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arXiv:1705.05423 · Physical review. B./Physical review. B

Excitation spectrum and density matrix renormalization group iterations

Natalia Chepiga, Frédéric Mila

Published Aug 21, 2017• 43 citations
arXivHTMLPDF

Abstract

We show that, in certain circumstances, exact excitation energies appear as locally site-independent (or flat) modes if one records the excitation spectrum of the effective Hamiltonian while sweeping through the lattice in the variational matrix-product-state formulation of the density matrix renormalization group, a remarkable property since the effective Hamiltonian is only constructed to target the ground state. Conversely, modes that are very flat over several consecutive iterations are systematically found to correspond to faithful excitations. We suggest to use this property to extract accurate information about excited states using the standard ground-state algorithm. The results are spectacular for critical systems, for which the low-energy conformal tower of states can be obtained very accurately at essentially no additional cost, as demonstrated by confirming the predictions of boundary conformal field theory for two simple minimal models: the transverse-field Ising model and the critical three-state Potts model. This approach is also very efficient to detect the quasidegenerate low-energy excitations in topological phases and to identify localized excitations in systems with impurities. Finally, using the variance of the Hamiltonian as a criterion, we assess the accuracy of the resulting matrix-product-state representations of the excited states.

DOI
10.1103/physrevb.96.054425

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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.
The density matrix renormalization group method is introduced in energy space to study Anderson impurity models. The method allows for calculations in the thermodynamic limit and is advantageous for studying not only the dynamical properties, but also the quantum entanglement of the ground state at the vicinity of an impurity quantum phase transition. This method is applied to obtain numerically exactly the ground-state phase diagram of the single-impurity Anderson model on the honeycomb lattice at half-filling. The calculation of local static quantities shows that the phase diagram contains two distinct phases, the local moment (LM) phase and the asymmetric strong coupling (ASC) phase, but no Kondo screening phase. These results are supported by the local spin and charge excitation spectra, which exhibit qualitatively different behavior in these two phases and also reveal the existence of the valence fluctuating point at the phase boundary. For comparison, we also study the low-energy effective pseudogap Anderson model using the method introduced here. Although the high-energy excitations are obviously different, we find that the ground-state phase diagram and the asymptotically low-energy excitations are in good quantitative agreement with those for the single-impurity Anderson model on the honeycomb lattice, thus providing a quantitative justification for the previous studies based on low-energy approximate approaches. Furthermore, we find that the lowest entanglement level is doubly degenerate for the LM phase, whereas it is singlet for the ASC phase and is accidentally threefold degenerate at the valence fluctuating point. This should be contrasted with the degeneracy of the energy spectrum because the ground state is found to be always singlet. Our results therefore clearly demonstrate that the low-lying entanglement spectrum can be used to determine with high accuracy the phase boundary of the impurity quantum phase transition.
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 use the formalism of tensor network states to 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. 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 we 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-TM’s eigenspectrum and give several arguments for the close relation between the structure of the low-energy spectrum of the system and the form of the static correlation functions. Finally, we discuss how the MPS-TM connects to the exact quantum transfer matrix of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of the MPS, which allows one to reinterpret variational MPS techniques (such as the density matrix renormalization group) as an application of Wilson’s numerical renormalization group along the virtual (imaginary time) dimension of the system.