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

TL;DR

This work analyzes interacting Majorana chains with finite-range four-fermion interactions, uncovering a robust Ising-to-TCI-to-gapped phase structure and an exact product ground state at $g/t=-0.5$ for odd $\delta$. By mapping the 1D variable-range chains onto 2D lattices of width $W=\frac{\delta+2}{2}$ with swirling helical boundary conditions, the authors reinterpret the $\delta$-dependent phase boundary as a (2+1)D finite-size effect and apply 2D finite-size scaling to extract the infinite-range limit $g_c(\infty)$. Numerical DMRG benchmarks for $\delta=5,7$ locate the tricritical Ising points via universal spectrum ratios, while larger $\delta$ values yield entanglement-based estimates consistent with a shrinking $|g_c|$ as $\delta$ grows. The resulting scaling form $g_c(\delta)=g_c(\infty)+C W^{-2}$ with $W=\frac{\delta+2}{2}$ provides a coherent framework linking 1D long-range physics to a (2+1)D tricritical universality class, offering a path to access infinite-range limits in other long-range 1D systems.

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.

Figures (5)

• Figure 1: Numerically calculated entanglement entropy $S$ as a function of coupling $g$ for several odd interaction ranges $\delta$. A clear feature is the drop to zero entropy at $g/t = -0.5$ for all odd $\delta$ considered, indicating a shared exact product ground state at this point.
• Figure 2: Geometric mapping from a 1D variable-range chain to a 2D locally alike system. (a) A 1D chain with interaction range $\delta$ as in our Hamiltonian. (b) The corresponding 2D swirling lattice constructed by wrapping the chain helically with a period of $\frac{\delta-1}{2}$, $\frac{\delta+1}{2}\cdots$. (c) Corresponding 2D model when apply anti-helical-boundary condition to the swirling chain in (b).
• Figure 3: Linear fit of the finite size critical coupling $\mathrm{g_c}(W)$ against the scaling width $W^{-2}$ and determination of the infinitely long range limit $\mathrm{g_c}(\infty)$
• Figure S1: Finite-size scaling analysis of the spectrum ratios for $\delta=5$ (top row) and $\delta=7$ (bottom row). The left panels show the scan of the selected ratio versus coupling $g$, while the right panels show the comparison of three different ratios against TCI CFT predictions at the determined critical points.
• Figure S2: Diagnostic plot for the central charge approximation. By plotting the subtracted entanglement entropy $S(L) - \frac{0.7}{6}\ln L$, we highlight the estimated bounds of the critical point, represented by the red (upper) and blue (lower) curves. The flat dashed line marks the theoretical reference value ($c=0.7$). The background green lines correspond to other interaction strengths $g$ with a separation of 0.1.