Laurens Vanderstraeten

Transfer matrices and excitations with matrix product states

We investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low energy excitations using the formalism of tensor network states. In particular, we show that the Matrix Product State Transfer Matrix (MPS-TM) - a central object in the computation of static correlation functions - provides important information about the location and magnitude of the minima of the low energy dispersion relation(s) and present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TMs eigenspectrum and give several arguments for the close relation between the structure of the low energy spectrum of the system and the form of static correlation functions. Finally, we discuss how the MPS-TM connects to the exact Quantum Transfer Matrix (QTM) of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of MPS, which allows to reinterpret variational MPS techniques (such as the Density Matrix Renormalization Group) as an application of Wilsons Numerical Renormalization Group along the virtual (imaginary time) dimension of the system.

Gradient methods for variational optimization of projected entangled-pair states

We present a conjugate-gradient method for the ground-state optimization of projected entangled-pair states (PEPS) in the thermodynamic limit, as a direct implementation of the variational principle within the PEPS manifold. Our optimization is based on an efficient and accurate evaluation of the gradient of the global energy functional by using effective corner environments, and is robust with respect to the initial starting points. It has the additional advantage that physical and virtual symmetries can be straightforwardly implemented. We provide the tools to compute static structure factors directly in momentum space, as well as the variance of the Hamiltonian. We benchmark our method on Ising and Heisenberg models, and show a significant improvement on the energies and order parameters as compared to algorithms based on imaginary-time evolution.

S Matrix from matrix product states

We use the matrix product state formalism to construct stationary scattering states of elementary excitations in generic one-dimensional quantum lattice systems. Our method is applied to the spin-1 Heisenberg antiferromagnet, for which we calculate the full magnon-magnon S matrix for arbitrary momenta and spin, the two-particle contribution to the spectral function, and higher order corrections to the magnetization curve. As our method provides an accurate microscopic representation of the interaction between elementary excitations, we envisage the description of low-energy dynamics of one-dimensional spin chains in terms of these particlelike excitations.

Particles, Holes, and Solitons: A Matrix Product State Approach

We introduce a variational method for calculating dispersion relations of translation invariant (1+1)-dimensional quantum field theories. The method is based on continuous matrix product states and can be implemented efficiently. We study the critical Lieb-Liniger model as a benchmark and excellent agreement with the exact solution is found. Additionally, we observe solitonic signatures of Liebs type II excitation. In addition, a nonintegrable model is introduced where a U(1)-symmetry breaking term is added to the Lieb-Liniger Hamiltonian. For this model we find evidence of a nontrivial bound-state excitation in the dispersion relation.

Symmetry breaking and the geometry of reduced density matrices

We demonstrate that the occurrence of symmetry breaking phase transitions together with the emergence of a local order parameter in classical statistical physics is a consequence of the geometrical structure of probability space. To this end we investigate convex sets generated by expectation values of certain observables with respect to all possible probability distributions of classical q-state spins on a two-dimensional lattice, for several values of q. The extreme points of these sets are then given by thermal Gibbs states of the classical q-state Potts model. As symmetry breaking phase transitions and the emergence of associated order parameters are signaled by the appearance ruled surfaces on these sets, this implies that symmetry breaking is ultimately a consequence of the geometrical structure of probability space. In particular we identify the different features arising for continuous and first order phase transitions and show how to obtain critical exponents and susceptibilities from the geometrical shape of the surface set. Such convex sets thus also constitute a novel and very intuitive way of constructing phase diagrams for many body systems, as all thermodynamically relevant quantities can be very naturally read off from these sets.The concept of symmetry breaking and the emergence of corresponding local order parameters constitute the pillars of modern day many body physics. The theory of quantum entanglement is currently leading to a paradigm shift in understanding quantum correlations in many body systems and in this work we show how symmetry breaking can be understood from this wavefunction centered point of view. We demonstrate that the existence of symmetry breaking is a consequence of the geometric structure of the convex set of reduced density matrices of all possible many body wavefunctions. The surfaces of those convex bodies exhibit non-analytic behavior in the form of ruled surfaces, which turn out to be the defining signatures for the emergence of symmetry breaking and of an associated order parameter. We illustrate this by plotting the convex sets arising in the context of three paradigmatic examples of many body systems exhibiting symmetry breaking: the quantum Ising model in transverse magnetic field, exhibiting a second order quantum phase transition; the classical Ising model at finite temperature in two dimensions, which orders below a critical temperature

Scattering particles in quantum spin chains

T_c

Excitations and the tangent space of projected entangled-pair states

; and a system of free bosons at finite temperature in three dimensions, exhibiting the phenomenon of Bose-Einstein condensation together with an associated order parameter

Quasiparticle interactions in frustrated Heisenberg chains

\langle\psi\rangle

Bridging Perturbative Expansions with Tensor Networks

. Remarkably, these convex sets look all very much alike. We believe that this wavefunction based way of looking at phase transitions demystifies the emergence of order parameters and provides a unique novel tool for studying exotic quantum phenomena.

Effective Particles in Quantum Spin Chains: Applications

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 nontrivial 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.