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

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.

• The paper demonstrates that quantum phase transitions at the boundaries of topological superconductors enable bosonic and fermionic modes to interrelate via emergent space-time supersymmetry.
• The paper employs numerical DMRG simulations and ε-expansion techniques to map the transition onto known supersymmetric models like the tricritical Ising model.
• The paper highlights practical experimental implications, suggesting tuned parameters in topological superconductors can reveal critical phenomena indicative of underlying supersymmetry.

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

The paper "Emergent Space-Time Supersymmetry at the Boundary of a Topological Phase", authored by Grover, Sheng, and Vishwanath, presents a rigorous theoretical investigation into the emergence of supersymmetry (SUSY) in topological superconductors (TSCs). The research articulates a remarkable connection between condensed matter systems' boundary properties and supersymmetric theories, providing clear experimental implications and theoretical insights into quantum critical phenomena.

Central to the study is the demonstration that quantum phase transitions (QPTs) at the boundary of TSCs, in both two- and three-dimensional systems, exhibit supersymmetry at critical points when observed over long distances and timescales. This research counters traditional views that supersymmetry is an exclusive property of high-energy physics by positioning it as emergent in certain condensed matter systems. Specifically, the authors explore the phase transition mechanisms on the surface of TSCs, particularly identifying how bosonic and fermionic modes become interrelated through SUSY transformations. The theoretical framework connects spontaneously broken symmetries at TSC surfaces to SUSY, offering predictions and insights for future empirical studies.

The authors employ a combination of numerical simulations and analytical methods to substantiate their claims. Particularly, the utilization of the Density Matrix Renormalization Group (DMRG) method confirms the presence of SUSY in 1+1-dimensional models, demonstrating that this supersymmetry emerges as a function of tuning a single parameter in the system. This aspect is crucial for realizing practical experiments as it simplifies the experimental scenarios to observe SUSY. Numerically, the research confirms that the transition between gapless and gapped boundary states on TSCs maps onto known supersymmetric models, including those characterized by the tricritical Ising model, known for hosting SUSY in 1+1 dimensions.

Extending to three dimensions, an $\epsilon$-expansion approach is applied to elucidate how the interactions between fermionic and bosonic fields on 2+1-D surfaces lead to emergent SUSY at QPTs. The symmetry emerges despite initial anisotropies in the velocities of bosonic and fermionic quasiparticles, suggesting that the supersymmetric fixed point dynamically enforces a balanced velocity at scales pertinent to measurement.

Experimentally, the research discusses potential tricks of trade by highlighting possible realizations in condensed matter systems, such as films of superfluid He$_3$-B, with magnetic fields applied strategically to manifest these supersymmetric phase transitions.

The implications of this paper are noteworthy for both theoretical and experimental physicists. Theoretically, it augments the understanding of topological phases and their critical phenomena while providing new contexts to study supersymmetry outside particle physics. Practically, it encourages exploration toward experimental observance, aided by exact relations concerning critical exponents and the behavior of surface states predicted by SUSY.

Future research pathways might include refining the experimental parameters for observing these transitions in various topological materials or investigating further links between different types of topological phases and possible emergent symmetries. Additionally, exploring related non-equilibrium dynamics could unveil further layers of complexity in supersymmetric-condensed systems, potentially contributing to the advancement of quantum computing technologies where robustness against external perturbations may benefit from such symmetries.

This paper stands as a substantial contribution to the field of quantum matter, presenting an avenue by which high-energy physics concepts like SUSY can reveal their relevance and validity within the domain of condensed matter physics.