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.
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.
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.
It is well known that the noninteracting Majorana chain is dual to the one-dimensional transverse-field Ising model, either through the Jordan–Wigner transformation or by gauging fermion parity. In this correspondence, the minimal translation of the Majorana chain maps to the celebrated Kramers–Wannier (KW) duality of the spin model, with the critical point mapped to the self-dual point. In this work, we generalize this mapping to two and higher dimensions by constructing a unitary equivalence between the parity-gauged fermionic system and a spin system defined on arbitrary polyhedral decompositions of space. Imposing the flatness condition on the gauge field yields a bosonization duality between the original (ungauged) fermionic system and a gauged spin system obeying a Gauss law. The dependence of the Gauss law in the spin system on the Kasteleyn orientation (and the discrete spin structure) of the fermionic system is made explicit. Applying this bosonization to one or two copies of Majorana fermions on translationally invariant lattices, we derive higher-dimensional analogs of KW (self-) dualities in spin systems arising from fermionic minimal translations. The KW (self-) dualities are non-invertible due to projections onto eigenspaces of higher-form symmetries in the associated symmetry operators. The bosonization framework we present is intuitive, general, and systematic, encompassing other known exact bosonization methods while offering a novel approach to establish new connections between fermionic and spin systems in arbitrary dimensions.
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.
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.
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.
Abstract Free theories are usually regarded as trivial examples of quantum systems. This chapter proves that this is not the case of the conformal field theories associated to the free bosonic and fermionic fields. The subject is not only full of beautiful mathematical identities but is also the source of deep physical concepts with far reaching applications. Chapter 12 also covers quantization of the bosonic field, vertex operators, the free bosonic field on a torus, modular transformations, the quantization of the free Majorana fermion, the Neveu–Schwarz and Ramond sectors, fermions on a torus, calculus for anti-commuting quantities and partition functions.