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

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Science  18 Apr 2014:
Vol. 344, Issue 6181, pp. 280-283
DOI: 10.1126/science.1248253

Hope for SUSY?

Supersymmetry (SUSY), the symmetry between fermions, particles that form matter, and bosons, which mediate the interactions between them, has been proposed as one of the more likely extensions of the Standard Model of particle physics; however, it has so far received little experimental support. Condensed matter systems, such as the superfluid helium-3, may save the concept. In preparation for experimentation, Grover et al. (p. 280, published online 3 April) develop a theoretical approach that suggests SUSY describes the quantum phase transition on the boundary of a topological superconductor between a magnetic phase characterized by a bosonic order parameter and a neighboring phase hosting Majorana fermions.

Abstract

In contrast to ordinary symmetries, supersymmetry (SUSY) interchanges bosons and fermions. Originally proposed as a symmetry of our universe, it still awaits experimental verification. Here, we theoretically show that SUSY emerges naturally in condensed matter systems known as topological superconductors. We argue that the quantum phase transitions at the boundary of topological superconductors in both two and three dimensions display SUSY when probed at long distances and times. Experimental consequences include 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 SUSY, indicating a deep relation between topological phases and SUSY.

Since the 1970s, space-time “supersymmetry” (SUSY) has been actively pursued by particle physicists to attack the long-standing hierarchy problem of fundamental forces (14). Unlike any other symmetry, SUSY interchanges bosons and fermions and, when applied twice, generates translations of space and time, which ultimately leads to the conservation of momentum and energy (3). But despite sustained effort, SUSY has yet to be experimentally established in nature.

Here, we theoretically show that certain condensed matter systems display phenomena of emergent SUSY; that is, space-time SUSY naturally emerges as an accurate description of these systems at low energy and at long distances, although the microscopic ingredients are not supersymmetric. The physical systems we mainly consider are topological superconductors (TSCs) (5, 6), in which pairs of fermions, which may be electrons or fermionic atoms such as helium-3, pair together in a special way. The resulting state has an energy gap to fermions in the bulk but gapless excitations at the surface. We consider the quantum phase transition at which the surface modes acquire a gap and establish emergent supersymmetry D = 1 + 1 and D = 2 + 1 dimensional surfaces (Fig. 1) by using a combination of numerical and analytical techniques (D denotes the space-time dimensionality).

Fig. 1 Supersymmetry in a 3D TSC.

Ising magnetic fluctuations (denoted by red arrows) at the boundary couple to the Majorana fermions (blue cone). When the tuning parameter Embedded Image, the Ising spins are ordered, leading to a gap for the Majorana fermions. The critical point that separates the two sides is supersymmetric, where bosons (Ising order parameter) and Majorana fermions transform into each other.

The study of SUSY at a phase transition point was initiated in (7), where it was shown that the 1 + 1 dimensional tricritical Ising model, which can be accessed by tuning two parameters, is supersymmetric. A few other proposals that realize SUSY by fine-tuning two or more parameters have been made as well (811). Here, we require that SUSY be achievable by tuning only a single parameter, akin to a conventional quantum critical point (12). This is crucial for our results to be experimentally realizable. We also require that our theory has full space-time SUSY rather than only a limited “quantum-mechanical” SUSY (13). This automatically ensures translation invariance in space and time and will lead to experimental consequences, as we discuss below. Perhaps most interesting, in contrast to the strategy adopted in (7), our approach is not restricted to 1 + 1 dimensional theories.

There has been an explosion of activity in the field of topological phases since the discovery of Embedded Image topological insulators (TIs) (1417). We will focus on a set of closely related phases, the time-reversal invariant TSCs (5, 6), which include the well-known B-phase of superfluid helium-3 (18). These phases exist in both two and three spatial dimensions (19, 20) and support Majorana modes at their boundary; the modes are protected by time-reversal symmetry from acquiring an energy gap. Spontaneous breaking of this symmetry provides a natural mechanism to gap them out. For example, electron-electron interactions at the surface could lead to magnetic order, which breaks time reversal symmetry. A natural question is how the surface modes evolve as the magnetic order sets in. We will see that space-time SUSY naturally emerges at the onset of magnetic order.

