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arXiv:cond-mat/0010418 · Physical review. B, Condensed matter

Nonperturbative effective-field theory for two-leg antiferromagnetic spin ladders

D. C. Cabra, A. Dobry, G. L. Rossini

Published Mar 19, 2001• 5 citations
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Abstract

We study the long wavelength limit of a spin-$\frac{1}{2}$ Heisenberg antiferromagnetic two-leg ladder, treating the interchain coupling in a nonperturbative way. We perform a mean field analysis and then include the fluctuations in an exact way. This allows for a discussion of the phase diagram of the system and provides an effective-field theory for the low-energy excitations. The coset fermionic Lagrangian obtained corresponds to a perturbed ${\mathrm{SU}(4)}_{1}/U(1)$ conformal field theory (CFT). This effective theory is naturally embedded in a ${\mathrm{SU}(2)}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ CFT, where perturbations are easily identified in terms of conformal operators in the two sectors. Crossed and zigzag ladders are also discussed using the same approach.

DOI
10.1103/physrevb.63.144408

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The spin-1/2 Heisenberg spin ladder model is studied by via a perturbation calculation and via bosonization. We mainly focus on the case of ferromagnetic interchain coupling in relation to the Haldane problem. Using the GKO coset construction of the conformal field theory, we bosonize the system in terms of the SU(2) (spin) sector and a residual Ising sector. The interchain interaction generically drives the system to strong-coupling massive phases (the Haldane phase or the dimerized phase). However, it is suggested that the criticality appears for particular choices of the couplings.
We apply perturbation theory to study quantum phase transitions in anisotropic two-leg spin ladders with antiferromagnetic rungs in the strong interchain coupling limit. The energy gap is expanded up to the fourth order in the ratio of the intrachain to interchain couplings. The transition point is determined by examing the disappearance of the energy gap. Present results for the anisotropic ferromagnetic-leg spin ladder are consistent qualitatively with the nonlinear -model analysis and in good agreement with the numerical results by the diagonalization method.
We propose a mechanism of a vector chiral long-range order in two-leg spin-12\frac{1}{2}21​ and spin-1 antiferromagnetic ladders with four-spin exchanges and a Zeeman term. It is known that for one-dimensional quantum systems, spontaneous breakdown of continuous symmetries is generally forbidden. Any vector chiral order hence does not appear in spin-rotationally [SU(2)]-symmetric spin ladders. However, if a magnetic field is added along the Sz{S}^{z}Sz axis of ladders and the SU(2) symmetry is reduced to the U(1) one, the zzz component of a vector chiral order can emerge with the remaining U(1) symmetry unbroken. Making use of Abelian bosonization techniques, we actually show that a certain type of four-spin exchange can yield a vector chiral long-range order in spin-12\frac{1}{2}21​ and spin-1 ladders under a magnetic field. In the chiral-ordered phase, the Z2{Z}_{2}Z2​ interchain-parity (i.e., chain-exchange) symmetry is spontaneously broken. We also consider effects of perturbations breaking the parity symmetry.
This study addresses low-energy properties of two-leg spin-1 ladders with antiferromagnetic (AF) intrachain coupling under a uniform or staggered external field HHH, and a few of their modifications. The generalization to spin-SSS ladders is also discussed. In the strong AF rung(interchain)-coupling ${J}_{\ensuremath{\perp}}$ region, degenerate perturbation theory applied to spin-SSS ladders predicts 2S2S2S critical curves in the parameter space $({J}_{\ensuremath{\perp}},H)$ for the staggered-field case, in contrast to 2S2S2S finite critical regions for the uniform-field case. All critical areas belong to a universality with central charge c=1c=1c=1. On the other hand, we employ Abelian and non-Abelian bosonization techniques in the weak rung-coupling region. They show that in the spin-1 ladder, a sufficiently strong uniform field engenders a c=1c=1c=1 critical state regardless of the sign of ${J}_{\ensuremath{\perp}}$, whereas the staggered field is expected not to yield any singular phenomena. From the bosonization techniques, new field-theoretical expressions of string order parameters in the spin-1 systems are also proposed.
We propose a low-energy effective theory of lightly doped two-leg $t\ensuremath{-}J$ ladders with the help of slave fermion technique. The continuum limit of this model consists of two kinds of Dirac fermions, which are coupled to the O(3) nonlinear sigma model in terms of the gauge coupling with opposite sign of ``charges.'' In addition to the gauge interaction, there is another kind of attractive force between these Dirac fermions, which arises from the short-ranged antiferromagnetic order. We show that the latter is essential to determine the low energy properties of lightly doped two-leg $t\ensuremath{-}J$ ladders. The effective Hamiltonian we obtain is a bosonic Gaussian model and the boson field basically describes the particle-density fluctuation. We also find two types of gapped spin excitations. Finally, we discuss the possible instabilities: charge density wave and singlet superconductivity (SC). We find that the SC instability dominates in our approximation. Our results indicate that lightly doped ladders fall into the universality class of Luther-Emery model.
The critical properties of the S=1/2S=1/2S=1/2 Heisenberg two-leg ladders are investigated in a magnetic field. Combining the exact diagonalization method and the finite-size-scaling analysis based on conformal field theory, we calculate the critical exponents of spin correlation functions numerically. For a strong interchain coupling, magnetization dependence of the critical exponents shows characteristic behavior depending on the sign of the interchain coupling. We also calculate the critical exponents for the S=1/2S=1/2S=1/2 Heisenberg two-leg ladder with a diagonal interaction, which is thought of as a model Hamiltonian of the organic spin ladder compound ${\mathrm{Cu}}_{2}(1,4\ensuremath{-}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{z}\mathrm{a}\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{h}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{e}{)}_{2}{\mathrm{Cl}}_{4}.$ Numerical results are compared with experimental results of temperature dependence of the NMR relaxation rate 1/T1.{1/T}_{1}.1/T1​.