Monodromy and excited-state quantum phase transitions in integrable systems: collective vibrations of nuclei

Excited-state quantum phase transitions and the entropy of the work distribution in the anharmonic Lipkin-Meshkov-Glick model

Studying the implications and characterizations of the excited-state quantum phase transitions (ESQPTs) would enable us to understand various phenomena observed in quantum many-body systems. In this work, we delve into the affects and characterizations of the ESQPTs in the anharmonic Lipkin-Meshkov-Glick (LMG) model by means of the entropy of the quantum work distribution. The entropy of the work distribution measures the complexity of the work distribution and behaves as a valuable tool for analyzing nonequilibrium work statistics. We show that the entropy of the work distribution captures salient signatures of the underlying ESQPTs in the model. In particular, a detailed analysis of the scaling behavior of the entropy verifies that it not only acts as a witness of the ESQPTs but also reveals the difference between different types of ESQPTs. We further demonstrate that the work distribution entropy also behaves as a powerful tool for understanding the features and differences of ESQPTs in the energy space. Our results provide further evidence of the usefulness of the entropy of the work distribution for investigating various phase transitions in quantum many-body systems and open up a promising way for experimentally exploring the signatures of ESQPTs.

Spectral kissing and its dynamical consequences in the squeeze-driven Kerr oscillator

Transmon qubits are the predominant element in circuit-based quantum information processing, such as existing quantum computers, due to their controllability and ease of engineering implementation. But more than qubits, transmons are multilevel nonlinear oscillators that can be used to investigate fundamental physics questions. Here, they are explored as simulators of excited state quantum phase transitions (ESQPTs), which are generalizations of quantum phase transitions to excited states. We show that the spectral kissing (coalescence of pairs of energy levels) experimentally observed in the effective Hamiltonian of a driven SNAIL-transmon is an ESQPT precursor. We explore the dynamical consequences of the ESQPT, which include the exponential growth of out-of-time-ordered correlators, followed by periodic revivals, and the slow evolution of the survival probability due to localization. These signatures of ESQPT are within reach for current superconducting circuits platforms and are of interest to experiments with cold atoms and ion traps.

Excited-state quantum phase transitions in the anharmonic Lipkin-Meshkov-Glick model: Static aspects

The basic Lipkin-Meshkov-Glick model displays a second-order ground-state quantum phase transition and an excited-state quantum phase transition (ESQPT). The inclusion of an anharmonic term in the Hamiltonian implies a second ESQPT of a different nature. We characterize this ESQPT using the mean field limit of the model. The alternative ESQPT, associated with the changes in the boundary of the finite Hilbert space of the system, can be properly described using the order parameter of the ground-state quantum phase transition, the energy gap between adjacent states, the participation ratio, and the quantum fidelity susceptibility.

Quantum transitions, ergodicity, and quantum scars in the coupled top model

We consider an interacting collective spin model known as coupled top (CT), exhibiting a rich variety of phenomena related to quantum transitions, ergodicity, and formation of quantum scars, discussed in our previous work [Mondal, Sinha, and Sinha, Phys. Rev. E 102, 020101(R) (2020)2470-004510.1103/PhysRevE.102.020101]. In this work, we present a detailed analysis of the different type of transitions in the CT model, and find their connection with the underlying collective spin dynamics. Apart from the quantum scarring phenomena, we also identify another source of deviation from ergodicity due to the presence of nonergodic multifractal states. The degree of ergodicity of the eigenstates across the energy band is quantified from the relative entanglement entropy as well as multifractal dimensions, which can be probed from nonequilibrium dynamics. Finally, we discuss the detection of nonergodic behavior and different types of quantum scars using "out-of-time-order correlators," which has relevance in the recent experiments.