The D = 2 + 1 dimensional TSC, protected by the time-reversal symmetry, provides the simplest setting to address this question. Whereas the bulk of the superconductor is gapped, the boundary, a Dedge = 1 + 1 dimensional system, contains a pair of Majorana modes Embedded Image that propagate in opposite directions. The aforementioned instability of the edge may be described by introducing an Ising field Embedded Image that changes sign under time reversal. The action is given byEmbedded Image (1)with Embedded Image, Embedded Image, and we have used the conventional Dirac gamma matrices for the relativistic fermion Embedded Image. vϕ and u are, respectively, the velocity and self-interaction of ϕ; g is the coupling between the fermions and ϕ, whereas r is the tuning parameter for the transition. The symmetry-broken phase is characterized by Embedded Image0, which leads to a mass gap Embedded Image for the fermions.

The mode count in the action Embedded Image is favorable for Embedded Image SUSY in D = 1 + 1 (21), with the bosons Embedded Image and Majorana fermions Embedded Image as superpartners. We now show that this is indeed the case by using a numerical simulation of a D = 1 + 1 lattice model that reproduces the action in Eq. 1 at low energies. The model is given byEmbedded Image whereEmbedded Image(2)Here, Embedded Image is a single Majorana fermion at site Embedded Image, whereas the Ising spins Embedded Image sit on bond centers. When the transverse magnetic field Embedded Image, Embedded Image and lattice translation symmetry ensures that the Majorana fermions are gapless. As Embedded Image decreases, at some point Embedded Image orders antiferromagnetically, leading to a mass gap for the fermions through the coupling Embedded Image, reproducing the field theory in Eq. 1 at and near the critical point. Embedded Image tunes the relative bare velocities between the boson and the fermion modes, similar to Embedded Image in Eq. 1.

We numerically simulate a spin version of Eq. 2 by using the density matrix renormalization group (DMRG) method (22).(Fig. 2). At larger Embedded Image, a gapless phase is obtained that is separated by a critical line from a gapped ordered phase at small Embedded Image. To characterize the critical theory, consider crossing the phase boundary along fixed Embedded Image, say, Embedded Image, while monitoring the central charge Embedded Image, which quantifies the amount of entanglement of a 1 + 1 – D critical system (Fig. 2B). At small Embedded Image, c is almost equal to zero, which indicates a gapped phase. At Embedded Image, the central charge saturates at Embedded Image, indicating that the symmetry is restored and a gapless Majorana mode is present. At the transition, Embedded Image, we find Embedded Image, which precisely corresponds to the SUSY tricritical Ising theory (7). Note that we access this transition by tuning a single variable. Even when the parameter Embedded Image, which controls relative bare velocities of boson and fermion, is varied between Embedded Image, and Embedded Image, we obtain a second-order transition in the same universality class (23). We now turn to 3 + 1 dimensional TSCs, whose two-dimensional (2D) surface supports gapless Majorana fermions, protected by time-reversal symmetry. As advertised earlier, we study spontaneous breaking of time-reversal symmetry that gaps out the surface states. In the absence of numerics, we resort to an analytical calculation within a Embedded Image expansion to unravel the nature of quantum criticality and argue that 2 + 1 dimensional SUSY arises here. The relevant action is again given by Eq. 1 with Embedded Image. At Embedded Image, this action has Embedded Image SUSY (21) in 2 + 1D only if one sets Embedded Image and Embedded Image. We ask whether SUSY emerges in the low energy limit as we tune to the quantum critical point, without enforcing these conditions microscopically. We find (23) that this is indeed the case: The terms that break SUSY are irrelevant at the critical point leading to a stable superconformal fixed point in the infrared. Additional support has recently emerged from a rather different, conformal bootstrap approach (24) that obtains the scaling dimension Embedded Image, which matches well with our Embedded Image expansion result, Embedded Image.

Fig. 2 Supersymmetry in a 2D TSC.

