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arXiv:2102.10982 · Physical review. B./Physical review. B

Efficient matrix product state methods for extracting spectral information on rings and cylinders

Maarten Van Damme, Robijn Vanhove, Jutho Haegeman, Frank Verstraete, Laurens Vanderstraeten

Published Sep 21, 2021• 36 citations
arXivHTMLPDF

Abstract

Based on the matrix product state (MPS) formalism, we introduce an ansatz for capturing excited states in finite systems with open boundary conditions, providing a very efficient method for computing, e.g., the spectral gap of quantum spin chains. This method can be straightforwardly implemented on top of an existing density-matrix renormalization group or MPS ground-state code. Although this approach is built on open-boundary MPSs, we also apply it to systems with periodic boundary conditions. Despite the explicit breaking of translation symmetry by the MPS representation, we show that momentum emerges as a good quantum number and can be exploited for labeling excitations on top of MPS ground states. We apply our method to the critical Ising chain on a ring and the classical Potts model on a cylinder. Finally, we apply the same idea to compute excitation spectra for 2D quantum systems on infinite cylinders. Again, despite the explicit breaking of translation symmetry in the periodic direction, we recover momentum as a good quantum number for labeling excitations. We apply this method to the 2D transverse-field Ising model and the half-filled Hubbard model; for the latter, we obtain accurate results for, e.g., the hole dispersion for cylinder circumferences up to eight sites.

