Bridging Fermionic and Bosonic Short Range Entangled States

In this paper we construct bosonic short range entangled (SRE) states in all spatial dimensions by coupling a $Z_2$ gauge field to fermionic SRE states with the same symmetries, and driving the $Z_2$ gauge field to its confined phase. We demonstrate that this approach allows us to construct many examples of bosonic SRE states, and we demonstrate that the previous descriptions of bosonic SRE states such as the semiclassical nonlinear sigma model field theory and the Chern-Simons field theory can all be derived using the fermionic SRE states.


Introduction -
A short range entangled (SRE) state is the ground state of a quantum many-body system that does not have bulk ground state degeneracy or topological entanglement entropy. However, these states can still have stable nontrivial edge states. Some of the SRE states need certain symmetry to protect the edge states, and these SRE states are also called symmetry protected topological (SPT) states. The most well-known SPT states include the Haldane phase of spin-1 chain [1,2], quantum spin Hall insulator [3,4], topological insulator [5][6][7], and topological superconductor such as Helium 3 -B phase [8,9]. All the free fermion SPT states have been well understood and classified in Ref. [10][11][12], and recent studies suggest that interaction may not lead to new SRE states, but it can reduce the classification of fermionic SRE states [13][14][15][16][17][18][19][20]. Unlike fermionic systems, bosonic SPT states do need strong interaction. Most bosonic SRE states can be classified by symmetry group cohomology [21,22], Chern-Simons theory [23] and semiclassical nonlinear sigma model [24].
In this work we demonstrate that there is a close relation between fermionic and bosonic SRE states, more precisely many bosonic SRE states can be constructed from fermionic SRE states with the same symmetry. All fermion systems have at least a Z 2 symmetry c i → −c i , where c i is a local fermion annihilation operator, thus we can couple all fermion Hamiltonians to a dynamical Z 2 gauge field, and microscopically this Z 2 gauge field commutes with the actual physical symmetry of the fermion system. Once the Z 2 gauge field is in its confined phase, the fermionic degree of freedom no longer exists in the spectrum of the Hamiltonian, and the system becomes a bosonic system. However, in many cases, confinement of a gauge field necessarily breaks certain symmetry of the system, thus we have to be very careful. In both 2d and 3d, a Z 2 gauge field has a confined phase and a deconfined phase. The deconfined phase is characterized by topological excitations of the Z 2 gauge field. In 2d, the Z 2 gauge field has a "vison" excitation, which corresponds to a π-flux seen by the matter fields. In 3d, the topological excitation is a "vison loop", which is a closed ring of π-flux. In 2d/3d, when the visons/vison loops pro-liferate (condense), the system enters the confined phase, i.e. fermions carrying Z 2 gauge charge cannot propagate freely in the bulk due to the phase fluctuations induced by the vison/vison loop condensation.
However, when the Z 2 gauge field is coupled to a fermionic SRE state, the vison and vison loop often carry nontrivial quantum numbers, or degenerate low-energy spectrum. In these cases, when visons and vison loops condense, the condensate would not be a fully gapped nondegenerate state that does not break any symmetry. Also, sometimes visons in 2d would have a nontrivial statistics, thus it cannot trivially condense. Thus only in certain specific cases can we confine the fermionic SRE states and obtain a fully gapped and symmetric bosonic state. Thus analysis of spectrum and quantum number carried by the vison and vison loop is the key of our study.
Our approach can also be viewed as a slave fermion construction of bosonic SRE states, which has been considered in Ref. 25-29. However, in all these previous studies the gauge group associated with the slave fermion is bigger than Z 2 , which means that when the gauge fluctuation is ignored, at the mean field level the slave fermion has a much larger symmetry than the boson system, and the analysis of gauge confined phase is much more complicated. In our case the gauge group is Z 2 , and since any fermion system has this Z 2 symmetry, the fermion SRE states would have the same symmetry as the bosonic states after gauge confinement. Thus in our case the nature of the confined phase can be analyzed reliably, and it only depends on the properties of visons and vison loops.
