Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods

Transfer matrices and matrix product operators play an ubiquitous role in the field of many body physics. This paper gives an ideosyncratic overview of applications, exact results and computational aspects of diagonalizing transfer matrices and matrix product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and of new results. Topics discussed are exact solutions of transfer matrices in equilibrium and non-equilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix product state methods for finding extremal eigenvectors of matrix product operators.Comment: Lecture notes from a course at Vienna Universit

Scattering particles in quantum spin chains

A variational approach for constructing an effective particle description of the low-energy physics of one-dimensional quantum spin chains is presented. Based on the matrix product state formalism, we compute the one- and two-particle excitations as eigenstates of the full microscopic Hamiltonian. We interpret the excitations as particles on a strongly-correlated background with non-trivial dispersion relations, spectral weights and two-particle S matrices. Based on this information, we show how to describe a finite density of excitations as an interacting gas of bosons, using its approximate integrability at low densities. We apply our framework to the Heisenberg antiferromagnetic ladder: we compute the elementary excitation spectrum and the magnon-magnon S matrix, study the formation of bound states and determine both static and dynamic properties of the magnetized ladder.Comment: published versio

Simulating excitation spectra with projected entangled-pair states

We develop and benchmark a technique for simulating excitation spectra of generic two-dimensional quantum lattice systems using the framework of projected entangled-pair states (PEPS). The technique relies on a variational ansatz for capturing quasiparticle excitations on top of a PEPS ground state. Our method perfectly captures the quasiparticle dispersion relation of the square-lattice transverse-field Ising model, and reproduces the spin-wave velocity and the spin-wave anomaly in the square-lattice Heisenberg model with high precision

Tangent-space methods for uniform matrix product states

In these lecture notes we give a technical overview of tangent-space methods for matrix product states in the thermodynamic limit. We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how matrix product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous matrix product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications

Post-Matrix Product State Methods: To tangent space and beyond

We develop in full detail the formalism of tangent states to the manifold of matrix product states, and show how they naturally appear in studying time-evolution, excitations and spectral functions. We focus on the case of systems with translation invariance in the thermodynamic limit, where momentum is a well defined quantum number. We present some new illustrative results and discuss analogous constructions for other variational classes. We also discuss generalizations and extensions beyond the tangent space, and give a general outlook towards post matrix product methods.Comment: 40 pages, 8 figure

Matrix product states and variational methods applied to critical quantum field theory

We study the second-order quantum phase-transition of massive real scalar field theory with a quartic interaction ($\phi^4$ theory) in (1+1) dimensions on an infinite spatial lattice using matrix product states (MPS). We introduce and apply a naive variational conjugate gradient method, based on the time-dependent variational principle (TDVP) for imaginary time, to obtain approximate ground states, using a related ansatz for excitations to calculate the particle and soliton masses and to obtain the spectral density. We also estimate the central charge using finite-entanglement scaling. Our value for the critical parameter agrees well with recent Monte Carlo results, improving on an earlier study which used the related DMRG method, verifying that these techniques are well-suited to studying critical field systems. We also obtain critical exponents that agree, as expected, with those of the transverse Ising model. Additionally, we treat the special case of uniform product states (mean field theory) separately, showing that they may be used to investigate non-critical quantum field theories under certain conditions.Comment: 24 pages, 21 figures, with a minor improvement to the QFT sectio

Renormalization group flows of Hamiltonians using tensor networks

A renormalization group flow of Hamiltonians for two-dimensional classical partition functions is constructed using tensor networks. Similar to tensor network renormalization ([G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)], [S. Yang, Z.-C. Gu, and X.-G Wen, Phys. Rev. Lett. 118, 110504 (2017)]) we obtain approximate fixed point tensor networks at criticality. Our formalism however preserves positivity of the tensors at every step and hence yields an interpretation in terms of Hamiltonian flows. We emphasize that the key difference between tensor network approaches and Kadanoff's spin blocking method can be understood in terms of a change of local basis at every decimation step, a property which is crucial to overcome the area law of mutual information. We derive algebraic relations for fixed point tensors, calculate critical exponents, and benchmark our method on the Ising model and the six-vertex model.Comment: accepted version for Phys. Rev. Lett, main text: 5 pages, 3 figures, appendices: 9 pages, 1 figur

Excitations and the tangent space of projected entangled-pair states

We develop tangent space methods for projected entangled-pair states (PEPS) that provide direct access to the low-energy sector of strongly-correlated two-dimensional quantum systems. More specifically, we construct a variational ansatz for elementary excitations on top of PEPS ground states that allows for computing gaps, dispersion relations, and spectral weights directly in the thermodynamic limit. Solving the corresponding variational problem requires the evaluation of momentum transformed two-point and three-point correlation functions on a PEPS background, which we can compute efficiently by using a contraction scheme. As an application we study the spectral properties of the magnons of the Affleck-Kennedy-Lieb-Tasaki model on the square lattice and the anyonic excitations in a perturbed version of Kitaev's toric code

Fermionic projected entangled-pair states and topological phases

We study fermionic matrix product operator algebras and identify the associated algebraic data. Using this algebraic data we construct fermionic tensor network states in two dimensions that have non-trivial symmetry-protected or intrinsic topological order. The tensor network states allow us to relate physical properties of the topological phases to the underlying algebraic data. We illustrate this by calculating defect properties and modular matrices of supercohomology phases. Our formalism also captures Majorana defects as we show explicitly for a class of $\mathbb{Z}_2$ symmetry-protected and intrinsic topological phases. The tensor networks states presented here are well-suited for numerical applications and hence open up new possibilities for studying interacting fermionic topological phases.Comment: Published versio