Geometric Entanglement and Quantum Phase Transition in Generalized Cluster-XY models

In this work, we investigate quantum phase transition (QPT) in a generic family of spin chains using the ground-state energy, the energy gap, and the geometric measure of entanglement (GE). In many of prior works, GE per site was used. Here, we also consider GE per block with each block size being two. This can be regarded as a coarse grain of GE per site. We introduce a useful parameterization for the family of spin chains that includes the XY models with n-site interaction, the GHZ-cluster model and a cluster-antiferromagnetic model, the last of which exhibits QPT between a symmetry-protected topological (SPT) phase and a symmetry-breaking antiferromagnetic phase. As the models are exactly solvable, their ground-state wavefunctions can be obtained and thus their GE can be studied. It turns out that the overlap of the ground states with translationally invariant product states can be exactly calculated and hence the GE can be obtained via further parameter optimization. The QPTs exhibited in these models are detected by the energy gap and singular behavior of geometric entanglement. In particular, the XzY model exhibits transitions from the nontrivial SPT phase to a trivial paramagnetic phase. Moreover, the halfway XY model exhibits a first-order transition across the Barouch-McCoy circle, on which it was only a crossover in the standard XY model.Comment: 29 pages, 12 figure

Weak ergodicity breaking transition in randomly constrained model

Experiments in Rydberg atoms have recently found unusually slow decay from a small number of special initial states. We investigate the robustness of such long-lived states (LLS) by studying an ensemble of locally constrained random systems with tunable range $\mu$. Upon varying $\mu$, we find a transition between a thermal and a weakly non-ergodic (supporting a finite number of LLS) phases. Furthermore, we demonstrate that the LLS observed in the experiments disappear upon the addition of small perturbations so that the transition reported here is distinct from known ones. We then show that the LLS dynamics explores only part of the accessible Hilbert space, thus corresponding to localisation in Hilbert space.Comment: 5 pages, 3 figures + Supp. Ma

AdS/CFT Correspondence with a 3D Black Hole Simulator

The AdS/CFT correspondence has been insightful for high-energy and condensed matter physics alike. An application of this correspondence is the duality between the entanglement entropy of Anti-de Sitter (AdS) black holes and lower-dimensional conformal field theories (CFT). To explicitly demonstrate this correspondence we simulate the effect a 3D black hole geometry has on Dirac fields by employing a square lattice of fermions with inhomogeneous tunnelling couplings. Simulating a 3D BTZ black hole horizon, we numerically obtain an area law behaviour that is in agreement with the corresponding 2D CFT with a central charge that depends on the cosmological constant of the AdS space. A systematic numerical investigation of various 3D black hole profiles suggests that all 3D black holes give an entropic behaviour that can be represented by the same CFT.Comment: 6 pages, 4 figure

Arresting classical many-body chaos by kinetic constraints

We investigate the effect of kinetic constraints on classical many-body chaos in a translationally-invariant Heisenberg spin chain using a classical counterpart of the out-of-time-ordered correlator (OTOC). The strength of the constraint drives a 'dynamical phase transition' separating a delocalised phase, where the classical OTOC propagates ballistically, from a localised phase, where the OTOC does not propagate at all and the entire system freezes. This is unexpected given that all spins configurations are dynamically connected to each other. We show that localisation arises due to the dynamical formation of frozen islands, contiguous segments of spins immobile due to the constraints, dominating over the melting of such islands.Comment: 6 pages, 4 figures + Supplementary Material (2 pages, 2 figures), version published in Phys. Rev. Let

AdS/CFT correspondence with a three-dimensional black hole simulator

One of the key applications of AdS/CFT correspondence is the duality it dictates between the entanglement entropy of anti–de Sitter (AdS) black holes and lower-dimensional conformal field theories (CFTs). Here we employ a square lattice of fermions with inhomogeneous tunneling couplings that simulate the effect rotationally symmetric three-dimensional (3D) black holes have on Dirac fields. When applied to 3D Banados-Teitelboim-Zanelli (BTZ) black holes we identify the parametric regime where the theoretically predicted two-dimensional CFT faithfully describes the black hole entanglement entropy. With the help of the universal simulator, we further demonstrate that a large family of 3D black holes exhibit the same ground-state entanglement entropy behavior as the BTZ black hole. The simplicity of our simulator enables direct numerical investigation of a wide variety of 3D black holes and the possibility to experimentally realize it with optical lattice technology