Signatures of excited-state quantum phase transitions in quantum many-body systems: Phase space analysis

Using the Husimi quasiprobability distribution, we investigate the phase space signatures of excited-state quantum phase transitions (ESQPTs) in the Lipkin-Meshkov-Glick and coupled top models. We show that the ESQPT is evinced by the dynamics of the Husimi function, that exhibits a distinct time dependence in the different ESQPT phases. We also discuss how to identify the ESQPT signatures from the long-time averaged Husimi function and its associated marginal distributions. Moreover, from the calculated second moment and Wherl entropy of the long-time averaged Husimi function, we estimate the critical points of the ESQPT in both models, obtaining a good agreement with analytical (mean field) results. We provide a firm evidence that phase space methods are both a new probe for the detection and a valuable tool for the study of ESQPTs.

Constant of Motion Identifying Excited-State Quantum Phases

We propose that a broad class of excited-state quantum phase transitions (ESQPTs) gives rise to two different excited-state quantum phases. These phases are identified by means of an operator C[over ^], which is a constant of motion in only one of them. Hence, the ESQPT critical energy splits the spectrum into one phase where the equilibrium expectation values of physical observables crucially depend on this constant of motion and another phase where the energy is the only relevant thermodynamic magnitude. The trademark feature of this operator is that it has two different eigenvalues ±1, and, therefore, it acts as a discrete symmetry in the first of these two phases. This scenario is observed in systems with and without an additional discrete symmetry; in the first case, C[over ^] explains the change from degenerate doublets to nondegenerate eigenlevels upon crossing the critical line. We present stringent numerical evidence in the Rabi and Dicke models, suggesting that this result is exact in the thermodynamic limit, with finite-size corrections that decrease as a power law.

Interferometric Order Parameter for Excited-State Quantum Phase Transitions in Bose-Einstein Condensates

Excited-state quantum phase transitions 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 excited-state quantum phase transitions poses, however, a major challenge. We introduce a topological order parameter that distinguishes excited-state phases in a large class of mean-field models and can be accessed by interferometry in current experiments with spinor Bose-Einstein condensates. Our work opens a way for the experimental characterization of excited-state quantum phases in atomic many-body systems.

Work statistics and symmetry breaking in an excited-state quantum phase transition

We examine how the presence of an excited-state quantum phase transition manifests in the dynamics of a many-body system subject to a sudden quench. Focusing on the Lipkin-Meshkov-Glick model initialized in the ground state of the ferromagnetic phase, we demonstrate that the work probability distribution displays non-Gaussian behavior for quenches in the vicinity of the excited-state critical point. Furthermore, we show that the entropy of the diagonal ensemble is highly susceptible to critical regions, making it a robust and practical indicator of the associated spectral characteristics. We assess the role that symmetry breaking has on the ensuing dynamics, highlighting that its effect is only present for quenches beyond the critical point. Finally, we show that similar features persist when the system is initialized in an excited state and briefly explore the behavior for initial states in the paramagnetic phase.

Characterizing the Lipkin-Meshkov-Glick model excited-state quantum phase transition using dynamical and statistical properties of the diagonal entropy

Using the diagonal entropy, we analyze the dynamical signatures of the Lipkin-Meshkov-Glick model excited-state quantum phase transition (ESQPT). We first show that the time evolution of the diagonal entropy behaves as an efficient indicator of the presence of an ESQPT. We also compute the probability distribution of the diagonal entropy values over a certain time interval and we find that the resulting distribution provides a clear distinction between the different phases of ESQPT. Moreover, we observe that the probability distribution of the diagonal entropy at the ESQPT critical point has a universal form, well described by a beta distribution, and that a reliable detection of the ESQPT can be obtained from the diagonal entropy central moments.

Chaos and Quantum Scars in Bose-Josephson Junction Coupled to a Bosonic Mode

We consider a model describing Bose-Josephson junction (BJJ) coupled to a single bosonic mode exhibiting quantum phase transition (QPT). Onset of chaos above QPT is observed from semiclassical dynamics as well from spectral statistics. Based on entanglement entropy, we analyze the ergodic behavior of eigenstates with increasing energy density which also reveals the influence of dynamical steady state known as π-mode on it. We identify the imprint of unstable π-oscillation as many body quantum scar (MBQS), which leads to the deviation from ergodicity and quantify the degree of scarring. Persistence of phase coherence in nonequilibrium dynamics of such initial state corresponding to the π-mode is an observable signature of MBQS which has relevance in experiments on BJJ.

