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arXiv:2002.02543 · Annales Henri Poincaré

Dimerization and Néel Order in Different Quantum Spin Chains Through a Shared Loop Representation

Michael Aizenman, Hugo Duminil-Copin, Simone Warzel

Published Jun 16, 2020• 10 citations
arXivHTMLPDF

Abstract

Abstract The ground-states of the spin-Santiferromagnetic chain $$H_{{\text {AF}}}$$ HAF with a projection-based interaction and the spin-1/2 XXZ-chain $$ H_{{\text {XXZ}}}$$ HXXZ at anisotropy parameter $$\Delta =\cosh (\lambda ) $$ Δ=cosh(λ) share a common loop representation in terms of a two-dimensional functional integral which is similar to the classical planarQ-state Potts model at $$ \sqrt{Q}= 2S+1 =2\cosh (\lambda )$$ Q=2S+1=2cosh(λ) . The multifaceted relation is used here to directly relate the distinct forms of translation symmetry breaking which are manifested in the ground-states of these two models: dimerization for $$H_{{\text {AF}}}$$ HAF at all $$S> 1/2$$ S>1/2 , and Néel order for $$ H_{{\text {XXZ}}} $$ HXXZ at $$\lambda >0$$ λ>0 . The results presented include: (i) a translation to the above quantum spin systems of the results which were recently proven by Duminil–Copin–Li–Manolescu for a broad class of two-dimensional random-cluster models, and (ii) a short proof of the symmetry breaking in a manner similar to the recent structural proof by Ray–Spinka of the discontinuity of the phase transition for $$Q>4$$ Q>4 . Altogether, the quantum manifestation of the change between $$Q=4$$ Q=4 and $$Q>4$$ Q>4 is a transition from a gapless ground-state to a pair of gapped and extensively distinct ground-states.

DOI
10.1007/s00023-020-00924-2

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Dimerization in O(n)O(n)O(n)-invariant quantum spin chains
We establish dimerization in O(n)O(n)O(n)-invariant quantum spin chains with large enough nnn, in a much larger region of the phase diagram than previous results. This entails identifying two distinct infinite volume ground states which are gapped, are translations by one unit of each other, and which both have exponentially decaying correlations. We show these states are the limits of finite-volume, finite temperature states as the graph tends to Z\mathbb{Z}Z and temperature tends to zero in any order. Our method relies on a probabilistic representation of the quantum system in terms of random loops, and an adaptation of a method developed for loop O(n)O(n)O(n) models on the hexagonal lattice by Duminil-Copin, Peled, Samotij and Spinka.
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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