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Physical Review Letters

Phase Transitions in Staggered Spin Ladders

Miguel Angel Martin-Delgado, R. Shankar, Germán Sierra

Published Oct 14, 1996• 67 citations
Publisher

Abstract

We map spin ladders with ${n}_{l}$ 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 ${n}_{l}$, and spin $S$. The phase diagram is rich and has some surprises: When two gapped chains are suitably coupled, the combination becomes gapless. With ${n}_{l}$, $\ensuremath{\gamma}$, 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.

DOI
10.1103/physrevlett.77.3443

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We map spin ladders with nl{n}_{l}nl​ legs and couplings J\ensuremath′{J}^{\ensuremath{'}}J\ensuremath′ across all rungs and J(1\ifmmode±\else\textpm\fi\ensuremathγ)J(1\ifmmode\pm\else\textpm\fi{}\ensuremath{\gamma})J(1\ifmmode±\else\textpm\fi\ensuremathγ) along the legs, staggered in both directions, to a sigma model. Setting its \ensuremathθ=(2m+1)\ensuremathπ\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(2m+1)\ensuremath{\pi}\ensuremathθ=(2m+1)\ensuremathπ (where it is known to be gapless), we locate the critical curves in the \ensuremathγ\ensuremath{\gamma}\ensuremathγ versus J\ensuremath′/J{J}^{\ensuremath{'}}/JJ\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​, \ensuremathγ\ensuremath{\gamma}\ensuremathγ, and J\ensuremath′/J{J}^{\ensuremath{'}}/JJ\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.
Motivated by near-term experiments with ultracold alkaline-earth atoms confined to optical lattices, we establish numerically and analytically the phase diagram of two-leg SU(N)\mathrm{SU}(N)SU(N) spin ladders. Two-leg ladders provide a rich and highly nontrivial extension of the single chain case on the way towards the relatively little explored two-dimensional situation. Focusing on the experimentally relevant limit of one fermion per site, antiferromagnetic exchange interactions, and $2\ensuremath{\le}N\ensuremath{\le}6$, we show that the phase diagrams as a function of the interchain (rung) to intrachain (leg) coupling ratio, ${J}_{\ensuremath{\perp}}/{J}_{\ensuremath{\parallel}}$, strongly differ for even versus odd NNN. For even N=4N=4N=4 and 6, we demonstrate that the phase diagram consists of a single valence bond crystal (VBC) with a spatial period of N/2N/2N/2 rungs. For odd N=3N=3N=3 and 5, we find surprisingly rich phase diagrams exhibiting three distinct phases. For weak rung coupling, we obtain a VBC with a spatial period of NNN rungs, whereas for strong coupling we obtain a critical phase related to the case of a single chain. In addition, we encounter intermediate phases for odd NNN, albeit of a different nature for N=3N=3N=3 as compared to N=5N=5N=5. For N=3N=3N=3, we find a novel gapless intermediate phase with ${J}_{\ensuremath{\perp}}$-dependent incommensurate spatial fluctuations in a sizeable region of the phase diagram. For N=5N=5N=5, there are strong indications for a narrow potentially gapped intermediate phase, whose nature is not entirely clear. Our results are based on (i) field theoretical techniques, (ii) qualitative symmetry considerations, and (iii) large-scale density matrix renormalization group (DMRG) simulations keeping beyond a million of states by fully exploiting and thus preserving the SU(N)\mathrm{SU}(N)SU(N) symmetry.
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