Chaos and quantum scars in a coupled top model

We consider a coupled top model describing two interacting large spins, which is studied semiclassically as well as quantum mechanically. This model exhibits a variety of interesting phenomena such as a quantum phase transition (QPT), a dynamical transition, and excited-state quantum phase transitions above a critical coupling strength. Both classical dynamics and entanglement entropy reveal ergodic behavior at the center of the energy density band for an intermediate range of coupling strength above QPT, where the level spacing distribution changes from Poissonian to Wigner-Dyson statistics. Interestingly, in this model we identify quantum scars as reminiscent of unstable collective dynamics even in the presence of an interaction. The statistical properties of such scarred states deviate from the ergodic limit corresponding to the random matrix theory and violate Berry's conjecture. In contrast to ergodic evolution, the oscillatory behavior in the dynamics of the unequal time commutator and survival probability is observed as the dynamical signature of a quantum scar, which can be relevant for its detection.

Complex Density of Continuum States in Resonant Quantum Tunneling

We introduce a complex-extended continuum level density and apply it to one-dimensional scattering problems involving tunneling through finite-range potentials. We show that the real part of the density is proportional to a real "time shift" of the transmitted particle, while the imaginary part reflects the imaginary time of an instantonlike tunneling trajectory. We confirm these assumptions for several potentials using the complex scaling method. In particular, we show that stationary points of the potentials give rise to specific singularities of both real and imaginary densities which represent close analogues of excited-state quantum phase transitions in bound systems.

Observation of Dynamical Quantum Phase Transitions with Correspondence in an Excited State Phase Diagram

Dynamical quantum phase transitions are closely related to equilibrium quantum phase transitions for ground states. Here, we report an experimental observation of a dynamical quantum phase transition in a spinor condensate with correspondence in an excited state phase diagram, instead of the ground state one. We observe that the quench dynamics exhibits a nonanalytical change with respect to a parameter in the final Hamiltonian in the absence of a corresponding phase transition for the ground state there. We make a connection between this singular point and a phase transition point for the highest energy level in a subspace with zero spin magnetization of a Hamiltonian. We further show the existence of dynamical phase transitions for finite magnetization corresponding to the phase transition of the highest energy level in the subspace with the same magnetization. Our results open a door for using dynamical phase transitions as a tool to probe physics at higher energy eigenlevels of many-body Hamiltonians.

Positive quantum Lyapunov exponents in experimental systems with a regular classical limit

Quantum chaos refers to signatures of classical chaos found in the quantum domain. Recently, it has become common to equate the exponential behavior of out-of-time order correlators (OTOCs) with quantum chaos. The quantum-classical correspondence between the OTOC exponential growth and chaos in the classical limit has indeed been corroborated theoretically for some systems and there are several projects to do the same experimentally. The Dicke model, in particular, which has a regular and a chaotic regime, is currently under intense investigation by experiments with trapped ions. We show, however, that for experimentally accessible parameters, OTOCs can grow exponentially also when the Dicke model is in the regular regime. The same holds for the Lipkin-Meshkov-Glick model, which is integrable and also experimentally realizable. The exponential behavior in these cases are due to unstable stationary points, not to chaos.

Collective performance of a finite-time quantum Otto cycle

We study the finite-time effects in a quantum Otto cycle where a collective spin system is used as the working fluid. Starting from a simple one-qubit system we analyze the transition to the limit cycle in the case of a finite-time thermalization. If the system consists of a large sample of independent qubits interacting coherently with the heat bath, then the super-radiant equilibration is observed. We show that this phenomenon can boost the power of the engine. Mutual interaction of qubits in the working fluid is modeled by the Lipkin-Meshkov-Glick Hamiltonian. We demonstrate that in this case the quantum phase transitions for the ground and excited states may have a strong negative effect on the performance of the machine. Conversely, by analyzing the work output we can distinguish between the operational regimes with and without a phase transition.