Construction of 3d bosonic SPT phases -Let us take the 3d topological superconductor (TSC) phase with time-reversal symmetry as an example. One example of such TSC is the 3 He B phase. Here instead of focusing on the real 3 He system, we are discussing a more general family of TSC phases defined on a lattice that are topologically equivalent to 3 He B. Close to the trivial-TSC phase transition, in the continuum limit this TSC phase can be described by the following universal real space Hamiltonian: where σ ij = σ i ⊗ σ j denotes the tensor product of Pauli matrices, and a = 1 · · · n is the flavor index. For each flavor index a, χ a is a four component Majorana fermion.
In this Hamiltonian m > 0 and m < 0 correspond to the TSC phase and the trivial phase respectively [43]. The time-reversal symmetry acts as Z T 2 : χ → iΓ 5 χ. Our conclusion is that, when we couple n-copies of this TSC to the same Z 2 gauge field, the Z 2 gauge field can have a fully gapped nondegenerate confined phase when and only when n is an integer multiple of 8. And when n = 8, the confined phase is the 3d bosonic topological superconductor with time-reversal symmetry first characterized in Ref. 30.
First of all, when n = 1, the vison loop must be gapless, and the gaplessness is protected by time-reversal symmetry [9]. For a vison line along x direction, the effective 1d Hamiltonian along the vison line reads: In this reduced 1d theory, time-reversal symmetry acts as Z T 2 : χ → iσ 2 χ. The only mass term χ ⊺ σ 2 χ in this vison line would break time-reversal symmetry, thus as long as time-reversal is preserved, the vison line is always gapless. This implies that when n = 1 the vison line definitely cannot drive the system into a fully gapped state by proliferation without breaking time-reversal.
For n > 1, the effective theory along the vison line becomes Then for even integer n, it appears that there is a timereversal symmetric mass term χ ⊺ a σ 1 A ab χ b , where A is an antisymmetric matrix in the flavor space. In the bulk theory Eq. (1), this mass term can correspond to several terms such as χ ⊺ a σ 13 A ab χ b , χ ⊺ a σ 10 A ab χ b , etc. However, none of these terms can gap out vison lines along all directions. For example, for vison loops along y direction, the modes moving along +y is an eigenstate of Γ 2 with Γ 2 = +1, and modes moving along −y direction have eigenvalue Γ 2 = −1. Because σ 13 commutes with Γ 2 = σ 10 , χ ⊺ a σ 13 A ab χ b can never back-scatter modes in the y vison line. In fact no flavor mixing time-reversal invariant fermion bilinear terms in the bulk would gap out the vison lines along all directions, while a Z 2 gauge confined phase requires dynamically condensing vison lines in all directions. Therefore the fermion bilinear flavor mixing terms in the bulk do not allow us to condense the vison lines in order to generate a fully gapped symmetric bosonic state.
Since no fermion bilinear term can gap out all the vison loops, we need to consider interaction effects. In Ref. 13,14, the authors studied the interaction effect on Eq. (3), and the conclusion is that for n = 8 there is an SO(7) invariant interaction term H 1 = dx A abcd χ ⊺ a σ 1 χ b χ ⊺ c σ 1 χ d that can gap out the 1d theory Eq. (3) without generating nonzero expectation value of any fermion bilinear operator. At the field theory level, Eq. (3) has an emergent U(1) symmetry χ → exp(iθσ 3 )χ, thus all the field theory analysis in Ref. 13,14 can be applied to the following interaction term can also gap out the 1d theory Eq. (3) without degeneracy. H 2 corresponds to the following term in the bulk: Since this term is rotationally invariant, it will gap out vison lines along all directions. Thus with n = 8, and with the interaction term H int in the bulk, all vison loops can be gapped out without breaking time-reversal symmetry, thus we can safely condense the vison loops and drive the system into a fully gapped, time-reversal invariant bosonic state. But this is only possible when n is an integral multiple of 8. In the following paragraphs we will argue that when n = 8 the confined bosonic state is a bosonic SPT state. Let us couple the 8 copies of 3 He B to a five-component unit vector n: where γ j are five 8 × 8 symmetric matrices in the flavor space that satisfy {γ i , γ j } = 2δ ij . Under time-reversal transformation, n → −n. Following the calculation in Ref. 31, we can show that for the 3 He B phase with m > 0, after integrating out the fermions, the effective field theory for the vector n contains a topological Θ-term at Θ = 2π: where Ω 4 is the volume of a four dimensional sphere with unit radius. Eq. (6) is precisely the field theory introduced in Ref. 24,30 to describe the 3d bosonic topological SC with time-reversal symmetry.
