Supersymmetric Free Fermions and Bosons: Locality, Symmetry and Topology

Supersymmetry, originally proposed in particle physics, refers to a dual relation that connects fermionic and bosonic degrees of freedom in a system. Recently, there has been considerable interest in applying the idea of supersymmetry to topological phases, motivated by the attempt to gain insights from the fermion side into the boson side and vice versa. We present a systematic study of this construction when applied to band topology in noninteracting systems. First, on top of the conventional ten-fold way, we find that topological insulators and superconductors are divided into three classes depending on whether the supercharge can be local and symmetric, must break a symmetry to preserve locality, or needs to break locality. Second, we resolve the apparent paradox between the nontriviality of free fermions and the triviality of free bosons by noting that the topological information is encoded in the identification map. We also discuss how to understand a recently revealed supersymmetric entanglement duality in this context. These findings are illustrated by prototypical examples. Our work sheds new light on band topology from the perspective of supersymmetry.


I. INTRODUCTION
Supersymmetric (SUSY) models play an important role in physics.Perhaps the most well-known is their use in relativistic quantum field theories, where they resolve a number of theoretical problems [1].SUSY in quantum mechanics was introduced to understand properties of SUSY theories [2], which led to profound understanding of SUSY breaking [3].SUSY can also appear in non-relativistic theories, for example, in statistical mechanics [4], or the Sachdev-Ye-Kitaev model [5].Besides, it is a powerful tool in the analysis of disordered systems [6] or to solve an array of problems in quantum and statistical physics [7].
Recently, the idea of SUSY has been applied to extract the topological indices of free-boson systems, such as photonic and magnonic crystals, from their free-fermion counterparts [16][17][18][19].This strategy indeed works well for individual bands, which may still be nontrivial despite the fact that the ground state of free bosons is always trivial [20].In seeming contradiction, one important measure of topology, the entanglement spectrum [21] has recently found to translate from the fermionic ground state to the bosonic ground state [22].
In this work, we analyze free-SUSY systems systematically.We start from a basic question concerning the existence of a SUSY construction for free-fermion topological phases.The answer turns out to be highly nontrivial: depending on the topological class, the SUSY generator, called supercharge, may be necessarily non-local, or local but necessarily asymmetric [23] [cf.Fig. 1(b)].The entanglement problem gets resolved as the SUSY map, called identification map, can lead to very strong squeezing, if it is locality-preserving.
< l a t e x i t s h a 1 _ b a s e 6 4 = " x w x 4 C / 5 Q 9 A K z Z / 5 X q x / n B F w P O m k = " > A A A B 8 H i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e x K U I 9 B L x 4 j m I c k S 5 i d z C Z D 5 r H M z A p h y V d 4 8 a C I V z / H m 3 / j J N m D J h Y 0 F F X d d H d F C W f G + v 6 3 V 1 h b 3 9 j c K m 6 X d n b 3 9 g / K h 0 c t o 1 J N a J M o r n Q n w o Z y J m n T M s t p J 9 E U i 4 j T d j S + n f n t J 6 o N U / L B T h I a C j y U L G Y E W y c 9 j v p Z T w s U T / v l i l / 1 5 0 C r J M h J B X I 0 p a F 9 X g s l q 7 r 1 X q N 3 k c R T i B U z i H A K 6 g D n f Q g C Y Q E P A M r / D m a e / F e / c + F q 0 F L 5 8 5 h j / w P n 8 A t z W Q X Q = = < / l a t e x i t > p a F 9 X g s l q 7 r 1 X q N 3 k c R T i B U z i H A K 6 g D n f Q g C Y Q E P A M r / D m a e / F e / c + F q 0 F L 5 8 5 h j / w P n 8 A s S G Q W Q = = < / l a t e x i t > 2D version supercharge q 4 B E 8 g 1 f w p j 1 p L 9 q 7 9 j F t L W i z m X 3 w B 9 r n D 9 r Z l X A = < / l a t e x i t > NL < l a t e x i t s h a 1 _ b a s e 6 4 = " d a 1  I).(c) Entanglement-spectrum (ES) duality for two subsystems related by the identification map.
For topological free-fermion phases, the subsystem on the bosonic side contains strongly squeezed modes.

