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.
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.
Two-dimensional topological paramagnets protected by Z3 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.
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 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.
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 study Z3 symmetry-protected topological (SPT) phases in one-dimensional spin systems with ${Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3}$ symmetry. We construct ground-state wave functions of the matrix product form for nontrivial Z3 phases and their parent Hamiltonian from a cocycle of the group cohomology ${H}^{2}({Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3},U(1))$. The Hamiltonian is an SU(3) version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, consisting of bilinear and biquadratic terms of su(3) generators in the adjoint representation. A generalization to the SU(N) case, the SU(N) AKLT Hamiltonian, is also presented, which realizes nontrivial ZN SPT phases. We use the infinite-size variant of the density matrix renormalization group (iDMRG) method to determine the ground-state phase diagram of the SU(3) bilinear-biquadratic model as a function of the parameter $\ensuremath{\theta}$ controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial Z3 SPT phase is found for a range of the parameter $\ensuremath{\theta}$ including the point of vanishing biquadratic term $(\ensuremath{\theta}=0)$ as well as the SU(3) AKLT point [$\ensuremath{\theta}=arctan(2/9)$]. A continuous phase transition to the SU(3) dimer phase takes place at $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.027\ensuremath{\pi}$, with a central charge $c\ensuremath{\approx}3.2$. For SU(3) symmetric cases, we define string order parameters for the Z3 SPT phases in a similar way to the conventional Haldane phase. We propose simple spin models that effectively realize the SU(3) and SU(4) AKLT models.
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.