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Journal of Physics Condensed Matter

Hidden Z 2 *Z 2 symmetry in quantum spin chains with arbitrary integer spin

Masaki Oshikawa

Published Sep 7, 1992• 222 citations
Publisher

Abstract

The author studies integer S>1 spin chains. He extends the Kennedy-Tasaki nonlocal unitary transformation for S=1 to arbitrary integer S. He shows the main results of Kennedy and Tasaki (1992) are maintained for S>1: Heisenberg-type Hamiltonians are transformed to Hamiltonians of nearest-neighbour interactions with Z 2 *Z 2 symmetry, and the den Nijs-Rommelse string observables are transformed to the ferromagnetic correlation observables. He asserts that in general values of integer S there exist several phases with the hidden Z 2 *Z 2 symmetry breaking. The den Nijs-Rommelse string order parameters, which measure the hidden Z 2 *Z 2 symmetry breaking, are calculated explicitly for several variants of the VBS-type states. In the standard VBS state, the hidden Z 2 *Z 2 symmetry breaks down when S is odd but remains unbroken when S is even. His results for partially dimerized VBS states suggest that the hidden Z 2 *Z 2 symmetry breaking can be used to detect the successive dimerization transitions predicted by Affleck and Haldane (1987). Some new anisotropic VBS-type states are investigated. The result suggests that there are successive phase transitions when he increases the uniaxial anisotropy in a Heisenberg-type model. Other new VBS-type states with long-range order are considered, and their relevance to the phase diagram of the Heisenberg XXZ model and the magnetization process of antiferromagnets is investigated. He introduces an extended string order parameter which possesses a characteristic behavior in the partially dimerized VBS states.

DOI
10.1088/0953-8984/4/36/019

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The phase-dependent generalized string order introduced by Oshikawa in the VBS (valence bond solid)-type states of the integer spin chain is extended to the spin-1/2 J - J ′ alternating Heisenberg chain with antiferromagnetic J . The numerical diagonalization and the bosonization analysis show that the generalized string order remains finite in the ground state of the present model for -∞< J ′ < J , taking the maximum at J ′ =0. The characteristic feature of the phase dependence of the generalized string order also coincides with that of the spin-1 VBS state. This reconfirms the notion that the Haldane state of the spin-1 chain and the dimerized state of the alternating chain belong to a single phase characterized by the local singlet-type correlation.

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Hidden Order and Dimerization Transition inS=2Chains

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Phase Diagram of S = 1 Bond-Alternating XXZ Chains Reflecting the Hidden Z2 × Z2 Symmetry

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The phase diagram of the spin-1 bond-alternating XXZ chain is studied numerically. This model is appropriate to study the VBS picture and the hidden Z_2\times Z_2 symmetry concerning to the Haldane gap problem. The possible phases are the ferromagnetic, the XY, the Haldane, the dimer, and the Neel ones. The critical properties of this model can be interpreted as the Ashkin-Teller type reflecting the hidden Z_2\times Z_2 symmetry. The phase transitions are of the Berezinskii-Kosterlitz-Thouless type (XY-Haldane and XY-dimer), of the 2D Gaussian type (Haldane-dimer), and of the 2D Ising type (Haldane-Neel and dimer-Neel). We determine the phase boundary and universality class using the recent developed techniques by the authors.

