We construct local generalizations of 3-state Potts models with exotic critical points. We analytically show that these are described by non-diagonal modular invariant partition functions of products of Z_3 Z 3 parafermion or u(1)_6 u ( 1 ) 6 conformal field theories (CFTs). These correspond either to non-trivial permutation invariants or block diagonal invariants, that one can understand in terms of anyon condensation. In terms of lattice parafermion operators, the constructed models correspond to parafermion chains with many-body terms. Our construction is based on how the partition function of a CFT depends on symmetry sectors and boundary conditions. This enables to write the partition function corresponding to one modular invariant as a linear combination of another over different sectors and boundary conditions, which translates to a general recipe how to write down a microscopic model, tuned to criticality. We show that the scheme can also be extended to construct critical generalizations of k k -state clock type models.
We construct local generalizations of 3-state Potts models with exotic critical points. We analytically show that these are described by non-diagonal modular invariant partition functions of products of Z_3 Z 3 parafermion or u(1)_6 u ( 1 ) 6 conformal field theories (CFTs). These correspond either to non-trivial permutation invariants or block diagonal invariants, that one can understand in terms of anyon condensation. In terms of lattice parafermion operators, the constructed models correspond to parafermion chains with many-body terms. Our construction is based on how the partition function of a CFT depends on symmetry sectors and boundary conditions. This enables to write the partition function corresponding to one modular invariant as a linear combination of another over different sectors and boundary conditions, which translates to a general recipe how to write down a microscopic model, tuned to criticality. We show that the scheme can also be extended to construct critical generalizations of k k -state clock type models.
Abstract We formulate a ℤk-parafermionization/bosonization scheme for one-dimensional lattice models and field theories on a torus, starting from a generalized Jordan-Wigner transformation on a lattice, which extends the Majorana-Ising duality atk= 2. The ℤk-parafermionization enables us to investigate the critical theories of parafermionic chains whose fundamental degrees of freedom are parafermionic, and we find that their criticality cannot be described by any existing conformal field theory. The modular transformations of these parafermionic low-energy critical theories as general consistency conditions are found to be unconventional in that their partition functions on a torus transform differently from any conformal field theory whenk >2. Explicit forms of partition functions are obtained by the developed parafermionization for a large class of critical ℤk-parafermionic chains, whose operator contents are intrinsically distinct from any bosonic or fermionic model in terms of conformal spins and statistics. We also use the parafermionization to exhaust all the ℤk-parafermionic minimal models, complementing earlier works on fermionic cases.
In this thesis exotic phases of models preserving the symmetries of the one-dimensional quantum Ising and three-state Potts chains are considered, the natural habitats of Majorana fermions and parafermions, respectively. Along with numerical techniques, conformal field theory (CFT) is used to study the resulting physics. Chapter 1 gives an introduction to the field and explains the basics of the numerical techniques to be used throughout the thesis. CFT is briefly introduced in Chapter 2 before the key results necessary for the thesis are given. In Chapter 3, self-dual interactions are added to the critical Ising model, giving a range of different phases. A tricritical Ising point separates the Ising phase from an order-disorder coexistence phase on the chirally symmetric line, while a supersymmetric line provides the transition to an incommensurate phase in the general model. The model is solved exactly on a free-fermionic line giving insight into the points where only numerical analysis can be performed. This is supplemented by Appendix A, where the qualitative difference between similar perturbations is considered. Chapters 4 and 5 consider models with the symmetries of the three-state Potts model. Various phases are found including an order-disorder coexistence phase, a second critical Potts phase, a "not-A" phase, and a representation symmetry protected topological phase. The effect of lattice momentum on the stability of phases and the connection with conformal field theory operators is studied. Lattice realizations of several CFTs are found, such as the tricritical Potts CFT and a CFT expressible in two different forms, either in terms of a boson and fermion, or Potts and tricritical Ising CFTs.
Finding the precise correspondence between lattice operators and the continuum fields that describe their long-distance properties is a largely open problem for strongly interacting critical points. Here, we solve this problem essentially completely in the case of the three-state Potts model, which exhibits a phase transition described by a strongly interacting 'parafermion' conformal field theory. Using symmetry arguments, insights from integrability, and extensive simulations, we construct lattice analogues of nearly all the relevant and marginal physical fields governing this transition. This construction includes chiral fields such as the parafermion. Along the way we also clarify the structure of operator product expansions between order and disorder fields, which we confirm numerically. Our results both suggest a systematic methodology for attacking non-free field theories on the lattice and find broader applications in the pursuit of exotic topologically ordered phases of matter.
Abstract We explore the topological defects of the critical three-state Potts spin system on the torus, Klein bottle and cylinder. A complete characterization is obtained by breaking down the Fuchs–Runkel–Schweigert construction of 2D rational CFT to the lattice setting. This is done by applying the strange correlator prescription to the recently obtained tensor network descriptions of string-net ground states in terms of bimodule categories (Lootens et al 2021 SciPost Phys. 10 053). The symmetries are represented by matrix product operators (MPO), as well as intertwiners between the diagonal tetracritical Ising model and the non-diagonal three-state Potts model. Our categorical construction lifts the global transfer matrix symmetries and intertwiners, previously obtained by solving Yang–Baxter equations, to MPO symmetries and intertwiners that can be locally deformed, fused and split. This enables the extraction of conformal characters from partition functions and yields a comprehensive picture of all boundary conditions.
We calculated the spectral properties of two related families of non-Hermitian free-particle quantum chains with N-multispin interactions (N=2,3,…). The first family have a Z(N) symmetry and are described by free parafermions. The second one has a U(1) symmetry and are generalizations of XX quantum chains described by free fermions. The eigenspectra of both free-particle families are formed by the combination of the same pseudo-energies. The models have a multicritical point with dynamical critical exponent z=1. The finite-size behavior of their eigenspectra, as well as the entanglement properties of their ground state wave function, indicate the models are conformally invariant. The models with open and periodic boundary conditions show quite distinct physics due to their non-Hermiticity. The models defined with open boundaries have a single conformal invariant phase while the XX multispin models show multiple phases with distinct conformal central charges in the periodic case. The critical exponents of the models are calculated for N=3,4,5 and 6.
In this work, we numerically study critical phases in translation-invariant ZN parafermion chains with both nearest- and next-nearest-neighbor hopping terms. The model can be mapped to a ZN spin model with nearest-neighbor couplings via a generalized Jordan-Wigner transformation and translational invariance ensures that the spin model is always self-dual. We first study the low-energy spectrum of chains with only nearest-neighbor coupling, which are mapped onto standard self-dual ZN clock models. For $3\ensuremath{\le}N\ensuremath{\le}6$, we match the numerical results to the known conformal field theory(CFT) identification. We then analyze in detail the phase diagram of a N=3 chain with both nearest and next-nearest-neighbor hopping and six critical phases with central charges being 4/5, 1, or 2 are found. We find continuous phase transitions between c=1 and 2 phases, while the phase transition between c=4/5 and 1 is conjectured to be of Kosterlitz-Thouless type.