A Supplementary Material

All Majorana Models with Translation Symmetry are Supersymmetric

Abstract

We establish results similar to Kramers and Lieb-Schultz-Mattis theorems but involving only translation symmetry and for Majorana modes. In particular, we show that all states are at least doubly degenerate in any one and two dimensional array of Majorana modes with translation symmetry, periodic boundary conditions, and an odd number of modes per unit cell. Moreover, we show that all such systems have an underlying supersymmetry and explicitly construct the generator of the supersymmetry. Furthermore, we establish that there cannot be a unique gapped ground state in such one dimensional systems with anti-periodic boundary conditions. These general results are fundamentally a consequence of the fact that translations for Majorana modes are represented projectively, which in turn stems from the anomalous nature of a single Majorana mode. An experimental signature of the degeneracy arising from supersymmetry is a zero-bias peak in tunneling conductance.

pacs:
71.10.Fd, 71.10.Pm, 73.20.-r, 11.30.Pb

A Majorana mode is a strange concept. Formally, it represents degree(s) of freedom because two Majorana modes constitute a single qubit or spinless fermion. By construction, the Majorana mode is its own anti-mode: its creation and annihilation operators are identical (1). While the mathematical existence of Majorana modes arises simply from a change of basis in the particle-hole space, the physical manifestations of the above properties are extremely nontrivial (2) and important for quantum computation purposes (3); (4); (5). The fact that a Majorana is only a fraction of a physical electron or qubit suggests the possibility of encoding information in two widely separated Majoranas, each of which is immune to local decoherence. Furthermore, the Hermitian nature of the Majorana mode forces it to exist at zero energy in the superconducting gap of physical systems, allowing experimentalists to zero in on finding a zero bias peak in tunneling, which is necessary if a Majorana mode is present.

There has been tremendous effort (9); (10); (6); (7); (8) toward realizing these Majorana modes at the endpoints of both one dimensional topological superconductors in nanowires (11); (12); (13); (14); (15) and atomic chains (16), as well as from two dimensional interfaces between topological insulators and superconductors (17); (18); (19). The increasingly compelling evidence for single Majorana modes and the substantial activity in this field suggest that scaling the system to realize multiple Majoranas along a line or in a two-dimensional grid may be realized in the near future. In such setups involving lattices of emergent Majorana modes, the low energy effective Hamiltonian involves interactions between such modes, and such models host abundant and fascinating phenomenology. For example, two (20) and three dimensional (21) lattices of Majorana zero modes may provide new architectures for quantum information processing and new topological phases.

Moreover, there have been several proposals which employ Majorana modes for realizing supersymmetry, which is a highly appealing concept from particle physics (22); (23); (24); (25), relating bosonic and fermionic modes to each other. Though signatures of it have yet to be observed, there are several condensed matter systems in which supersymmetry may emerge at long time and distance scales (“scaling limit”), especially close to a critical point (26); (27); (28); (29); (30). In particular, the supersymmetric tricritical Ising model may be realized at a critical point of Majorana systems (28); (31). Furthermore, time-reversal acts as a supersymmetry on vortices of topological superconductors (32). However, exact supersymmetry in lattice models typically requires fine-tuned Hamiltonians (33); (34); (35); (36); (38); (37).

In this work, we show that all Majorana systems with translation symmetry and an odd number of Majorana modes per unit cell exhibit supersymmetry and we explicitly construct its generator and identify its experimental consequences. For one dimensional systems with periodic boundary conditions, we establish a Kramers-like theorem (39) but for translation, not time-reversal symmetry: we show that every energy level is at least doubly degenerate. With anti-periodic boundary conditions, we establish a result along the lines of Lieb-Schultz-Mattis (40) and rule out the possibility of a unique gapped ground state, using recent results for spin chains (42); (41); (43); (44). For two dimensional systems, we also establish at least twofold degeneracy for all states and for all system dimensions. The essence of all these results is the fractional nature of the Majorana mode. Each unit cell, with an odd number of Majoranas, cannot exist intrinsically, and therefore the symmetry group involving translations and fermion parity is represented projectively. We will now illustrate this in detail and conclude by mentioning several experimental venues for our results, in which a striking signature of supersymmetry is a zero-bias peak in tunneling experiments.