Using the field theory Eq. (6), we can demonstrate that the 2d boundary of this 3d bosonic SPT state could be a 2d Z 2 topological order, whose mutually semionic excitations e and m are both Kramers' doublet [30] (The so called eT mT state) [44]. Ref. 19,20,32 argued that the boundary of 8 copies of 3 He B is the (fermionized) eT mT state. For the sake of completeness, we will repeat this argument. Based on the field theory Eq. (6), the e and m excitations at the 2d boundary of the 3d bosonic SPT phase correspond to the vortex of boson field b 1 ∼ n 1 + in 2 , and vortex of b 2 ∼ n 3 + in 4 respectively [45], which can be considered as surface terminations of bulk vortex lines. By solving the Bogoliubov-de Gennes equation with a vortex at the boundary, we can demonstrate that there are four Majorana fermion zero modes located at each vortex core. These four Majorana fermion zero modes can in total generate four different states. Under interaction, time-reversal symmetry [46] guarantees that these four states split into two degenerate doublets with opposite fermion number parity. Thus in the bulk each vortex line is effectively four copies of 1d Kitaev's Majorana chain. Since we are in a Z 2 gauge confined phase, we are only allowed to consider states with even number of fermions, thus after gauge projection, only one of the two doublets survives, which according to the supplementary material and Ref. 20 is a Kramers doublet. Also the vortex of b 1 carries charge ±1/2 of b 2 , and vortex of b 2 carries ±1/2 charge of b 1 , thus these two vortices are both Kramers doublet, and they have mutual semion statistics. This means that boundary of the confined phase is really the eT mT state.
Combining all the results together, we conclude that the Z 2 confined phase of 8 copies of 3 He B is really the bosonic SPT phase with time-reversal symmetry. We can also give the 8 copies of 3 He B phase various flavor symmetries, and we can construct many 3d bosonic SPT phases with symmetry that contains Z T 2 as a normal subgroup by confining the bulk Z 2 gauge field. Since all the free fermion SPT states in 3d require the time-reversal symmetry, thus so far our approach does not allow us to construct 3d bosonic SPT phases without Z T 2 . Construction of 2d bosonic SPT phases -Now let us look at 2d examples. In 2d the simplest fermionic SRE state is the p + ip topological superconductor (TSC) that does not require any symmetry, and the simplest bosonic SRE state is the so called "E 8 " state with chiral central charge c − = 8 at its boundary [33,34]. In the following we will prove that if we couple n copies of p+ip TSC to a Z 2 gauge field, the Z 2 gauge field can confine to a gapped bosonic state when and only when n is a integral multiple of 16. And when n = 16, the confined phase is precisely the bosonic E 8 SRE state [35]. First of all, when n = 1, the vison of the Z 2 gauge field carries a Majorana fermion zero mode, which grants the vison a nonabelian statistics, thus when n = 1 (and generally for odd integer n) the Z 2 gauge field cannot enter its confined phase by condensing the vison. When n is even, n-copies of p + ip TSC is equivalent to an integer quantum Hall (IQH) state with Hall conductivity ν = n/2, thus a vison (half flux quantum) would carry charge n/4, and has statistics angle πn/8 under exchange. Thus the smallest n that makes vison a boson is 16, and when n = 16, the Z 2 gauge field can enter a confined phase by condensing the bosonic vison.
