Emergence of supersymmetry on the surface of three dimensional topological insulators

We propose two possible experimental realizations of a 2+1 dimensional spacetime supersymmetry at a quantum critical point on the surface of three dimensional topological insulators. The quantum critical point between the semi-metallic state with one Dirac fermion and the s-wave superconducting state on the surface is described by a supersymmetric conformal field theory within $\epsilon$-expansion. We predict the exact voltage dependence of the differential conductance at the supersymmetric critical point.

For the past forty years, supersymmetry has been studied intensively in high energy physics because of its attractive features, e.g. as a possible solution to the hierarchy problem [1]. Although there is so far no experimental evidence for our universe to be supersymmetric, there is some expectation that supersymmetry may be revealed in the large hadron collider (LHC) in a near future. Condensed matter systems provide alternative ways to realize supersymmetry in nature through emergence [2]. Namely, supersymmetry can dynamically emerge in the low energy limit of some condensed matter systems although the microscopic Hamiltonians explicitly break it. Because supersymmetry is a symmetry between boson and fermion, it is essential to have a same number of low energy modes for boson and fermion in order to realize supersymmetry. Although there are examples of emergent spacetime supersymmetry in 1+1 dimensions [3,4,5], where the distinction between boson and fermion is rather obscure, it is not easy to realize supersymmetry in lattice models in higher dimensions [6]. Because of the fermion doubling problem, which is actually an intrinsic feature rather than a 'problem' in condensed matter systems, there are usually more fermionic degrees of freedom than bosonic degress of freedom unless there is a special symmetry or dynamical mechanism that protect multiple gapless bosonic modes at low energies [7,8,9]. On the contrary, there is no such problem in continuum model [10].
Topological insulator [16,17,18,19,20,11,12,13,14,15] is a topological phase of matter where gapless edge or surface modes are protected by time reversal symmetry [21,22,23]. In topological insulators, there is no fermion doubling problem because the second set of fermionic modes is located on the other edge or surface of a sample. For example, on the surface of a semi infinite three dimensional topological insulator one can have only one 2+1 dimensional Dirac fermion, which is worth of one complex boson in terms of counting the number of propagating modes. Therefore topological insulator provides a platform to realize interesting critical states [24], incluing states with emergent supersymmetry. In this paper, we consider a superconducting quantum critical point on the surface of a three dimensional topological insulator. It is likely that the critical point exhibits an emergent supersymmetry because there are the same number of propagating modes for boson and fermion which are strongly mixed with each other at low energies.
We consider a three dimensional topological insulator which has a gap in the bulk and a gapless Dirac fermion at the Γ-point of the surface Brillouin zone. For example, Bi 2 Se 3 has the desired properties [25,19]. This is an ideal material due to the large band gap in the bulk (0.3 eV) and the possibility of manipulating the Fermi energy of the bulk and the surface by chemical modifications [26]. We consider the case where the chemical potential is tuned to the Dirac point. Since the dispersion relation is linear near the Dirac point, the low-energy excitations are described by a two-component massless Dirac fermion,  The two dimensional electrons on the surface are subject to the electrostatic Coulomb repulsion and the attractive interaction mediated by phonons. If the attractive interaction is strong enough, the semi-metallic state can become unstable, undergoing a quantum phase transition to a superconducting state. In order to access the critical point by tuning the strength of the Coulomb repulsion, we consider a substrate placed at a distance d above the topological insulator ( Fig. 1). The substrate consists of a two-dimensional metal with dispersion relation ǫ = |p| 2 2m − µ F , where µ F is the chemical potential. The long range Coulomb repulsion between electrons on the surface of the topological insulator is screened by the metallic substrate. The strength of the residual short range repulsion can be controlled by changing d, which can be used to drive the system to the critical point. Here we assume that the attractive interaction due to phonon is sufficiently strong so that the semi-metallic state is unstable to the swave superconducting state without the Coulomb repulsion. Now we examine how the screened Coulomb interaction depends on the distance between the substrate and the topological insulator.
The system composed of the topological insulator and the substrate is described by the partition function where Here a 0 is the temporal component of the 3+1D electromagnetic field. We choose to work in the Coulomb gauge ∇ · a = 0, and neglect the spatial components of the electromagnetic gauge field whose contribution is down by c f /c, where c is the speed of light. The z-coordinates of the surface of the topological insulator and the substrate are 0 and d respectively. We neglect tunneling between the substrate and the topological insulator. By integrating out the fermions of the substrate, we obtain the effective action for the gauge field, where Here p = (ω, p) denotes 2+1 dimensional energy-momentum vector. The bare propagator and the polarization (in the limit |p| ≪ p F ) is given by It is noted that the presence of the substrate breaks the translation invariance along the z-direction, and the momentum along this direction is not conserved. Inverting the dressed propagator, we obtain the two dimensional screened Coulomb repulsion between electrons on the surface of the topological insulator, The static effective potential is not singular as |p| → 0. It is given by V 0 = e 2 2 d + 1 e 2 N (µ F ) , where N(µ F ) is the density of states of the substrate at the Fermi energy.
