Optimising, Renormalising, Evolving and Quantising Tensor Networks

Matrix product density operators (MPDOs) are an important class of states with interesting properties. Consequently, it is important to understand how to prepare these states experimentally. One possible way to do it is to design an open system such that it evolves only towards desired states. In this work we develop an algorithm that for a given (small) linear subspace of MPDOs determines if this subspace can be the stable space for some Lindbladian consisting of only local terms and, if so, outputs a desired Lindbladian. The related question of how many stable states does Lindbladian for a 1d system consisting of generic nearest neighbour terms without translational invariance can have is being discussed.

As the pure Ashkin-Teller chain, the random quantum Ashkin-Teller chain undergoes a single quantum phase transition at weak intra-chain coupling. This transition line splits into two at strong intra-chain coupling. Analysis by the Strong Disorder Renormalization Group has shown that the critical behavior is described by the same Infinite Disorder Fixed Point as the random Ising chain in a transverse chain. Only the tricritical point belongs to a different universality class. These critical lines are surrounded by Griffiths phases where rare macroscopic regions leads to a singular behavior under a magnetic or electric field. Theses phases have been studied using DMRG, a technique which becomes inefficient at large randomness. The problem is reconsidered using a recently proposed numerical renormalization of the Hamiltonian expressed as a Matrix Product Operator.

A quantum state obeying the area law for entanglement on an infinite 2D lattice can be represented by a tensor network ansatz -- known as the projected entangled pair state (PEPS) -- with a finite bond dimension $D$. Its -- either imaginary or real -- time evolution can be split into small time steps. A straightforward application of a time step generates a new PEPS with a bond dimension $k$ times the original one. The new PEPS does not make an optimal use of its greater bond dimension $kD$, hence in principle it can be represented accurately by a more compact ansatz, favorably with the original $D$. In this work we show how the more compact PEPS can be optimized variationally to maximize its overlap with the new PEPS [1]. The key point is an efficient calculation of the overlap with the corner matrix renormalization group. We test our algorithm in the transverse field quantum Ising model where we simulate both imaginary time evolution generating thermal states and real time evolution after a sudden quench of the transverse field [1]. [1] J.Dziarmaga, P.Czarnik, "Time Evolution of a Projected Entangled Pair State", in prep.

Since the introduction of the Density Matrix Renormalisation Group (DMRG) in 1992, Tensor networks (TN) have proven to be a successful tool to investigate quantum many-body systems. They have been applied using different geometries, such as Matrix Product States (MPS) or Tree-TN, to address several problems in one and two dimensions. Taking advantage of certain area laws, TN provides the possibility to compactly compress the systems’ information while giving an accurate approximation of the induced error. Additionally exploiting symmetries of the system to investigate offers a higher efficiency in the TN algorithms and a precise targeting of specific symmetry sectors. The underlying principle is to restrict the systems’ degrees of freedom with respect to the corresponding symmetries. For global symmetries, this can be realised by decomposing each Tensor into smaller, independent subspaces. This incorporation of symmetries reduces the numerical resources needed and thereby helps to push forward the computational limitations of TNs. Based on this concept we aim to implement a general TN-framework for any number of abelian and non-abelian symmetries. In this poster, a detailed view on the analysis of the SU(2)-symmetric spin-1 Heisenberg model is presented. We illustrate the numerical insights gained out of an imaginary time evolution via Time-Evolving Block Decimation (TEBD) algorithm in the presence of the SU(2)-symmetry and compare the results to the DMRG investigation by S. White [1]. References [1] S. White, Physical Review B 48.6, 3844 (1993) [2] A. Weichselbaum, Annals of Physics 327, 2972-3047 (2012) [3] S. Singh, PhD thesis, arXiv:1203.2222v2 [quant-ph] (2012) [4] G. Vidal, Physical Review Letters 91.14, 147902 (2003)

