Emergence of supersymmetry at a critical point of a lattice model

TL;DR

The paper investigates whether supersymmetry can emerge dynamically at a quantum critical point in a condensed-matter lattice system. It introduces a 2+1D lattice model with interacting fermions and bosons, derives the low-energy theory with two Dirac fermions and two bosonic modes, and shows that a finite Yukawa coupling drives the system to a fixed point with emergent Lorentz invariance and decoupled sectors. At this fixed point, the theory consists of two copies of the ${\cal N}=2$ Wess-Zumino theory, yielding exact leading-order anomalous dimensions $\eta_\phi=\eta_\psi=\epsilon/3$ and suggesting the SUSY fixed point persists to all orders in $\epsilon$ for small $\epsilon$. This provides a concrete route to emergent space-time SUSY in a condensed-matter setting and predicts a second-order normal-to-Bose-condensed transition with universal exponents, while also noting caveats about potential multi-criticality or first-order obstructions in 2+1D.

Abstract

Supersymmetry is a symmetry between a boson and a fermion. Although there is no apparent supersymmetry in nature, its mathematical consistency and appealing property have led many people to believe that supersymmetry may exist in nature in the form of a spontaneously broken symmetry. In this paper, we explore an alternative possibility by which supersymmetry is realized in nature, that is, supersymmetry dynamically emerges in the low energy limit of a non-supersymmetric condensed matter system. We propose a 2+1D lattice model which exhibits an emergent space-time supersymmetry at a quantum critical point. It is shown that there is only one relevant perturbation at the supersymmetric critical point in the $ε$-expansion and the critical theory is the two copies of the Wess-Zumino theory with four supercharges. Exact critical exponents are predicted.

Figures (5)

• Figure 1: (a) The lattice structure in the real space. Fermions are defined on the honeycomb lattice and the bosons, on the dual triangular lattice. ${\bf a}_1$, ${\bf a}_2$ are the lattice vectors with length $a$, and ${\bf b}_1$, ${\bf b}_2$, two independent vectors which connect a site on the triangular lattice to the nearest neighbor sites on the honeycomb lattice. (b) The phases of a fermion pair in the real space. (c) The first Brillouin zone in the momentum space. $A$ and $B$ indicate two inequivalent points with momenta ${\bf k}_A = \frac{2 \pi}{a} ( \frac{1}{3}, \frac{1}{\sqrt{3}} )$ and ${\bf k}_B = \frac{2 \pi}{a} ( \frac{2}{3}, 0 )$ where the low energy modes are located. $\psi_1$, $\phi_2$ are located at ${\bf k}_A$ and $\psi_2$, $\phi_1$, at ${\bf k}_B$.
• Figure 2: One-loop diagrams. (a) and (b) are the self-energy corrections of fermion and boson respectively. (c), (d), (e) and (f) contribute to the vertex correction of $\lambda_1$ and (f), (g) and (h), to the vertex correction of $\lambda_2$.
• Figure 3: The schematic RG flow of the bosonic couplings in the subspace of $m=h=0$.
• Figure 4: The schematic RG flows of (a) the velocities with $h \neq 0$ and (b) $\lambda_1$, $\lambda_2$ and $h$ in the subspace of $m=0$. In (b), the solid lines represent the flow in the plane of $(h,\lambda_1)$ and the dashed lines, the flow outside the plane.
• Figure 5: The schematic phase diagram as a function of the ratio of the boson hopping $t_b$ to the on-site boson repulsion energy $U$ for a generic value of $h_0$.