Emergent Space-time Supersymmetry at the Boundary of a Topological Phase

TL;DR

This work demonstrates that space-time supersymmetry can emerge at the boundary quantum critical points of time-reversal invariant topological superconductors and related topological insulators. It develops a field-theoretic framework for 1+1D and 2+1D boundaries and validates the 1+1D SUSY scenario with DMRG on a lattice model that exhibits the tricritical Ising universality with central charge $c=7/10$. In 2+1D, an ε-expansion RG analysis uncovers a SUSY fixed point with anomalous dimensions $ ext{η}_χ= ext{η}_φ= ext{ε}/7$ and emergent Lorentz invariance; for TI surfaces, exact anomalous dimensions $ ext{η}_φ= ext{η}_ψ=1/3$ are predicted within a $ ext{N}=2$ Wess-Zumino framework. The results point to experimental routes, such as He$_3$-B films, and reveal a deep connection between topological phases and supersymmetry with precise, testable predictions.

Abstract

In contrast to ordinary symmetries, supersymmetry interchanges bosons and fermions. Originally proposed as a symmetry of our universe, it still awaits experimental verification. Here we theoretically show that supersymmetry emerges naturally in topological superconductors, which are well-known condensed matter systems. Specifically, we argue that the quantum phase transitions at the boundary of topological superconductors in both two and three dimensions display supersymmetry when probed at long distances and times. Supersymmetry entails several experimental consequences for these systems, such as, exact relations between quantities measured in disparate experiments, and in some cases, exact knowledge of the universal critical exponents. The topological surface states themselves may be interpreted as arising from spontaneously broken supersymmetry, indicating a deep relation between topological phases and SUSY. We discuss prospects for experimental realization in films of superfluid He$_3$-B.

Figures (5)

• Figure 1: Phase diagram of a three dimensional topological superconductor, as Ising magnetic fluctuations (denoted by red arrows) at the boundary couple to the Majorana fermions (blue cone). When the tuning parameter $r < r_c$, the Ising spins order leading to a gap for the Majorana fermions. In the main text, it is argued that the critical point that separates the two sides is supersymmetric, where bosons (Ising order parameter) and Majorana fermions transform into each other. Similar phase diagram is obtained for two-dimensional topological superconductors (Fig.\ref{['fig:phasedia2d']}).
• Figure 2: (a) The phase diagram of the Hamiltonian $H$ in Eqn.\ref{['eq:lattice1d']}, which realizes the Majorana edge of a two dimensional time-reversal invariant topological superconductor coupled to Ising magnetic fluctuations. The vertical axis $g$ is the coupling between the Majorana fermions and the Ising order parameter. At large $h$, the Ising spins disorder and the counter-propagating Majorana modes remain gapless. As $h$ decreases, the ordering of Ising spins leads to a gap for the Majorana modes. The black arrows indicate the region of the phase diagram detailed in Fig.\ref{['fig:phasedia2d']}(b). (b) Central charge $c$ as a function of $h$, for fixed $g = 0.5$. For $h > h_c\,(=1.62)$, one finds $c = 1/2$, consistent with gapless Majorana modes whole for $h < h_c$, one finds $c = 0$ indicating the gapped phase. The critical point separating the two phases is characterized by $c = 7/10$ which corresponds to supersymmetric tri-critical Ising point. The inset shows the von Neumann entropy $S$ and the Renyi entropies $S_2, S_3$ at the critical point, which were used to deduce the central charge.
• Figure 3: Entanglement entropy at the critical point for the 1+1-D lattice model for the parameter $\mathcal{J} = 4$. The paramater $\mathcal{J}$ equals the ratio of the bare verlocity of the Majorana fermion to that of the boson $\phi$. The above curve shows that the supersymmetric critical point with central charge $c= 7/10$ survives even when the velocity anisotropy is four. The red crosses are the numerical data while the green curve is the theoretical expected result for central charge $c = 7/10$. The inset shows the Renyi entropies $S_n$ which also fit perfectly to $c = 7/10$.
• Figure 4: (a) Mean-field phase diagram for a thin film of superfluid He$_3$-B as a magnetic field parallel to the surface is applied. The $\hat{\bf n}$ vector (green arrow) is oriented along the vertical direction, perpendicular to the surface. A weak field applied in plane does not open a gap to the majorana fermions on the boundary, since a residual symmetry, composed of 180 degree rotations and time reversal is present, that preserves the gapless surface state. However, on increasing the field above $H_c\approx 30 {\rm Gauss}$, the $\hat{\bf n}$ vector spontaneously tilts into the plane leading to a gap on the surface and hence a compensating energy gain. (b) For this transition to occur spontaneously on the two surfaces independently, one needs to consider films much thicker than the $\hat{\bf n}$ healing length. Then, in the bulk the $\hat{\bf n}$ vector is pinned along the field. In principle this breaks symmetry, but since this is sufficiently far away from the surface is expected to have a negligible effect.
• Figure 5: The top figure shows the scaling of the correlation function $\langle \mu^z_r \mu^z_{r'} \rangle$ while the bottom one shows the bond-bond correlation function $\langle \mu^z_r \mu^z_{r+1} \mu^z_{r'} \mu^z_{r'+1} \rangle - \langle \mu^z_r \mu^z_{r+1}\rangle^2$ (we denote the bond operator $B^z_{r}=\mu^z_{r}\mu^z_{r+1}$ in the label of the figure) at the critical point between the two phases in Fig.2 in the main text. Emergent supersymmetry at the critical point implies that the difference in the power-law exponents for these two correlation functions is exactly two while the precise values of the exponents themselves are also found to be consistent with the tricritical Ising model (see the main text for details).