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arXiv:1711.00036 · Physical review. B./Physical review. B

Phase transitions of a two-dimensional deformed Affleck-Kennedy-Lieb-Tasaki model

Nicholas Pomata, Ching-Yu Huang, Tzu-Chieh Wei

Published Jul 27, 2018• 6 citations
arXivHTMLPDF

Abstract

We study spin-2 deformed-AKLT models on the square lattice, specifically a two-parameter family of $O(2)$-symmetric ground-state wave functions as defined by Niggemann, Kl\"umper, and Zittartz, who found previously that the phase diagram consists of a N\'eel-ordered phase and a disordered phase which contains the AKLT point. Using tensor-network methods, we not only confirm the N\'eel phase but also find an XY phase with quasi-long-range order and a region adjacent to it, within the AKLT phase, with very large correlation length, and investigate the consequences of a perfectly factorizable point at the corner of that phase.

DOI
10.1103/physrevb.98.014432

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Using tensor network methods, we perform finite-size scaling analysis to study the parameter-induced phase transitions of two-dimensional deformed Affleck-Kennedy-Lieb-Tasaki (AKLT) states. We use the higher-order tensor renormalization group method to evaluate the moments and the correlations. Then, the critical point and critical exponents are determined simultaneously by collapsing the data. Alternatively, the crossing points of the dimensionless ratios are used to determine the critical point, and the scaling at the critical point is used to determine the critical exponents. For the transition between the disordered AKLT phase and the ferromagnetic (FM) ordered phase, we demonstrate that both the critical point and the exponents can be determined accurately. Furthermore, the values of the exponents confirm that the AKLT-FM transition belongs to the two-dimensional Ising universality class. We also investigate the Berezinskii-Kosterlitz-Thouless transition from the AKLT phase to the critical $\mathit{XY}$ phase. In this case we show that the critical point can be located by the crossing point of the correlation ratio.

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arXiv:1912.10327•Physical review. B./Physical review. B•Jul 10, 2020•1 citation
We study spin-2 deformed-AKLT models on the square lattice, specifically a two-parameter family of O(2)O(2)O(2)-symmetric ground-state wave functions as defined by Niggemann, Kl"umper, and Zittartz, who found previously that the phase diagram consists of a N'eel-ordered phase and a disordered phase which contains the AKLT point. Using tensor-network methods, we not only confirm the N'eel phase but also find an XY phase with quasi-long-range order and a region adjacent to it, within the AKLT phase, with very large correlation length, and investigate the consequences of a perfectly factorizable point at the corner of that phase.
Using tensor network methods, we perform finite-size scaling analysis to study the parameter-induced phase transitions of two-dimensional deformed Affleck-Kennedy-Lieb-Tasaki (AKLT) states. We use the higher-order tensor renormalization group method to evaluate the moments and the correlations. Then, the critical point and critical exponents are determined simultaneously by collapsing the data. Alternatively, the crossing points of the dimensionless ratios are used to determine the critical point, and the scaling at the critical point is used to determine the critical exponents. For the transition between the disordered AKLT phase and the ferromagnetic (FM) ordered phase, we demonstrate that both the critical point and the exponents can be determined accurately. Furthermore, the values of the exponents confirm that the AKLT-FM transition belongs to the two-dimensional Ising universality class. We also investigate the Berezinskii-Kosterlitz-Thouless transition from the AKLT phase to the critical XY\mathit{XY}XY phase. In this case we show that the critical point can be located by the crossing point of the correlation ratio.
Affleck, Kennedy, Lieb, and Taski (AKLT) constructed an exemplary spin-3/2 valence-bond solid (VBS) state on the hexagonal lattice, which is the ground state of an isotropic quantum antiferromagnet and possesses no spontaneous magnetization but finite correlation length. This is distinct from the N\'eel ordered state of the spin-3/2 Heisenberg model on the same lattice. Niggemann, Kl\"umper, and Zittartz then generalized the AKLT Hamiltonian to one family invariant under spin rotation about the zzz axis. The ground states of this family can be parameterized by a single parameter that deforms the AKLT state, and this system exhibits a quantum phase transition between the VBS and N\'eel phases, as the parameter increases from the AKLT point to large anisotropy. We investigate the opposite regime when the parameter decreases from the AKLT point and find that there appears to be a Berezinskii-Kosterlitz-Thouless-like transition from the VBS phase to an XY phase. Such a transition also occurs in the deformation of other types of AKLT states with triplet-bond constructions on the same lattice. However, we do not find such an XY-like phase in the deformed AKLT models on other trivalent lattices, such as square-octagon, cross, and star lattices. On the star lattice, the deformed family of AKLT states remain in the same phase as the isotropic AKLT state throughout the whole region of the parameter. However, for two triplet-bond generalizations, the triplet VBS phase is sandwiched between two ferromagnetic phases (for large and small deformation parameters, respectively), which are characterized by spontaneous magnetizations along different axes. Along the way, we also discuss how various deformed AKLT states can be used for the purpose of universal quantum computation.
The two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) model on a honeycomb lattice has been shown to be a universal resource for quantum computation. In this valence bond solid, however, the spin interactions involve higher powers of the Heisenberg coupling (S[over →](I)·S[over →](j))(n), making these states seemingly unrealistic on bipartite lattices, where one expects a simple antiferromagnetic order. We show that those interactions can be generated by orbital physics in multiorbital Mott insulators. We focus on t(2g) electrons on the honeycomb lattice and propose a physical realization of the spin-3/2 AKLT state. We find a phase transition from the AKLT to the Néel state on increasing Hund's rule coupling, which is confirmed by density matrix renormalization group simulations. An experimental signature of the AKLT state consists of protected, free S=1/2 spins on lattice vacancies, which may be detected in the spin susceptibility.
The models constructed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) [Phys. Rev. Lett. 59, 799 (1987)] describe a family of quantum antiferromagnets on arbitrary lattices, where the local spin SSS is an integer multiple MMM of half the lattice coordination number. The equal-time quantum correlations in their ground states may be computed as finite temperature correlations of a classical O(3)O(3)O(3) model on the same lattice, where the temperature is given by T=1/MT=1/MT=1/M. In dimensions d=1d=1d=1 and 2 this mapping implies that all AKLT states are quantum disordered. We consider AKLT states in d=3d=3d=3 where the nature of the AKLT states is now a question of detail depending upon the choice of lattice and spin; for sufficiently large SSS some form of N\'eel order is almost inevitable. On the unfrustrated cubic lattice, we find that all AKLT states are ordered, while for the unfrustrated diamond lattice the minimal S=2S=2S=2 state is disordered while all other states are ordered. On the frustrated pyrochlore lattice, we find (conservatively) that several states starting with the minimal S=3S=3S=3 state are disordered. The disordered AKLT models we report here are a significant addition to the catalog of magnetic Hamiltonians in d=3d=3d=3 with ground states known to lack order on account of strong quantum fluctuations.
We study S=1 quantum spin systems on the infinite chain with short ranged Hamiltonians that have a certain rotational and discrete symmetry. We define a Z_{2} index for any gapped unique ground state, and prove that it is invariant under a smooth deformation. By using the index, we provide the first rigorous proof of the existence of a "topological" phase transition, which cannot be characterized by any conventional order parameters, between the Affleck-Kennedy-Lieb-Tasaki (AKLT) model and trivial models. This rigorously establishes that the AKLT model is in a nontrivial symmetry protected topological phase.