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arXiv:1801.08548 · Physical review. B./Physical review. B

Nontopological parafermions in a one-dimensional fermionic model with even multiplet pairing

Leonardo Mazza, Fernando Iemini, Marcello Dalmonte, Christophe Mora

Published Nov 14, 2018• 30 citations
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Abstract

Topological order and spontaneous symmetry breaking are typically thought of as paradigmatic, yet antipodal, phenomena in correlated quantum matter. Here, the authors show instead that they can concomitantly act to determine the ground-state properties of interacting systems, providing a novel angle for the discussion of topological phenomena and parafermions in one-dimensional fermionic systems. They show that topological Majorana modes can coexist with \ensuremath{\mathbb{Z}}${}_{n}$ spontaneous symmetry breaking by discussing a lattice model of one-dimensional spinless fermions and thoroughly characterize the presence of nontopological parafermions.

DOI
10.1103/physrevb.98.201109

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•Jan 1, 2016•0 citations
Topological order and spontaneous symmetry breaking are typically thought of as paradigmatic, yet antipodal, phenomena in correlated quantum matter. Here, the authors show instead that they can concomitantly act to determine the ground-state properties of interacting systems, providing a novel angle for the discussion of topological phenomena and parafermions in one-dimensional fermionic systems. They show that topological Majorana modes can coexist with \ensuremath{\mathbb{Z}}n{}_{n}n​ spontaneous symmetry breaking by discussing a lattice model of one-dimensional spinless fermions and thoroughly characterize the presence of nontopological parafermions.
Parafermions are the simplest generalizations of Majorana fermions that realize topological order. We propose a less restrictive notion of topological order in one-dimensional open chains, which generalizes the seminal work by Fendley [J. Stat. Mech. (2012) P11020]. The first essential property is that the ground states are mutually indistinguishable by local, symmetric probes, and the second is a generalized notion of zero edge modes which cyclically permute the ground states. These two properties are shown to be topologically robust, and applicable to a wider family of topologically ordered Hamiltonians than has been previously considered. As an application of these edge modes, we formulate a notion of twisted boundary conditions on a closed chain, which guarantees that the closed-chain ground state is topological, i.e., it originates from the topological manifold of the open chain. Finally, we generalize these ideas to describe symmetry-breaking phases with a parafermionic order parameter. These exotic phases are condensates of parafermion multiplets, which generalize Cooper pairing in superconductors. The stability of these condensates is investigated on both open and closed chains.
We discuss a one-dimensional fermionic model with a generalized ZN\mathbb{Z}_{N}ZN​ even multiplet pairing extending Kitaev Z2\mathbb{Z}_{2}Z2​ chain. The system shares many features with models believed to host localized edge parafermions, the most prominent being a similar bosonized Hamiltonian and a ZN\mathbb{Z}_{N}ZN​ symmetry enforcing an NNN-fold degenerate ground state robust to certain disorder. Interestingly, we show that the system supports a pair of parafermions but they are non-local instead of being boundary operators. As a result, the degeneracy of the ground state is only partly topological and coexists with spontaneous symmetry breaking by a (two-particle) pairing field. Each symmetry-breaking sector is shown to possess a pair of Majorana edge modes encoding the topological twofold degeneracy.
Parafermions are emergent excitations that generalize Majorana fermions and can also realize topological order. In this Letter, we present a nontrivial and quasi-exactly-solvable model for a chain of parafermions in a topological phase. We compute and characterize the ground-state wave functions, which are matrix-product states and have a particularly elegant interpretation in terms of Fock parafermions, reflecting the factorized nature of the ground states. Using these wave functions, we demonstrate analytically several signatures of topological order. Our study provides a starting point for the nonapproximate study of topological one-dimensional parafermionic chains with spatial inversion and time-reversal symmetry in the absence of strong edge modes.