(A) The phase diagram of the Hamiltonian Embedded Image in Eq. 2, which realizes the Majorana edge coupled to Ising magnetic fluctuations in this system. At large Embedded Image, the Ising spins disorder, and the counterpropagating Majorana modes (blue arrows) remain gapless. As Embedded Image decreases, the ordering of Ising spins (red arrows) leads to a gap for the Majorana modes. (B) The region of the phase diagram indicated by the black arrows in (A). Plotted is the central charge Embedded Image as a function of Embedded Image for fixed Embedded Image. For Embedded Image, c equals 1/2, consistent with gapless Majorana modes; for Embedded Image, c is 0, which indicates the gapped phase. At the critical point, c is 7/10, which corresponds to supersymmetric tricritical Ising point. (Inset) von Neumann entropy Embedded Image and the Renyi entropies Embedded Image at the critical point, which were used to deduce the central charge. L is the system size, and l is the size of the entanglement subsystem.

To explore an experimental realization, consider a film of superfluid He3-B, with surface perpendicular to the Embedded Image direction and a magnetic field parallel to the surface. Despite breaking time-reversal symmetry, this does not introduce an energy gap (24, 25) because of a symmetry Embedded Image, that combines time reversal with 180° rotation about the Embedded Image axis. However, the system can lower its energy by spontaneously breaking rotation symmetry and gapping the surface modes. Indeed, the order parameter of He3-B includes a rotation matrix Embedded Image that depends on an angle Embedded Image and a direction Embedded Image. In the absence of a magnetic field, microscopic dipolar interactions pin the vector Embedded Image. However, if the Embedded Image vector develops an in-plane component Embedded Image, it can break the protecting symmetry Embedded Image. The energetic incentive for an in-plane component increases with the applied magnetic field, because the surface gap m is proportional to the field: Embedded Image. Thus, for fields Embedded Image we have a gapless symmetric state, whereas for Embedded Image spontaneous symmetry breaking occurs, leading to a gap for surface Majorana modes. In (24), the critical field was predicted to be Embedded Image G, which is readily accessible. However, we note the challenges to realizing SUSY in this physical system (23). The low temperatures of the superfluid transition restricts the temperature window for experiments, and the bare boson and fermion velocities are expected to be rather different, with a ratio controlled by Embedded Image.

We now discuss the possibility of emergent SUSY in 3D TIs. The surface states of these materials (1417) consist of Dirac fermions at a chemical potential Embedded Image. Let us fine-tune Embedded Image to zero and consider the instability of these surface modes to an s-wave superconductor. This multicritical point can potentially be driven by intrinsic interactions and may also be realized by patterning the surface with a Josephson junction array. We restrict ourselves to particle-hole symmetric superconducting fluctuations. The effective near the transition is given byEmbedded Image (3)where Embedded Image is the superconducting order parameter, the fermion field Embedded Image, and Embedded Image is the 2D antisymmetric tensor. g, r, and u have meaning analogous to Eq. 1, whereas c is the velocity of Embedded Image. The above action is known to flow to a Embedded Image superconformal fixed point (26, 27).

Because SUSY rotates fermions into bosons, the universal critical exponents satisfy nontrivial relations between them, thus relating results from disparate experiments. In particular, the scaling dimensions Embedded Image of Embedded Image satisfy Embedded Image. In an electronic system, Embedded Image may be determined via tunneling experiments, whereas Embedded Image may be obtained from neutron scattering. As an example, consider the boundary of a 2D TSC, for which we showed that the transition lies in the tricritical Ising universality class. The exact critical exponents in this case are Embedded Image. Similarly, at the boundary of 3D TSC, we find Embedded Image at order Embedded Image in the renormalization group. Lastly, in the case of 3D TI, because of the larger (Embedded Image) SUSY, one can determine the scaling dimensions of the boson and Fermi fields exactly, with (28) Embedded Image. Another consequence of SUSY is that the velocity for the fermion and the boson at low energies become equal, despite the fact that their bare velocities will be generically different.

