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Physical review. B, Condensed matter

Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S =1 Heisenberg chain

Steven R. White, David A. Huse

Published Aug 1, 1993• 605 citations
Publisher

Abstract

We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be ${\mathit{e}}_{0}$\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.

DOI
10.1103/physrevb.48.3844

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Physical review. B, Condensed matter•Jul 1, 1993•123 citations
We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be e0{\mathit{e}}_{0}e0​\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.
The S=1 antiferromagnetic Heisenberg chain both with and without single-ion anisotropy is studied. Using the recently proposed density-matrix renormalization-group technique of White, we calculate the energy gaps as well as several different correlation functions. The two gaps, ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{\ensuremath{\parallel}}}$ and ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{\ensuremath{\perp}}}$, along with associated correlation lengths and velocities are determined. The numerical results are shown to be in good agreement with theoretical predictions derived from the nonlinear \ensuremath{\sigma} model and a free-boson model. We also study the S=1/2 excitations that occur at the ends of open chains; in particular we study the behavior associated with open boundary conditions, using a model of S=1/2 spins coupled to the free bosons.
The ground state energy, the correlation functions and the excitation energy of a Heisenberg antiferromagnetic chain of spin S are investigated by the real-space renormalization-group method. The asymptotic form ( S →∞) of the ground state energy is compared with the result obtained by the spin wave theory. The calculated correlation function < S ( i ) · S ( i + l ) > shows no long range order for a system of finite S . The excitation spectrum is found to have singular dips at q =(odd integer) ×π/3 n a . This is associated with the shortcoming of the three-site renormalization.
Low-lying magnon dispersion in a S=1S=1S=1 Heisenberg antiferromagnetic (AF) chain is analyzed using the non-Abelian density-matrix-renormalization-group (DMRG) method. The scattering length ab{a}_{\mathrm{b}}ab​ of the boundary coupling and the intermagnon scattering length aaa are determined. The scattering length ab{a}_{\mathrm{b}}ab​ is found to exhibit a characteristic diverging behavior at the crossover point. In contrast, the Haldane gap $\ensuremath{\Delta}$, the magnon velocity vvv, and aaa remain constant at the crossover. Our method allowed estimation of the gap of the S=2S=2S=2 AF chain to be $\ensuremath{\Delta}=0.0891623(9)$ using a chain length longer than the correlation length $\ensuremath{\xi}$.
The excitation energies of magnons and solitons as well as the ground-state energy of the spin-1 antiferromagnetic Heisenberg-Ising ring are studied by the numerical method. The method of Vanden Broeck and Schwartz is used to estimate the infinite-lattice limits accurately. The result confirms Haldane's prediction that a novel singlet phase with a nonzero excitation gap appears between the gapless XY phase and the doublet ground-state N\'eel phase. It extends over the range of anisotropy (${\ensuremath{\Delta}}_{l}$\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}${\ensuremath{\Delta}}_{u}$) around the isotropic Heisenberg point \ensuremath{\Delta}=0.5. It is ascertained that at the upper transition point ${\ensuremath{\Delta}}_{u}$=0.540\ifmmode\pm\else\textpm\fi{}0.001 (1) the longitudinal magnon gap closes, (2) the soliton existing in the Ising side softens, and (3) the second derivative of the ground-state energy diverges. Around the lower transition point of ${\ensuremath{\Delta}}_{l}$=0.38\ifmmode\pm\else\textpm\fi{}0.02, the transverse magnon gap exhibits an exponential singularity accompanying with the slight anomaly of the second derivative of the ground-state energy.
The manifestations of topological effects in finite antiferromagnetic Heisenberg chains are examined by density-matrix renormalization-group techniques. We find that the difference between integer and half-integer spin chains shows up in ground-state energy per site when the length of the spin chain is longer than \ensuremath{\sim}\ensuremath{\xi}, where $\ensuremath{\xi}\ensuremath{\sim}\mathrm{exp}(\ensuremath{\pi}S)$ is a spin-spin correlation length, for spin magnitudes S up to 5/2.5/2.5/2. For open chains with spin magnitudes S=5/2S=5/2S=5/2 to S=5,S=5,S=5, we verify that end states with fractional spin quantum numbers ${S}^{\ensuremath{'}}$ exist and are visible in the structure of the low-energy excitation spectrum even when the chain length is much smaller than the correlation length \ensuremath{\xi}. The robustness of end states points to the possibility of the existence of general hidden topological order in quantum spin chains with spin magnitudes S> 1.S>~1.S> 1.
Properties of the ground state of open chains of the S=1 Heisenberg antiferromagnet are studied by a quantum Monte Carlo method. Size (length) dependences of the ground-state energy, the energy gap, the staggered magnetization 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and the spin-correlation function 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 are investigated under free, fixed, and periodic boundary conditions. The size dependences are significant in chains with less than 25 spins, where the magnitude of the staggered magnetization is rather large. However, the fourfold degeneracy of the ground state and the exponential decay of correlations that are inherent to the Haldane system are recovered when the chain length becomes long enough. Both the staggered magnetization and the spin-correlation function decay exponentially. The correlation length is about six in lattice spacing, which agrees with the value obtained in the bulk.