We introduce a tensor-network approach to calculate the statistics of work done on one-dimensional quantum lattice systems initially prepared in thermal equilibrium states. In this approach, the dynamics is simulated with time-evolving block decimation (TEBD), and the initial thermal equilibrium state is prepared either directly with TEBD or with minimally entangled typical thermal states, which generates a set of typical states representing the Gibbs canonical ensemble. As an illustrative example, we apply this approach to the Ising chain in mixed transverse and longitudinal fields. Under a prescribed protocol, the moment generating function for work distribution can be calculated, from which the quantum Jarzynski equality and the generalized quantum work relation involving a functional of an arbitrary observable are tested.
We introduce a tensor-network approach to calculate the statistics of work done on one-dimensional quantum lattice systems initially prepared in thermal equilibrium states. In this approach, the dynamics is simulated with time-evolving block decimation (TEBD), and the initial thermal equilibrium state is prepared either directly with TEBD or with minimally entangled typical thermal states, which generates a set of typical states representing the Gibbs canonical ensemble. As an illustrative example, we apply this approach to the Ising chain in mixed transverse and longitudinal fields. Under a prescribed protocol, the moment generating function for work distribution can be calculated, from which the quantum Jarzynski equality and the generalized quantum work relation involving a functional of an arbitrary observable are tested.
We introduce a numerical approach to calculate the statistics of work done on 1D quantum lattice systems initially prepared in thermal equilibrium states. This approach is based on two tensor-network techniques: Time Evolving Block Decimation (TEBD) and Minimally Entangled Typical Thermal States (METTS). The former is an efficient algorithm used to simulate the dynamics of 1D quantum lattice systems, while the latter a finite-temperature algorithm for generating a set of typical states representing the Gibbs canonical ensemble. As an illustrative example, we apply this approach to the Ising chain in mixed transverse and longitudinal fields. Under an arbitrary protocol, the moment generating function of the work can be obtained, from which the work moments are numerically calculated and the quantum Jarzynski equality can be tested. Moreover, the numerical approach is also adapted to test the universal quantum work relation involving a functional of an arbitrary observable.
Tensor networks are adopted to calculate the responses for one-dimensional quantum spin systems that are initially in thermal equilibrium. The Ising chain in mixed transverse and longitudinal fields is used as the benchmarking system. The linear and second-order responses of the magnetization in z-direction induced by the time-dependent force conjugated with the magnetization in x-direction are calculated. In addition, the magnetization in z-direction is also exactly calculated in response to this excitation. As expected, the first two responses are shown to be excellent corrections to the equilibrium magnetization in z-direction when the excitation is weak. This result represents an illustrative example of the response theory for nontrivial quantum many-body systems.
We introduce a general numerical method to compute the dynamics and multitime correlations of chains of quantum systems, where each system may couple strongly to a structured environment. The method combines the process tensor formalism for general (possibly non-Markovian) open quantum systems with time-evolving block decimation for one-dimensional chains. It systematically reduces the numerical complexity originating from system-environment correlations before integrating them into the full many-body problem, making a wide range of applications numerically feasible. We illustrate the power of this method by studying two examples. First, we study the thermalization of individual spins of a short XYZ Heisenberg chain with strongly coupled thermal leads. Our results confirm the complete thermalization of the chain when coupled to a single bath, and they reveal distinct effective temperatures in low-, mid-, and high-frequency regimes when the chain is placed between a hot and a cold bath. Second, we study the dynamics of diffusion in a longer XY chain, when each site couples to its own bath.
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.
The infinite time-evolving block decimation algorithm [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)] allows to simulate unitary evolution and to compute the ground state of one-dimensional (1D) quantum lattice systems in the thermodynamic limit. Here we extend the algorithm to tackle a much broader class of problems, namely, the simulation of arbitrary one-dimensional evolution operators that can be expressed as a (translationally invariant) tensor network. Relatedly, we also address the problem of finding the dominant eigenvalue and eigenvector of a one-dimensional transfer matrix that can be expressed in the same way. New applications include the simulation, in the thermodynamic limit, of open (i.e., master equation) dynamics and thermal states in 1D quantum systems, as well as calculations with partition functions in two-dimensional (2D) classical systems, on which we elaborate. The present extension of the algorithm also plays a prominent role in the infinite projected entangled-pair states approach to infinite 2D quantum lattice systems.
[eng] Quantum many-body dynamics is a vast field of physics at the intersection of statistical mechanics, condensed matter, quantum information, and computational physics. Understanding the time evolution of observable correlations—such as those involved in material responses—is both a milestone for technological advancement and a corner-stone for generating fundamental knowledge. At the intersection of emergent collective behavior in many-body systems, quantum physics, and information theory lies the classical simulation of out-of-equilibrium phenomena. Without numerical tools, progress in disciplines that require heavy computations with high-dimensional vectors and matrices would be limited to purely analytical predictions and constrained by the available expertise and resources needed to build benchmarking experiments. The goal of this thesis is to advance the classical simulation of quantum many-body systems using Tensor Networks. This technique has proven fruitful for predicting equilibrium ground states and thermal properties of one-dimensional systems, yet it still faces challenges when applied to out-of-equilibrium scenarios. In brief, the dynamics of experimentally accessible expectation values can be represented as a 2D Tensor Network, which must be contracted using iterative methods such as Time-Evolving Block Decimation. However, the memory and computational time required for these calculations—which reflect the methods’ complexity—are known to grow exponentially with the simulated physical time in generic systems. This limitation is often referred to in the literature as the entanglement barrier. This manuscript tackles the problem through the three inequivalent ways in which the 2D Tensor Network can be contracted. In Chapter 3, we study how Schrödinger-picture contraction from the state edge of the network can be modified to improve the convergence of iterative contraction and reduce truncation errors in small systems. To this end, we explore physical decoherence in optimized bases and artificial generalized decoherence, finding that both alternatives show promise for new protocols that overcome the entanglement barrier. In Chapter 4, we explore Heisenberg-picture contraction from the observable edge, combining the concept of generalized decoherence with Pauli Weight truncation. By thoroughly characterizing the Pauli weight requirements to accurately describe the time evolution of local expectation values from initial product states, we conclude that truncating off-diagonal Pauli Weights is an effective scheme for simulating long-time dynamics—and we successfully implement it. We conclude with a spatial characterization of Pauli Weight spreading, bridging Tensor Network simulations and the field of operator hydrodynamics. Finally, in Chapter 5, we focus on the transverse contraction of the 2D Tensor Net-work using the light-cone algorithm, analyzing its variants in terms of truncation strategies and convergence across different models. This leads to a systematic characterization of the method based on the integrability of the underlying Hamiltonians and single-spin noise models.