The presence of strong-interactions and entanglement in a quantum many-body system can lead to several exotic states of matter. Quantum Spin liquids are one such set of exotic states that have attracted significant interest due to their interesting properties and possible technological applications. The rapid progress in cold atom experiment setups, particularly the Rydberg atom simulators, has opened up new avenues to study these spin liquids in one and two-dimensions. This calls for advanced theoretical techniques to investigate the non-equilibrium properties of isolated pure quantum states. This thesis aims to develop numerical algorithms to study the non-equilibrium properties of spin liquids on a square lattice following a quantum quench using the framework of tensor networks. A variational form of Projected Entangled Pair State (PEPS) ansatz is used to describe the spin liquids. A new symmetric form of the Corner Transfer Matrix Renormalization group (CTMRG), the algorithm necessary to construct the reduced density matrix of the wavefunction in the thermodynamic limit for a complex-valued PEPS wavefunction, is described. To study the quench dynamics, two algorithms called simple-update and full update, along with the necessary modifications to adapt them to study spin liquid wavefunctions, are explained. The tools are then used to study the non-equilibrium dynamics of the resonating valence bond (RVB) state with nearest-neighbor (NN) valence bond configurations, a spin-liquid state that is known to exhibit critical dimer-dimer correlations, within the PEPS manifold.
The presence of strong-interactions and entanglement in a quantum many-body system can lead to several exotic states of matter. Quantum Spin liquids are one such set of exotic states that have attracted significant interest due to their interesting properties and possible technological applications. The rapid progress in cold atom experiment setups, particularly the Rydberg atom simulators, has opened up new avenues to study these spin liquids in one and two-dimensions. This calls for advanced theoretical techniques to investigate the non-equilibrium properties of isolated pure quantum states. This thesis aims to develop numerical algorithms to study the non-equilibrium properties of spin liquids on a square lattice following a quantum quench using the framework of tensor networks. A variational form of Projected Entangled Pair State (PEPS) ansatz is used to describe the spin liquids. A new symmetric form of the Corner Transfer Matrix Renormalization group (CTMRG), the algorithm necessary to construct the reduced density matrix of the wavefunction in the thermodynamic limit for a complex-valued PEPS wavefunction, is described. To study the quench dynamics, two algorithms called simple-update and full update, along with the necessary modifications to adapt them to study spin liquid wavefunctions, are explained. The tools are then used to study the non-equilibrium dynamics of the resonating valence bond (RVB) state with nearest-neighbor (NN) valence bond configurations, a spin-liquid state that is known to exhibit critical dimer-dimer correlations, within the PEPS manifold.
Within the Projected Entangled Pair State (PEPS) tensor network formalism, a simple update (SU) method has been used to investigate the time evolution of a two-dimensional U(1) critical spin-1/2 spin liquid under Hamiltonian quench [Phys. Rev. B 106, 195132 (2022)]. Here we introduce two different variational frameworks to describe the time dynamics of SU(2)-symmetric translationally-invariant PEPS, aiming to improve the accuracy. In one approach, after using a Trotter-Suzuki decomposition of the time evolution operator in term of two-site elementary gates, one considers a single bond embedded in an environment approximated by a Corner Transfer Matrix Renormalization Group (CTMRG). A variational update of the two tensors on the bond is performed under the application of the elementary gate and then, after symmetrization of the site tensors, the environment is updated. In the second approach, a cluster optimization is performed on a finite (periodic) cluster, maximizing the overlap of the exact time-evolved state with a symmetric finite-size PEPS ansatz. Observables are then computed on the infinite lattice contracting the infinite-PEPS (iPEPS) by CTMRG. We show that the variational schemes outperform the SU method and remain accurate over a significant time interval before hitting the entanglement barrier. Studying the spectrum of the transfer matrix, we find that the asymptotic correlations are very well preserved under time evolution, including the critical nature of the singlet correlations, as expected from the Lieb-Robinson (LR) bound theorem. Consistently, the system (asymptotic) boundary is found to be described by the same Conformal Field Theory of central charge c = 1 c = 1 during time evolution. We also compute the time-evolution of the short distance spin-spin correlations and estimate the LR velocity.
Strongly correlated quantum systems are one of the most interesting and challenging phenomena in modern condensed matter physics. Unfortunately, the theoretical access even to their low energy properties is limited by the exponential growth of the Hilbert space in a many-body setting. To mitigate this computational barrier, tensor network methods are one of the most competitive approaches to classically simulate complex quantum systems. In this thesis we will discuss recent developments in the field of tensor networks to efficiently and accurately simulate two-dimensional quantum many-body states in the thermodynamic limit. Hereby, we introduce the variational update globally optimizing the wave-function, encoded as tensor network in the form of an infinite projected entangled pair state (iPEPS), in order to obtain ground states of many-body Hamiltonian models. We focus in the discussion on the well-established method to approximately contract this two-dimensional structure, the corner transfer matrix renormalization group (CTMRG), and discuss its native application on square lattice and triangular lattice based models. Furthermore, we report the development of a refined numerical algorithm, the split-CTMRG, to significantly reduce its computational complexity, pushing the limits of feasible simulations within modern hardware capabilities. Finally, the method advancements are applied to a currently widely studied model of frustrated magnetism, the Heisenberg maple-leaf antiferromagnet. We are able to find a large phase of vanishing magnetization in contradiction to previous works and, to the best of our knowledge, the lowest reported variational ground state energy at the isotropic point of the model. As an outlook, we describe a potential adaption of the previously developed methods to variationally simulate two-dimensional many-body thermal states.
