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arXiv:2312.15095 · SciPost Physics

Two-dimensional topological paramagnets protected by $\mathbb{Z}_3$ symmetry: Properties of the boundary Hamiltonian

Hrant Topchyan, Vasilii Iugov, Mkhitar Mirumyan, Tigran Hakobyan, Tigran Sedrakyan, A. Sedrakyan

Published Feb 25, 2025• 1 citation
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Abstract

We systematically study gapless edge modes corresponding to \mathbb{Z}_3 ℤ 3 symmetry-protected topological (SPT) phases of two-dimensional three-state Potts paramagnets on a triangular lattice. First, we derive microscopic lattice models for the gapless edge and, using the density-matrix renormalization group (DMRG) approach, investigate the finite-size scaling of the low-lying excitation spectrum and the entanglement entropy. Based on the obtained results, we identify the universality class of the critical edge, namely the corresponding conformal field theory and the central charge. Finally, we discuss the inherent symmetries of the edge models and the emergent winding number symmetry. As a result, one-dimensional chains with this symmetry form a model that supports gapless excitations due to its tricritical symmetry. Numerically, we show that low-energy states in the continuous limit of the edge model can be described by conformal field theory (CFT) with central charge c=1 c = 1 , given by the coset SU_k(3)/SU_k(2) S U k ( 3 ) / S U k ( 2 ) CFT at level k=1.