If N(µ F ) is large and d is small, the Coulomb repulsion can be made weak so that the attractive interaction mediated by phonon dominates. If the strength of the attractive interaction is sufficiently strong, one can tune the system across the superconducting phase transition by changing d.
In the presence of one Dirac point located at the Γ point, one can, in general, have a pairing of the form ∆ α,β (k)ψ α,k ψ β,−k , where α, β = 1, 2 are pseudospin indices and ∆ α,β (k) = −∆ β,α (−k). The gap function can be decomposed as ∆ α,β (k) = ǫ α,β ∆ s (k) + [σ y σ] α,β · ∆ t (k), where ∆ s (k) and ∆ t (k) are pseudospin singlet and triplet order parameters, respectively. It is expected that the triplet state is energetically less favourable than the singlet state because the gap vanishes at k = 0 for the triplet state. However, this ultimately depends on the microscopic details of the systems. Here we proceed with the assumption that the s-wave singlet superconducting state is the dominant instability channel in the presence of the strong attractive interaction.
Suppose that electrons are interacting through a net attractive interaction Emergence of supersymmetry on the surface of three dimensional topological insulators5 To decouple the four fermion interaction we introduce a complex boson field through the Hubbard-Stratonovich transformation, Here ε is the 2 × 2 antisymmetric matrix with ε 12 = −ε 21 = 1. Although the complex boson (Cooper pair) has no bare kinetic term, it is generated by fermions at low energies. The low energy effective theory becomes where c b is the velocity of bosons which may be different from c f . Note that the dynamics of bosons is guaranteed to be relativistic with the dynamical critical exponent z = 1 as far as fermions are relativistic because Cooper pairs are formed out of the relativistic Dirac fermions. As d is tuned, the mass of the boson is changed. In the superconducting state with m 2 < 0, the boson is condensed, and the Dirac fermion becomes gapped. At a critical distance d c , bosons are massless and the theory flows to an interacting fixed point in the low energy limit. The field theory which has two copies of the present theory has been studied, where each set of modes describes one Dirac fermion and one complex boson defined at one of the two distinct momentum points (K and K ′ ) on the honeycomb lattice [8]. In the ǫ expansion, it has been shown that the theory flows to a supersymmetric critical point with four emergent supercharges where two sets of modes are decoupled in the low energy limit. Therefore the same conclusion can be drawn for the present case. In the low energy limit, the critical point is described by the N = 2 Wess-Zumino theory with one chiral multiplet [27]. In order for an intrinsic superconducting state to be stable, one needs to have a sufficiently strong electron-phonon interaction, which may or may not be the case for real materials. In cold atom systems, strength of interaction can be easily tuned. It will be of interest to realize 3D topological insulators and the supersymmetric quantum critical point in cold atom systems by tuning attractive interaction between particles [30,31]. Here we propose a second scenario which realizes supersymmetry in the low energy limit. The system consists of a Josephson junction array deposited on the top of a topological insulator (Fig. 2). A Josephson junction array (JJA) consists of a regular network of superconducting islands coupled by tunnel junctions. The Dirac electrons of the topological insulator can tunnel to the JJA to form Cooper pairs and vice versa. The JJA and the Dirac fermion are also described by the same Lagrangian in Eq. 13 at low energies if the average number of Cooper pairs within each island is tuned to be an integer. As the Josephson coupling is tuned, the JJA can undergo a phase transition to a superconducting state. The quantum critical point is again described by the N = 2 Wess-Zumino theory with one chiral multiplet.
At the critical point, the scaling dimensions of the chiral primary fields, including the Dirac fermion and the Cooper pair field, are constrained by the superconformal algebra [28]. As a result, the exact anomalous dimensions of the fermion and boson are given by η φ = η ψ = 1 3 [29]. It is of interest to provide a clear experimental signature for the emergent supersymmetry. Here we consider tunneling spectroscopy which measures the local density of states. The single particle Green's function of electron at the supersymmetric critical point is given by where σ = (σ 3 , c f σ 1 , c f σ 2 ). Integrating the spectral function over momentum, we obtain that the local density of states As a result, we expect the differential conductance dI dV ∼ V 4/3 at the supersymmetric critical point. This provides a clear experimental signature of the supersymmetric state. Note that the exact exponent is predicted thanks to the superconformal symmetry, although the critical point is described by the strongly interacting theory.
In summary, we propose that a 2+1D superconformal field theory can be realized at the quantum critical point between the semi-metallic state and the swave superconducting state on the surface of three dimensional topological insulators. The critical point is described by the N = 2 Wess-Zumino model with one chiral multiplet. At the critical point, the local density of states obeys the power law behavior ρ(ω) ∼ ω 4/3 , which can be measured by scanning tunneling microscopes. Recently, it has been shown that the supersymmetric critical point is not stable in the presence of disorder [32]. However, the critical behaviour governed by the putative supersymmetric critical point can be observed within a finite temperature range in the weak disorder limit.