SUSY has nontrivial consequences for the proximate phases as well. At the TSC boundary in the time-reversal broken phase, SUSY implies that the masses of the fermion and the boson are equal to each other as one moves away from the critical point. Mathematically, mϕ and mχ are equal to α(rrc)v, where Embedded Image is the critical value of the tuning parameter, α is a constant, and, owing to SUSY, Embedded Image.

Emergent SUSY also provides insight into the topological excitations on either side of the transition. First, consider the time-reversal invariant boundary of a TSC in either two or three spatial dimensions, which supports a gapless Majorana fermion. This is conventionally viewed as resulting from the topological band structure. However, a completely different viewpoint is obtained by realizing that the flow from SUSY critical point (Embedded Image) to a phase where Embedded Image is gapped (Embedded Image) corresponds to spontaneous breaking of SUSY. Because SUSY is a fermionic symmetry, this leads to the generation of a massless Majorana fermion, the “goldstino” mode (23, 29, 30), which precisely corresponds to the topologically protected Majorana fermion. Thus, one can model the boundary modes with a continuum model in the same dimension if SUSY is assumed, which serves to protect their gaplessness. We provide details of this phenomenon in (23).

The dichotomy between SUSY and topology runs deeper: It has robust, physical manifestation in the time-reversal broken phase as well. As discussed in (23) and hinted above, the SUSY is unbroken as the boson Embedded Image condenses, resulting in equal mass for Embedded Image and Embedded Image. It is well known that, in the symmetry broken boundary phase, topological defects—kinks in 1D and line domain walls in 2D—carry fermion modes on them (31). We interpret these also as Goldstino modes of a type of SUSY breaking. First consider kinks, which spontaneously break translational symmetry in 1D. Because of the deep connection between translation symmetry and SUSY, this ultimately leads to a spontaneous breaking of half of the SUSY (23, 32, 33). Thus, in addition to the coordinate of the domain wall, Embedded Image, which can be viewed as a gapless 0 + 1 – D Goldstone boson of translation breaking, there is also a single Majorana Goldstino mode Embedded Image localized on the kink, which is the superpartner of Embedded Image. The low-energy theory of the kink is Embedded Image, which one might have anticipated independently on topological grounds (31). This argument also applies to TSCs coupled via Josephson junction discussed in (34).

The generalization to line domain walls at the time-reversal breaking boundary of 3D TSC is straightforward. The partial SUSY breaking leads to 1 + 1 – D chiral Majorana fermion Embedded Image propagating along the domain line; Embedded Image is a superpartner of the Goldstone mode Embedded Image associated with the translational symmetry breaking, Embedded Image. SUSY not only predicts the anticipated chiral Majorana fermions on the domain walls, it also predicts equal velocity for the superpartners Embedded Image and an exact “Fermi-Bose” degeneracy associated with the whole excitation spectra of the domain wall or kink. These predictions hold in the ordered phase close to the criticality. In general, we expect that the boundary modes in a topological phase can be thought of as Goldstinos unless the tuning parameter breaks SUSY explicitly; an example of the latter possibility is provided by the SUSY critical point in 3D TIs mentioned above.

Supplementary Materials

www.sciencemag.org/content/344/6181/280/suppl/DC1

Supplementary Text

References (3739)

References and Notes

  1. Quantum mechanical SUSY corresponds to SUSY only in the time-direction; see, e.g., (35) and (10).
  2. Supplementary materials are available on Science Online.
  3. A slightly different theory, which might also exhibit emergent SUSY, arises in the context of time-reversal breaking 1 + 1 – D topological superconductors coupled via a Josephson junction (36).
  4. Acknowledgments: We thank M. Fisher, T. Mizushima. D. Huse, R. Mong, and S. Trivedi for discussions. This work is supported by Army Research Office Multidisciplinary University Research Initiative grant W911-NF-12-0461 and NSF Division of Materials Research grant 0645691 (A.V.) and by U.S. Department of Energy Office of Basic Energy Sciences under grant DE-FG02-06ER46305 (D.N.S.).
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