II. SUPERSYMMETRIC BANDS
We consider a pair of fermion and boson systems in d dimensions (dD).For simplicity, we assume that fermions and bosons live on identical hypercubic lattices Ω ⊂ Z d with a set of internal states I at each unit cell and subject to periodic boundary conditions.Denoting the fermionic / bosonic modes by ĉrs / brs (r ∈ Ω, s ∈ I), we can write down a general translation invariant supercharge where ĉk cardinality of a set).The 2|I| × 2|I| matrix q(k) satisfies arXiv:2112.07527v3[quant-ph] 14 Mar 2022 q(k + 2πe j ) = q(k) (e j : unit vector in the jth direction) ∀j = 1, 2, ..., d, and ( Here, σ µ (µ = x) is the Pauli matrix and 1 1 is the identity acting on the internal states.This symmetry (2) arises from the Hermiticity of the supercharge.If the fermion-boson coupling in Q is short-ranged, i.e., decays no slower than exponentially, we know that q(k) is analytic in k and vice versa [24].We will call such a supercharge local.By taking the square of the supercharge (1), we obtain a SUSY pair of fermion and boson bands [cf.Fig. 1(a)] where the Bogoliubov-de Gennes (BdG) Hamiltonians h f (k) and h b (k) are determined by q(k) via The full derivation of the above results are given in Appendix A. As an important implication of SUSY, the dynamical matrices h f (k) and Zh b (k) share exactly the same spectrum [25].This can be easily seen from the fact that the spectrum of AB is identical to that of BA for any two matrices.
It is also clear that the necessary and sufficient condition for a nonzero band gap (at zero energy) is det q(k) = 0.

III. TOPOLOGICAL OBSTRUCTION FROM LOCALITY AND SYMMETRY
To discuss band topology, we should impose the locality such that h f,b (k) is a smooth map from T d ≡ (2πR/Z) d (dD torus) to a matrix space constrained by the gap condition and symmetries.We define two Hamiltonians to be topologically equivalent if they can be smoothly interpolated within the map space, possibly with the assistance of some auxiliary bands [10].
As pointed out in Ref. [20], h b (k) is always trivial when requiring a gap at zero energy in the dynamical matrix.This can be seen from the fact that it can always be smoothly deformed into the identity via a linear interpolation, which preserves any additional symmetries.This result is consistent with the triviality of short-range correlated bosonic Gaussian states [26].Moreover, h b (k) can always be generated by a local and symmetric supercharge with q(k) = h b (k).
On the other hand, h f (k) can be nontrivial.Remarkably, if we assume that the supercharge is local and has a symmetry, we will find that not all topological fermion bands can be generated.In fact, the SUSY construction can only cover "disentanglable" topological phases, which can be transformed into  [11][12][13] and their realizability by the SUSY construction.Cell uncolored: the supercharge can be local and symmetric.Cells in dark red: the supercharge necessarily breaks the locality.Cells in light red: the supercharge, if local, necessarily breaks the symmetry.Cells in light blue: a subgroup 2Z out of Z can be generated by local and symmetric supercharges.Otherwise, the supercharge necessarily breaks the symmetry, provided that it is local.
A broader class of topological phases can be created, if we loosen the constraints on the supercharge.If we do not require the supercharge to obey the same symmetries as the phase but still impose the locality, then those phases that become disentanglable upon removing the symmetries can be generated.In particular, all the symmetry-protected topological phases that become trivial in the absence of symmetries can arise from a local supercharge, since trivial states are in the same phase as product states and they are connected by some trivial Gaussian operations [26].We summarize the results for all the tenfold fundamental symmetry classes in Table I.While the full derivation is a bit involved (see Appendices B and C), simple explanations are available in specific cases.For example, 2D Chern insulators (in class A) cannot be generated by local supercharges since the Wannier functions cannot be exponentially localized [28].
Forgoing locality as well, one can obtain all topological phases from the SUSY construction.Given an arbitrary h f (k), the corresponding supercharge can be chosen as k) may not be continuous.

