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Investigation of supersymmetry on a strongly-interacting Majorana zero mode chain

Mobin Shakeri

Published Jan 1, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.14288/1.0402631

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•Jan 1, 2021•0 citations
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.
A strongly interacting chain of Majorana fermions realizes the supersymmetric tricritical Ising phase, with supersymmetry (SUSY) extending into a symmetry-broken ordered phase adjacent to the tricritical point. Although the signatures of SUSY at the tricritical point are well understood, their behavior in the gapped phase remains less clear. Here, we address two key questions: how SUSY manifests in the gapped phase and what is the nature of the excitations in this phase. We show that, in the thermodynamic limit, a conventional SUSY diagnostic that remains finite at the tricritical point diverges immediately on the Ising side, yet decays continuously to zero deeper in the gapped phase, signaling the persistence of SUSY. Focusing on the lowest excited states in the supersymmetric gapped regime, we find that the excitations consist of soliton-antisoliton pairs separating distinct ordered regions. Each soliton binds an emergent localized Majorana mode, and together the pair forms a nonlocal Dirac fermion. The occupation of this Dirac mode distinguishes eigenstates with even and odd fermion parity.
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.
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.
Lattice models with supersymmetry are known to exhibit a variety of remarkable properties that do not exist in the relativistic models. In this paper, we introduce an interacting generalization of the Kitaev chain of Majorana fermions with N=1\mathcal{N}=1N=1 supersymmetry and investigate its low-energy properties, paying particular attention to the ground-state degeneracy and low-lying fermionic excitations. First, we establish the existence of a phase with spontaneously broken supersymmetry and a phase transition out of it with the help of variational arguments and the exact ground state. We then develop, based on the superfield formalism, a simple mean-field theory, in which the order parameters detect supersymmetry breaking, to understand the ground-state phases and low-lying Nambu-Goldstone fermions. At the solvable point (frustration-free point), the exact ground state of an open chain exhibits large degeneracy of the order of the system size, which is attributed to the existence of a zero-energy domain wall (dubbed kink or skink) separating the topological and trivial states of Majorana fermions. Our results may shed new light on the intriguing ground-state properties of supersymmetric lattice models.
Recent decades have seen a proliferation of unconventional quasiparticles in condensed matter systems. Majorana fermions, theoretically predicted in several setups and of great interest in topological quantum computation, are a focus of intense research efforts. Higher-spin moments, relevant for both solid-state compounds and cold atoms, are of great theoretical interest as they interpolate between the quantum and classical limits but sometimes show surprising behavior. In the meantime, successful fabrication and characterization of low-dimensional systems have brought new phenomenology and physical insights. In this dissertation I will theoretically explore a few low-dimensional models using Majorana fermions and higher-spin moments as building blocks. I will first discuss a generalized family of the celebrated Sachdev-Ye-Kitaev model, a zero-dimensional all-to-all Majorana model that exhibits non-Fermi liquid behavior and is holographically dual to a black hole. The generalized model has a phase transition between a non-Fermi liquid and a disordered Fermi liquid. Then I will discuss the Heisenberg model with higher spins, with a focus on chaos and information scrambling. Using matrix-product-state-based methods, we are able to obtain numerical results for spin up to 4 and characterize the Lyapunov growth. After that I will discuss a generalization of the Hubbard model to Majorana fermions on the honeycomb lattice. Unlike previous similar models, we find topological phases with (anti-)chiral edge modes for weak interaction. Finally I will show a construction of explicit supersymmetric Majorana model on the kagome lattice, where a family of exact solutions is found and the nature of supersymmetry breaking is explored.