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We introduce the concept of ergodicity and explore its deviation caused by quantum scars in an isolated quantum system, employing a pedagogical approach based on a toy model. Quantum scars, originally identified as traces of classically unstable orbits in certain wavefunctions of chaotic systems, have recently regained interest for their role in non-ergodic dynamics, as they retain memory of their initial states. We elucidate these features of quantum scars within the same framework of this toy model. The integrable part of the model consists of two large spins, with a classical counterpart, which we combine with a random matrix to induce ergodic behavior. Scarred states can be selectively generated from the integrable spin Hamiltonian by protecting them from the ergodic states using a projector method. Deformed projectors mimic the 'quantum leakage' of scarred states, enabling tunable mixing with ergodic states and thereby controlling the degree of scarring. In this simple model, we investigate various properties of quantum scarring and shed light on different aspects of many-body quantum scars observed in more complex quantum systems. Notably, the underlying classicality can be revealed through the entanglement spectrum and the dynamics of 'out-of-time-ordered correlators'.

Ergodicity, a fundamental concept in statistical mechanics, is not yet a fully understood phenomena for closed quantum systems, particularly its connection with the underlying chaos. In this review, we consider a few examples of collective quantum systems to unveil the intricate relationship of ergodicity as well as its deviation due to quantum scarring phenomena with their classical counterpart. A comprehensive overview of classical and quantum chaos is provided, along with the tools essential for their detection. Furthermore, we survey recent theoretical and experimental advancements in the domain of ergodicity and its violations. This review aims to illuminate the classical perspective of quantum scarring phenomena in interacting quantum systems., Comment: 28 pages, 15 figures, Topical Review

We investigate the formation of paired states of bosons in an optical lattice, namely, pair superfluid (PSF) and pair supersolid (PSS) in the presence of pair hopping as well as the next nearest neighbor (NNN) interaction mimicking long-range forces. Both the zero and finite temperature phase diagrams are obtained using the cluster mean field theory, which includes the effect of correlations systematically. We also compute the low-energy excitations which capture the characteristic features of such paired states and their transitions. Apart from the gapless sound mode due to the PSF order, a gapped mode also appears in the PSF phase, similar to the Higgs mode of the usual atomic superfluid (ASF). The PSF to ASF transition exhibits an intriguing behavior due to the existence of a `tri-critical' point, where the nature of transition changes. As a consequence of the continuous PSF-ASF transition, the gapped mode of both the phases becomes gapless at the critical point. For sufficiently strong NNN interaction strength, a PSS phase appears with coexisting pair superfluidity and stripe density order. The softening of the roton mode as a precursor of density ordering and the appearance of a low-energy gapped mode serve as robust features related to the formation of the PSS phase. We also investigate the melting of PSF and PSS phases to normal fluid at finite temperatures, particularly the melting pathway of PSS which occurs in atleast two steps due to the coexisting orders. Finally, we discuss the possibility of emulating such exotic phases in the ongoing cold atom experiments., Comment: 14 pages, 15 figures

Recently created self-bound quantum droplets of binary Bose mixtures open intriguing possibilities for the study of impurity physics. We show that the properties of impurities embedded in quasi-one-dimensional droplets are determined by the interplay between back-action and quantum fluctuations. Due to such back-action, repulsive impurities may form a metastable quasi-bound state inside the droplet. In contrast, attractive impurities remain bound to the droplet, leading to the hybridization of droplet and impurity excitations, as well as to peculiar scattering resonances. Interestingly, impurity trapping may result solely from the effect of quantum fluctuations. These results may readily be probed experimentally by doping the currently available droplets of binary mixtures., Comment: 10 pages, 4 figures

We study the finite temperature phases and collective excitations of hardcore as well as softcore bosons on a square lattice with nearest and next nearest neighbor interactions, focusing on the formation of various types of supersolid (SS) phases and their stability under thermal fluctuations. The interplay between the on-site, nearest, and next nearest neighbor interactions leads to various density ordering and structural transitions, which we have plotted out. Thermodynamic properties and phase diagrams are obtained by cluster mean field theory at finite temperatures, which includes quantum effects systematically, and they are compared with the single-site mean field results. We investigate the melting process of the SS phase to normal fluid (NF), which can occur in at least two steps due to the presence of two competing orders in the SS. A tetra-critical point exists at finite temperature and exhibits intriguing behavior, which is analyzed for different regimes of interactions. The phase diagrams reveal the different pathways of the thermal transition of SSs to the NF phase, for different interaction regimes, which can be accessible by thermal quench protocols used in recent experiments. We show how the phases and the transitions between them can be identified from the characteristic features of the excitation spectrum. We analyze the appearance of a low-energy gapped mode apart from the gapless sound mode in the SS phase, which is analogous to the gapped mode recently studied for dipolar SS phases. Finally, we discuss the relevance of the results of the present work in the context of ongoing experiments on ultracold atomic gases and newly observed SS phases., Comment: 30 pages, 15 figures

We consider a Bose-Josephson junction (BJJ) formed by a binary mixture of ultracold atoms to investigate the manifestation of coherent collective dynamics on ergodicity and quantum scars, unfolding the connection between them. By tuning the inter- and intra-species interaction, we demonstrate a rich variety of Josephson dynamics and transitions between them, which plays a crucial role in controlling the overall ergodic behavior. The signature of underlying classicality is revealed from the entanglement spectrum, which also elucidates the formation of quantum scars of unstable steady states and of periodic orbits leading to athermal behavior. The degree of ergodicity across the energy band and scarring phenomena can be probed from the auto-correlation function as well from the phase fluctuation of the condensates, which has relevance in cold atom experiments.