IV. IDENTIFICATION MAP AND ENTANGLEMENT DUALITY
The supercharge induces a pair of identification maps which, in a canonical way, identify fermionic subsystems with bosonic subsystems, and vice versa.Subsystems identified in this way were recently shown to obey an entanglement duality [22] which, in the following, we review briefly and investigate in the context of translation invariant systems.
The identification maps are most easily defined in terms of their action on the eigenmodes of the system.Let f k = [ fk1 , fk2 , ..., fk|I| ] and φk = [ φk1 , φk2 , ..., φk|I| ] be such that they diagonalize the Hamiltonians (3), i.e., Then the identification maps identify bosonic and fermionic eigenmodes with identical excitation energies.Specifically, the first map acts as L 1 ( f k ) = φk and the second acts as L 2 ( φk ) = −i f k .In the original basis, the action of the identification map is given by with the matrices (6) Hence, if the fermion-boson coupling is short-ranged, i.e., q(k) is analytic, so are L 1 (k) and L 2 (k).Therefore, for a short-range supercharge Q both L 1 and L 2 are locality preserving mappings between the fermionic and bosonic lattices.
The identification maps preserve the expectation value of the commutator and anti-commutator of linear operators, i.e., for arbitrary fermionic linear operators f1 , f2 (7) and accordingly for the anti-commutator, and for L 2 .Also, the identification maps fully encode the ground state of the Hamiltonians (1).This is because their compositions yield the linear complex structures J f = L 2 • L 1 of the fermionic ground state |Ψ f , and [22,29,30].The linear complex structures relate the expectation values of anti-commutators and commutators in the ground states as for any bosonic linear operators φ1 , φ2 , and The linear complex structures also encode the entanglement content of the ground states.To this end, consider subsystems consisting of a single fermionic / bosonic mode ĉ / b.If the reduced state of a mode is pure, i.e., if the mode is in a product state with the complement of its ground state |Ψ f/b , then the restriction of J f/b to the linear space spanned by ĉ / b and ĉ † / b † has eigenvalues ±i [31].However, if < l a t e x i t s h a 1 _ b a s e 6 4 = " u E a G j 4 Z N l N 3 s t q M u C H F a d 1 u 8 N N + P 9 q 3 X E G M 4 s g x 9 h f H w C 8 W W n P g = = < / l a t e x i t > n < l a t e x i t s h a _ b a s e = " B n C I a P G v q X x P h e C y e H S = " T e e U m 1 r U r M M X p j t P w I G 8 L q k < / l a t e x i t > / |r| 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " H 5 r A O e g C D B 6 t D W v H 6 t g z 9 p b t 2 c F T q 2 1 N Z z 6 B F 2 F / + w d y E b 1 1 < / l a t e x i t > diagonal < l a t e x i t s h a 1 _ b a s e 6 4 = " C P y H k r S p y Q l r R h H q a / D 7 w j k W 3 5 8 A r e r C f r x X q 3 P q a l S 9 a s 5 w D 8 g f X 1 A 2 + N o k 8 = < / l a t e x i t > |[q n ] SS 0 | < l a t e x i t s h a 1 _ b a s e 6 4 = " e g I 5 i X v c z O 3 4 i h 9 FIG. 2. Real-space profiles of the diagonal (blue) and off-diagonal (green) components in the supercharge for (a) the Kitaev chain ( 9) with µ = 1, t = 0.7 (system size: 400) and (b) the 2D chiral superconductor (10) with m = 1 (direction: r ∝ (1, 1), system size: 400 × 400).In (a), both components decay exponentially.In (b), the diagonal/off-diagonal decays algebraically as |r| −2 / |r| −3 .Here qr = |Ω| −1 k q(k)e ik•r and S = ±s (s ∈ I), where "±" arises from the creation or annihilation half of the basis [cf.Eq. ( 1)].
the reduced state is mixed, i.e., the mode is entangled with the complement, then the restriction of J f/b to the subsystem has eigenvalues The von Neumann entropy of the mode is then s(λ f/b ) with Gaussian, subsystems consisting of several modes can be decomposed in terms of the subsystems' normal modes and then the above applies mode by mode [22,29,30].
The entanglement duality [22] asserts that if a fermionic and a bosonic subsystem are related by an identification map, e.g., L 1 (ĉ) = b or L 2 ( b) = ĉ, then the eigenvalues of the restricted complex linear structures are inverses of each other, This means that, on the one hand, modes in pure states are mapped to modes in pure states, and, on the other hand, highly entangled modes are mapped to highly entangled modes.If the fermion system is topological, the reduced state on some region of the lattice is known to contain some (almost) maximally entangled modes [21].For such modes the eigenvalue λ f approaches zero and its entanglement entropy approaches log 2. On the bosonic side, such a system is mapped to a subsystem with divergent values of λ b , and thus a divergent entanglement entropy.However, because the identification maps preserve locality, the real-space profile does not change much under the mapping from the fermionic to the bosonic lattice.Hence, the divergent entanglement in the dual bosonic mode necessarily is due to high squeezing.In fact, this can be seen by considering a highly entangled fermionic mode operator ĉ , i.e., with Through L 1 this mode is identified with the bosonic mode defined by b = L (ĉ ) / √ .The rescaling with √ , which is necessary due to (7) to obtain a proper bosonic normalization, causes the coefficients in the expansion of b to diverge as → 0.
In contrast to the fermionic mode, it is not mirror-symmetric and, counterintuitively, at the left edge it is localized outside the shaded region.(Note that L1 (ĉ ) is not a normalized bosonic ladder operator, but † ] ≈ 1.78 • 10 −4 , hence after normalizing the operator we obtain a highly squeezed bosonic mode.)