I 1D, periodic boundary conditions

Figure 1: (left) Translationally invariant Majorana modes, with periodic boundary conditions, have at least twofold degeneracy in the energy spectrum. The underlying supersymmetry requires that each energy level contains pairs of fermionic and bosonic superpartners. (right) The same system, with anti-periodic boundary conditions (depicted by a slash through a bond), cannot have a unique gapped ground state in the thermodynamic limit.

Consider (even, to have a well-defined total Hilbert space) Majorana modes localized at sites spaced around a ring, and consider general Hamiltonians invariant under translation by one site and with periodic boundary conditions:

(1)

We make no assumptions on the structure of other than translation invariance and the conservation of fermion parity. In particular, let be the fermion parity operator, for which we have .

Let be translation by one site:

(2)

By assumption,

(3)
(4)

However, translation and fermion parity anti-commute:

(5)

because is even and is anti-commuted through an odd number of Majoranas to return to the beginning. We note that the fact that and can anti-commute has been used to classify fermionic phases protected by translation symmetry (45). It follows that every eigenstate of is at least doubly degenerate: if and (), then and , and thus is also an eigenstate with energy and orthogonal to . A similar algebraic structure (but not involving translation) was used in (46) to establish spectrum doubling.

Ii Degeneracy as a consequence of Supersymmetry

We now show that all translationally invariant Majorana Hamiltonians in 1D with periodic boundary conditions and an odd number of Majoranas per unit cell are supersymmetric. The two-fold degeneracy of the spectra found in the previous section can then be thought of as a consequence of this underlying supersymmetry.

We first shift all the eigenvalues of the Hamiltonian by a constant so that they are all non-negative. Then we define the following fermionic, non-Hermitian operator :

(6)

where is the identity operator. Clearly, commutes with the Hamiltonian : . Most importantly, due to the relation , one finds

(7)
(8)

Therefore, acts as the generator of an supersymmetry ( equals two because is a non-Hermitian operator and can be decomposed as where and are Hermitian) (47). Thus, all Majorana Hamiltonians in 1D which have an odd number of Majoranas per unit cell with periodic boundary conditions furnish an supersymmetry.

Supersymmetry naturally explains the results derived in the previous section on the nature of spectra. All energy eigenvalues are doubly degenerate, and the corresponding eigenstates can be chosen as fermion parity eigenstates with opposite parity. Explicitly, given an eigenstate with energy and fermion parity , its fermionic partner eigenstate, which has the same energy eigenvalue but opposite parity, is given by:

(9)

where we have normalized so that . The opposite fermion parity of follows because .

Due to the factor of in Eq. (9), in a general supersymmetric theory, the existence of supersymmetric partner eigenstates is guaranteed only when . However, in our case, the zero of energy plays no special role (recall that, generically, we already have to shift all energy levels by a constant so that ), and therefore, the supersymmetric partner eigenstates exist for all , including the ground state. Therefore, the Witten index, which is defined as the difference between the number of bosonic and fermionic groundstates, is zero.

Iii 1D, anti-periodic boundary conditions

Local Hamiltonians of the above type but with anti-periodic boundary conditions commute with the twisted translation operator , which has the action

(10)
(11)

Since commutes with , the degeneracy found above is not required here.

However, we now show that for such Hamiltonians with anti-periodic boundary conditions, it is not possible for there to be both a unique ground state and a finite excitation gap in the thermodynamic limit. Such constraints, with origins in the Lieb-Schultz-Mattis theorem, have been recently established (42); (41) for spin chains in which each unit cell transforms under a projective representation of a global symmetry (e.g., time reversal). We will now make contact with these recent results by doubling the Majorana system and reinterpreting it as a spin system with additional symmetry from the doubling construction.