•Jan 1, 2008•0 citations
The author studies integer S>1 spin chains. He extends the Kennedy-Tasaki nonlocal unitary transformation for S=1 to arbitrary integer S. He shows the main results of Kennedy and Tasaki (1992) are maintained for S>1: Heisenberg-type Hamiltonians are transformed to Hamiltonians of nearest-neighbour interactions with Z 2 *Z 2 symmetry, and the den Nijs-Rommelse string observables are transformed to the ferromagnetic correlation observables. He asserts that in general values of integer S there exist several phases with the hidden Z 2 *Z 2 symmetry breaking. The den Nijs-Rommelse string order parameters, which measure the hidden Z 2 *Z 2 symmetry breaking, are calculated explicitly for several variants of the VBS-type states. In the standard VBS state, the hidden Z 2 *Z 2 symmetry breaks down when S is odd but remains unbroken when S is even. His results for partially dimerized VBS states suggest that the hidden Z 2 *Z 2 symmetry breaking can be used to detect the successive dimerization transitions predicted by Affleck and Haldane (1987). Some new anisotropic VBS-type states are investigated. The result suggests that there are successive phase transitions when he increases the uniaxial anisotropy in a Heisenberg-type model. Other new VBS-type states with long-range order are considered, and their relevance to the phase diagram of the Heisenberg XXZ model and the magnetization process of antiferromagnets is investigated. He introduces an extended string order parameter which possesses a characteristic behavior in the partially dimerized VBS states.
We investigate the string correlation functions proposed by den Nijs and Rommelse for S=1 Heisenberg antiferromagnets in one dimension. The Hamiltonian with D${\mathrm{\ensuremath{\Sigma}}}_{\mathit{i}}$(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})2 term is diagonalized by the Lanczos method to obtain the ground state. We calculate both the usual spin-spin correlation functions and the string correlation functions not only in the z direction (quantized direction) but also in the x direction to investigate the Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ symmetry breaking recently proposed by Kennedy and Tasaki. We find that the long-range string correlation, which is argued to exist in the Haldane disordered phase, in fact, exists at the Heisenberg point D=0 by a finite-size analysis. We can show explicitly that the string correlation in the x direction signifies the difference between the Haldane phase and the N\'eel phase, which appears for the D0, \ensuremath{\Vert}D\ensuremath{\Vert}\ensuremath{\gtrsim}1 case. In the large-D (D\ensuremath{\gtrsim}1) phase, all spin-spin correlations are of short range as expected. There is a significant enhancement in the usual and string correlations in the x direction at the boundary between the Haldane phase and the large-D phase.
The phase-dependent generalized string order introduced by Oshikawa in the VBS (valence bond solid)-type states of the integer spin chain is extended to the spin-1/2 J - J ′ alternating Heisenberg chain with antiferromagnetic J . The numerical diagonalization and the bosonization analysis show that the generalized string order remains finite in the ground state of the present model for -∞< J ′ < J , taking the maximum at J ′ =0. The characteristic feature of the phase dependence of the generalized string order also coincides with that of the spin-1 VBS state. This reconfirms the notion that the Haldane state of the spin-1 chain and the dimerized state of the alternating chain belong to a single phase characterized by the local singlet-type correlation.
We study ground state properties of the S=2S=2S=2 quantum antiferromagnetic chain with a bond alternation \[ H = \sum_{j} [ 1 + δ(-1)^j ] \mbox{\boldmath $S$}_{j} \cdot \mbox{\boldmath $S$}_{j+1} \] by a Quantum Monte Carlo calculation. We find that the hidden Z2×Z2Z_2 \times Z_2Z2​×Z2​ symmetry is broken for 0.3<∣δ∣<0.50.3 < |δ| < 0.50.3<∣δ∣<0.5 while it is unbroken in the other regions. This confirms the successive dimerization transitions first predicted by Affleck and Haldane. Our result shows that these transitions can be understood in terms of the hidden Z2×Z2Z_2 \times Z_2Z2​×Z2​ symmetry breaking, as was discussed using the Valence-Bond-Solid states. Furthermore, we find that the behavior of the generalized string correlation is qualitatively very similar to that in the Valence-Bond-Solid states, including the location of zeroes as a function of the angle parameter.
The qualitative difference in low-energy properties of spin S quantum antiferromagnetic chains with integer S and half-odd-integer S discovered by Haldane [F. D. M. Haldane, arXiv:1612.00076 (1981); Phys. Lett. A 93, 464–468 (1983); Phys. Rev. Lett. 50, 1153–1156 (1983)] and Tasaki [Tasaki, Graduate Texts in Physics (Springer, 2020)] can be intuitively understood in terms of the valence-bond picture proposed by Affleck et al. [I. Affleck, Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477–528 (1988)]. Here, we develop a similarly intuitive diagrammatic explanation of the qualitative difference between chains with odd S and even S, which is at the heart of the theory of symmetry-protected topological (SPT) phases. (There is a 24 min video in which the essence of the present work is discussed: https://youtu.be/URsf9e_PLlc.) More precisely, we define one-parameter families of states, which we call the asymmetric valence-bond solid (VBS) states, that continuously interpolate between the Affleck–Kennedy–Lieb–Tasaki (AKLT) state and the trivial zero state in quantum spin chains with S = 1 and 2. The asymmetric VBS state is obtained by systematically modifying the AKLT state. It always has exponentially decaying truncated correlation functions and is a unique gapped ground state of a short-ranged Hamiltonian. We also observe that the asymmetric VBS state possesses the time-reversal, the Z2×Z2, and the bond-centered inversion symmetries for S = 2 but not for S = 1. This is consistent with the known fact that the AKLT model belongs to the trivial SPT phase if S = 2 and to a nontrivial SPT phase if S = 1. Although such interpolating families of disordered states were already known, our construction is unified and is based on a simple physical picture. It also extends to spin chains with general integer S and provides us with an intuitive explanation of the essential difference between models with odd and even spins.
We generalize the hidden symmetry-breaking picture of symmetry-protected topological (SPT) order developed by Kennedy and Tasaki in the context of the Haldane phase. Our generalization applies to a wide class of SPT phases in one-dimensional spin chains, protected by an on-site representation of a finite Abelian group. This generalization takes the form of a nonlocal unitary map that relates local symmetry-respecting Hamiltonians in an SPT phase to local Hamiltonians in a symmetry-broken phase. Using this unitary, we establish a relation between the two-point correlation functions that characterize fully symmetry-broken phases with the string-order correlation functions that characterize the SPT phases, therefore establishing the perspective in these systems that SPT phases are characterized by hidden symmetry breaking. Our generalization is also applied to systems with continuous symmetries, including SO(2k+1)\mathrm{SO}(2k+1)SO(2k+1) and SU(k)\mathrm{SU}(k)SU(k).
The phase diagram of the spin-1 bond-alternating XXZ chain is studied numerically. This model is appropriate to study the VBS picture and the hidden Z_2\times Z_2 symmetry concerning to the Haldane gap problem. The possible phases are the ferromagnetic, the XY, the Haldane, the dimer, and the Neel ones. The critical properties of this model can be interpreted as the Ashkin-Teller type reflecting the hidden Z_2\times Z_2 symmetry. The phase transitions are of the Berezinskii-Kosterlitz-Thouless type (XY-Haldane and XY-dimer), of the 2D Gaussian type (Haldane-dimer), and of the 2D Ising type (Haldane-Neel and dimer-Neel). We determine the phase boundary and universality class using the recent developed techniques by the authors.