Error estimates for extrapolations with matrix-product states

We introduce a new error measure for matrix-product states without requiring the relatively costly two-site density matrix renormalization group (2DMRG). This error measure is based on an approximation of the full variance $\langle \psi | ( \hat H - E )^2 |\psi \rangle$. When applied to a series of matrix-product states at different bond dimensions obtained from a single-site density matrix renormalization group (1DMRG) calculation, it allows for the extrapolation of observables towards the zero-error case representing the exact ground state of the system. The calculation of the error measure is split into a sequential part of cost equivalent to two calculations of $\langle \psi | \hat H | \psi \rangle$ and a trivially parallelized part scaling like a single operator application in 2DMRG. The reliability of the new error measure is demonstrated at four examples: the $L=30, S=\frac{1}{2}$ Heisenberg chain, the $L=50$ Hubbard chain, an electronic model with long-range Coulomb-like interactions and the Hubbard model on a cylinder of size $10 \times 4$. Extrapolation in the new error measure is shown to be on-par with extrapolation in the 2DMRG truncation error or the full variance $\langle \psi | ( \hat H - E )^2 |\psi \rangle$ at a fraction of the computational effort.Comment: 10 pages, 11 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

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

A matrix product state based algorithm for determining dispersion relations of quantum spin chains with periodic boundary conditions

We study a matrix product state (MPS) algorithm to approximate excited states of translationally invariant quantum spin systems with periodic boundary conditions. By means of a momentum eigenstate ansatz generalizing the one of \"Ostlund and Rommer [1], we separate the Hilbert space of the system into subspaces with different momentum. This gives rise to a direct sum of effective Hamiltonians, each one corresponding to a different momentum, and we determine their spectrum by solving a generalized eigenvalue equation. Surprisingly, many branches of the dispersion relation are approximated to a very good precision. We benchmark the accuracy of the algorithm by comparison with the exact solutions of the quantum Ising and the antiferromagnetic Heisenberg spin-1/2 model.Comment: 13 pages, 11 figures, 5 table

Calculus of continuous matrix product states

We discuss various properties of the variational class of continuous matrix product states, a class of Ansatz states for one-dimensional quantum fields that was recently introduced as the direct continuum limit of the highly successful class of matrix product states. We discuss both attributes of the physical states, e.g., by showing in detail how to compute expectation values, as well as properties intrinsic to the representation itself, such as the gauge freedom. We consider general translation noninvariant systems made of several particle species and derive certain regularity properties that need to be satisfied by the variational parameters. We also devote a section to the translation invariant setting in the thermodynamic limit and show how continuous matrix product states possess an intrinsic ultraviolet cutoff. Finally, we introduce a new set of states, which are tangent to the original set of continuous matrix product states. For the case of matrix product states, this construction has recently proven relevant in the development of new algorithms for studying time evolution and elementary excitations of quantum spin chains. We thus lay the foundation for similar developments for one-dimensional quantum fields

Entanglement renormalization for quantum fields

It is shown how to construct renormalization group flows of quantum field theories in real space, as opposed to the usual Wilsonian approach in momentum space. This is achieved by generalizing the multiscale entanglement renormalization ansatz to continuum theories. The variational class of wavefunctions arising from this RG flow are translation invariant and exhibit an entropy-area law. We illustrate the construction for a free non-relativistic boson model, and argue that the full power of the construction should emerge in the case of interacting theories.Comment: 4 pages: completely revised. Focus on a single non-relativistic model for clarit

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 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.Comment: Improved version, extra supplemental materia

Fermionic matrix product states and one-dimensional topological phases

We develop the formalism of fermionic matrix product states (fMPS) and show how irreducible fMPS fall in two different classes, related to the different types of simple Z(2) graded algebras, which are physically distinguished by the absence or presence of Majorana edge modes. The local structure of fMPS with Majorana edge modes also implies that there is always a twofold degeneracy in the entanglement spectrum. Using the fMPS formalism, we make explicit the correspondence between the Z(8) classification of time-reversal-invariant spinless superconductors and the modulo 8 periodicity in the representation theory of real Clifford algebras. Studying fMPS with general onsite unitary and antiunitary symmetries allows us to define invariants that label symmetry-protected phases of interacting fermions. The behavior of these invariants under stacking of fMPS is derived, which reveals the group structure of such interacting phases. We also consider spatial symmetries and show how the invariant phase factor in the partition function of reflection-symmetric phases on an unorientable manifold appears in the fMPS framework

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