Understanding the behavior of nonequilibrium stochastic systems in the thermodynamic limit remains a key goal in the statistical physics community. Large deviation theory has provided the framework for recent progress in this front, providing practitioners with the theoretical scaffolding to determine steady-state properties analogous to the free energy and entropy used in equilibrium systems. While the bulk of studies rely upon either analytical solutions or Monte Carlo methods in determining large deviation properties, recent work has utilized what is referred to as the quantum Hamiltonian formulation of the generator of nonequilibrium driven diffusive models to approach these problems via methods normally used to study quantum systems. Density Matrix Renormalization Group (DMRG), in particular, has been used to determine the current and its fluctuations in models such as the one-dimensional Simple Exclusion Process (SEP) and its variants. In this work, I recap and demonstrate the utility of studying one-dimensional driven diffusive systems via DMRG. Additionally, I show how these calculations can be modified to determine the properties of two-dimensional stochastic nonequilibrium models, such as the Weakly Asymmetric SEP (WASEP).

An impurity in a Tomonaga-Luttinger liquid leads to a crossover between short- and long-distance regime which describes many physical phenomena. However, calculation of the entire crossover of correlation functions over different length scales has been difficult. We develop a powerful numerical method based on infinite DMRG utilizing a finite system with infinite boundary conditions, which can be applied to correlation functions near an impurity. For the S=1/2 chain, we demonstrate that the full crossover can be precisely obtained, and that their limiting behaviors show a good agreement with field-theory predictions.

The ground state of the spin-1/2 bond-alternating Heisenberg Chain shows symmetry fractionalization which is manifested in the odd and even Haldane phases known as symmetry-protected topological (SPT) phases. We are using this model as a primer for studying the behavior of SPT order driven out-of equilibrium. The question which arises naturally is if it is possible to restore the symmetry fractionalization at non-zero temperatures, which we would like to refer as symmetry restoration. For our study we are using the Density-Matrix Renormalization Group (DMRG) in its formalism of uniform Matrix Product States (uMPS) to obtain the ground state in the thermodynamic limit and a multisite Time Evolving Block Decimation (TEBD) for the time evolution of the obtained ground state. Further investigations of Dynamical Quantum Phase Transition (DQPT) connected to the underlying symmetries of the ground state are given.

Recently, long-range Ising models have been simulated experimentally with ions confined in a Penning trap. Motivated by these experiments, we report the results of a study of the 2D transverse-field Ising model with long-range power-law interactions using the infinite cylinder geometry. For small transverse field, we find either a striped/columnar phase, or a clock ordered phase depending on the exponent of the long-range interaction. We also present some first results of a global quench, using a new Krylov-based algorithm for infinite systems.

We develop a quantum Langevin equation for matrix product states, describing the evolution of an open quantum system including entanglement. This consists of an extension of the time-dependent variational principle with dissipation and noise terms obeying a temperature dependent fluctuation-dissipation relation. We report the results of preliminary investigations of how the environmental coupling shapes the entanglement structure on different timescales.

We introduce an approach to efficiently represent long-range interactions in two-dimensional tensor network states in terms of projected entangled-pair operators (PEPOs). The isotropic, long-range interaction of interest is generated via a sum over a set of correlation functions of an auxiliary, fictitious system with purely local interactions, e.g., the classical Ising model at different temperatures. The long-range PEPO is thus constructed as a sum of local PEPOs, each of which can be exactly represented with a small, constant bond dimension. For a given level of accuracy, it is argued that the number of terms in the sum is independent of the system size, which is the distinctive feature of this approach compared with other existing constructions for long-range interactions. It allows for the operator to be used effectively in both the finite and infinite projected entangled-pair ansatzes for two-dimensional quantum systems, and is readily generalized to higher dimensions. The power and generality of the approach are illustrated on a finite two-dimensional spin system, where the interactions between spins are of long-range Coulomb type.