The Majorana lattice gauge theory purely composed of Majorana fermions on square lattice is studied throughly. The ground state is obtained exactly and exhibits the coexistence of symmetry breaking and topological order. The Z_2 Z 2 symmetry breaking of matter fields leads to the intertwined antiferromagnetic spin order and \eta η -pairing order. The topological order is reflected in the Z_2 Z 2 quantum spin liquid ground state of gauge fields. The Majorana lattice gauge theory, alternatively can be viewed as interacting Majorana fermion model, is possibly realized on a Majorana-zero-mode lattice.
Parafermion zero modes are generalizations of Majorana modes that underlie comparatively rich non-Abelian-anyon properties. We introduce exact mappings that connect parafermion chains, which can emerge in two-dimensional fractionalized media, to strictly one-dimensional fermionic systems. In particular, we show that parafermion zero modes in the former setting translate into symmetry-enriched Majorana modes that intertwine with a bulk order parameter---yielding braiding and fusion properties that are impossible in standard Majorana platforms. Fusion characteristics of symmetry-enriched Majorana modes are directly inherited from the associated parafermion setup and can be probed via two kinds of anomalous pumping cycles that we construct. Most notably, our mappings relate Z4{\mathbb{Z}}_{4}Z4​ parafermions to conventional electrons with time-reversal symmetry. In this case, one of our pumping protocols entails fairly minimal experimental requirements: Cycling a weakly correlated wire between a trivial phase and time-reversal-invariant topological superconducting state produces an edge magnetization with quadrupled periodicity. Our work highlights new avenues for exploring beyond-Majorana physics in experimentally relevant one-dimensional electronic platforms, including proximitized ferromagnetic chains.
The concept of topology in condensed matter physics has led to the discovery of rich and exotic physics in recent years. Especially when strong correlations are included, phenomenons such as fractionalization and anyonic particle statistics can arise. In this thesis, we study several systems hosting topological phases of interacting fermions. In the first part, we consider one-dimensional systems of parafermions, which are generalizations of Majorana fermions, in the presence of a Z_N charge symmetry. We classify the symmetry-protected topological (SPT) phases that can occur in these systems using the projective representations of the symmetries and find a finite number of distinct phases depending on the prime factorization of N. The different phases exhibit characteristic degeneracies in their entanglement spectrum (ES). Apart from these SPT phases, we report the occurrence of parafermion condensate phases for certain values of N. When including an additional Z_N symmetry, we find a non-Abelian group structure under the addition of phases. In the second part of the thesis, we focus on two-dimensional lattice models of spinless fermions. First, we demonstrate the detection of a fractional Chern insulator (FCI) phase in the Haldane honeycomb model on an infinite cylinder by means of the density-matrix renormalization group (DMRG). We report the calculation of several quantities characterizing the topological order of the state, i.e., (i)~the Hall conductivity, (ii)~the spectral flow and level counting in the ES, (iii)~the topological entanglement entropy, and (iv)~the charge and topological spin of the quasiparticles. Since we have access to sufficiently large system sizes without band projection with DMRG, we are in addition able to investigate the transition from a metal to the FCI at small interactions which we find to be of first order. In a further study, we consider a time-reversal symmetric model on the honeycomb lattice where a Chern insulator (CI) induced by next-nearest neighbor interactions has been predicted by mean field theory. However, various subsequent studies challenged this picture and it was still unclear whether the CI would survive quantum fluctuations. We therefore map out the phase diagram of the model as a function of the interactions on an infinite cylinder with DMRG and find evidence for the absence of the CI phase. However, we report the detection of two novel charge-ordered phases and corroborate the existence of the remaining phases that had been predicted in mean field theory. Furthermore, we characterize the transitions between the various phases by studying the behavior of correlation length and entanglement entropy at the phase boundaries. Finally, we develop an improvement to the DMRG algorithm for fermionic lattice models on cylinders. By using a real space representation in the direction along the cylinder and a real space representation in the perpendicular direction, we are able to use the momentum around the cylinder as conserved quantity to reduce computational costs. We benchmark the method by studying the interacting Hofstadter model and report a considerable speedup in computation time and a severely reduced memory usage.