The vison condensation can be formulated by the Chern-Simons theory. [36] Let us start from the Chern-Simons description for n-copies of p + ip TSC with even n = 2ν (i.e. ν layers of IQH), and couple the fermion currents da I (I = 1, · · · , ν) to the Z 2 gauge field A. The Lagrangian density can be written as whereÃ field couples to the vison current in the Z 2 gauge theory. The field A can be treated as a Lagrangian multiplier and integrated out first, which leads to the constraint I a I + 2Ã = 0. This constraint can be solved by the following reparameterization Substituting Eq. (8) into Eq. (7), we arrive at a bosonic theory in terms of the new set of gauge fieldsã I , as is the Cartan matrix of the so(n) Lie algebra(for even n > 2). For n = 16, the K-matrix reads which gives the SO(16) 1 Chern-Simons theory. We now extend K SO(16) by a block of trivial boson, given by the K-matrix σ 1 [37], and define K ext = K SO(16) ⊕ σ 1 . One finds a transform W , with det W = 1, given by such that The last 2 × 2 block describes a Z 2 topological order. The fermion excitations of this K-matrix corresponds to the original fermion in the p + ip TSC. The vison couples to the last gauge field, i.e. it corresponds to the charge vector (0, 0, 0, 0, 0, 0, 0, 0, 0, 1), and is a boson ready to condense. Thus after the vison condensation, the Z 2 topological order is destroyed and the original fermion is confined.  17. In this system the fermions with zero Z 2 charge form a p + ip TSC, while fermions carrying Z 2 charge form a p − ip TSC. This Z 2 global symmetry is different from the Z 2 gauge symmetry, since all the fermions in our system carry Z 2 gauge charge. For one copy of the p ± ip TSC coupled to the Z 2 gauge field, the vison carries two independent Majorana fermion zero modes χ 1 and χ 2 , and the global Z 2 symmetry acts Z 2 : χ → σ z χ. There is no nontrivial Hamiltonian for these two Majorana fermion modes that preserves the Z 2 symmetry, thus the spectrum of the vison is always two fold degenerate, and hence condensing the vison will not lead to a nondegenerate state.
Two copies of the p ± ip TSC is formally equivalent to a quantum spin Hall (QSH) insulator: fermions that carry global Z 2 charge 0 and 1 form ν = 1 and −1 integer quantum Hall states respectively. Then after coupling to the Z 2 gauge field, the vison would carry two complex localized fermion modes c 1 and c 2 , and a vison would carry charge ±1/2 of the Z 2 global symmetry, which corresponds to n 2 = c † 2 c 2 = 1, 0 respectively. Thus the condensate of the vison always spontaneously breaks the Z 2 symmetry. This situation is very similar to the case discussed in Ref. 38. The universality class of the confinement transition is the so-called 3d XY * transition, namely at the quantum critical point the Z 2 symmetry order parameter has an anomalous dimension η ∼ 1.49 [39,40].
Eventually for four copies of this p ± ip TSC, a vison carries four complex fermion modes c 1A , c 1B , c 2A , c 2B . The vison now can be a boson that does not carry any Z 2 global charge, for example the state with n 2A = 1 and n 2B = 0 is a Z 2 charge neutral boson. Thus condensing this vison would lead to a fully gapped nondegenerate bosonic state that preserves the global Z 2 symmetry. Now let us couple four copies of the p ± ip TSC to a four-component unit vector n: with γ 1 = σ 2100 , γ 2 = σ 2221 , γ 3 = σ 2223 , γ 4 = σ 2202 . The global Z 2 symmetry acts as Z 2 : χ → σ 0300 χ, and n → −n. After integrating out the fermions, the resulting theory is a (2 + 1)d O(4) NLSM with a topological Θ-term at Θ = 2π: where Ω 3 = 2π 2 is the volume of a three dimensional sphere with unit radius, and this is precisely the field theory describing the 2d bosonic SPT phase with Z 2 symmetry, which was first studied in Ref. 41. This field theory was studied in Ref. 24, 42. Finally we condense the vison in this system to confine the fermions. Similar to our previous K-matrix calculation, we couple the four copies of p ± ip TSC to the Z 2 gauge field, as described by the Lagrangian density where the matrix K QSH is diagonal with the diagonal elements (1, 1, −1, −1). In the theory, the global Z 2 symmetry charge is given by the charge vector q Z2 = (0, 0, 1, 1).
