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arXiv:2512.04180

2D Helical Twist Controls Tricritical Point in an Interacting Majorana Chain

Zhao, Hekai, Philip Phillips

Published Dec 3, 2025• 0 citations
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Abstract

We analyze a series of interacting Majorana Fermion chains with finite range pair interactions with coupling strength $g$ 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, $δ$, 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 $g=-0.5$ where the entanglement entropy vanishes. By introducing a swirling geometrical twist, we map our 1D $δ$-range chains to a series of 2D $δ/2$-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 $g_c-δ$ behavior is controlled by a 2D tri-critical universality class at $δ\to\infty$ limit and is predicted by finite-size scaling theory.

DOI
10.48550/arxiv.2512.04180

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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.
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.
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.
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 explore critical properties of a chain of interacting Majorana fermions, particles that are their own antiparticles. We study the combined effect of two competing interaction terms of the shortest possible range and show this results in a very rich phase diagram with nine different phases, five of which are critical. In addition, we report a wide variety of quantum phase transitions: the tri-critical Ising lines; the Lifshitz critical line characterized by the dynamical critical exponent z=3z=3z=3; two Kosterlitz-Thouless transitions; and an exotic first-order transition between the floating and the gapped phases. However, the most surprising result is the emergence of the commensurate line at which the floating phases collapse into direct transition. We provide numerical evidences that the resulting multicritical point belongs to the universality class of the eight-vertex model. Implications in the context of supersymmetric properties of the Majorana chain are briefly discussed.
The twofold twist defects in the D(Zk)D({\mathbb{Z}}_{k})D(Zk​) quantum double model (Abelian topological phase) carry non-Abelian fractional Majorana-like characteristics. We align these twist defects in a line and construct a one-dimensional Hamiltonian which only includes the pairwise interaction. For the defect chain with an even number of twist defects, it is equivalent to the Zk{\mathbb{Z}}_{k}Zk​ clock model with a periodic boundary condition (up to some phase factor for the boundary term), while for the odd number case, it maps to the Zk{\mathbb{Z}}_{k}Zk​ clock model with a duality twisted boundary condition. At the critical point, for both cases, the twist defect chain enjoys an additional translation symmetry, which corresponds to the Kramers-Wannier duality symmetry in the Zk{\mathbb{Z}}_{k}Zk​ clock model and can be generated by a series of braiding operators for twist defects. We further numerically investigate the low energy excitation spectrum for k=3,4,5k=3,4,5k=3,4,5, and 6. For the even-defect chain, the critical points are the same as the Zk{\mathbb{Z}}_{k}Zk​ clock conformal field theories (CFTs), while for the odd-defect chain, when $k\ensuremath{\ne}4$, the critical points correspond to orbifolding a Z2{\mathbb{Z}}_{2}Z2​ symmetry of CFTs of the even-defect chain. For k=4k=4k=4 case, we numerically observe some similarity to the Z4{\mathbb{Z}}_{4}Z4​ twist fields in the SU(2)1/D4SU{(2)}_{1}/{D}_{4}SU(2)1​/D4​ orbifold CFT.