V. EXAMPLES
As a canonical example we consider the 1D Kitaev chain [32] under periodic boundary conditions: Following the construction given in the Appendices B and D, one finds that this Hamiltonian is generated by the supercharge ).This supercharge generates the bosonic Hamiltonian with h b (k) = k σ 0 , which, in contrast to the Kitaev chain, is particlenumber conserving, and thus has a trivial ground state.
Both the supercharge [cf.Fig. 2(a)] and the bosonic Hamiltonian are short range.However, whereas both Ĥf and Ĥb are mirror-symmetric, the supercharge is not.This asymmetry shows most clearly in the flat band case, µ = 0, where the fermionic eigenmodes are formed by pairing Majorana modes on adjacent sides, fj = (ĉ j + ĉj+1 + ĉ † j − ĉ † j+1 )/2.Note that L 1 ( fj ) = bj , implying that the bosonic operator on the jth site is identified with a fermionic mode residing equally on the jth and (j + 1)th sites, thus breaking mirror-symmetry.
This asymmetry persists in the topological phase of the Ki-taev chain also for non-zero values of µ.Fig. 3 demonstrates this by showing the localization of the fermionic entanglement edge mode in a subsystem and that of the bosonic dual mode.
The figure also conveys that the dual bosonic mode is highly squeezed.This behaviour is also related to the very different scaling of the entanglement entropy of fermionic subsytems and their bosonic dual subsystems, also given in Fig. 3. On the fermion side, the entanglement entropy approaches a finite value as the subsystem size l increases.Meanwhile, as l increases, the minimal λ f decays exponentially.Due to the duality (8), this leads to dual bosonic eigenvalue which is increasing exponentially, thus leading to a scaling of O(l) of the entanglement entropy on the boson side.
We turn to consider a 2D chiral superconductor (class D), which is also described by a two-band BdG Hamiltonian For 0 < |m| < 2, the system is in a topological phase characterized by a nontrivial Chern number [33].According to the previous general analysis, such a topological phase cannot be generated by a local supercharge.Nevertheless, we can still construct a nonlocal supercharge whose diagonal (hopping)/off-diagonal (pairing) component follows a powerlaw decay |r| −2 /|r| −3 [cf.Fig. 2b].These algebraic tails arise from the non-analyticity of q(k) at specific points in the Brillouin zone (see Appendix B 1).It might be interesting to ask whether the supercharge could be more localized than |r| −2 decay.
As a final example, we mention that all the time-reversalsymmetric topological insulators (class AII), both in 2D and 3D, can be generated by local time-reversal-symmetric supercharges that break the U(1) symmetry (see Appendix B 3).

