Emergent spacetime supersymmetry at 2D fractionalized quantum criticality

TL;DR

The paper shows that emergent spacetime supersymmetry can arise at a fractionalized quantum critical point in a two-dimensional Kitaev honeycomb model coupled to SSH-type spin-phonon interactions. By combining Lieb's theorems to constrain the adiabatic ground state with a low-energy field theory that includes phonon fluctuations, the authors identify a continuous transition from a gapless Dirac spin liquid to a topologically ordered VBS phase, which, at the critical point, flows to an $\mathcal{N}=2$ spacetime SUSY^* fixed point. This fractionalized QCP (the SUSY^* universality class) exhibits distinct finite-size spectra and universal experimental signatures, notably in thermal transport and viscosity. The work provides a concrete lattice realization of emergent supersymmetry in 2D and outlines measurable probes for Kitaev-like materials with strong spin-lattice coupling.

Abstract

While experimental evidence for spacetime supersymmetry (SUSY) in particle physics remains elusive, condensed matter systems offer a promising arena for its emergence at quantum critical points (QCPs). Although there have been a variety of proposals for emergent SUSY at symmetry-breaking QCPs, the emergence of SUSY at fractionalized QCPs remains largely unexplored. Here, we demonstrate emergent space-time SUSY at a fractionalized QCP in the Kitaev honeycomb model with Su-Schrieffer-Heeger (SSH) spin-phonon coupling. Specifically, through numerical computations and analytical analysis, we show that the anisotropic SSH-Kitaev model hosts a fractionalized QCP between a Dirac spin liquid and an incommensurate/commensurate valence-bond-solid phase coexisting with $\mathbb{Z}_2$ topological order. A low-energy field theory incorporating phonon quantum fluctuations reveals that this fractionalized QCP features an emergent $\mathcal{N}=2$ spacetime SUSY. We further discuss their universal experimental signatures in thermal transport and viscosity, highlighting the concrete lattice realization of emergent SUSY at a fractionalized QCP in 2D.

Figures (2)

• Figure 1: (a) The quantum phase diagram in the adiabatic limit. The horizontal and vertical axes represent the anisotropy parameter $J_z/J_{xy}$ and the dimensionless spin-phonon coupling $\lambda$, respectively. For weak coupling $\lambda < \lambda_c^{1}$ (red line), the ground state is a gapless spin liquid with a single Dirac cone. As $\lambda$ increases, the system first enters an incommensurate VBS phase with topological order, and eventually transitions into a columnar VBS phase when $\lambda > \lambda_c^{2}$ (dark blue dashed line). In the incommensurate regime, there inevitably exist regions of commensurate VBS phases whose phase boundaries are difficult to resolve numerically when the periodicity is large. Here we explicitly show the case of period three, corresponding to a Kekulé VBS order, with $\lambda_c^{3}$ marking its phase boundary. The numerical calculations are performed on a finite lattice with $2\times 240 \times 120$ sites, and phase boundaries $\lambda_c^{2,3}$ are identified by comparing the energies of competing phases. $\lambda_c^1$ are identified by the linear extrapolation of the order parameter $\Delta_\text{ivbs}$. (b) The incommensurate VBS order parameter $\Delta_\text{ivbs}\equiv |\tfrac{1}{N}\sum_{i\in A} e^{-2i \tilde{\mathbf{K}} \cdot\mathbf r_i} X_{\langle ij\rangle\in z}|$ near the critical point $\lambda_c^1$, which clearly exhibits a continuous phase transition. The linear onset indicates the presence of $|\Delta_\text{ivbs}|^3$ terms in the free energy, originating from the Dirac cone. (c) Schematic phase diagram at a finite phonon frequency $\omega_{\text{ph}}$. The fractionalized quantum critical line between the gapless QSL and the incommensurate VBS QSL belongs to the $\text{SUSY}^*$ universality class. In the limit $\omega_{\text{ph}} \to \infty$, the spin degrees of freedom are decoupled from phonons.
• Figure S1: Schematic of the honeycomb lattice and its mirror symmetry. $\mathcal{M}_l$ denotes the mirror plane that maps the upper half of the system onto the lower half. According to Lieb’s theorem, the bonds along the vertical direction are equivalent, so the lattice retains translational symmetry in the vertical direction, but with an enlarged unit cell containing four sites. The six bonds in a unit cell are reduced to four independent degrees of freedom, as Lieb’s theorem constrains the values of the $xy$-bonds. When periodic boundary conditions are applied in the vertical direction, the mirror plane appears in pairs ($\mathcal{M}_l, \mathcal{M}_l'$) and also divide the system into two parts that are mapped onto each other by the mirror symmetry.