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arXiv:cond-mat/9706104 · Journal of Physics A Mathematical and General

Dualities in spin ladders

Germán Sierra, Miguel A Martin-Delgado

Published Feb 13, 1998• 1 citation
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Abstract

We introduce a set of discrete modular transformations and in order to study the relationships between the different phases of the Heisenberg ladders obtained with all possible exchange coupling constants. For the two-legged ladder we show that the resonating valence bond (RVB) phase is invariant under the transformation, while the Haldane phase is invariant under . These two phases are related by . Moreover, there is a `mixed' phase, that is invariant under , and which under becomes the RVB phase, while under becomes the Haldane phase. For odd ladders there exists only the transformation which, for strong coupling, maps the effective antiferromagnetic spin chain onto the spin chain. Our work is based on a combination of approximate methods such as bosonization, perturbation theory and the nonlinear sigma model, adapted to the different regimes of coupling constants.

DOI
10.1088/0305-4470/31/6/013

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We introduce a set of discrete modular transformations and in order to study the relationships between the different phases of the Heisenberg ladders obtained with all possible exchange coupling constants. For the two-legged ladder we show that the resonating valence bond (RVB) phase is invariant under the transformation, while the Haldane phase is invariant under . These two phases are related by . Moreover, there is a `mixed' phase, that is invariant under , and which under becomes the RVB phase, while under becomes the Haldane phase. For odd ladders there exists only the transformation which, for strong coupling, maps the effective antiferromagnetic spin chain onto the spin chain. Our work is based on a combination of approximate methods such as bosonization, perturbation theory and the nonlinear sigma model, adapted to the different regimes of coupling constants.
The well known Haldane map from spin chains into the O(3) nonlinear sigma model is generalized to the case of spin ladders. This map allows us to explain the different qualitative behaviour between even and odd ladders, in exactly the same way as it explains the difference between integer and half-integer spin chains. Namely, for even ladders the topological term in the sigma model action is absent, while for odd ladders the parameter, which multiplies the topological term, is equal to , where S is the spin of the ladder. Hence even ladders should have a dynamically generated spin gap, while odd ladders with half-integer spin should stay gapless and physically equivalent to a perturbed Wess - Zumino - Witten model in the infrared regime. We also derive some consequences from the dependence of the sigma model coupling constant on the ladder Heisenberg couplings constants.
We map spin ladders with nl{n}_{l}nl​ legs and couplings ${J}^{\ensuremath{'}}$ across all rungs and $J(1\ifmmode\pm\else\textpm\fi{}\ensuremath{\gamma})$ along the legs, staggered in both directions, to a sigma model. Setting its $\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(2m+1)\ensuremath{\pi}$ (where it is known to be gapless), we locate the critical curves in the $\ensuremath{\gamma}$ versus ${J}^{\ensuremath{'}}/J$ plane at each nl{n}_{l}nl​, and spin SSS. The phase diagram is rich and has some surprises: When two gapped chains are suitably coupled, the combination becomes gapless. With nl{n}_{l}nl​, $\ensuremath{\gamma}$, and ${J}^{\ensuremath{'}}/J$ to control, the prospects for experimentally observing any one of these equivalent transitions are enhanced. We interpret our results in the framework of the resonating valence bond description of ladders.
This thesis investigates the ground-state properties and quantum phase diagrams of low-dimensional spin systems, with a focus on spin–1/2 ladder chains and related models. These systems provide a fertile ground for exploring the interplay between magnetic frus- tration, quantum correlations, topology, and external fields. By combining advanced nu- merical methods, primarily the density matrix renormalization group (DMRG), with ana- lytical approaches, we analyze how different coupling geometries and magnetic fields give rise to distinct quantum phases, including magnetization plateaus, symmetry-protected and topological phases, and first-order transitions. The study covers both uniform and alternating ladders, as well as coupled ladder systems that interpolate between one- and two-dimensional behavior. In particular, we examine how frustration influences the stabil- ity of collective spin states, and how interladder couplings can drive transitions between resonating valence bond (RVB) states, Haldane phases, and trivial phases hosting collec- tive spin excitations. Although approximate analytical models, such as hard-core boson mappings and spin-wave theory, capture some qualitative aspects of these systems, our results highlight their limitations in accurately describing critical behavior and phase boundaries, reinforcing the importance of high-precision numerical analysis.
We present a manifestly rotational invariant formulation of the matrix product method valid for spin chains and ladders. We apply it to two-legged spin ladders with spins , 1 and and different magnetic structures labelled by the exchange coupling constants, which can be ferromagnetic or antiferromagnetic along the legs and the rungs of the ladder. We compute ground-state energy densities, correlation lengths and string order parameters. We present numerical evidence of the duality properties of the three different nonferromagnetic spin ladders. We show that the long-range topological order characteristic of isolated spin 1 chains is broken by the interchain coupling. The string order correlation function decays exponentially with a finite correlation length that we compute. A physical picture of the spin 1 ladder is given in terms of a collection of resonating spin 1 chains. Finally, for ladders with spin equal to or greater than we define a class of AKLT states whose matrix product coefficients are given by 9- j symbols.
We analyze the possible existence of topological phases in two-legged spin ladders, considering a staggered interaction in both chains. When the staggered interaction in one chain is shifted by one site with respect to the other chain, the model can be mapped, in the continuum limit, into a nonlinear sigma model $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ plus a topological term which is nonvanishing when the number of legs is two. This implies the existence of a critical point which distinguishes two phases. We perform a numerical analysis of energy levels, parity, and string nonlocal order parameters, correlation functions between x,y,zx,y,zx,y,z components of spins at the edges of an open ladder, the degeneracy of the entanglement spectrum, and the entanglement entropy to characterize these two different phases. We identify one phase with a Mott insulator and the other one with a Haldane insulator.
We study two similar spin ladder systems with the ferromagnetic leg coupling. The first model includes two sorts of spins s=1∕2s=1∕2s=1∕2 and 1, and the second model comprises only s=1∕2s=1∕2s=1∕2 legs coupled by a ``triangular'' rung exchange. We show that the antiferromagnetic (AF) rung coupling destroys the long-range order and eventually makes the systems equivalent to the AF s=1∕2s=1∕2s=1∕2 Heisenberg chain. We study the crossover from the weak- to strong-coupling regime by different methods, particularly by comparing the results of the spin-wave theory and the bosonization approach. We argue that the crossover regime is characterized by the gapless spectrum and nonuniversal critical exponents are different from those in XXZXXZXXZ model.