VI. DISCUSSIONS
Strictly local systems, whose coupling ranges are finite, constitute an important subclass of short-range systems.In particular, all the examples of Ĥf above fall into this category.It is natural to ask whether the supercharge can also be chosen to be strictly local for them.While we do not have a complete answer, we can show this is possible at least for classes BDI, CII and AIII in any dimensions (see Appendix B 2).This is to be contrasted to zero-length correlated topological phases, i.e., those with strictly local (compactly supported) Wannier functions, which do not exist in d ≥ 2D for all the fundamental symmetry classes [34].
A remarkable observation in the numerical demonstration is the spatial asymmetry in the identification map for the Kitaev chain, which is mirror-symmetric.In fact, the identification map, or equivalently the supercharge can never be mirrorsymmetric under the locality constraint.The most convenient way to see this is to forget the time-reversal symmetry and regard the Kitaev chain as a nontrivial phase in class D, which is classified by Z 2 in 1D.This Z 2 index is determined by the parity of the winding number of q(k).Provided that the supercharge is mirror-symmetric, its winding number will be enforced to be even and thus cannot generate a nontrivial Kitaev chain.It would be interesting to explore the topological constraints of additional symmetries in a more general setting.
Finally, we note that the exemplified boson Hamiltonians respect U(1) particle-number symmetry, although the corresponding fermion Hamiltonians and supercharges do not.In fact, given a general local supercharge, we can always gauge transform it in a locality-preserving manner such that h f (k) (h b (k)) remains invariant, while h b (k) (h f (k)) becomes particle-conserving (see Appendix D).In particular, this result implies that all the fermion topological phases can be mapped into the boson vacuum, the ground state of an arbitrary U(1) symmetric boson Hamiltonian.

VII. SUMMARY AND OUTLOOK
We have examined the role of topology in the SUSY construction of quadratic fermion and boson Hamiltonians.We have found that not all the topological fermion bands can be generated by local and symmetric supercharges.We have also clarified that the boson bands are always trivial and the topological information is encoded in the SUSY map.The apparent inconsistency with the entanglement duality can be resolved by noting that the bosonic subsystem could be highly squeezed.
Our work raises many open problems for future studies.On top of those in the discussion part, it would be interesting to see whether and how the SUSY construction can be extended to unstable and meta-stable [35,36] bosonic systems.Probably the supercharge and the fermion Hamiltonian should be non-Hermitian [24,37].It might also be interesting to consider the generalization to a full open-system setting (described by Lindbladians) and to systems in the continuum rather than lattices.
The result of {r k , r † k } turns out to be from which we know Combining all the results above, we end up with and With the matrix in Eq. (A2) denoted as q(k), one can check that Eqs.(A12) and (A13) are nothing but Eq. ( 4).
Appendix B: Constructing supercharges out of fermion Hamiltonians

