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Physical Review Letters

Quantum liquid in antiferromagnetic chains: A stochastic geometric approach to the Haldane gap

Hal Tasaki

Published Feb 11, 1991• 172 citations
Publisher

Abstract

The S=1 quantum antiferromagnetic chain with the Hamiltonian H= ${\mathcal{J}}_{\mathit{i}}$ ${\mathrm{S}}_{\mathit{i}}^{\mathit{x}}$${\mathit{S}}_{\mathit{i}+1}^{\mathit{x}}$+${\mathit{S}}_{\mathit{i}}^{\mathit{y}}$${\mathit{S}}_{\mathit{i}+1}^{\mathit{y}}$+\ensuremath{\lambda}${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$ ${\mathit{S}}_{\mathit{i}+1}^{\mathit{z}}$+D(${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$${)}^{2}$ is studied. By developing a (path-integral-type) stochastic geometric representation and using the ideas of percolation, we find that the Haldane phase with a unique disordered ground state and a gap can be regarded as ``liquid''. The large-D phase and the N\'eel ordered phase are identified as ``gas'' and ``solid'', respectively. We introduce a new order parameter that distinguishes the Haldane phase from other disordered phases such as the large-D phase or the dimer phase.

DOI
10.1103/physrevlett.66.798

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The phase diagram of the quantum spin-12\frac{1}{2}21​ antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XXZ chain was obtained by Haldane using bosonization techniques [Haldane, Phys. Rev. B 25, 4925 (1982); Phys. Rev. B 26, 5257 (1982)]. It supports three distinct phases for $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$, i.e., a gapless algebraic spin-liquid phase, a gapped long-range ordered Neel phase, and a gapped long-range ordered dimer phase. Even though the Neel and dimer phases are not related hierarchically by a pattern of symmetry breaking, it was shown that they meet along a line of quantum critical points with a U(1) symmetry and central charge c=1c=1c=1. Here, we extend the analysis made by Haldane on the quantum spin-12\frac{1}{2}21​ antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XYZ chain using both bosonization and numerical techniques. We show that there are three Neel phases and the dimer phase that are separated from each other by six planes of phase boundaries realizing Gaussian criticality when $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$. We also show that each long-range ordered phase harbors topological point defects (domain walls) that are dual to those across the phase boundary in that a defect in one ordered phase locally binds the other type of order around its core. By using the bosonization approach, we identify the critical theory that describes simultaneous proliferation of these dual point defects, and show that it supports an emergent U(1) symmetry that originates from the discrete symmetries of the XYZ model. To confirm this numerically, we perform exact diagonalization and density-matrix renormalization-group calculations and show that the critical theory is characterized by the central charge c=1c=1c=1 with critical exponents that are consistent with those obtained from the bosonization approach. Furthermore, we generalize the field-theoretic description of direct continuous phase transition to higher dimensions, especially in d=3d=3d=3, by using a nonlinear sigma model (NLSM) with a topological term. In particular, we propose the $\ensuremath{\pi}$-flux phase on the cubic lattice with local quartic interactions as a platform for direct continuous phase transition and deconfined criticality. We discuss possible phase diagrams for the $\ensuremath{\pi}$-flux phase on the cubic lattice with these quartic interactions from the renormalization flow of NLSMs.
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