DOI
10.1103/physrevb.104.115142

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Based on the matrix product state (MPS) formalism, we introduce an ansatz for capturing excited states in finite systems with open boundary conditions, providing a very efficient method for computing, e.g., the spectral gap of quantum spin chains. This method can be straightforwardly implemented on top of an existing density-matrix renormalization group or MPS ground-state code. Although this approach is built on open-boundary MPSs, we also apply it to systems with periodic boundary conditions. Despite the explicit breaking of translation symmetry by the MPS representation, we show that momentum emerges as a good quantum number and can be exploited for labeling excitations on top of MPS ground states. We apply our method to the critical Ising chain on a ring and the classical Potts model on a cylinder. Finally, we apply the same idea to compute excitation spectra for 2D quantum systems on infinite cylinders. Again, despite the explicit breaking of translation symmetry in the periodic direction, we recover momentum as a good quantum number for labeling excitations. We apply this method to the 2D transverse-field Ising model and the half-filled Hubbard model; for the latter, we obtain accurate results for, e.g., the hole dispersion for cylinder circumferences up to eight sites.
We demonstrate an efficient method that allows for simultaneous determination of the ground state, low energy excitation properties and excitation gap in quantum many body systems. To this aim we first use the \textit{ab-initio} optimization principle of tensor networks (TN), to show that the infinite density matrix renormalization group (iDMRG) in the real space is associated in a natural manner to the infinite time-evolving block decimation (iTEBD) implemented on a continuous matrix product state (MPS), and defined in imaginary time. We illustrate this association showing that the (imaginary) time matrix product state (MPS) in iTEBD reproduces accurately the properties of the two-dimensional (2D) classical Ising model, verifying in this way that the time MPS corresponds to a well-defined physical state. We apply then our scheme to the one-dimensional (1D) quantum Ising chain, where the time MPS is defined in continuous imaginary time. It is found that the time MPS at or close to the critical point is always maximally entangled, but gives essentially different correlations than those obtained from iDMRG; we argue that these correlations describe the dynamic correlations for the ground state.
We establish that a Bloch-state ansatz based on periodic uniform matrix product states (PUMPS), originally designed to capture single-quasiparticle excitations in gapped systems, is in fact capable of accurately approximating all low-energy eigenstates of critical quantum spin chains on the circle. When combined with the methods of [Milsted and Vidal, Phys. Rev. B 96, 245105PRBMDO2469-995010.1103/PhysRevB.96.245105] based on the Koo-Saleur formula, PUMPS Bloch states can then be used to identify each low-energy eigenstate of a chain made of up to hundreds of spins with its corresponding scaling operator in the emergent conformal field theory (CFT). This enables the following two tasks that we demonstrate using the quantum Ising model and a recently proposed generalization thereof due to O'Brien and Fendley [Phys. Rev. Lett. 120, 206403]. (i) From the spectrum of low energies and momenta we extract conformal data (specifying the emergent CFT) with unprecedented numerical accuracy. (ii) By changing the lattice size, we investigate nonperturbatively the renormalization group flow of the low-energy spectrum between two CFTs. In our example, where the flow is from the tricritical Ising CFT to the Ising CFT, we obtain excellent agreement with an analytical result [Klassen and Melzer, Nucl. Phys. B370, 51110.1016/0550-3213(92)90422-8] conjectured to describe the flow of the first spectral gap directly in the continuum.
We introduce a simple and efficient variation of the tangent-space excitation Ansatz used to compute elementary excitation spectra of one-dimensional quantum lattice systems using matrix product states (MPS). A small basis for the excitation tensors is formed based on a single diagonalization analogous to a single site DMRG step but for multiple states. Once overlap and Hamiltonian matrix elements are found, obtaining the excitation for any momentum only requires diagonalization of a tiny matrix, akin to a non-orthogonal band-theory diagonalization. The approach is based on an infinite MPS description of the ground state, and we introduce an extremely simple alternative to variational uniform matrix product states (VUMPS) based on finite system DMRG. For the S=1 S = 1 Heisenberg chain, our method—site basis excitation Ansatz (SBEA)—efficiently produces the one-magnon dispersion with high accuracy. We also examine the role of MPS gauge choices, finding that not imposing a gauge condition—leaving the basis nonorthogonal—is crucial for the approach, whereas imposing a left-orthonormal gauge (as in prior work) severely hampers convergence. We also show how one can construct Wannier excitations, analogous to the Wannier functions of band theory, where one Wannier excitation, translated to all sites, can reconstruct the single magnon modes exactly for all momenta.
We develop variational matrix product state (MPS) methods with symmetries to determine dispersion relations of one-dimensional quantum lattices as a function of momentum and preset quantum number. We test our methods on the XXZ spin chain, the Hubbard model, and a nonintegrable extended Hubbard model and determine the excitation spectra with a precision similar to the one of the ground state. The formulation in terms of quantum numbers makes the topological nature of spinons and holons very explicit. In addition, the method also enables an easy and efficient direct calculation of the necessary magnetic field or chemical potential required for a certain ground state magnetization or particle density.
During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. The underlying matrix product state (MPS) ansatz is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS controls the size of the corner of the many-body Hilbert space that can be reached. Whereas the MPS ansatz will only yield an efficient description for noncritical one-dimensional systems, it can still be used as a variational ansatz for other finite-size systems. Rather large virtual dimensions are then required. The two most important aspects to reduce the corresponding computational cost are a proper choice and ordering of the active space orbitals, and the exploitation of the symmetry group of the Hamiltonian. By taking care of both aspects, DMRG becomes an efficient replacement for exact diagonalization in quantum chemistry. DMRG and Hartree-Fock theory have an analogous structure. The former can be interpreted as a self-consistent mean-field theory in the DMRG lattice sites, and the latter in the particles. It is possible to build upon this analogy to introduce post-DMRG methods. Based on an approximate MPS, these methods provide improved ansätze for the ground state, as well as for excitations. Exponentiation of the single-particle (single-site) excitations for a Slater determinant (an MPS with open boundary conditions) leads to the Thouless theorem for Hartree-Fock theory (DMRG), an explicit nonredundant parameterization of the entire manifold of Slater determinants (MPS wavefunctions). This gives rise to the configuration interaction expansion for DMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliary field quantum Monte Carlo for Slater determinants. An analogous transformation for spin-lattice Hamiltonians allows to formulate a promising variant for MPSs.
We introduce a method based on matrix product states (MPS) for computing spectral functions of (quasi-) one-dimensional spin chains, working directly in momentum space in the thermodynamic limit. We simulate the time evolution after applying a momentum operator to an MPS ground state by working with the momentum superposition of a window MPS. We show explicitly for the spin-1 Heisenberg chain that the growth of entanglement is smaller in momentum space, even inside a two-particle continuum, such that we can attain very accurate spectral functions with relatively small bond dimension. We apply our method to compute spectral lineshapes of the gapless XXZ chain and the square-lattice ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on a six-leg cylinder.