The primary goal of this paper is to prove a near-martingale optional stopping theorem and establish solvability and large deviations for a class of anticipating linear stochastic differential equations. We prove the existence and uniqueness of solutions using two approaches: (1) Ayed-Kuo differential formula using an ansatz, and (2) a novel braiding technique by interpreting the integral in the Skorokhod sense. We establish a Freidlin-Wentzell type large deviations result for solution of such equations., Comment: 23 pages, 2 figures

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 [Phys. Rev. E 102, 020101(R) (2020)]. In this work, we present a detailed analysis of the different type of transitions in 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 non-ergodic 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 non-equilibrium dynamics. Finally, we discuss the detection of non-ergodic behavior and different types of quantum scars using `out-of-time-order correlators', which has relevance in the recent experiments.

Unlike classical system, understanding ergodicity from phase space mixing remains unclear for interacting quantum systems due to the absence of phase space trajectories. By considering an interacting spin model known as kicked coupled top, we elucidate the manifestation of phase space dynamics on local ergodic behavior of its quantum counterpart and quantum scarring phenomena. A transition to chaos occurs by increasing the kicking strength, and in the mixed phase space, the islands of regular motions within the chaotic sea clearly exhibit deviation from ergodicity, which we quantify from entanglement entropy and survival probability. Interestingly, the reminiscence of unstable orbits and fixed points can be identified as scars in quantum states, exhibiting athermal behavior and violation of Berry's conjecture for ergodic states. We also discuss the detection of quantum scars by a newly developed method of 'out-of-time-order correlators', which has experimental relevance.

We investigate the onset of chaos in a periodically kicked Dicke model (KDM), using the out-of-time-order correlator (OTOC) as a diagnostic tool, in both the oscillator and the spin subspaces. In the large spin limit, the classical Hamiltonian map is constructed, which allows us to investigate the corresponding phase space dynamics and to compute the Lyapunov exponent. We show that the growth rate of the OTOC for the canonically conjugate coordinates of the oscillator is able to capture the Lyapunov exponent in the chaotic regime. The onset of chaos is further investigated using the saturation value of the OTOC, that can serve as an alternate indicator of chaos in a generic interacting quantum system. This is also supported by a system independent effective random matrix model. We further identify the quantum scars in KDM and detect their dynamical signature by using the OTOC dynamics. The relevance of the present study in the context of ongoing cold atom experiments is also discussed.

Lead and tin-based chalcogenide semiconductors like PbTe or SnSe have long been known to exhibit an unusually low thermal conductivity that makes them very attractive thermoelectric materials. An apparently unrelated fact is that the excitonic bandgap in these materials increases with temperature, whereas for most semiconductors one observes the opposite trend. These two anomalous features are also seen in a very different class of photovoltaic materials, namely the halide-perovskites such as CsPbBr3. It has been previously proposed that emphanisis, a local symmetry-breaking phenomenon, is the one common origin of these unusual features. Discovered a decade ago, emphanisis is the name given to the observed displacement of the lead or the tin ions from their cubic symmetry ground state to a locally distorted phase at high temperature. This phenomenon has been puzzling because it is unusual for the high-temperature state to be of a lower symmetry than the degenerate ground state. Motivated by the celebrated vibration-inversion resonance of the ammonia molecule, we propose a quantum tunneling-based model for emphanisis where decoherence is responsible for the local symmetry breaking with increasing temperature. From the analytic expression of the temperature dependence of the tunnel splitting (which serves as an order parameter), we provide three-parameter fitting formulae which capture the observed temperature dependence of the ionic displacements as well as the anomalous increase of the excitonic bandgap in all the relevant materials., Comment: The following article has been submitted to Applied Physics Letters

We consider a coupled top model describing two interacting large spins, which is studied semiclassically as well as quantum mechanically. This model exhibits variety of interesting phenomena such as quantum phase transition (QPT), dynamical transition and excited state quantum phase transitions above a critical coupling strength. Both classical dynamics and entanglement entropy reveals ergodic behavior at the center of 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 reminiscence of unstable collective dynamics even in presence of interaction. Statistical properties of such scarred states deviate from ergodic limit corresponding to random matrix theory and violate Berry's conjecture. In contrast to ergodic evolution, oscillatory behavior in dynamics of unequal time commutator and survival probability is observed as dynamical signature of quantum scar, which can be relevant for its detection.

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 $\pi$-mode on it. We identify the imprint of unstable $\pi$-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 $\pi$-mode is an observable signature of MBQS which has relevance in experiments on BJJ.

We investigate the effect of dissipation in a Bose-Josephson junction (BJJ) coupled to bath of bosons at two sites. Apart from the dynamical transition due to repulsive interactions, the BJJ undergoes a quantum phase transition by increasing the coupling strength with the bath modes. We analyze this system by mapping to an equivalent spin model coupled to the bosonic modes. The excitation energies and fluctuation of number imbalance are obtained within Holstein-Primakoff approximation, which exhibit vanishing of energy gap and enhanced quantum fluctuations at the critical coupling. We study the dynamics of BJJ using time dependent variational method and analyze stability of different types of steady states. As a special case we study in details the phase space dynamics of BJJ coupled to a single mode, which reveals diffusive and incoherent behaviour with increasing coupling to the bath mode. The dynamical steady states corresponding to the Pi-oscillation and self-trapped state become unstable in the region where their oscillation frequencies are in resonance with the bath modes. We investigate the time evolution of number imbalance and relative phase in presence of Ohmic bath with Gaussian noise to incorporate thermal fluctuations. Apart from damping of Josephson oscillations and transition to symmetry broken state for strong coupling we observe decay of Pi-oscillation and self-trapped state to the ground state as a result of dissipation. Variation of phase fluctuation with temperature of the bath shows similar behaviour as observed in experiment. Finally we discuss the experimental setup to study the observable effects of dissipation in BJJ.