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Modern Physics Letters B

TWO WEAKLY COUPLED HEISENBERG CHAINS—SOLUTION IN CONTINUOUS LIMIT

Alexei M. Tsvelik

Published Dec 30, 1991• 1 citation
Publisher

Abstract

Using the bosonization procedure proposed for the spin-1/2 Heisenberg chain in Ref. 8 we obtain the effective Hamiltonian for the system of two weakly coupled Heisenberg chains with the XXZ anisotropy as the sum of two sine-Gordon Hamiltonians. It appears that this simple system reveals a rather rich variety of properties in different regions of anisotropy. We calculate the spin-spin correlation functions; two-spin correlation functions decay exponentially but there are regions where the four-spin correlation functions decay as a power law. In particular, if both the inter- and intrachain exchange interactions are isotropic and the interchain exchange is ferromagnetic correlations of the chirality field χ= S 1× S 2 (the indexes 1 and 2 numerate the chains) are enhanced and their correlation functions follow a power law. In this case coupling between pairs of chains can lead to chiral ordering in three dimensions.

DOI
10.1142/s0217984991002379

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Using the bosonization procedure proposed for the spin-1/2 Heisenberg chain in Ref. 8 we obtain the effective Hamiltonian for the system of two weakly coupled Heisenberg chains with the XXZ anisotropy as the sum of two sine-Gordon Hamiltonians. It appears that this simple system reveals a rather rich variety of properties in different regions of anisotropy. We calculate the spin-spin correlation functions; two-spin correlation functions decay exponentially but there are regions where the four-spin correlation functions decay as a power law. In particular, if both the inter- and intrachain exchange interactions are isotropic and the interchain exchange is ferromagnetic correlations of the chirality field χ= S 1× S 2 (the indexes 1 and 2 numerate the chains) are enhanced and their correlation functions follow a power law. In this case coupling between pairs of chains can lead to chiral ordering in three dimensions.
An L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} system of an odd number of coupled Heisenberg spin chains is studied using a degenerate perturbation theory, where L is the number of coupled chains. An effective chain Hamiltonian is derived explicitly in terms of two spin-1/2 degrees of freedom of a closed chain of L sites, valid in the regime where the interchain coupling is stronger than the intrachain coupling. The spin gap has been calculated numerically using the effective Hamiltonian for L=3, 5, 7, and 9 for a finite chain up to ten sites. It is suggested that the ground state of the effective Hamiltonian is correlated, by examining various variational trial states for the effective-spin chain Hamiltonian.
Calculations are presented for the on-site dynamical spin pair correlation of the s=1/2 Heisenberg chain at T=∞. The result for the isotropic chain is consistent with short time analysis. The long time behavior may be interpreted as compatible with exponential decay as well as with power law decay interpretations. The chain with small Ising-like anisotropy also exhibits exponential-like long time decay as found in recent Mössbauer effect spectroscopy work. Preliminary calculations are also presented for the mathematically challenging case of extreme Ising-like anisotropy.
The ground-state phase diagram of a spin S=12XXZS=\frac{1}{2}\phantom{\rule{4pt}{0ex}}\text{XXZ}S=21​XXZ Heisenberg chain with spatially modulated Dzyaloshinskii-Moriya interaction $\mathcal{H}={\ensuremath{\sum}}_{n}{J\left({S}_{n}^{x}{S}_{n+1}^{x}+{S}_{n}^{y}{S}_{n+1}^{y}\right)+{J}_{z}{S}_{n}^{z}{S}_{n+1}^{z}+[{D}_{0}+{(\ensuremath{-}1)}^{n}{D}_{1}]({S}_{n}^{x}{S}_{n+1}^{y}\ensuremath{-}{S}_{n}^{y}{S}_{n+1}^{x})}$ is studied using the continuum-limit bosonization approach and extensive density-matrix renormalization group computations. It is shown that the effective continuum-limit bosonized theory of the model is given by the double-frequency sine-Gordon model (DSG) where the frequencies, i.e., the scaling dimensions of the two competing cosine perturbation terms, depend on the effective anisotropy parameter ${\ensuremath{\gamma}}^{*}={J}_{z}/\sqrt{{J}^{2}+{D}_{0}^{2}+{D}_{1}^{2}}$. Exploring the ground-state properties of the DSG model we show that the zero-temperature phase diagram contains the following four phases: (i) the ferromagnetic phase at ${\ensuremath{\gamma}}^{*}\ensuremath{\le}\ensuremath{-}1$; (ii) the gapless Luttinger-liquid (LL) phase at $\ensuremath{-}1<{\ensuremath{\gamma}}^{*}<{\ensuremath{\gamma}}_{\text{C1}}^{*}=\ensuremath{-}1/\sqrt{2}$; (iii) the gapped composite (C1) phase characterized by coexistence of the long-range-ordered (LRO) dimerization pattern $\ensuremath{\epsilon}\ensuremath{\sim}{(\ensuremath{-}1)}^{n}({\mathbf{S}}_{n}{\mathbf{S}}_{n+1})$ with the LRO alternating spin chirality pattern $\ensuremath{\kappa}\ensuremath{\sim}{(\ensuremath{-}1)}^{n}\left({S}_{n}^{x}{S}_{n+1}^{y}\ensuremath{-}{S}_{n}^{y}{S}_{n+1}^{x}\right)$ at ${\ensuremath{\gamma}}_{\text{C1}}^{*}<{\ensuremath{\gamma}}^{*}<{\ensuremath{\gamma}}_{\text{C2}}^{*}$; and (iv) at ${\ensuremath{\gamma}}^{*}>{\ensuremath{\gamma}}_{\text{C2}}^{*}>1$ the gapped composite (C2) phase characterized in addition to the coexisting spin dimerization and alternating chirality patterns, by the presence of LRO antiferromagnetic order. The transition from the LL to the C1 phase at ${\ensuremath{\gamma}}_{\text{C1}}^{*}$ belongs to the Berezinskii-Kosterlitz-Thouless universality class, while the transition at ${\ensuremath{\gamma}}^{*}={\ensuremath{\gamma}}_{\text{C2}}^{*}$ from C1 to C2 phase is of the Ising type.
In quasi-one-dimensional (q1D) quantum antiferromagnets, the complicated interplay of intrachain and interchain exchange couplings may give rise to rich phenomena. Motivated by recent progress on field-induced phase transitions in the q1D antiferromagnetic (AFM) compound YbAlO3, we study the phase diagram of spin-1/2 Heisenberg chains with Ising anisotropic interchain couplings under a longitudinal magnetic field via large-scale quantum Monte Carlo simulations, and investigate the role of the spin anisotropy of the interchain coupling on the ground state of the system. We find that the Ising anisotropy of the interchain coupling can significantly enhance the longitudinal spin correlations and drive the system to an incommensurate AFM phase at intermediate magnetic fields, which is understood as a longitudinal spin density wave (LSDW). With increasing field, the ground state changes to a canted AFM order with transverse spin correlations. We further provide a global phase diagram showing how the competition between the LSDW and the canted AFM states is tuned by the Ising anisotropy of the interchain coupling.
The properties of antiferromagnetic Heisenberg S= 1/2 ladders with two, three and four chains are expanded in the ratio of the intrachain and interchain coupling constants. A simple mapping procedure is introduced to relate the four- and two-chain ladders, which keeps to moderate values of the expansion parameters. A second-order calculation of the spin gap to the lowest triplet excitation in the two- and four-chain ladders is found to be quite accurate, even at the isotropic point where the couplings are equal. Similar expansions and mapping procedures are presented for the three-chain ladders, which are in the same universality class as single chains.