General construction
We introduce a general and simple way to extract the supercharge and construct the SUSY boson band for a given gapped fermion BdG Hamiltonian h f (k), although this construction may not guarantee the locality.
Having in mind that h f (k) is gapped and respects the particle-hole symmetry (i.e., Xh f (k) * X = −h f (−k)), we know that there should be exactly n = |I| positive eigenenergy kα > 0 for each k.We can thus identify the corresponding normalized eigenstates: (B1) By again using the PHS, we have implying that Xu * −kα is also an eigenstate but with a negative eigenenergy − −kα .Obviously, {u kα , Xu * −kα } n α=1 form the complete eigenbasis of h f (k), indicating the following spectral decomposition (diagonalization): Such a choice of the eigenbasis has appeared in Ref. [14] (cf.Eq. (5.2)) and plays an important role in calculating PHSprotected topological invariants.The privilege of this eigenbasis is rooted in the following symmetry property: Accordingly, we can choose q(k) to be such that Eq. ( 2) is fulfilled.For a general two-band BdG Hamiltonian h f (k) = µ=x,y,z d µ (k)σ µ , one can check that a valid choice reads where variable k is dropped for simplicity, |d| = Note that the (non-local) supercharge for the chiral superconductor in the main text follows this construction.For this example, one can check that e iφ is not well-defined at the highsymmetry points Γ = (0, 0), (0, π), (π, 0) and (π, π), while |d| ± d z is well-defined but may exhibit a linear singularity ∼ |k − Γ| nearby.After the Fourier transform, these singularities are turned into the algebraic tails of the supercharge in real space, as shown in Fig. 2(b).Similar phenomena have been observed for the parent Hamiltonians of chiral Gaussian fermionic projected entangled pair states [38].
Remarkably, the corresponding boson BdG Hamiltonian of Eq. (B5) is diagonalized: This Hamiltonian is always local and conserves the particle number.One can also easily write down the identification maps: which are both unitary.
While the above general construction may not be local, we can actually construct a local supercharge from another local one for a different Hamiltonian, provided that Hamiltonian is in the same (in the homotopy sense) topological phase as the target one.To see this, we consider h f (k; 0) and h f (k; 1) which can be smoothly interpolated by h f (k; λ) (λ ∈ [0, 1]).Suppose that h f (k; 0) can be generated by a local supercharge q 0 (k), then we can construct q 1 (k) = v 1 (k)q 0 (k) that generates h f (k; 1), where v 1 (k) is obtained by solving According to Eq. (B9), this construction is also compatible with any additional symmetries.

