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Tricritical Ising edge modes in a Majorana-Ising ladder

Chengshu Li

Published Jan 1, 2017• 0 citations
Publisher

Abstract

While Majorana fermions remain at large as fundamental particles, they emerge in condensed matter systems with peculiar properties. Grover et al. proposed a Majorana-Ising chain model, or the GSV model, where the system undergoes a tricritical Ising transition by tuning just one parameter. In this work, we generalize this model to a ladder with inter-chain Majorana couplings. From a mean field analysis, we argue that the tricritical Ising transition will also occur with inter-chain couplings that allow the system to be gapless in the non-interacting case. More crucially, based on analysis of the interacting chain model and the non-interacting ladder model, we expect the tricritical Ising modes to appear on the edges, a feature that might persist when going to 2d. We carry out extensive DMRG calculations to verify the theory in the ladder model. Finally, we discuss possible numerical probes of a 2d model.

DOI
10.14288/1.0349175

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•Jan 1, 2021•0 citations
While Majorana fermions remain at large as fundamental particles, they emerge in condensed matter systems with peculiar properties. Grover et al. proposed a Majorana-Ising chain model, or the GSV model, where the system undergoes a tricritical Ising transition by tuning just one parameter. In this work, we generalize this model to a ladder with inter-chain Majorana couplings. From a mean field analysis, we argue that the tricritical Ising transition will also occur with inter-chain couplings that allow the system to be gapless in the non-interacting case. More crucially, based on analysis of the interacting chain model and the non-interacting ladder model, we expect the tricritical Ising modes to appear on the edges, a feature that might persist when going to 2d. We carry out extensive DMRG calculations to verify the theory in the ladder model. Finally, we discuss possible numerical probes of a 2d model.
We introduce a two-ladder lattice model with interacting Majorana fermions that could be realized on the surfaces of a topological insulator film. We study this model with a combination of analytical and numerical techniques, and we find a phase diagram that features both gapless and gapped phases as well as interesting phase transitions including a quantum critical point in the tricritical Ising (TCI) universality class. The latter occurs at an intermediate-coupling strength at a meeting point of a first-order transition line and an Ising critical line, and it is known to be described by a superconformal field theory with central charge c=710c=\frac{7}{10}c=107​. We discuss the experimental feasibility of constructing the model and tuning parameters to the vicinity of the TCI point where signatures of the elusive supersymmetry can be observed.
Tricritical Ising (TCI) phase transition is known to occur in several interacting spin and Majorana fermion models and is described in terms of a supersymmetric conformal field theory (CFT) with central charge c=7/10c=7/10c=7/10. The field content of this CFT is highly nontrivial and includes among its primary fields the Fibonacci anyon, making it of potential interest to strategies seeking to implement fault-tolerant topological quantum computation with non-Abelian phases of matter. In this paper we explore the possibility that a TCI CFT can occur at the edge of a gapped two-dimensional topological state as a stable phase. We discuss a possible realization of this 2D phase based on a coupled-wire construction using the Grover-Sheng-Vishwanath chain model of Majorana zero modes coupled to Ising spins which is known to undergo the TCI phase transition. From the combined analysis using mean-field theory, conformal field theory, and density matrix renormalization group (DMRG) on two- and four-leg ladders, we find that the left- and right-moving gapless TCI modes become spatially separated and reside on two opposite edges of the system, forming a precursor of the required 2D topological phase.
The Hubbard and spinless fermion chains are paradigms of strongly correlated systems, very well understood using the Bethe ansatz, density matrix renormalization group (DMRG), and field theory/renormalization group (RG) methods. They have been applied to one-dimensional materials and have provided important insights for understanding higher-dimensional cases. Recently, an interacting fermion model has been introduced, with possible applications to topological materials. It has a single Majorana fermion operator on each lattice site and interactions with the shortest possible range that involve four sites. We present a thorough analysis of the phase diagram of this model in one dimension using field-theory/RG and DMRG methods. It includes a gapped supersymmetric region and a gapless phase with coexisting Luttinger liquid and Ising degrees of freedom. In addition to a first-order transition, three critical points occur: tricritical Ising, Lifshitz, and a generalization of the commensurate-incommensurate transition. We also survey various gapped phases of the system that arise when the translation symmetry is broken by dimerization and find both trivial and topological phases with 0, 1, and 2 Majorana zero modes bound to the edges of the chain with open boundary conditions.
We analyze a series of interacting Majorana Fermion chains with finite range pair interactions with coupling strength ggg that all exhibit a tri-critical point that separates an Ising critical phase from a supersymmetric gapped phase. We first notice that the interacting models exhibit an even-odd asymmetry depending on the number of sites, δδδ, over which the interaction ranges. The even case exhibits competing order, thereby making it numerically untractable while the odd case exhibits an exactly solvable point at g=−0.5g=-0.5g=−0.5 where the entanglement entropy vanishes. By introducing a swirling geometrical twist, we map our 1D δδδ-range chains to a series of 2D δ/2δ/2δ/2-width models. Our new 2D models possess a unique helical boundary condition, constructed from 1D chains with the end of one connected to the start of another. We propose that the phase transition in the 1D system can be understood as a finite-system size transition in 2D. That is, the gc−δg_c-δgc​−δ behavior is controlled by a 2D tri-critical universality class at δ→∞δ\to\inftyδ→∞ limit and is predicted by finite-size scaling theory.
Majorana fermions have been an important subject of research for the past few years in the field of condensed matter physics. After the realization of Majorana zero mode (MZM) in a Kitaev-chain, studies on the systems of many-body MZMs have been increased. Throughout the previous research, It was found that a few of the Majorana zero mode 1-dimensional chain models possess a Tricritical Ising model conformal field theory as a critical point in their phase diagram. These critical points are supersymmetric and the supersymmetric behaviour remains at a near vicinity of them in their neighbouring gapped phase. This particular behaviour of these 1D strongly-interacting MZMs is believed to be interesting since the system possesses a supersymmetric phase whilst having a nite correlation length in the gapped phase. In this thesis, we are going to investigate the supersymmetric realization of these models in their gapped phase, try to understand their excitation levels with the concept of soliton/antisoliton states and try to find traces of supersymmetry in this system of a condensed matter.