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arXiv:2011.02625 · Physical review. B./Physical review. B

Matrix product wave function of the ground state and elementary excitations in the spin- 12 chain

Jintae Kim, Minsoo Kim, Pramod Padmanabhan, Jung Hoon Han, Hyun‐Yong Lee

Published Mar 29, 2021• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.103.125157

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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.
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 introduce a family of Jastrow pair product states for quasi-one-dimensional spin systems. Depending on a parameter they interpolate between the resonating valence-bond ground state of the Haldane-Shastry model describing a spin liquid and the (dimerized) valence-bond solid ground states of the Majumdar-Ghosh spin chain. These states are found to form an excellent basis for variational studies of Heisenberg chains with next-nearest-neighbor interactions and bond alternation as realized in the spin-Peierls system CuGeO3{}_{3}3​.
Following the discovery of high Tc superconductivity in the copper oxides, there has been a great deal of interest in the RVB wave function proposed by Anderson [1]. As a warm-up exercise we have considered a valence-bond wave function for the one dimensional spin-1/2 Heisenberg chain. The main virtue of our work is to propose a new variational singlet wavefunction which is almost analytically tractable by a transfer-matrix technique. We have obtained the ground state energy for finite as well as infinite chains, in good agreement with exact results. Correlation functions, excited states, and the effects of other interactions (e.g., spin-Peierls) are also accessible within this scheme [2]. Since the ground state of the chain is known to be a singlet (Lieb & Mattis [3]), we write the appropriate wave function as a superposition of valence-bond singlets, [Formula: see text], where | k > is a spin configuration obtained by pairing all spins into singlet pairs, in a way which is common in valence-bond calculations of large molecules. As in that case, each configuration, | k >, can be represented by a Rümer diagram, with directed bonds connecting each pair of spins on the chain. The c k 's are variational co-efficients, the form of which is determined as follows: Each singlet configuration (Rümer diagram) is divided into "zones", a "zone" corresponding to the region between two consecutive sites. Each zone is indexed by its distance from the end of the chain and by the number of bonds crossing it. Our procedure assigns a variational parameter, x ij , to the j th zone, when crossed by i bonds. The resulting wavefunction for an N-site chain is written as [Formula: see text] where m ij(k) equals 1 when zone j is crossed by i bonds and zero otherwise. To make the calculation tractable we reduce the number of variational parameters by disallowing configurations with bonds connecting any two sites separated by more than 2M lattice points. (For simplicity, we have limited ourselves to M=3, but the scheme can be used for any M). With the simple ansatz, matrix elements can be calculated by a transfer-matrix method. To understand the transfer-matrix method note that since only local zone parameters appear in the description of each state | k >, matrix elements and overlaps, [Formula: see text] and < k | k '>, are completely specified by a small number of "local states" associated with each zone. Within a given zone a local state is defined by (i) the number of bonds crossing the zone and (ii), by whether the bonds originate from the initial (| k >) or final (| k '>) state. It is then easy to see that "local states" of consecutive zones are connected by a 15 × 15 transfer matrix (for the case M=3). Furthermore, the overlap matrix element can be written as a product of transfer-matrices associated with each zone of the chain. When calculating matrix elements of the Hamiltonian, an additional matrix, U , must be defined, to represent the particular zone involving the two spins connected by the Heisenberg interaction. The relevant details as well as the comparison with exact results will be given elsewhere. We are planning to ultimately apply this method to the two dimensional case, and hope to include the effects of holes.
We present a method of constructing matrix-product (MP) states and apply it to the ground state of the Heisenberg spin chain with alternating spins 1 and and antiferromagnetic exchange interactions between nearest neighbors (the simplest example of a quantum ferrimagnet). The elementary matrix state is constructed in a way that ensures given transformational properties under rotations, which allows one to fix the total spin, and its z projection, of the entire MP wave function. We compare the variational MP results with the numerical results obtained through a quantum Monte Carlo method; the agreement is found to be within 0.4% for the ground-state energy and 5% for the correlation functions.
The matrix product method (MPM) has been used in the past to generate variational Ans\"atze of the ground state (GS) of spin chains and ladders. In this paper we apply the MPM to study the GS of conjugated polymers in the valence bond basis, exploiting the charge and spin conservation as well as the electron-hole and spin-parity symmetries. We employ the U-VVV-$\ensuremath{\delta}$ Hamiltonian, which is a simplified version of the Pariser-Parr-Pople Hamiltonian. For several coupling constants U and V and dimerizations $\ensuremath{\delta},$ we compute the GS energy per monomer, which agrees within a 2--4 % accuracy with the density-matrix renormalization group results. We also show the evolution of the MP-variational parameters in the weak and strong dimerization regimes.
Using the matrix product formalism, we define a multiparameter family of spin models on one-dimensional chains, with nearest- and next-nearest-neighbor interactions for which exact analytical expressions can be found for its doubly degenerate ground states. The family of Hamiltonians depends on five continuous parameters and the Majumdar-Ghosh model is a particular point in this parameter space. The model can also be represented as a zigzag spin ladder in which there are extra three-particle couplings between spins on vertices of triangles. The doubly degenerate ground states of these models models have a very simple structure, they are the product of entangled (dimer) states on adjacent sites (or consecutive rungs of the ladder in the zigzag ladder picture). At the Majumdar-Ghosh point, these entangled states become the spin-singlets pertaining to this model. We will also calculate, in closed form, the two point correlation functions, both for finite sizes of the chain and in the thermodynamic limit.