Determination of Dynamical Quantum Phase Transitions in Strongly Correlated Many-Body Systems Using Loschmidt Cumulants

Dynamical phase transitions extend the notion of criticality to nonstationary settings and are characterized by sudden changes in the macroscopic properties of time-evolving quantum systems. Investigations of dynamical phase transitions combine aspects of symmetry, topology, and nonequilibrium physics; however, progress has been hindered by the notorious difficulties of predicting the time evolution of large, interacting quantum systems. Here, we tackle this outstanding problem by determining the critical times of interacting many-body systems after a quench using Loschmidt cumulants. Specifically, we investigate dynamical topological phase transitions in the interacting Kitaev chain and in the spin-1 Heisenberg chain. To this end, we map out the thermodynamic lines of complex times, where the Loschmidt amplitude vanishes, and identify the intersections with the imaginary axis, which yield the real critical times after a quench. For the Kitaev chain, we can accurately predict how the critical behavior is affected by strong interactions, which gradually shift the time at which a dynamical phase transition occurs. We also discuss the experimental perspectives of predicting the first critical time of a quantum many-body system by measuring the energy fluctuations in the initial state, and we describe the prospects of implementing our method on a near-term quantum computer with a limited number of qubits. Our work demonstrates that Loschmidt cumulants are a powerful tool to unravel the far-from-equilibrium dynamics of strongly correlated many-body systems, and our approach can immediately be applied in higher dimensions.Dynamical phase transitions extend the notion of criticality to nonstationary settings and are characterized by sudden changes in the macroscopic properties of time-evolving quantum systems. Investigations of dynamical phase transitions combine aspects of symmetry, topology, and nonequilibrium physics; however, progress has been hindered by the notorious difficulties of predicting the time evolution of large, interacting quantum systems. Here, we tackle this outstanding problem by determining the critical times of interacting many-body systems after a quench using Loschmidt cumulants. Specifically, we investigate dynamical topological phase transitions in the interacting Kitaev chain and in the spin-1 Heisenberg chain. To this end, we map out the thermodynamic lines of complex times, where the Loschmidt amplitude vanishes, and identify the intersections with the imaginary axis, which yield the real critical times after a quench. For the Kitaev chain, we can accurately predict how the critical behavior is affected by strong interactions, which gradually shift the time at which a dynamical phase transition occurs. We also discuss the experimental perspectives of predicting the first critical time of a quantum many-body system by measuring the energy fluctuations in the initial state, and we describe the prospects of implementing our method on a near-term quantum computer with a limited number of qubits. Our work demonstrates that Loschmidt cumulants are a powerful tool to unravel the far-from-equilibrium dynamics of strongly correlated many-body systems, and our approach can immediately be applied in higher dimensions.Peer reviewe

USING SOME POME FRUIT TREES IN LANDSCAPE DESIGNS

Landscape; when viewed from a point of view, natural and cultural beings that are able to enter into the frame of view are brought together to form a fountain. The materials that make up the live decor of the areas consist of especially the large trees of the plant kingdom, shrubs, undergrowths, ivies, single annual, biennial or perennial herbaceous plants, that is, roots consist of onion, lumpy or rhizomaceous herbaceous plants, grass plants and water plants which can be kept on the ground continuously. Among these, wild and cultured forms of soft-seeded fruits constitute an important place. In this study, the functional and visual use of wild plants such as wild pear, pear, apple, quince and their wild forms in different landscape designs have been investigated. In plantation studies, plants can be used in esthetic, functional or both ways to be more effective. It can also be growth for economic reasons. Economically cultivated species are particularly high economic values. However, they are often used for esthetic purposes outside of commercial assets, such as in other fruit trees. For this reason, the most common uses are to take advantage of both fruit and to benefit from the visual effect of flowers and fruit