The question of the origin time's arrow is a major outstanding problem in physics. Here we present a possible mechanism for the generation of a cosmological arrow via spontaneous symmetry breaking in a theory of quantum gravity. In arXiv:1401.3736 Chen and Vishwanath have shown that a local notion of time-reversal symmetry can be encoded into a gauge field living on a tensor network. Using the fact that tensor networks can be used to describe the state space of Loop Quantum Gravity (arXiv:1309.6282, arXiv:1610.02134) we show that a spontaneous symmetry breaking mechanism operating on such tensor networks can lead to the spontaneous emergence of an arrow of time corresponding to the generation of non-zero order parameter for the Chen-Vishwanath gauge field.

We study one dimensional models of molecules and solids where both the electrons and nuclei are treated as quantum particles, going beyond the usual Born-Oppenheimer approximation. The continuous system is approximated by a grid which computationally resembles a ladder, with the electrons living on one leg and the nuclei on the other. To simulate DMRG well with this system, a three-site algorithm has been implemented. We also use a compression method to treat the long-range interactions between charged particles. We find that 1D diatomic molecules ("H2") with spin-1/2 nuclei in the spin triplet state will unbind when the mass of the nuclei reduces to only a few times larger than the electron mass. The molecule with nuclei in the singlet state always binds. The case of spin-0 bosonic nuclei is investigated as well.

Lattice models of gapless domain walls between twisted and untwisted gauge theories of finite group G are constructed systematically. As simple examples, we numerically studied the gapless domain walls between twisted and untwisted $Z_N$ (with N<6) gauge models in 2+1D using the state-of-art loop optimization of tensor network renormalization algorithm. We also studied the physical mechanism for these gapless domain walls and obtained quantum field theory descriptions that agree perfectly with our numerical results. By taking the advantage of the systematic classification and construction of twisted gauge models using group cohomology theory, we systematically construct general lattice models to realize gapless domain walls for arbitrary finite symmetry group G. Such constructions can be generalized into arbitrary dimensions and might provide us a systematical way to study gapless domain walls and topological quantum phase transitions. Refs: Chenfeng Bao, Shuo Yang, Chenjie Wang, Zheng-Cheng Gu, arXiv:1801.00719. Shuo Yang, Zheng-Cheng Gu, Xiao-Gang Wen, Phys. Rev. Lett. 118, 110504 (2017).

Due to the potentially exponential growth in entanglement during time evolution, computing the dynamics of a system accurately is difficult for long simulation times. By solving the Nakajima-Zwanzig (NZ) equation, one is able to exactly calculate the dynamics of a system of interest, provided that one can knows the memory kernel, which contains all of the relevant dynamics of the surrounding environment. While computing the memory kernel is also computationally intensive, it goes to zero relatively quickly for most systems. The lifetime of the memory kernel can potentially be used as a metric to classify the simulation complexity of different kinds of Hamiltonians. Here, we write the NZ equation in the tensor network language, allowing one to efficiently calculate the memory kernel for arbitrary Hamiltonians and obtain approximate solutions for those that could not be solved for using previous methods.

In the Fock representation, we propose the generalized matrix product states to describe one-dimensional topological phases of fermions/parafermions. The defining feature of these topological phases is the presence of Majorana/parafermion zero modes localized at the edges. It is shown that the single-block bipartite entanglement spectrum and its entanglement Hamiltonian are described by the effective coupling between two edge quasiparticles. Furthermore, we demonstrate that sublattice bulk bipartition can create an extensive number of edge quasiparticles in the reduced subsystem, and the symmetric couplings between the nearest neighbor edge quasiparticles lead to the critical entanglement spectra, characterizing the topological phase transitions from the fermionic/parafermionic topological phases to its adjacent trivial phase. The corresponding entanglement Hamiltonians for the critical entanglement spectra can also be derived. To conclude, we provide a novel recipe to extract the topological quantum critical phases adjacent to a topological phase, and this indicates generalizing the well-known bulk-edge correspondence of topological phases to the possible bulk-edge-criticality correspondence.