Transfer matrices and matrix product operators play a ubiquitous role in the field of many-body physics. This review gives an idiosyncratic 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 new results. Topics discussed are exact solutions of transfer matrices in equilibrium and nonequilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix product state methods for finding extremal eigenvectors of matrix product operators.
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.
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 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.
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.
We provide a generalization of the matrix product operator formalism for string-net projected entangled pair states (PEPS) to include nonunitary solutions of the pentagon equation. These states ...provide the explicit lattice realization of the Galois conjugated counterparts of (2+1)-dimensional topological quantum field theories, based on tensor fusion categories. Although the parent Hamiltonians of these renormalization group fixed point states are gapless, these states can still be the topological ground states of a gapped non-Hermitian Hamiltonian. We show by example that the topological sectors of the Yang-Lee theory (the nonunitary counterpart of the Fibonacci fusion category) can be constructed, even in the absence of closure under Hermitian conjugation of the basis elements of the Ocneanu tube algebra. The topological sector construction is demonstrated by applying the concept of strange correlators to the Yang-Lee model, giving rise to a nonunitary version of the classical hard hexagon model in the Yang-Lee universality class and obtaining all generalized twisted boundary conditions on a finite cylinder of the Yang-Lee edge singularity. Finally, we construct the PEPS transfer matrix and show that taking the Hermitian conjugate changes the topological phase for these nonunitary string-net models.
We study the second-order quantum phase transition of massive real scalar field theory with a quartic interaction in (1 + 1) dimensions on an infinite spatial lattice using matrix product states. We ...introduce and apply a naive variational conjugate gradient method, based on the time-dependent variational principle 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 density matrix normalization group 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 noncritical quantum field theories under certain conditions.
Several tensor networks are built of isometric tensors, i.e. tensors
satisfying
W\dagger W = \mathbb{1}
W
†
W
=
1
.
Prominent examples include matrix product states (MPS) in canonical
form, the ...multiscale entanglement renormalization ansatz (MERA), and
quantum circuits in general, such as those needed in state preparation
and quantum variational eigensolvers. We show how gradient-based
optimization methods on Riemannian manifolds can be used to optimize
tensor networks of isometries to represent e.g. ground states of 1D
quantum Hamiltonians. We discuss the geometry of Grassmann and Stiefel
manifolds, the Riemannian manifolds of isometric tensors, and review how
state-of-the-art optimization methods like nonlinear conjugate gradient
and quasi-Newton algorithms can be implemented in this context. We apply
these methods in the context of infinite MPS and MERA, and show
benchmark results in which they outperform the best previously-known
optimization methods, which are tailor-made for those specific
variational classes. We also provide open-source implementations of our
algorithms.
We extend the concept of strange correlators, defined for symmetry-protected phases in You et al. Phys. Rev. Lett. 112, 247202 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.247202, to topological ...phases of matter by taking the inner product between string-net ground states and product states. The resulting two-dimensional partition functions are shown to be either critical or symmetry broken, since the corresponding transfer matrices inherit all matrix product operator symmetries of the string-net states. For the case of critical systems, these nonlocal matrix product operator symmetries are the lattice remnants of topological conformal defects in the field theory description. Following Aasen et al. J. Phys. A 49, 354001 (2016)JPAMB51751-811310.1088/1751-8113/49/35/354001, we argue that the different conformal boundary conditions can be obtained by applying the strange correlator concept to the different topological sectors of the string net obtained from Ocneanu's tube algebra. This is demonstrated on the lattice by calculating the conformal field theory spectra in the different topological sectors for the Fibonacci (hard-hexagon) and Ising string net. Additionally, we provide a complementary perspective on symmetry-preserving real-space renormalization by showing how known tensor network renormalization methods can be understood as the approximate truncation of an exactly coarse-grained strange correlator.
A
bstract
We construct a Hamiltonian lattice regularisation of the
N
-flavour Gross-Neveu model that manifestly respects the full O(2
N
) symmetry, preventing the appearance of any unwanted marginal ...perturbations to the quantum field theory. In the context of this lattice model, the dynamical mass generation is intimately related to the Coleman-Mermin-Wagner and Lieb-Schultz-Mattis theorems. In particular, the model can be interpreted as lying at the first order phase transition line between a trivial and symmetry-protected topological (SPT) phase, which explains the degeneracy of the elementary kink excitations. We show that our Hamiltonian model can be solved analytically in the large
N
limit, producing the correct expression for the mass gap. Furthermore, we perform extensive numerical matrix product state simulations for
N
= 2, thereby recovering the emergent Lorentz symmetry and the proper non-perturbative mass gap scaling in the continuum limit. Finally, our simulations also reveal how the continuum limit manifests itself in the entanglement spectrum. As expected from conformal field theory we find two conformal towers, one tower spanned by the linear representations of O(4), corresponding to the trivial phase, and the other by the projective (i.e. spinor) representations, corresponding to the SPT phase.
PDF
