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Physical review. B, Condensed matter

Real-space renormalization-group study of the phase diagram of two coupled quantum spin-1/2 chains

Ulrich Schollwöck, D. Y. K. Ko

Published Jan 1, 1996• 8 citations
Publisher

Abstract

We investigate the phase diagram of two coupled quantum spin-1/2 chains, a square ladder, allowing for anisotropies both in the interchain and intrachain coupling, using the recently introduced density-matrix renormalization-group (DMRG) algorithm. We consider open chains up to a length of 2\ifmmode\times\else\texttimes\fi{}200, while retaining up to 120 states in the truncated basis of the Hilbert space. The phase diagram is discussed in connection with exact results and bosonization analysis. Numerically, we show that the crossover from the critical behavior of the uncoupled disordered spin-1/2 chain to a gapped quantum liquid sets in for infinitesimal interchain coupling, both for ferromagnetic and antiferromagnetic interchain coupling. The Haldane phase of the coupled chains is, even for small interchain couplings, generically described by the Affleck-Kennedy-Lieb-Tasaki model. We also discuss typical shortcomings of the DMRG for some choices of the interaction parameters. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.240

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We study the opening of the Haldane gap in two-leg and four-leg anisotropic spin ladders using bosonization and renormalization group methods, and we determine the phase diagram as a function of the interchain coupling and relative anisotropy Jz/Jxy.{J}^{z}{/J}^{\mathrm{xy}}.Jz/Jxy. It is found that the opening of the Haldane gap is qualitatively different for the two cases considered. For the two-leg ladder the Haldane gap opens for arbitrarily small interchain coupling, independent of Jz/Jxy,{J}^{z}{/J}^{\mathrm{xy}},Jz/Jxy, and the Haldane phase exists in a large region of parameter space. For the four-leg ladder the opening of the Haldane gap is strongly dependent on both the interchain coupling as well as Jz/Jxy,{J}^{z}{/J}^{\mathrm{xy}},Jz/Jxy, and the Haldane phase exists only in a narrow region about the isotropic antiferromagnet.
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The effect of antiferromagnetic interchain coupling in alternating spin-(1,1/2) chains is studied by means of spin-wave theory and density-matrix renormalization group (DMRG). In particular, two limiting cases are investigated, the two-leg ladder and its two-dimensional (2D) generalization. Results of the ground-state properties like energy, spin gap, magnetizations, and correlation functions are reported for the whole range of the interchain coupling ${J}_{\ensuremath{\perp}}.$ For the 2D case the spin-wave results predict a smooth dimensional crossover from 1D to 2D keeping the ground state always ordered. For the ladder system, the DMRG results show that any ${J}_{\ensuremath{\perp}}>0$ drives the system to a gapped ground state. Furthermore the behavior of the correlation functions closely resemble the uniform spin-1/2 ladder. For ${J}_{\ensuremath{\perp}}$ lower than 0.3, however, the gap behaves quadratically as $\ensuremath{\Delta}\ensuremath{\sim}{0.6J}_{\ensuremath{\perp}}^{2}.$ Finally, it is argued that the behavior of the spin gap for an arbitrary number of mixed coupled spin chains is analogous to that of the uniform spin-1/2 chains.
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We consider hardcore bosons in two coupled chains of one dimensional lattices at half filling with repulsive intrachain interaction and interchain attraction. This can be mapped onto a coupled chain of spin-1/2 XXZXXZXXZ model with interchain ferromagnetic coupling. We investigate various phases of hardcore bosons (and related spin model) at zero temperature by a density matrix renormalization group method. Apart from the usual superfluid and density wave phases, pairing of interchain bosons leads to the formation of phases like pair superfluid and density wave of strongly bound pairs. We discuss the possible experimental realization of such correlated phases in the context of cold dipolar gas.
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