Experimental Observation of Classical Dynamical Monodromy

A Hamiltonian system is said to have nontrivial monodromy if its fundamental action-angle loops do not return to their initial topological state at the end of a closed circuit in angular momentum-energy space. This process has been predicted to have consequences which can be seen in dynamical systems, called dynamical monodromy. Using an apparatus consisting of a spherical pendulum subject to magnetic potentials and torques, we observe nontrivial monodromy by the associated topological change in the evolution of a loop of trajectories.

Analogies of the classical Euler top with a rotor to spin squeezing and quantum phase transitions in a generalized Lipkin-Meshkov-Glick model

We show that the classical model of Euler top (freely rotating, generally asymmetric rigid body), possibly supplemented with a rotor, corresponds to a generalized Lipkin-Meshkov-Glick (LMG) model describing phenomena of various branches of quantum physics. Classical effects such as free precession of a symmetric top, Feynman's wobbling plate, tennis-racket instability and the Dzhanibekov effect, attitude control of satellites by momentum wheels, or twisting somersault dynamics, have their counterparts in quantum effects that include spin squeezing by one-axis twisting and two-axis countertwisting, transitions between the Josephson and Rabi regimes of a Bose-Einstein condensate in a double-well potential, and other quantum critical phenomena. The parallels enable us to expand the range of explored quantum phase transitions in the generalized LMG model, as well as to present a classical analogy of the recently proposed LMG Floquet time crystal.

From thermal to excited-state quantum phase transition: The Dicke model

We study the thermodynamics of the full version of the Dicke model, including all the possible values of the total angular momentum j, with both microcanonical and canonical ensembles. We focus on both the excited-state quantum phase transition, appearing in the microcanonical description of the maximum angular momentum sector, j=N/2, and the thermal phase transition, which occurs when all the sectors are taken into account. We show that two different features characterize the full version of the Dicke model. If the system is in contact with a thermal bath and is described by means of the canonical ensemble, the parity symmetry becomes spontaneously broken at the critical temperature. In the microcanonical ensemble, and despite that all the logarithmic singularities which characterize the excited-state quantum phase transition are ruled out when all the j sectors are considered, there still exists a critical energy (or temperature) dividing the spectrum into two regions: one in which the parity symmetry can be broken, and another in which this symmetry is always well defined.

Defect in the Joint Spectrum of Hydrogen due to Monodromy

In addition to the well-known case of spherical coordinates, the Schrödinger equation of the hydrogen atom separates in three further coordinate systems. Separating in a particular coordinate system defines a system of three commuting operators. We show that the joint spectrum of the Hamilton operator, the z component of the angular momentum, and an operator involving the z component of the quantum Laplace-Runge-Lenz vector obtained from separation in prolate spheroidal coordinates has quantum monodromy for energies sufficiently close to the ionization threshold. The precise value of the energy above which monodromy is observed depends on the distance of the focus points of the spheroidal coordinates. The presence of monodromy means that one cannot globally assign quantum numbers to the joint spectrum. Whereas the principal quantum number n and the magnetic quantum number m correspond to the Bohr-Sommerfeld quantization of globally defined classical actions a third quantum number cannot be globally defined because the third action is globally multivalued.

Exceptional points near first- and second-order quantum phase transitions

We study the impact of quantum phase transitions (QPTs) on the distribution of exceptional points (EPs) of the Hamiltonian in the complex-extended parameter domain. Analyzing first- and second-order QPTs in the Lipkin-Meshkov-Glick model we find an exponentially and polynomially close approach of EPs to the respective critical point with increasing size of the system. If the critical Hamiltonian is subject to random perturbations of various kinds, the averaged distribution of EPs close to the critical point still carries decisive information on the QPT type. We therefore claim that properties of the EP distribution represent a parametrization-independent signature of criticality in quantum systems.

