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arXiv:2005.10980 · Physical review. B./Physical review. B

Construction of variational matrix product states for the Heisenberg spin-1 chain

Jintae Kim, Minsoo Kim, Naoki Kawashima, Jung Hoon Han, Hyun‐Yong Lee

Published Aug 10, 2020• 8 citations
arXivHTMLPDF

Abstract

We propose a simple variational wave function that captures the correct ground-state energy of the spin-1 Heisenberg chain model to within 0.04%. The wave function is written in the matrix product state (MPS) form with the bond dimension $D=8$ and is characterized by three fugacity parameters. The proposed MPS generalizes the Affleck-Kennedy-Lieb-Tasaki state by dressing it with dimers, trimers, and general $q$-mers. The fugacity parameters control the number and the average size of the $q$-mers. Furthermore, the $D=8$ variational MPS state captures the ground states of the entire family of the bilinear-biquadratic Hamiltonian belonging to the Haldane phase to high accuracy. The 2-4-2 degeneracy structure in the entanglement spectrum of our MPS state is found to match well the results of the density matrix renormalization group (DMRG) calculation, which is computationally much heavier. Spin-spin correlation functions also find an excellent fit with those obtained by DMRG.

DOI
10.1103/physrevb.102.085117

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We propose a simple variational wave function that captures the correct ground-state energy of the spin-1 Heisenberg chain model to within 0.04%. The wave function is written in the matrix product state (MPS) form with the bond dimension D=8D=8D=8 and is characterized by three fugacity parameters. The proposed MPS generalizes the Affleck-Kennedy-Lieb-Tasaki state by dressing it with dimers, trimers, and general qqq-mers. The fugacity parameters control the number and the average size of the qqq-mers. Furthermore, the D=8D=8D=8 variational MPS state captures the ground states of the entire family of the bilinear-biquadratic Hamiltonian belonging to the Haldane phase to high accuracy. The 2-4-2 degeneracy structure in the entanglement spectrum of our MPS state is found to match well the results of the density matrix renormalization group (DMRG) calculation, which is computationally much heavier. Spin-spin correlation functions also find an excellent fit with those obtained by DMRG.
We present a variational matrix product state (vMPS) for the ground state of the spin-1/2 Heisenberg model. The MPS effectively organizes the various dimer configurations, in faithful reflection of the resonating valence bond picture of the spin liquid, with the energy only 0.024% higher than the exact value given by Bethe ansatz. Building on the ground-state vMPS, the one-spin wave function is constructed in a simple manner with the dispersion that matches well with the exact spectrum. The vMPS scheme is applied to the family of Hamiltonian extrapolating between the Heisenberg model and the Majumdar-Ghosh model.
An alternative method of writing down the path integral for spin-1 Heisenberg antiferromagnetic chain is introduced. In place of the conventional coherent-state basis that leads to the nonlinear $\ensuremath{\sigma}$ model, we use a basis called the fluctuating matrix product states which embodies intersite entanglement from the outset. It forms an overcomplete set spanning the entire Hilbert space of the spin-1 chain. Saddle-point analysis performed for the bilinear-biquadratic spin model predicts the Affleck-Kennedy-Lieb-Tasaki (AKLT) state as the ground state in the vicinity of the AKLT Hamiltonian. Quadratic effective action derived by gradient expansion around the saddle point is free from constraints that plagued the nonlinear $\ensuremath{\sigma}$ model and exactly solvable. The obtained excitation modes agree precisely with the single-mode approximation result for the AKLT Hamiltonian. Excitation spectra for other bilinear-biquadratic Hamiltonians are obtained as well by diagonalizing the quadratic action.
We exactly calculate the reduced density matrix of matrix product states (MPS). Our compact result enables one to perform analytic studies of entanglement in MPS. In particular, we consider the MPS ground states of two anisotropic spin chains. One is a q -deformed Affleck-Kennedy-Lieb-Tasaki (AKLT) model and the other is a general spin-1 quantum antiferromagnet with nearest-neighbor interactions. Our analysis shows how anisotropy affects entanglement on different continuous parameter manifolds. We also construct an effective boundary spin model that describes a block of spins in the ground state of the q -deformed AKLT Hamiltonian. The temperature of this effective model is given in terms of the deformation parameter q .
This work presents a method to quantize low energy excitations of systems whose ground state is best represented by a matrix product state (MPS) such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) model. The approach is similar to the spin-wave model of excitations in typical magnetic materials, but includes a more general set of fluctuations in entanglement structure. We argue that the quasilocal action of tensor fluctuations facilitates a description in terms of bosonic modes. We apply this approach to compute fluctuation corrections to the bilinear-biquadratic Heisenberg model, whose ground state we expect to be close to the exact bond dimension 2 AKLT state in a certain parameter range. Our results show significant improvements in energy approximations, highlighting both the qualitative and quantitative potential of this paradigm for studying complex entangled systems. This approach paves the way for new insights into the low-energy physics of correlated materials and the development of effective field theories beyond traditional semiclassical methods.
In this paper we study a frustrated antiferromagnetic isotropic Heisenberg S=1 quantum spin chain H=${\ensuremath{\sum}}_{\mathrm{i}}$Si{\mathbf{S}}_{\mathrm{i}}Si​Si+1{\mathbf{S}}_{\mathrm{i}+1}Si+1​+\ensuremath{\alpha}${\ensuremath{\sum}}_{\mathrm{i}}$Si{\mathbf{S}}_{\mathrm{i}}Si​Si+2{\mathbf{S}}_{\mathrm{i}+2}Si+2​, using a variational ansatz starting from valence bond states and the density-matrix renormalization group. We find both methods to give results in very good qualitative and good quantitative agreement, which clarify the phase diagram as follows: At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{D}}$=0.284(1), there is a disorder point of the first kind, marking the onset of incommensurate spin-spin correlations in the chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{L}}$=0.3725(25) there is a Lifshitz point, at which the excitation spectrum is found to develop a particular doubly degenerate structure. These points are the quantum remnants of the transition from antiferromagnetic to spiral order in the classical frustrated chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{T}}$=0.7444(6) there is a first-order phase transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT) phase characterized by nonvanishing string order to a phase which can be understood as a next-nearest-neighbor generalization of the AKLT model. At the transition, the string order parameter shows (to numerical precision) a discontinuous jump of 0.085 to zero; the correlation length and the gap are both finite at the transition. The problem of edge states in open frustrated chains is discussed at length.
Matrix product states (MPS) and ``dressed'' ground states of quadratic mean fields (e.g., Gutzwiller projected Slater determinants) are both important classes of variational wave functions. This latter class has played important roles in understanding superconductivity and quantum spin liquids. We present a method to obtain both the finite and infinite MPS (iMPS) representation of the ground state of an arbitrary fermionic quadratic mean-field Hamiltonian (which in the simplest case is a Slater determinant and in the most general case is a Pfaffian). We also show how to represent products of such states (e.g., determinants times Pfaffians). From this representation one can project to single occupancy and evaluate the entanglement spectra after Gutzwiller projection. We then obtain the MPS and iMPS representation of Gutzwiller projected mean-field states that arise from the variational slave-fermion approach to the S=1S=1S=1 bilinear-biquadratic quantum spin chain. To accomplish this, we develop an approach to orthogonalize degenerate iMPS to find all the states in the degenerate ground-state manifold. We find the energies of the MPS and iMPS states match the variational energies closely, indicating the method is accurate and there is minimal loss due to truncation error. We then present an exploration of the entanglement spectra of projected slave-fermion states, exploring their qualitative features and finding good qualitative agreement with the respective exact ground-state spectra found from density matrix renormalization group.