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Register a new userExcited State Quantum Phase Transitions Studied from a Non-Hermitian Perspective
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A main distinguishing feature of non-Hermitian quantum mechanics is the presence of exceptional points (EPs). They correspond to the coalescence of two energy levels and their respective eigenvectors. Here, we use the Lipkin-Meshkov-Glick (LMG) model as a testbed to explore the strong connection between EPs and the onset of excited state quantum phase transitions (ESQPTs). We show that for finite systems, the exact degeneracies (EPs) obtained with the non-Hermitian LMG Hamiltonian continued into the complex plane are directly linked with the avoided crossings that characterize the ESQPTs for the real (physical) LMG Hamiltonian. The values of the complex control parameter $alpha$ that lead to the EPs approach the real axis as the system size $Nrightarrow infty$. This happens for both, the EPs that are close to the separatrix that marks the ESQPT and also for those that are far away, although in the latter case, the rate the imaginary part of $alpha$ reduces to zero as $N$ increases is smaller. With the method of Pade approximants, we can extract the critical value of $alpha$.
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Excited-state quantum phase transitions (ESQPTs) extend the notion of quantum phase transitions beyond the ground state. They are characterized by closing energy gaps amid the spectrum. Identifying order parameters for ESQPTs poses however a major challenge. We introduce spinor Bose-Einstein condensates as a versatile platform for studies of ESQPTs. Based on the mean-field dynamics, we define a topological order parameter that distinguishes between excited-state phases, and discuss how to interferometrically access the order parameter in current experiments. Our work opens the way for the experimental characterization of excited-state quantum phases in atomic many-body systems.
Background: Composed systems have became of great interest in the framework of the ground state quantum phase transitions (QPTs) and many of their properties have been studied in detail. However, in these systems the study of the so called excited-state quantum phase transitions (ESQPTs) have not received so much attention. Purpose: A quantum analysis of the ESQPTs in the two-fluid Lipkin model is presented in this work. The study is performed through the Hamiltonian diagonalization for selected values of the control parameters in order to cover the most interesting regions of the system phase diagram. [Method:] A Hamiltonian that resembles the consistent-Q Hamiltonian of the interacting boson model (IBM) is diagonalized for selected values of the parameters and properties such as the density of states, the Peres lattices, the nearest-neighbor spacing distribution, and the participation ratio are analyzed. Results: An overview of the spectrum of the two-fluid Lipkin model for selected positions in the phase diagram has been obtained. The location of the excited-state quantum phase transition can be easily singled out with the Peres lattice, with the nearest-neighbor spacing distribution, with Poincare sections or with the participation ratio. Conclusions: This study completes the analysis of QPTs for the two-fluid Lipkin model, extending the previous study to excited states. The ESQPT signatures in composed systems behave in the same way as in single ones, although the evidences of their presence can be sometimes blurred. The Peres lattice turns out to be a convenient tool to look into the position of the ESQPT and to define the concept of phase in the excited states realm.
We analyze the scaling behavior of the fidelity, and the corresponding susceptibility, emerging in finite-size many-body systems whenever a given control parameter $lambda$ is varied across a quantum phase transition. For this purpose we consider a finite-size scaling (FSS) framework. Our working hypothesis is based on a scaling assumption of the fidelity in terms of the FSS variables associated to $lambda$ and to its variation $delta lambda$. This framework entails the FSS predictions for continuous transitions, and meanwhile enables to extend them to first-order transitions, where the FSS becomes qualitatively different. The latter is supported by analytical and numerical analyses of the quantum Ising chain along its first-order quantum transition line, driven by an external longitudinal field.
We describe how to characterize dynamical phase transitions in open quantum systems from a purely dynamical perspective, namely, through the statistical behavior of quantum jump trajectories. This approach goes beyond considering only properties of the steady state. While in small quantum systems dynamical transitions can only occur trivially at limiting values of the controlling parameters, in many-body systems they arise as collective phenomena and within this perspective they are reminiscent of thermodynamic phase transitions. We illustrate this in open models of increasing complexity: a three-level system, a dissipative version of the quantum Ising model, and the micromaser. In these examples dynamical transitions are accompanied by clear changes in static behavior. This is however not always the case, and in general dynamical phase behavior needs to be uncovered by observables which are strictly dynamical, e.g. dynamical counting fields. We demonstrate this via the example of a class of models of dissipative quantum glasses, whose dynamics can vary widely despite having identical (and trivial) stationary states.
We study the nonequilibrium dynamics of the extended toric code model (both ordered and disordered) to probe the existence of the dynamical quantum phase transitions (DQPTs). We show that in the case of the ordered toric code model, the zeros of Loschmidt overlap (generalized partition function) occur at critical times when DQPTs occur, which is confirmed by the nonanalyticities in the dynamical counter-part of the free-energy density. Moreover, we show that DQPTs occur for any non-zero field strength if the initial state is the excited state of the toric code model. In the disordered case, we show that it is imperative to study the behavior of the first time derivative of the dynamical free-energy density averaged over all the possible configurations, to characterize the occurrence of a DQPTs in the disordered toric code model since the disorder parameter itself acts as a new artificial dimension. We also show that for the case where anyonic excitations are present in the initial state, the conditions for a DQPTs to occur are the same as what happens in the absence of any excitation.
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