Probing the excited-state quantum phase transition through statistics of Loschmidt echo and quantum work

By analyzing the probability distributions of the Loschmidt echo (LE) and quantum work, we examine the nonequilibrium effects of a quantum many-body system, which exhibits an excited-state quantum phase transition (ESQPT). We find that depending on the value of the controlling parameter the distribution of the LE displays different patterns. At the critical point of the ESQPT, both the averaged LE and the averaged work show a cusplike shape. Furthermore, by employing the finite-size scaling analysis of the averaged work, we obtain the critical exponent of the ESQPT. Finally, we show that at the critical point of ESQPT the eigenstate is a highly localized state, further highlighting the influence of the ESQPT on the properties of the many-body system.

Excited-state quantum phase transitions in the two-spin elliptic Gaudin model

We study the integrability of the two-spin elliptic Gaudin model for arbitrary values of the Hamiltonian parameters. The limit of a very large spin coupled to a small one is well described by a semiclassical approximation with just one degree of freedom. Its spectrum is divided into bands that do not overlap if certain conditions are fulfilled. In spite of the fact that there are no quantum phase transitions in each of the band heads, the bands show excited-state quantum phase transitions separating a region in which the parity symmetry is broken from another region in which time-reversal symmetry is broken. We derive analytical expressions for the critical energies in the semiclassical approximation, and confirm the results by means of exact diagonalizations for large systems.

Classical chaos in atom-field systems

The relation between the onset of chaos and critical phenomena, like quantum phase transitions (QPTs) and excited-state quantum phase transitions (ESQPTs), is analyzed for atom-field systems. While it has been speculated that the onset of hard chaos is associated with ESQPTs based in the resonant case, the off-resonant cases, and a close look at the vicinity of the QPTs in resonance, show clearly that both phenomena, ESQPTs and chaos, respond to different mechanisms. The results are supported in a detailed numerical study of the dynamics of the semiclassical Hamiltonian of the Dicke model. The appearance of chaos is quantified calculating the largest Lyapunov exponent for a wide sample of initial conditions in the whole available phase space for a given energy. The percentage of the available phase space with chaotic trajectories is evaluated as a function of energy and coupling between the qubit and bosonic part, allowing us to obtain maps in the space of coupling and energy, where ergodic properties are observed in the model. Different sets of Hamiltonian parameters are considered, including resonant and off-resonant cases.

Entropy, chaos, and excited-state quantum phase transitions in the Dicke model

We study nonequilibrium processes in an isolated quantum system-the Dicke model-focusing on the role played by the transition from integrability to chaos and the presence of excited-state quantum phase transitions. We show that both diagonal and entanglement entropies are abruptly increased by the onset of chaos. Also, this increase ends in both cases just after the system crosses the critical energy of the excited-state quantum phase transition. The link between entropy production, the development of chaos, and the excited-state quantum phase transition is more clear for the entanglement entropy.

Delocalization and quantum chaos in atom-field systems

Employing efficient diagonalization techniques, we perform a detailed quantitative study of the regular and chaotic regions in phase space in the simplest nonintegrable atom-field system, the Dicke model. A close correlation between the classical Lyapunov exponents and the quantum Participation Ratio of coherent states on the eigenenergy basis is exhibited for different points in the phase space. It is also shown that the Participation Ratio scales linearly with the number of atoms in chaotic regions and with its square root in the regular ones.

Effective time-independent analysis for quantum kicked systems

We present a mapping of potentially chaotic time-dependent quantum kicked systems to an equivalent approximate effective time-independent scenario, whereby the system is rendered integrable. The time evolution is factorized into an initial kick, followed by an evolution dictated by a time-independent Hamiltonian and a final kick. This method is applied to the kicked top model. The effective time-independent Hamiltonian thus obtained does not suffer from spurious divergences encountered if the traditional Baker-Cambell-Hausdorff treatment is used. The quasienergy spectrum of the Floquet operator is found to be in excellent agreement with the energy levels of the effective Hamiltonian for a wide range of system parameters. The density of states for the effective system exhibits sharp peaklike features, pointing towards quantum criticality. The dynamics in the classical limit of the integrable effective Hamiltonian shows remarkable agreement with the nonintegrable map corresponding to the actual time-dependent system in the nonchaotic regime. This suggests that the effective Hamiltonian serves as a substitute for the actual system in the nonchaotic regime at both the quantum and classical level.

