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arXiv:cond-mat/9610097 · Physical review. B, Condensed matter

Matrix-product-states approach to Heisenberg ferrimagnetic spin chains

Alexei K. Kolezhuk, H.-J. Mikeska, Shoji Yamamoto

Published Feb 1, 1997• 121 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.55.r3336

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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.
Second-order spin-wave expansions are used to compute the ground-state energy and sublattice magnetizations of the quantum one-dimensional Heisenberg ferrimagnet with nearest-neighbor antiferromagnetic interactions and two types of alternating sublattice spins S1>S2.{S}_{1}>{S}_{2}.S1​>S2​. It is found that in the extreme quantum cases (S1,S2){(S}_{1}{,S}_{2})(S1​,S2​)=(1,1/2), (3/2,1), and (3/2,1/2), the estimates for the ground-state energy and sublattice magnetizations differ less than 0.03% for the energy and 0.2% for the sublattice magnetizations from the recently published density-matrix renormalization-group numerical calculations. The reported results strongly suggest that the quantum Heisenberg ferrimagnetic chains give another example of a low-dimensional quantum spin system where the spin-wave approach demonstrates a surprising efficiency.
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.
Using the matrix product formalism, we find exact ground states for two new spin-(1/2) and spin-(3/2) quantum chains. On a lattice of 3 N sites, these ground states have a total spin of N /2, and hence have ferrimagnetic character. The novel point of this construction is that we construct a matrix product representation, not for a state which is invariant under rotation as has been hitherto considered, but for a state which transforms to other states under this symmetry operation. The method can be directly generalized to other symmetry groups.
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.
The ground-state properties of the ferrimagnetic spin chain consisting of (5/2,1/2,1/2) trimers with antiferromagnetic exchange between the 5/2 and 1/2 spins and ferromagnetic exchange between the 1/2 and 1/2 spins are analysed in the framework of the quantum renormalization group in real space and the matrix product method. The results are compared with a spin-wave calculation for a spin chain with two types of spin, (5/2,1). The ground-state energy, the average magnetizations and the correlation length are found as functions of the ratio between the antiferromagnetic and ferromagnetic exchange. The analysis shows that the trimer spin chain is characterized by strong quantum fluctuations reducing the average magnetizations and possesses an extremely short correlation length.