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Journal of Physics A Mathematical and Theoretical

Duality and ground-state phase diagram for the quantum XYZ model with arbitrary spin s in one spatial dimension

Qian‐Qian Shi, Sheng-Hao Li, Huan-Qiang Zhou

Published Feb 21, 2020• 4 citations
Publisher

Abstract

Abstract Five duality transformations are unveiled for the quantum XYZ model with arbitrary spin s in one spatial dimension. The presence of these duality transformations, together with an extra Hamiltonian symmetry, drastically reduce the entire ground-state phase diagram to two finite regimes—the principal regimes, with all the other ten regimes dual to them. Combining with the determination of critical points from the conventional order parameter approach and/or the fidelity approach to quantum phase transitions, we are able to map out the ground-state phase diagram for the quantum XYZ model with arbitrary spin s . This is explicitly demonstrated for s = 1 / 2 , 1 , 3 / 2 and 2. As it turns out, all the critical points, with central charge c = 1 , are self-dual for half-integer as well as integer spin s . However, in the latter case, the presence of the Haldane phase results in extra lines of critical points with central charge c = 1 / 2 , which are not self-dual.

DOI
10.1088/1751-8121/ab78cd

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Abstract Five duality transformations are unveiled for the quantum XYZ model with arbitrary spin s in one spatial dimension. The presence of these duality transformations, together with an extra Hamiltonian symmetry, drastically reduce the entire ground-state phase diagram to two finite regimes—the principal regimes, with all the other ten regimes dual to them. Combining with the determination of critical points from the conventional order parameter approach and/or the fidelity approach to quantum phase transitions, we are able to map out the ground-state phase diagram for the quantum XYZ model with arbitrary spin s . This is explicitly demonstrated for s = 1 / 2 , 1 , 3 / 2 and 2. As it turns out, all the critical points, with central charge c = 1 , are self-dual for half-integer as well as integer spin s . However, in the latter case, the presence of the Haldane phase results in extra lines of critical points with central charge c = 1 / 2 , which are not self-dual.
Duality and ground-state phase diagram for the quantum XYZ model with arbitrary spin sss in one spatial dimension
Five duality transformations are unveiled for the quantum XYZ model with arbitrary spin sss in one spatial dimension. The presence of these duality transformations drastically reduces the entire ground-state phase diagram to two {\it finite} regimes - the principal regimes, with all the other ten regimes dual to them. Combining with the determination of critical points from the conventional order parameter approach and/or the fidelity approach to quantum phase transitions, we are able to map out the ground-state phase diagram for the quantum XYZ model with arbitrary spin sss. This is explicitly demonstrated for s=1/2,1,3/2s=1/2,1,3/2s=1/2,1,3/2 and 2. As it turns out, all the critical points, with central charge c=1c=1c=1, are self-dual under a respective duality transformation for half-integer as well as integer spin sss. However, in the latter case, the presence of the so-called symmetry protected topological phase, i.e., the Haldane phase, results in extra lines of critical points with central charge c=1/2c=1/2c=1/2, which is not self-dual under any duality transformation.
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.
The phase diagrams of general qqq-state planar spin models on a square lattice are studied for $q=4\ensuremath{-}7$. The phase diagrams are calculated by a variational method and the infinitesimal Migdal renormalization-group (RG) transformations. Since the Migdal RG commutes with the self-dual transformations of Z(q)Z(q)Z(q) models, one expects reliable phase diagrams (but not accurate critical exponents). In fact all exactly known critical points are reproduced by our calculations together with a fair numerical simulation of lines of critical points. Our results clearly indicate the existence of massless phases for q=6and7q=6 \mathrm{and} 7q=6and7 but are inconclusive for q=5q=5q=5. More accurate methods are needed to determine the phase structure of the Z(5)Z(5)Z(5) model; see, for example, the finite-lattice approximation of Roomany and Wyld (preceding paper).
The ground-state phase diagram is obtained for an antiferromagnetic spin-1 anisotropic biquadratic model. With the help of symmetry and duality transformations, three symmetry-protected trivial phases and one dimerized symmetry-breaking phase are found. Local and nonlocal order parameters are identified to characterize these phases. Quantum phase transitions between the symmetry-protected phases belong to the Gaussian universality class with central charge c=1c=1c=1, and quantum phase transitions from the symmetry-protected trivial phases to the dimerized phase belong to the Ising universality class with central charge c=1/2c=1/2c=1/2. In addition, the model admits three characteristic lines of factorized ground states, which are located in the symmetry-protected trivial phases instead of a symmetry-breaking phase, in sharp contrast to other known cases.
Abstract We provide several examples and an intuitive diagrammatic representation demonstrating the use of two-qubit unitary transformations for mapping coupled spin Hamiltonians to simpler ones and vice versa . The corresponding dualities may be exploited to identify phase transition points or to aid the diagonalization of such Hamiltonians. For example, our method shows that a suitable one-parameter family of coupled Hamiltonians whose ground states transform from an initially factorizing state to a final cluster state on a lattice of arbitrary dimension is dual to a family of trivial decoupled Hamiltonians containing local on-site terms only. As a consequence, the minimum energy gap (which determines the adiabatic run-time) does not scale with system size, which facilitates an efficient and simple adiabatic preparation of e.g. the two-dimensional cluster state used for measurement-based quantum computation.
In quantum spin-1 chains, there is a nonlocal unitary transformation known as the Kennedy-Tasaki transformation UKT{U}_{\mathrm{KT}}UKT​, which defines a duality between the Haldane phase and the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry-breaking phase. In this paper, we find that UKT{U}_{\mathrm{KT}}UKT​ also defines a duality between a topological Ising critical phase and a trivial Ising critical phase, which provides a ``hidden symmetry breaking'' interpretation of the topological criticality. Moreover, since the duality relates different phases of matter, we argue that a model with self-duality (i.e., invariant under UKT{U}_{\mathrm{KT}}UKT​) is natural to be at a critical or multicritical point. We study concrete examples to demonstrate this argument. In particular, when HHH is the Hamiltonian of the spin-1 antiferromagnetic Heisenberg chain, we prove that the self-dual model H+UKTHUKTH+{U}_{\mathrm{KT}}H{U}_{\mathrm{KT}}H+UKT​HUKT​ is exactly equivalent to a gapless spin-1/21/21/2 XY chain, which also implies an emergent quantum anomaly. On the other hand, we show that the topological and trivial Ising critical phases that are dual to each other meet at a multicritical point which is indeed self-dual.