Transport as a sensitive indicator of quantum criticality

We consider bosonic transport through one-dimensional spin systems. Transport is induced by coupling the spin systems to bosonic reservoirs kept at different temperatures. In the limit of weak-coupling between spins and bosons we apply the quantum-optical master equation to calculate the energy transmitted from source to drain reservoirs. At large thermal bias, we find that the current for longitudinal transport becomes independent of the chain length and is also not drastically affected by the presence of disorder. In contrast, at small temperatures, the current scales inversely with the chain length and is further suppressed in the presence of disorder. We also find that the critical behaviour of the ground state is mapped to critical behaviour of the current--even in configurations with infinite thermal bias.

Quantum criticality and dynamical instability in the kicked-top model

We investigate precursors of critical behavior in the quasienergy spectrum due to the dynamical instability in the kicked top. Using a semiclassical approach, we analytically obtain a logarithmic divergence in the density of states, which is analogous to a continuous excited state quantum phase transition in undriven systems. We propose a protocol to observe the cusp behavior of the magnetization close to the critical quasienergy.

Excited-state quantum phase transitions in Dicke superradiance models

We derive analytical results for various quantities related to the excited-state quantum phase transitions in a class of Dicke superradiance models in the semiclassical limit. Based on a calculation of a partition sum restricted to Dicke states, we discuss the singular behavior of the derivative of the density of states and find observables such as the mean (atomic) inversion and the boson (photon) number and its fluctuations at arbitrary energies. Criticality depends on energy and a parameter that quantifies the relative weight of rotating versus counterrotating terms, and we find a close analogy to the logarithmic and jump-type nonanalyticities known from the Lipkin-Meshkov-Glick model.

Excited-state phase transition and onset of chaos in quantum optical models

We study the critical behavior of excited states and its relation to order and chaos in the Jaynes-Cummings and Dicke models of quantum optics. We show that both models exhibit a chain of excited-state quantum phase transitions demarcating the upper edge of the superradiant phase. For the Dicke model, the signatures of criticality in excited states are blurred by the onset of quantum chaos. We show that the emergence of quantum chaos is caused by the precursors of the excited-state quantum phase transition.

Classical analysis of quantum phase transitions in a bilayer model

In this Brief Report we extend the classical analysis performed on the schematic model proposed in [T. Moreira, G. Q. Pellegrino, J. G. Peixoto de Faria, M. C. Nemes, F. Camargo, and A. F. R. Toledo Piza, Phys. Rev. E 77, 051102 (2008)] concerning quantum phase transitions in a bilayer system. We show that appropriate integrations along the classical periodic orbits reproduce with excellent agreement both the quantum spectrum and the expected mean value for the number of excitons in the system, quantities which are directly related to the observed boson-fermion quantum phase transition.

Impact of quantum phase transitions on excited-level dynamics

The influence of quantum phase transitions on the evolution of excited levels in the critical parameter region is discussed. The analysis is performed for one- and two-dimensional systems with first- and second-order ground-state transitions. Examples include the cusp and nuclear collective Hamiltonians. Applications in systems of higher dimensions are possible.

Entanglement and classical instabilities: fingerprints of electron-hole-to-exciton phase transition in a simple model

We propose a schematic model to study the formation of excitons in bilayer electron systems. The phase transition is signalized both in the quantum and classical versions of the model. In the present contribution we show that not only the quantum ground state but also higher energy states, up to the energy of the corresponding classical separatrix orbit, "sense" the transition. We also show two types of one-to-one correspondences in this system: On the one hand, between the changes in the degree of entanglement for these low-lying quantum states and the changes in the density of energy levels; on the other hand, between the variation in the expected number of excitons for a given quantum state and the behavior of the corresponding classical orbit.

Coulomb analogy for non-Hermitian degeneracies near quantum phase transitions

Degeneracies near the real axis in a complex-extended parameter space of a Hermitian Hamiltonian are studied. We present a method to measure distributions of such degeneracies on the Riemann sheet of a selected level and apply it in classification of quantum phase transitions. The degeneracies are shown to behave similarly as complex zeros of a partition function.