in the short-range correlators. The location of the dynamical critical point matches that of the quantum critical point and can be identified using a finite-time scaling method. We extend this scaling picture to systems near integrability and demonstrate the continued existence of a dynamical critical point detectable at prethermal timescales. We quantify the difference in the locations of the dynamical and quantum critical points away from (but near) integrability. Thus, we demonstrate that this method can be used to approximately locate the quantum critical point near integrability. The scaling method is also relevant to experiments with finite time and system size, and our predictions are testable in near-term experiments with trapped ions and Rydberg atoms.