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

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

doi:10.1103/physrevb.98.014432

Cited by 7 publications

(14 citation statements)

References 43 publications

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“…The point corresponding to the AKLT model sits well inside a non-critical phase, thus confirming that the model is gapped. While many deformations of the AKLT model have been proposed in the literature 9,20,21 , to the best of our knowledge these deformations do not give rise to boundary states related to these loop models, which ap-pear instead for some recently proposed AKLT models on decorated lattices 22,23 .…”

Section: Introductionmentioning

confidence: 80%

Deformations of the Boundary Theory of the Square Lattice AKLT Model

Martyn,

Kato,

Lucia

2019

Preprint

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The 1D AKLT model is a paradigm of antiferromagnetism, and its ground state exhibits symmetry-protected topological order. On a 2D lattice, the AKLT model has recently gained attention because it too displays symmetry-protected topological order, and its ground state can act as a resource state for measurement-based quantum computation if the model is gapped. While the 1D model has been shown to be gapped, it remains an open problem to prove the existence of a spectral gap on the 2D square lattice, which would guarantee the robustness of the resource state. Recently, it has been shown that one can deduce this spectral gap by analyzing the model's boundary theory via a tensor network representation of the ground state. In this work, we express the boundary state of the 2D AKLT model in terms of a classical loop model, where loops, vertices, and crossings are each given a weight. We use numerical techniques to sample configurations of loops and subsequently evaluate the boundary state and boundary Hamiltonian on a square lattice. As a result, we evidence a spectral gap in the square lattice AKLT model. In addition, by varying the weights of the loops, vertices, and crossings, we indicate the presence of three distinct phases exhibited by the classical loop model.

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Section: Introductionmentioning

confidence: 80%

Deformations of the Boundary Theory of the Square Lattice AKLT Model

Martyn,

Kato,

Lucia

2019

Preprint

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“…This results in a two parameters family of states, while the AKLT point corresponds to (a 2 , a 1 , a 0 ) = ( √ 6, 3/2, 1). It is known that there are parameter induced phase transitions as on tunes one of the as within the parameter space [17,18]. The AKLT point is inside the gapped, disordered phase with SPT order and we will denote this phase as the AKLT-phase.…”

Section: Model and Methodsmentioning

confidence: 99%

“…It is straightforward to generalize the valence bond construction of the 1D AKLT state to higher dimensions. In recent years, the 2D generalizations of the AKLT states and their parent Hamiltonians have been actively investigated to understand the nature of the symmetry protected topological order [13][14][15][16][17]. Specifically, the "deformed-AKLT" family of states is a two-parameter family of states on the 2D square lattice, which can be obtained by deforming locally the 2D AKLT state.…”

Section: Introductionmentioning

confidence: 99%

“…As one varies the parameters within the parameter space, the state exhibits parameterinduced phase transitions. It is known that the phase diagram consists of a disordered AKLT phase, a ordered Ferromagnetic (FM) phase (or equivalently a Néel phase), and a critical XY phase [17,18]. Specifically, we use higher order tensor renormalization group (HOTRG) [19] method to coarse-grain the tensor network and to calculate the correlations as well as the higher moments [8].…”

Section: Introductionmentioning

confidence: 99%

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Finite-size scaling analysis of two-dimensional deformed Affleck-Kennedy-Lieb-Tasaki states

Huang

Chen

2020

Phys. Rev. B

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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 states. We use 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 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 2D Ising universality class. We also investigate the Berezinskii-Kosterlitz-Thouless transition from the AKLT phase to the critical 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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“…The point corresponding to the AKLT model sits well inside a noncritical phase, thus confirming that the model is gapped. While many deformations of the AKLT model have been proposed in the literature [9,23,24], to the best of our knowledge these deformations do not give rise to boundary states related to these loop models, which appear instead for some recently proposed AKLT models on decorated lattices [25,26].…”

Section: Introductionmentioning

confidence: 97%

Deformations of the boundary theory of the square-lattice AKLT model

2020

View full text Add to dashboard Cite

The one-dimensional (1D) Affleck-Kennedy-Lieb-Tasaki (AKLT) model is a paradigm of antiferromagnetism, and its ground state exhibits symmetry-protected topological order. On a two-dimensional (2D) lattice, the AKLT model has recently gained attention because it too displays symmetry-protected topological order, and its ground state can act as a resource state for measurement-based quantum computation. While the 1D model has been shown to be gapped, it remains an open problem to prove the existence of a spectral gap on the 2D square lattice, which would guarantee the robustness of the resource state. Recently, it has been shown that one can deduce this spectral gap by analyzing the model's boundary theory via a tensor network representation of the ground state. In this work, we express the boundary state of the 2D AKLT model in terms of a classical loop model, where loops, vertices, and crossings are each given a weight. We use numerical techniques to sample configurations of loops and subsequently evaluate the boundary state and boundary Hamiltonian on a square lattice. As a result, we evidence a spectral gap in the square-lattice AKLT model. In addition, by varying the weights of the loops, vertices, and crossings, we indicate the presence of three distinct phases exhibited by the classical loop model.

show abstract

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

Cited by 7 publications

References 43 publications

Deformations of the Boundary Theory of the Square Lattice AKLT Model

Deformations of the Boundary Theory of the Square Lattice AKLT Model

Finite-size scaling analysis of two-dimensional deformed Affleck-Kennedy-Lieb-Tasaki states

Deformations of the boundary theory of the square-lattice AKLT model

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