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arXiv:1706.02538 · Physical review. B./Physical review. B

Dimer correlation amplitudes and dimer excitation gap in spin- 12 XXZ and Heisenberg chains

Toshiya Hikihara, Akira Furusaki, Sergei L. Lukyanov

Published Oct 26, 2017• 21 citations
arXivHTMLPDF

Abstract

Correlation functions of dimer operators, the product operators of spins on two adjacent sites, are studied in the spin-$\frac{1}{2}$ XXZ chain in the critical regime. The amplitudes of the leading oscillating terms in the dimer correlation functions are determined with high accuracy as functions of the exchange anisotropy parameter and the external magnetic field through the combined use of bosonization and density-matrix renormalization-group methods. In particular, for the antiferromagnetic Heisenberg model with SU(2) symmetry, logarithmic corrections to the dimer correlations due to the marginally irrelevant operator are studied, and the asymptotic form of the dimer correlation function is obtained. The asymptotic form of the spin-Peierls excitation gap including logarithmic corrections is also derived.

DOI
10.1103/physrevb.96.134429

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Correlation functions of dimer operators, the product operators of spins on two adjacent sites, are studied in the spin-12\frac{1}{2}21​ XXZ chain in the critical regime. The amplitudes of the leading oscillating terms in the dimer correlation functions are determined with high accuracy as functions of the exchange anisotropy parameter and the external magnetic field through the combined use of bosonization and density-matrix renormalization-group methods. In particular, for the antiferromagnetic Heisenberg model with SU(2) symmetry, logarithmic corrections to the dimer correlations due to the marginally irrelevant operator are studied, and the asymptotic form of the dimer correlation function is obtained. The asymptotic form of the spin-Peierls excitation gap including logarithmic corrections is also derived.
We present accurate numerical estimates for the correlation amplitudes of leading and main subleading terms of the two- and four-spin correlation functions in the one-dimensional spin-1/21/21/2 XXZ\mathrm{XXZ}XXZ model under a magnetic field. These data are obtained by fitting the correlation functions, computed numerically with the density-matrix renormalization-group method, to the corresponding correlation functions in the low-energy effective theory. For this purpose we have developed the Abelian bosonization approach to the spin chain under the open-boundary conditions. We use the numerical data of the correlation amplitudes to quantitatively estimate spin gaps induced by a transverse staggered field and by exchange anisotropy.
The density-matrix renormalization-group technique is used to calculate the spin-correlation functions 〈SjxSkx〉〈{S}_{j}^{x}{S}_{k}^{x}〉〈Sjx​Skx​〉 and 〈SjzSkz〉〈{S}_{j}^{z}{S}_{k}^{z}〉〈Sjz​Skz​〉 of the one-dimensional S=12S=\frac{1}{2}S=21​ XXZ\mathrm{XXZ}XXZ model in the gapless regime. The numerical results for open chains of 200 spins are analyzed by comparing them with correlation functions calculated from a low-energy field theory. This gives precise estimates of the amplitudes of the correlation functions in the thermodynamic limit. The exact amplitude recently conjectured by Lukyanov and Zamolodchikov and the logarithmic correction in the Heisenberg model are confirmed numerically.
We present a simple method, combining the density-matrix renormalization-group algorithm with finite-size scaling, which permits the study of critical behavior in quantum spin chains. Spin moments and dimerization are induced by boundary conditions at the chain ends and these exhibit power-law decay at critical points. Results are presented for the spin-1/21/21/2 Heisenberg antiferromagnet; an analytic calculation shows that logarithmic corrections to scaling can sometimes be avoided. We also examine the spin-1 chain at the critical point separating the Haldane gap and dimerized phases. Exponents for the dimer-dimer and the spin-spin correlation functions are consistent with results obtained from bosonization.