DOI
10.21468/scipostphys.18.2.068

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Two-dimensional topological paramagnets protected by Z3\mathbb{Z}_3Z3​ symmetry: Properties of the boundary Hamiltonian
We systematically study gapless edge modes corresponding to \mathbb{Z}_3 ℤ 3 symmetry-protected topological (SPT) phases of two-dimensional three-state Potts paramagnets on a triangular lattice. First, we derive microscopic lattice models for the gapless edge and, using the density-matrix renormalization group (DMRG) approach, investigate the finite-size scaling of the low-lying excitation spectrum and the entanglement entropy. Based on the obtained results, we identify the universality class of the critical edge, namely the corresponding conformal field theory and the central charge. Finally, we discuss the inherent symmetries of the edge models and the emergent winding number symmetry. As a result, one-dimensional chains with this symmetry form a model that supports gapless excitations due to its tricritical symmetry. Numerically, we show that low-energy states in the continuous limit of the edge model can be described by conformal field theory (CFT) with central charge c=1 c = 1 , given by the coset SU_k(3)/SU_k(2) S U k ( 3 ) / S U k ( 2 ) CFT at level k=1.
We systematically study gapless edge modes corresponding to Z(3) symmetry-protected topological (SPT) phases of two-dimensional three-state Potts paramagnets on a triangular lattice. First, we derive microscopic lattice models for the gapless edge and, using the density-matrix renormalization group (DMRG) approach, investigate the finite-size scaling of the low-lying excitation spectrum and the entanglement entropy. Based on the obtained results, we identify the universality class of the critical edge, namely the corresponding conformal field theory and the central charge. Finally, we discuss the inherent symmetries of the edge models and the emergent winding number symmetry. As a result, one-dimensional chains with this symmetry form a model that supports gapless excitations due to its tricritical symmetry. Numerically, we show that low-energy states in the continuous limit of the edge model can be described by conformal field theory (CFT) with central charge c=1, given by the coset SUk(3)/SUk(2) CFT at level k=1.
Abstract We identify two-dimensional three-state Potts paramagnets with gapless edge modes on a triangular lattice protected by (×Z3)3 ≡ Z3 × Z3 × Z3 symmetry and smaller Z3 symmetry. We derive microscopic models for the gapless edge, uncover their symmetries and analyze the conformal properties. We study the properties of the gapless edge by employing the numerical density-matrix renormalization group (DMRG) simulation and exact diagonalization. We discuss the corresponding conformal field theory, its central charge, and the scaling dimension of the corresponding primary field. We argue, that the low energy limit of our edge modes defined by the SUk(3)/SUk(2) coset conformal field theory with the level k = 2. The discussed two-dimensional models realize a variety of symmetry-protected topological phases, opening a window for studies of the unconventional quantum criticalities between them.
We construct a two-dimensional bosonic symmetry-protected topological (SPT) paramagnet protected by an on-site G = Z 4 × 3 symmetry, starting from a three-component Z 4 Potts paramagnet on a triangular lattice. Within the group-cohomology framework, H 3 [ G , U ( 1 ) ] ≅ Z 4 × 7 , we focus on a colorless cocycle representative obtained by antisymmetrizing the basic Z 4 three-cocycle, and generate the corresponding SPT Hamiltonian via a cocycle-induced local unitary transformation H 3 [ G , U ( 1 ) ] ≅ Z 4 × 7 followed by symmetry averaging. For open geometry, we derive the boundary theory explicitly: one color sector decouples, while the nontrivial edge reduces to an interacting Z 4 chain with next-nearest-neighbor constraints that admits a compact dressed-Potts form. Using DMRG we show that the boundary model is gapless, with the lowest gap scaling as 1 / L and an entanglement-entropy scaling consistent with a conformal field theory of central charge c = 2.191 ( 4 ) ≃ 11 / 5 . The rational value c = 11 / 5 matches the coset S U ( 3 ) 3 / S U ( 2 ) 3 , making it a candidate for the continuum description of the Z 4 × 3 edge; we outline spectral and symmetry-resolved diagnostics needed to test this identification at the level of conformal towers beyond the central charge.
We investigate the continuum limit of the edge-state Hamiltonian of a two-dimensional symmetry-protected topological (SPT) model with (×Z₃)³ = Z³ × Z³ × Z³ symmetry. Starting from the lattice Hamiltonian, we first reformulate the model in terms of S3 generators and subsequently introduce a constrained Schwinger-like fermionic representation. Within this formulation, the relevant operators are expressed as fermionic bilinears, while non-local Jordan–Wigner string operators do not appear explicitly. At the critical point, λ = λ̄, we show that the continuum Hamiltonian supports gapless excitations with a linear dispersion relation, consistent with a conformal description of the edge theory. Using a mean-field treatment of the interaction terms, we further obtain a preliminary estimate of the finite-size correction to the ground-state energy. The corresponding effective central charge, c = √3 ≈ 1.73, is remarkably close to the numerical value previously obtained for the (×Z₃)³ SPT edge model and to the value predicted by the SU(3)2/SU(2)2 coset conformal field theory. Although this derivation remains heuristic, our results suggest that the constrained fermionic formulation provides a natural framework for describing the low-energy continuum limit of (Z₃)³ SPT edge states and may serve as a useful starting point for a more systematic derivation of the underlying conformal field theory.
Effective field theory is a very useful technique for understanding quantum many body systems. We use this approach to study a certain class of critical quantum spin-1 chains and symmetry protected topological phases of matter in two spatial dimensions. In the first context we consider the SU(3) symmetric spin-1 chain with additional interactions. We use bosonization to demonstrate that the effective field theories describing the critical behaviour of these spin chains can be mapped to free compact boson conformal field theories (CFTs) with central charge c=2. We also describe how some predictions from field theory can be verified in numerical calculations using exact diagonalization (ED) and the density matrix renormalization group (DMRG) algorithm. In particular, the bosonization method gives a formula for the evolution of four Tomonaga-Luttinger liquid (TLL) parameters as a function of the lattice parameters. Using the analytic formulae for the scaling dimensions in terms of the TLL parameters and matching of lowest scaling dimensions, we numerically calculate these field-theoretic parameters and confirm that their evolution agrees with the prediction using bosonization. We also tackle aspects of the challenging problem of understanding interacting topological phases. In this context, we aim to understand the effects if interactions in certain classes of symmetry protected topological (SPT) phases of matter. We consider non-chiral SPT phases in two spatial dimensions protected by a discrete symmetry such as Z_K or Z_K x Z_K symmetry. We argue that modular invariance/noninvariance of the partition function of the one-dimensional edge theory can be used to diagnose whether, by adding a suitable potential, the edge theory can be gapped or not without breaking the symmetry. By taking bosonic phases described by Chern-Simons K-matrix theories and fermionic phases relevant to topological superconductors as examples, we demonstrate explicitly that when modular invariance is achieved, we can construct an interaction potential that is consistent with the symmetry and can completely gap out the edge. We also briefly discuss preliminary results of a numerical approach for simulating the 3 dimensional Landau Level problem. This includes a brief description of a potential application of variational Monte Carlo to spin-orbit coupled ab initio systems.
We report that a spin-1/2 tetrahedral Heisenberg chain realizes a gapless symmetry-protected topological (gSPT) phase characterized by the coexistence of the Tomonaga-Luttinger-liquid criticality due to chirality degrees of freedom and the symmetry-protected edge state due to spin degrees of freedom. This gSPT phase has an interesting feature that no symmetry forbids the trivial spin gap opening but a discrete symmetry, Z3×Z2T\mathbb Z_3\times\mathbb Z_2^TZ3​×Z2T​, forbids the unique gapped ground state. In the first part of the paper, we numerically show the coexistence of a critical entanglement entropy and a nontrivially degenerate entanglement spectrum based on the density-matrix renormalization group (DMRG) method.Next, we clarify that chirality degrees of freedom form the Tomonaga-Luttinger liquid while spin degrees of freedom form the spin-1 Haldane state based on a degenerate perturbation theory. Last but not least, we discuss the Lieb-Schultz-Mattis-type ingappability in the gSPT phase, using a local Z3\mathbb{Z}_3Z3​ rotation. We can thus characterize our gSPT phase as a symmetry-protected critical phase protected by the Z3\mathbb{Z}_3Z3​ on-site symmetry, the Z2T\mathbb{Z}_2^TZ2T​ time-reversal symmetry, the lattice translation symmetry, and the U(1) spin-rotation symmetry.