The Ising model with a transverse field. II. Ground state properties

Monte Carlo Based Techniques for Quantum Magnets with Long-Range Interactions

Long-range interactions are relevant for a large variety of quantum systems in quantum optics and condensed matter physics. In particular, the control of quantum-optical platforms promises to gain deep insights into quantum-critical properties induced by the long-range nature of interactions. From a theoretical perspective, long-range interactions are notoriously complicated to treat. Here, we give an overview of recent advancements to investigate quantum magnets with long-range interactions focusing on two techniques based on Monte Carlo integration. First, the method of perturbative continuous unitary transformations where classical Monte Carlo integration is applied within the embedding scheme of white graphs. This linked-cluster expansion allows extracting high-order series expansions of energies and observables in the thermodynamic limit. Second, stochastic series expansion quantum Monte Carlo integration enables calculations on large finite systems. Finite-size scaling can then be used to determine the physical properties of the infinite system. In recent years, both techniques have been applied successfully to one- and two-dimensional quantum magnets involving long-range Ising, XY, and Heisenberg interactions on various bipartite and non-bipartite lattices. Here, we summarise the obtained quantum-critical properties including critical exponents for all these systems in a coherent way. Further, we review how long-range interactions are used to study quantum phase transitions above the upper critical dimension and the scaling techniques to extract these quantum critical properties from the numerical calculations.

Exactly solvable one-dimensional quantum models with gamma matrices

In this paper we write exactly solvable generalizations of one-dimensional quantum XY and Ising-like models by using 2^{d}-dimensional gamma matrices as the degrees of freedom on each site. We show that these models result in quadratic Fermionic Hamiltonians with Jordan-Wigner-like transformations. We illustrate the techniques using a specific case of four-dimensional gamma matrices and explore the quantum phase transitions present in the model.

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.

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.

Off-diagonal expansion quantum Monte Carlo

We propose a Monte Carlo algorithm designed to simulate quantum as well as classical systems at equilibrium, bridging the algorithmic gap between quantum and classical thermal simulation algorithms. The method is based on a decomposition of the quantum partition function that can be viewed as a series expansion about its classical part. We argue that the algorithm not only provides a theoretical advancement in the field of quantum Monte Carlo simulations, but is optimally suited to tackle quantum many-body systems that exhibit a range of behaviors from "fully quantum" to "fully classical," in contrast to many existing methods. We demonstrate the advantages, sometimes by orders of magnitude, of the technique by comparing it against existing state-of-the-art schemes such as path integral quantum Monte Carlo and stochastic series expansion. We also illustrate how our method allows for the unification of quantum and classical thermal parallel tempering techniques into a single algorithm and discuss its practical significance.

Test of quantum thermalization in the two-dimensional transverse-field Ising model

We study the quantum relaxation of the two-dimensional transverse-field Ising model after global quenches with a real-time variational Monte Carlo method and address the question whether this non-integrable, two-dimensional system thermalizes or not. We consider both interaction quenches in the paramagnetic phase and field quenches in the ferromagnetic phase and compare the time-averaged probability distributions of non-conserved quantities like magnetization and correlation functions to the thermal distributions according to the canonical Gibbs ensemble obtained with quantum Monte Carlo simulations at temperatures defined by the excess energy in the system. We find that the occurrence of thermalization crucially depends on the quench parameters: While after the interaction quenches in the paramagnetic phase thermalization can be observed, our results for the field quenches in the ferromagnetic phase show clear deviations from the thermal system. These deviations increase with the quench strength and become especially clear comparing the shape of the thermal and the time-averaged distributions, the latter ones indicating that the system does not completely lose the memory of its initial state even for strong quenches. We discuss our results with respect to a recently formulated theorem on generalized thermalization in quantum systems.

Eigenstate thermalization in the two-dimensional transverse field Ising model

We study the onset of eigenstate thermalization in the two-dimensional transverse field Ising model (2D-TFIM) in the square lattice. We consider two nonequivalent Hamiltonians: the ferromagnetic 2D-TFIM and the antiferromagnetic 2D-TFIM in the presence of a uniform longitudinal field. We use full exact diagonalization to examine the behavior of quantum chaos indicators and of the diagonal matrix elements of operators of interest in the eigenstates of the Hamiltonian. An analysis of finite size effects reveals that quantum chaos and eigenstate thermalization occur in those systems whenever the fields are nonvanishing and not too large.

Quantum-to-classical crossover near quantum critical point

A quantum phase transition (QPT) is an inherently dynamic phenomenon. However, while non-dissipative quantum dynamics is described in detail, the question, that is not thoroughly understood is how the omnipresent dissipative processes enter the critical dynamics near a quantum critical point (QCP). Here we report a general approach enabling inclusion of both adiabatic and dissipative processes into the critical dynamics on the same footing. We reveal three distinct critical modes, the adiabatic quantum mode (AQM), the dissipative classical mode [classical critical dynamics mode (CCDM)], and the dissipative quantum critical mode (DQCM). We find that as a result of the transition from the regime dominated by thermal fluctuations to that governed by the quantum ones, the system acquires effective dimension d + zΛ(T), where z is the dynamical exponent, and temperature-depending parameter Λ(T) ∈ [0, 1] decreases with the temperature such that Λ(T = 0) = 1 and Λ(T → ∞) = 0. Our findings lead to a unified picture of quantum critical phenomena including both dissipation- and dissipationless quantum dynamic effects and offer a quantitative description of the quantum-to-classical crossover.

Vertical density matrix algorithm: a higher-dimensional numerical renormalization scheme based on the tensor product state ansatz

We present a new algorithm to calculate the thermodynamic quantities of three-dimensional (3D) classical statistical systems, based on the ideas of the tensor product state and the density matrix renormalization group. We represent the maximum-eigenvalue eigenstate of the transfer matrix as the product of local tensors that are iteratively optimized by the use of the "vertical density matrix" formed by cutting the system along the transfer direction. This algorithm, which we call vertical density matrix algorithm (VDMA), is successfully applied to the 3D Ising model. Using the Suzuki-Trotter transformation, we can also apply the VDMA to 2D quantum systems, which we demonstrate for the 2D transverse field Ising model.