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
Emergent phenomena of interacting quantum many-body systems arguably pose some of the greatest challenges to modern theoretical physics. Many of these systems are characterized by drastic changes in their properties as they are driven into a regime where quantum effects and interactions become relevant. Examples are the fractional quantum Hall effect and high-temperature superconductors, to name just two. New theoretical tools are necessary to study such systems, as even simple model Hamiltonians are not fully understood after decades of intense research. Numerical simulations are emerging as an indispensable tool alongside more traditional analytic approaches and have seen immense progress in recent years. In this thesis, we study tensor network states, a new class of numerical methods that have been developed in recent years drawing on insights from quantum information theory. Combining renormalization group methods with knowledge about the entanglement structure of ground states of quantum systems, tensor network states aim to efficiently describe ground states of strongly correlated quantum systems. Center stage is taken by the projected entangled-pair states. This ansatz for two-dimensional systems is discussed in detail and its capabilities are assessed by comparison to established methods and by exploring a frustrated system. We then develop a general formalism to exploit Abelian symmetries in a tensor network algorithm and demonstrate its application to projected entangled-pair states. A large part of this thesis is devoted to the discussion of a class of supersymmetric models for interacting lattice fermions. After introducing the concept of supersymmetry in quantum mechanics and an extensive overview of conformal field theory and its relation to numerical simulations, we study the critical theory of the supersymmetric model on the square ladder. We use two tensor network state algorithms, the density-matrix renormalization group and multi-scale entanglement renormalization along with exact diagonalization to explore this challenging system. In the final two chapters, we discuss two applications of tensor network states, namely simulations of the SU(3) Heisenberg model in two dimensions using two different tensor network state algorithms with the goal of illuminating the nature of its ground state, and the indistinguishability, a measure to detect phase transitions and classify wave functions which is particularly suitable for calculations based on tensor network states.
Implementation of the hard-core-dimer Hilbert space in cold-Rydberg-atom simulators opens a route of investigating real-time dynamics of dimer liquids under Hamiltonian quench. Here, we consider an initial resonating valence bond state on the square lattice realizing a critical Coulomb phase with algebraic and dipolar correlations. Using its representation as a special point of a broad manifold of SU(2)-symmetric, translationally invariant, projected entangled pair states (PEPS), we compute its nonequilibrium dynamics upon turning on intersite Heisenberg interactions. We show that projecting the time evolution onto the PEPS manifold remains accurate at small timescales. We also find that the state evolves within a PEPS submanifold characterized by a U(1) gauge symmetry, suggesting that the Coulomb phase is stable under such unitary evolution.
The field of tensor networks, kicked off in 1992 by Steve White's invention of the spectacularly successful density matrix renormalization group (DMRG) algorithm, has exploded in popularity in recent years. Tensor networks are poised to play a role in helping us solve some of the greatest open physics problems of our time, such as understanding the nature of high-temperature superconductivity and illuminating a theory of quantum gravity. DMRG and extensions based on a class of variational states known as tensor network states have been indispensable tools in helping us understand both numerically and theoretically the properties of complicated classical and quantum many-body systems. However, practical challenges to these techniques still remain, and algorithmic developments are needed before tensor network algorithms can be applied to more physics problems. In this thesis we present a variety of recent advancements to tensor network algorithms. First we describe a DMRG-like algorithm for noninteracting fermions. Noninteracting fermions, naturally being gapless and therefore having high levels of entanglement, are actually a challenging setting for standard DMRG algorithms, and we believe this new algorithm can help with tensor network calculations in that setting. Next we explain a new algorithm called the variational uniform matrix product state (VUMPS) algorithm that is a DMRG-like algorithm that works directly in the thermodynamic limit, improving upon currently available MPS-based methods for studying infinite 1D and quasi-1D quantum many-body systems. Finally, we describe a variety of improvements to algorithms for contracting 2D tensor networks, a common problem in tensor network algorithms, for example for studying 2D classical statistical mechanics problems and 2D quantum many-body problems with projected entangled pair states (PEPS). One is a new variant of the corner transfer matrix renormalization group (CTMRG) algorithm of Nishino and Okunishi that improves the numerical stability for contracting asymmetric two-dimensional tensor networks compared to the most commonly used method. Another is the application of the VUMPS algorithm to contracting 2D tensor networks. The last is a new alternative to CTMRG, where the tensors are solved for with eigenvalue equations instead of a power method, which we call the fixed point corner method (FPCM). We present results showing the transfer matrix VUMPS algorithm and FPCM significantly improve upon the convergence time of CTMRG. We expect these algorithms will play an important role in expanding the set of 2D classical and 2D quantum many-body problems that can be addressed with tensor networks.