Lee-Yang theory and large deviation statistics of interacting many-body systems

The collective behavior of large numbers of interacting particles may give rise to a phase transition. A continuing challenge is to identify the underlying principles of this phenomenon emerging in many important systems in nature and characterize the critical behavior of interacting many-body systems. In this thesis, we present a theoretical and methodological framework for predicting the phase properties of a macroscopic system based on the behavior of just a few of its constituents. To this end, we devise a direct pathway from the detection of partition function zeros by measuring or simulating fluctuating observables in systems of finite size to the characterization of criticality and large deviation statistics in interacting many-body systems. Our approach combines ideas and concepts from the finite-size scaling analysis with the Lee-Yang formalism and theories of high cumulants and large deviations, and it can be applied in a wide range of critical systems from physics, chemistry, and biology, both in theory and experiment. The thesis consists of four publications. In publications I and II, we report a novel method that makes it possible to extract the partition function zeros in interacting many-body systems of finite size solely from the fluctuations of thermodynamic observables without any prior knowledge of the partition function. To illustrate the feasibility of our approach, we use the Fisher zeros and their relation to the energy fluctuations as a tool for probing criticality in Ising models in two and three dimensions. In particular, we suggest an alternative way of extracting the universal critical exponents from measured fluctuations in finite-size systems away from the phase transition. In publications III and IV, we develop a scaling analysis of the partition function zeros to investigate the criticality in higher dimensions where the hyperscaling breaks down. We also show that even if the system does not exhibit a sharp phase transition, the partition function zeros carry important information about the large-deviation statistics of the system and its symmetry properties. To this end, we determine the rare magnetization fluctuations from the asymptotic behavior of the Lee-Yang zeros, i.e., from the Yang-Lee edge singularities. This finding may constitute a profound connection between Lee-Yang theory and large-deviation statistics

Determination of universal critical exponents using Lee-Yang theory

Lee-Yang zeros are points in the complex plane of an external control parameter at which the partition function vanishes for a many-body system of finite size. In the thermodynamic limit, the Lee-Yang zeros approach the critical value on the real axis, where a phase transition occurs. Partition function zeros have for many years been considered a purely theoretical concept; however, the situation is changing now as Lee-Yang zeros have been determined in several recent experiments. Motivated by these developments, we here devise a direct pathway from measurements of partition function zeros to the determination of critical points and universal critical exponents of continuous phase transitions. To illustrate the feasibility of our approach, we extract the critical exponents of the Ising model in two and three dimensions from the fluctuations of the total energy and the magnetization in lattices of finite size. Importantly, the critical exponents can be determined even if the system is away from the phase transition. Moreover, in contrast to standard methods based on Binder cumulants, it is not necessary to drive the system across the phase transition. As such, our method provides an intriguing perspective for investigations of phase transitions that may be hard to reach experimentally, for instance at very low temperatures or at very high pressures.Peer reviewe

Lee-Yang theory of the Curie-Weiss model and its rare fluctuations

Phase transitions are typically accompanied by nonanalytic behaviors of the free energy, which can be explained by considering the zeros of the partition function in the complex plane of the control parameter and their approach to the critical value on the real axis as the system size is increased. Recent experiments have shown that partition function zeros are not just a theoretical concept. They can also be determined experimentally by measuring fluctuations of thermodynamic observables in systems of finite size. Motivated by this progress, we investigate here the partition function zeros for the Curie-Weiss model of spontaneous magnetization using our recently established cumulant method. Specifically, we extract the leading Fisher and Lee-Yang zeros of the Curie-Weiss model from the fluctuations of the energy and the magnetization in systems of finite size. We develop a finite-size scaling analysis of the partition function zeros, which is valid for mean-field models and which allows us to extract both the critical values of the control parameters and the critical exponents, even for small systems that are away from criticality. We also show that the Lee-Yang zeros carry important information about the rare magnetic fluctuations as they allow us to predict many essential features of the large-deviation statistics of the magnetization. This finding may constitute a profound connection between Lee-Yang theory and large-deviation statistics.Peer reviewe