Assume for the sake of contradiction that the Hamiltonian has a unique gapped ground state . Consider the doubled system where is simply a second copy of with Majorana operators represented by . Since each subsystem has a unique gapped ground state, the composite also has a unique gapped ground state . We now Jordan-Wigner transform into a spin system:

(12)
(13)

Care must be taken because the spin system with fixed boundary conditions only corresponds to a fixed fermion parity sector of the fermionic system. It is straightforward to check that the spin system with periodic boundary conditions corresponds to the fermion system with anti-periodic boundary conditions and even fermion parity. This sector includes the composite ground state (whose parity is the square of the parity of ).

Through the doubling construction, has a set of discrete symmetries involving swapping the two chains. There is a group generated by and a group generated by . In the spin language, these correspond respectively to the symmetries

(14)
(15)

and

(16)
(17)
(18)

Altogether, we have an onsite group of rotations which is represented projectively (by spin ).

Hence, the arguments in (42); (41); (43) rule out a gapped unique ground state of the spin system; in brief, a gapped unique ground state in one dimension is short-range entangled, and this local structure leads to incompatibility between the projective representation of each unit cell and translation symmetry. By contradiction, the original fermion chain cannot have a unique ground state with a gap in the thermodynamic limit.

Iv Two and Higher Dimensions

In this section, we first consider two-dimensional systems with translation symmetry in both directions, periodic boundary conditions, and a single Majorana mode per unit cell. If the system has one odd length, then degeneracy of all energy levels follows by bundling all Majoranas along the odd length direction into a supercell and applying the 1D argument above. However, this method does not apply to systems with two even dimensions, but nevertheless the degeneracy holds. The fundamental reason is the fact, established below, that the two translations and along the two directions and anticommute when both dimensions and are even. This implies (in conjunction with ) that all states have at least twofold degeneracy.

We label the array of Majorana modes by their row and column positions. Translation has projective representation given by the product of translations for each row:

(19)
(20)

See the Supplementary Material for an explanation of why the translation operator has the above form; the essential feature is the Majorana operator (, an antisymmetric matrix, is not important for our purposes). Then

(21)
(22)

because each involves an odd number of distinct Majorana operators and there are an odd number () of anti-commutations required to return to the original ordering of . Thus, , which ensures all states have at least twofold degeneracy. We note that in this case, the degeneracy is not due to supersymmetry(48).

The above results for periodic boundary conditions readily generalize to three dimensional systems with at least one dimension of odd length, but we note that systems with three even length dimensions need not be degenerate. As a simple counterexample, consider a array of Majorana modes with four-Majorana interactions on each face; this Hamiltonian has a unique ground state.

V Applications and Phenomenology

Such Hamiltonians involving interacting Majorana modes serve as effective models for either the boundaries or vortex lattices of topological superconductors. For example, a stack of topological superconducting wires hosts an array of Majorana modes localized at the ends of the wires (see Fig. 2). The low energy physics of such systems is thus described by the interactions of the Majorana modes, for which our work is relevant.

The particular interactions between Majorana modes depend on how the wires are coupled to each other, and as an example, a natural effective Hamiltonian for such systems is considered in (31); (49):

(23)

For a particular ratio of , the above system is in the (supersymmetric) tricritical Ising universality class (31). However, our work demonstrates that for all values of , the above system exhibits supersymmetry, and as a consequence all energy levels are at least doubly degenerate.

Figure 2: Our work applies to effective Hamiltonians (green) describing the Majorana modes (blue) emerging at the endpoints of topological superconductors (vertical chains). Each oval is a spinless fermion consisting of two Majorana fermions (black dots).

Such degeneracy between states of opposite fermion parity has a distinct signature in tunneling experiments: for a point contact located near the endpoint of one topological superconducting wire, there will be a zero bias peak in the tunneling conductance. However, because the operator (which connects a state to its superpartner) is generically non-local, the zero-bias peak will be harder to observe as system size grows. More precisely, for a point contact to a non-interacting normal lead near the endpoint of wire , the tunneling conductance is if the temperature and the voltage bias are both smaller than , where and are the degenerate ground-state superpartners (50); (51). For the Hamiltonian in Eq. (23), we find that for any system size in the non-interacting case of . Furthermore, by means of exact diagonalization, we verify that with an exponent in the range of for all parameter values . For relatively small numbers of superconducting wires (which are realistic for experiments), we therefore expect the zero-bias peak to be observable. Note that while the zero-bias peak is expected for a single Majorana mode, it is generically not present for an even number of modes; the translation/supersymmetry is crucial here.