Comparing numerically evaluated excitation gaps of dimerized spin-12\frac{1}{2}21​ XXZXXZXXZ chains with the gap formula for the low-energy effective sine-Gordon theory, we determine coefficients dxy{d}_{xy}dxy​ and dz{d}_{z}dz​ of bosonized dimerization operators in spin-12\frac{1}{2}21​ XXZXXZXXZ chains, which are defined as ${(\ensuremath{-}1)}^{j}({S}_{j}^{x}{S}_{j+1}^{x}+{S}_{j}^{y}{S}_{j+1}^{y})={d}_{xy}\text{ }\text{sin}[\sqrt{4\ensuremath{\pi}}\ensuremath{\phi}(x)]+\ensuremath{\cdots}$ and ${(\ensuremath{-}1)}^{j}{S}_{j}^{z}{S}_{j+1}^{z}={d}_{z}\text{ }\text{sin}[\sqrt{4\ensuremath{\pi}}\ensuremath{\phi}(x)]+\ensuremath{\cdots}$. We also calculate the coefficients of both spin and dimer operators for the spin-12\frac{1}{2}21​ Heisenberg antiferromagnetic chain with a nearest-neighbor coupling JJJ and a next-nearest-neighbor coupling J2=0.2411J{J}_{2}=0.2411JJ2​=0.2411J. As applications of these coefficients, we present ground-state phase diagrams of dimerized spin chains in a magnetic field and antiferromagnetic spin ladders with a four-spin interaction. The optical conductivity and electric polarization of one-dimensional Mott insulators with Peierls instability are also evaluated quantitatively.
In this thesis the quantum Monte Carlo loop algorithm is applied to study the properties of correlation functions in low-dimensional quantum antiferromagnets. Two different types of lattice models are investigated. The first system is the quantum critical XXZ chain. The second class of models emerges when coupling the square lattice Heisenberg model to phonons. In case of the quantum critical XXZ chain, the loop algorithm is used to compute the two-point spin and dimer correlation functions at low temperatures. Starting from the well-known asymptotic expressions from conformal field theory for the decay of the correlations, the crossover from the finite-temperature to the ground state behavior is investigated. In addtion, the Monte Carlo data are used to extract numerical estimates for the correlation amplitudes. The deviations from asymptotic scaling at low finite temperatures and intermediate distances are examined. The results are also used to study the algebraic and logarithmic corrections in the ground state. The method is further applied to compute the low-temperature spin correlations of the square lattice Heisenberg model coupled to dispersionless Einstein phonons. Also, statically dimerized Heisenberg models in two dimensions are investigated. Here the quantum Monte Carlo technique is used to determine the specific pattern of dimerized bonds which has the largest gain in the magnetic energy. A direct comparison of ground state energies is only possible for a small number of dimerized systems. For this reason an alternative approach is developed, which allows to study a significantly larger class of models. Starting from a Landau expansion of the free energy, this amounts to a computation of specific structure factors of the dimer correlations of the uniform Heisenberg model. As a consequence, a Landau theory for the two-dimensional counterpart of the spin-Peierls transition is established. This adiabatic approach also applies to the Heisenberg model coupled to dispersive phonons. In this case the data for the dimer correlations are used to study the impact of the spin-phonon coupling on the phonon dispersion relations. In particular, the softening of phonon modes is investigated, which is relevant for experimental studies on substances with a quasi two-dimensional magnetic structure.
We use the density-matrix renormalization-group technique developed by White to calculate the spin correlation functions 〈Sn+lz{\mathit{S}}_{\mathit{n}+\mathit{l}}^{\mathit{z}}Sn+lz​ Snz{\mathit{S}}_{\mathit{n}}^{\mathit{z}}Snz​〉=(-1)l{)}^{\mathit{l}})l\ensuremath{\omega}(l,N) for isotropic Heisenberg rings up to N=70 sites. The correlation functions for large l and N are found to obey the scaling relation \ensuremath{\omega}(l,N)=\ensuremath{\omega}(l,\ensuremath{\infty})${\mathit{f}}_{\mathit{XY}}^{\mathrm{\ensuremath{\alpha}}}$(l/N) proposed by Kaplan et al., which is used to determine \ensuremath{\omega}(l,\ensuremath{\infty}). The asymptotic correlation function \ensuremath{\omega}(l,\ensuremath{\infty}) and the magnetic structure factor S(q=\ensuremath{\pi}) show logarithmic corrections consistent with \ensuremath{\omega}(l,\ensuremath{\infty})\ensuremath{\sim}a\ensuremath{\surd}lncl /l, where c is related to the cut-off dependent coupling constant geff{\mathit{g}}_{\mathrm{eff}}geff​(l0{\mathit{l}}_{0}l0​)=1/ln(cl0{\mathit{cl}}_{0}cl0​), as predicted by field theoretical treatments.