Integrating out A leads to the constraint I a I +2Ã = 0, which can be solved by Correspondingly, the global Z 2 charge is transformed tõ q Z2 = W ⊺ q Z2 = (1, −1, 0, 0), with the transformation matrix W taken from the first 4 rows of the matrix in Eq. (15). In K SPT * , the lower 2 × 2 block describes the Z 2 topological order, which contains the bosonic vison with neutral global Z 2 charge (as seen fromq Z2 ). As the vison condenses, the Z 2 topological order is removed, leaving the upper 2×2 block, i.e. the σ 1 matrix, as the Kmatrix describing a SRE bosonic state, with the global Z 2 charge q = (1, −1) (as taken fromq Z2 ). Such a K-matrix equipped with the Z 2 symmetry matches [23] the Chern-Simons description of the Z 2 SPT state. Therefore after confining the fermions in four copies of p ± ip TSC, we obtain the bosonic SPT state with Z 2 global symmetry. Extra symmetries can be added to the four copies of p ± ip TSC discussed above, and other 2d bosonic TSC can be constructed in the same way. Construction of 1d bosonic SPT phases is much more obvious, which will be discussed in the supplementary material.
Summary -In this paper we demonstrate that many bosonic SRE phases can be constructed by fermionic SRE phases with the same symmetry. The fermionic SRE states and the Z 2 gauge field can all be defined on a lattice, thus our method has provided a projective construction of the lattice wave function of these bosonic SRE states. Also, our method provides a full lattice regularization of the CS field theory [23] and semiclassical NLSM field theory [24] description of bosonic SPT phases. However, some bosonic SPT phases cannot be constructed using the method discussed in the current paper, for example, there is one bosonic SPT phase with U (1)⋊Z 2 symmetry in 3d, while there is no free fermion SPT phase with the same symmetry. We will leave the construction of these bosonic SPT phases to future study.
with γ 1 = σ 332 , γ 2 = σ 320 , γ 3 = σ 312 . The time reversal symmetry acts as Z T 2 : χ → σ 300 χ and n → −n followed by the complex conjugation (denoted K). Note that the time reversal operator T = Kσ 300 behaves as T 2 = 1 on the Majorana fermions χ. After integrating out the fermions, the resulting theory is a (1 + 1)d O(3) NLSM with a topological Θ-term at Θ = 2π: where Ω 2 = 4π is the volume of a two dimensional sphere with unit radius, and this is precisely the field theory describing the 1d bosonic SPT phase with Z T 2 symmetry, i.e. the Haldane phase of 1d spin chain [1,2]. Then we can couple the fermions to a Z 2 gauge field, namely we impose the following gauge constraint on every site: χ i0 χ i1 χ i2 χ i3 = 1. The same gauge constraint is imposed on the edge Majorana fermion zero modes. The edge Majorana fermion zero modes may be arranged in a matrix as F = 1 2 (χ 0 σ 0 + iχ 1 σ 1 + iχ 2 σ 2 + iχ 3 σ 3 ).
Under time-reversal transformation, Z T 2 : F → F * = (iσ 2 )F (−iσ 2 ). Two three-component vector operators can be conveniently constructed with these edge Majorana operators (a = 1, 2, 3): In fact, the boundary Majorana fermions have an emergent SO(4) symmetry, and the two vectors correspond to the two independent SU(2) subgroups of the SO(4). The full SO(4) rotational symmetry among the four flavors of Majorana fermions is decomposed to SU(2) spin ×SU(2) gauge , generated by S and K respectively. For the fermions in F , the SU(2) spin rotation corresponds to a left rotation F → U † F with U ∈SU(2) spin , while the SU(2) gauge rotation corresponds to a right rotation F → F G with G ∈SU(2) gauge . Under the constraint χ 0 χ 1 χ 2 χ 3 = 1, which is equivalent to the requirement of gauge neutrality, i.e. K = 0. Therefore under the gauge constraint, the physical state of the boundary is only two fold degenerate, and these states are invariant under SU(2) gauge . This means that we are free to combine time-reversal symmetry with a SU(2) gauge transformation. For example, we can define a new time-reversal transformation T : F → F * (iσ 2 ) = −iσ 2 F , this new time-reversal transformation satisfies T 2 = −1, and it is exactly the same time-reversal transformation for spin-1/2 object. Thus we conclude that under gauge constraint, four copies of Kitaev's chain is equivalent to the Haldane's phase.