There are other routes toward experimental realization of the many models of the above type, including Abrikosov vortex lattices on the surface of topological insulators (52) and Josephson-coupled topological superconductor islands (53), in which charging energy mediates interactions between the emergent Majorana modes.

Vi Summary and Discussion

We have shown that one and two dimensional systems of Majorana modes with translation symmetry, periodic boundary conditions, and odd number of modes per unit cell, have at least twofold degeneracy for every state in the energy spectrum and that this is a reflection of an underlying supersymmetry. Moreover, we have shown that such a one dimensional system with anti-periodic boundary conditions cannot have a unique gapped ground state in the thermodynamic limit. Such Majorana systems may be realized at the boundaries or vortex lattices of topological superconductors, and the degeneracy arising from supersymmetry is potentially manifest as a zero-bias peak in tunneling experiments.

Our results motivate the conjecture that for all translationally invariant Majorana systems with an odd number of modes per unit cell, there cannot be a unique gapped ground state in the thermodynamic limit, regardless of dimension or boundary condition. This leaves the possibilities of gaplessness and symmetry breaking in one dimension, and the additional possibility of topological order in higher dimensions. Furthermore, while our one-dimensional anti-periodic boundary conditions doubling analysis conveniently makes use of recent spin system results, it would be very enlightening to find a direct proof of the result without having to double the system. It is not obvious to us how to apply the flux insertion arguments given by Oshikawa (54) and Hastings (55) in our case since the symmetries are discrete.

Translation symmetry is only one of many crystal symmetries that can be considered. Other natural extensions include mirror reflection, inversion, and perhaps non-symmorphic symmetries as well. The effect of these symmetries and their interplay with on-site symmetries such as time-reversal is an intriguing direction for future work. For now, we note as a small extension of our work that mirror reflection and inversion each anti-commute with fermion parity if the number of Majoranas that are transformed into other Majoranas (and not themselves) is for . We focused on translation symmetry because it enables the simplest manifestation of supersymmetry.

Finally, we note that the double degeneracy of the full spectrum discussed in our paper can be thought of as ergodicity breaking that exists at all temperatures. This is because the degenerate eigenstates that differ in fermion parity cannot be connected by a local operator in the thermodynamic limit (note that the operator in Eq. (6) is non-local in general). Furthermore, if the system does not spontaneously break translational symmetry, finite energy density degenerate eigenstates and will satisfy and for all local operators , which is reminiscent of topological order (4); (56).

Acknowledgements.
Acknowledgements: We thank Leon Balents for helpful comments. All authors are supported by a fellowship from the Gordon and Betty Moore Foundation (Grant 4304). TG also acknowledges startup funds from UCSD.

Appendix A Supplementary Material

Here we review the Majorana representation of rotations in the orthogonal group and explain the explicit form of the translation operator used in the main text. Consider Majoranas labeled by . We will hereafter assume that is even. A generic rotation in the space of these Majoranas is given by , where is a real orthogonal matrix satisfying . We are interested in projective representations of such that for all . If is a proper rotation such that , then it can always be written as , where is a real antisymmetric matrix. In this case, the projective representation corresponding to is

(24)

If is an improper rotation such that , it can always be written as a product of a proper rotation and a reflection given by

(25)

Note that because is even. The projective representation corresponding to is simply because for any but . The projective representation corresponding to is then

(26)

For a one dimensional ring with an even number of Majorana modes, the action of translation by one site on the Majorana modes is:

(27)

This is an improper rotation when is even, and therefore the corresponding projective representation reads

(28)

where is an antisymmetric matrix satisfying . It can be determined by looking at the eigenvalues and eigenvectors of , but its precise form is not important for our purposes.

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