Strictly local and symmetric constructions for chiral symmetry classes
In the previous subsection, we introduced a way to construct q(k) out of h f (k).Provided that q(k) turns out to be local, it generally exhibits exponential tails even for strictly local h f (k).Here we show that, for strictly local Hamiltonians in the chiral symmetry classes AIII, BDI and CII, which are known to be fully disentanglable [26], we can always construct strictly local and symmetric supercharges in any dimensions, no matter whether h f (k) is topological or not.
Let us start from class BDI, which respects a spinless timereversal symmetry (TRS) alone: The general form of the supercharge is given by where q ± (k) * = q ± (−k), and that of the Hamiltonian reads where h y,z (k The supercharge (B11) generates the fol-lowing Hamiltonian: (B13) To make the above equation the target Hamiltonian (B12), we should require: which admits the following symmetry-preserving (spinless TRS) and strictly local solution: The strict locality follows from the fact that the entries of q(k) are linear combinations of those of h f (k), which is strictly local by assumption.We move on to class AIII, which respects a spin-1/2 TRS: as well as a U(1) spin-rotation symmetry along z-axis: The general form of the supercharge is given by where q 1 (k) and q 2 (k) are arbitrary (as long as det q(k) = 0), and that of the Hamiltonian reads B19) where h 1,2 (k) † = h 1,2 (k).The supercharge (B18) generates the following Hamiltonian: (q 1 q † 1 − q 2 q † 2 )(k) 0 0 −(q 1 q † 2 + q 2 q † 1 )(k) 0 (q * 1 q T 1 − q * 2 q T 2 )(−k) (q * 1 q T 2 + q * 2 q T 1 )(−k) 0 0 (q * 1 q T 2 + q * 2 q T 1 )(−k) (q * 2 q T 2 − q * 1 q T 1 )(−k) 0 −(q 1 q † 2 + q 2 q † 1 )(k) 0 0 (q 2 q † 2 − q 1 q † 1 )(k) (B20) To make the above equation equal to the target Hamiltonian (B19), we should require: q 1 (k)q 1 (k) † − q 2 (k)q † 2 (k) = h 1 (k), q 1 (k)q 2 (k) † + q 2 (k)q 1 (k) † = h 2 (k), (B21) 1. U(1)-symmetric h b with fixed h f We first note that a gauge transformation in the supercharge q(k) → q(k)S(k), where S(k) is a symplectic matrix satisfying S(k)ZS(k) † = Z, XS(k)X = S(−k) * , (D1) does not change h f (k).On the other hand, h b (k) = q(k) † q(k) is turned into h b (k) = S(k) † q(k) † q(k)S(k).where we have used the symplecticity (D7) of S(k).This identity indeed holds true: Finally, to see the uniqueness of S(k), we only have to rewrite Eq. (D10) into Noting that both q(k)S(k) 2 q(k) † and |h f (k)| are positivedefinite, the only solution should be Eq.(D6) [42].As a simple exercise, let us consider a two-band fermion system in class BDI and choose the original supercharge to be given by Eqs.(B11) and (B15) with h y,z (k) ∈ R.After some straightforward calculations, one can obtain the gauge transformed supercharge q(k)S(k) to be √ where k = h y (k) 2 + h z (k) 2 .Note that the supercharge for the Kitaev chain in the main text follows this construction.

U(1)-symmetric h f with fixed h b
We finally turn to the converse.For a fixed h b (k), we have a gauge transformation q(k) → W (k)q(k), where W (k) is a unitary matrix satisfying XW (k) * X = W (−k). (D14) To make the transformed fermion Hamiltonian particlenumber conserving, i.e., we can choose the involutory unitary W (k) † ZW (k) to be the flattened Hamiltonian (before transformation): We claim that a possible solution is given by h f < l a t e x i t s h a 1 _ b a s e 6 4 = " e 3 x Y a O G n + d G 6 + p t F h J I e 0 X P V F B c = " > A A A B 8 H i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e x K U I 9 B L x 4 j m I c k S 5 i d z C Z D 5 r H M z A p h y V d 4 8 a C I V z / H m 3 / j J N m D J h Y 0 F F X d d H d F C W f G + v 6 3 V 1 h b 3 9 j c K m 6 X d n b 3 9 g / K h 0 c t o 1 J N a J M o r n Q n w o Z y J m n T M s t p J 9 E U i 4 j T d j S + n f n t J 6 o N U / L B T h I a C j y U L G Y E Wy c 9 j v p Z T w s U T f v l i l / 1 5 0

FIG. 1 .
FIG. 1. (a)A translation-invariant supercharge generates a pair of SUSY fermion and boson systems.The SUSY pair can be related to each other by the identification maps.(b) Free-fermion topological phases can be categorized into three classes, depending on whether their parent supercharges (i) are necessarily non-local (NL); (ii) can be local but then necessarily asymmetric (LAS); (iii) can be both local and symmetric (LS) (see TableI).(c) Entanglement-spectrum (ES) duality for two subsystems related by the identification map.For topological free-fermion phases, the subsystem on the bosonic side contains strongly squeezed modes.

12 < l a t e x i t s h a 1 _ b a s e 6 4 =
r t 3 p b Y 1 6 1 k H j 8 z + 9 g + U U 8 H Y < / l a t e x i t > " u / X 8 b 2 b

TABLE I .
Periodic table of topological insulators and superconductors