Lattice supersymmetry and order-disorder coexistence in the tricritical Ising model

TL;DR

This work constructs and analyzes a lattice spin/Majorana chain in the tricritical Ising universality class, achieving an explicit lattice realization of supersymmetry alongside order–disorder coexistence. The Hamiltonian is written as $H=(Q^+)^2+(Q^-)^2=2\lambda_I H_I+\lambda_3 H_3+E_0$, with explicit lattice SUSY generators $Q^\pm$ and, on the lattice, currents $G_j$, $\overline{G}_j$, $\psi_j$, and $\overline{\psi}_j$ whose correlators reproduce the continuum scaling dimensions at the tricritical point (e.g., $\Delta_G=\Delta_{\overline G}=3/2$, $\Delta_\psi=\Delta_{\overline\psi}=7/10$). A self-dual, frustration-free point yields exact three ground states, including two ordered and one disordered, demonstrating robust order–disorder coexistence in a gapped phase. A chiral deformation produces a line where lattice supersymmetry is exact, and the tricritical point merges with this line, signaling a transition toward an incommensurate phase with $p^3$ dispersion. The results provide a concrete, nonperturbative lattice realization of tricritical Ising SUSY and offer a platform to study SUSY, duality, and topological order on the lattice.

Abstract

We introduce and analyze a quantum spin/Majorana chain with a tricritical Ising point separating a critical phase from a gapped phase with order-disorder coexistence. We show that supersymmetry is not only an emergent property of the scaling limit, but manifests itself on the lattice. Namely, we find explicit lattice expressions for the supersymmetry generators and currents. Writing the Hamiltonian in terms of these generators allows us to find the ground states exactly at a frustration-free coupling. These confirm the coexistence between two (topologically) ordered ground states and a disordered one in the gapped phase. Deforming the model by including explicit chiral symmetry breaking, we find the phases persist up to an unusual chiral phase transition where the supersymmetry becomes exact even on the lattice.

Figures (6)

• Figure 1: Phase diagram and RG flows of (\ref{['Hdef']}).
• Figure 2: The ratios $R_1$,$R_2$ and $R_3$ from the DMRG. Black, green, red and mauve points are $\lambda_3/\lambda_I=0,0.8,0.856,0.87$ respectively, while blue and yellow in the inset are $0.855$ and $0.857$. The green and purple lines denote the theoretical values for the Ising and TCI CFTs respectively.
• Figure 3: Dependence of $C_{jk}=\langle\psi_j\psi_k\rangle$ (yellow crosses) and $\langle G_jG_k\rangle$ (blue crosses) on $|j-k|$ at the tricritical point $\lambda_3/\lambda_I=.856$. The purple and red lines correspond to $A|j-k|^{-1.4}$ and $B|j-k|^{-3}$ respectively, where $A$ and $B$ are fitting parameters.
• Figure 4: Phase diagram including chiral interactions, with $\lambda_I+\lambda_3+\lambda_c=1$. The diagram is invariant under sending $\lambda_c\to -\lambda_c$.
• Figure 5: Locations of the TCI transition in phase space for $\lambda_3$ as a function of $\lambda_R$ with $\lambda_y=0$ (blue dots), and a function of $\lambda_y$ with $\lambda_R=0$ (orange crosses).