# Bridging Fermionic and Bosonic Short Range Entangled States

###### Abstract

In this paper we construct bosonic short range entangled (SRE) states in all spatial dimensions by coupling a gauge field to fermionic SRE states with the same symmetries, and driving the 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.

###### pacs:

*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 Haldane (1983a, b), quantum spin Hall insulator Kane and Mele (2005a, b), topological insulator Fu et al. (2008); Moore and Balents (2007); Roy (2009), and topological superconductor such as Helium-B phase Roy (2008); Qi et al. (2009). All the free fermion SPT states have been well understood and classified in Ref. Schnyder et al., 2009; Ryu et al., 2010; Kitaev, 2009, and recent studies suggest that interaction may not lead to new SRE states, but it can reduce the classification of fermionic SRE states Fidkowski and Kitaev (2010, 2011); Qi (2013); Ryu and Zhang (2012); Gu and Levin (2013); Yao and Ryu (2013); Fidkowski et al. (2013); Wang and Senthil (2014). Unlike fermionic systems, bosonic SPT states do need strong interaction. Most bosonic SRE states can be classified by symmetry group cohomology Chen et al. (2013, 2012), Chern-Simons theory Lu and Vishwanath (2012) and semiclassical nonlinear sigma model Bi et al. (2015).

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 symmetry
, where is a local fermion
annihilation operator, thus we can couple all fermion Hamiltonians
to a dynamical gauge field, and microscopically this
gauge field commutes with the actual physical symmetry of
the fermion system. Once the 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 and , a gauge
field has a confined phase and a deconfined phase. The deconfined
phase is characterized by topological excitations of the
gauge field. In , the gauge field has a “vison”
excitation, which corresponds to a -flux seen by the matter
fields. In , the topological excitation is a “vison loop”,
which is a closed ring of -flux. In /, when the
visons/vison loops proliferate (condense), the system enters the
confined phase, *i.e.* fermions carrying gauge
charge cannot propagate freely in the bulk due to the phase
fluctuations induced by the vison/vison loop condensation.

However, when the 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 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. Ye and Wen, 2013, 2014; Lu and Lee, 2014; Grover and Vishwanath, 2013; Oon et al., 2013. However, in all these previous studies the gauge group associated with the slave fermion is bigger than , 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 , and since any fermion system has this 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 He B phase. Here instead of focusing on the real He system, we are discussing a more general family of TSC phases defined on a lattice that are topologically equivalent to He-B. One typical Hamiltonian of such TSC defined on the cubic lattice reads

(1) |

Here plays the same role as the chemical potential in real He system: is the trivial-TSC transition critical point. The time-reversal symmetry acts as . Close to the trivial-TSC phase transition, in the continuum limit this TSC phase can be described by the following universal real space Hamiltonian:

(2) | |||

(3) | |||

(4) |

where denotes the tensor product of Pauli matrices, and is the flavor index. This is a widely used approximate form for this class of TSC (For example, Ref. Wang et al., 2011; Ryu et al., 2012). For each flavor index , is a four component Majorana fermion. In this Hamiltonian and correspond to the TSC phase and the trivial phase respectively. The time-reversal symmetry acts as . Our conclusion is that, when we couple -copies of this TSC to the same gauge field, the gauge field can have a fully gapped nondegenerate confined phase when and only when is an integer multiple of . And when , the confined phase is the bosonic SPT state with time-reversal symmetry first characterized in Ref. Vishwanath and Senthil, 2013.

First of all, when , the vison loop must be gapless, and the gaplessness is protected by time-reversal symmetry Qi et al. (2009). On a vison line along direction, there will be a pair of counter-propagating Majorana modes, so the effective Hamiltonian along the vison line reads (see Appendix B for derivation):

(5) |

In this reduced theory, time-reversal symmetry acts as . The only mass term 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 the vison line definitely cannot drive the system into a fully gapped state by proliferation without breaking time-reversal.

For , the effective theory along the vison line becomes

(6) |

Then for even integer , it appears that there is a time-reversal symmetric mass term , where is an antisymmetric matrix in the flavor space. In the bulk theory Eq. (4), this mass term can correspond to several terms such as (see Appendix B). However, none of these terms can gap out vison lines along all directions. For example, for vison loops along direction, the modes moving along is an eigenstate of with , and modes moving along direction have eigenvalue . Because commutes with , can never back-scatter modes in the 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 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. Fidkowski and Kitaev, 2010, 2011, the authors studied the
interaction effect on Eq. (6), and the conclusion is that
for there is an SO(7) invariant interaction term
that can gap out the
theory Eq. (6) without generating nonzero expectation
value of any fermion bilinear operator, where is some
coefficient tensor specified in
Ref. Fidkowski and Kitaev, 2010, 2011. The same field theory
analysis applies here: the effective interaction
can gap out the theory Eq. (6)
along the vison loop without degeneracy.
corresponds^{1}^{1}1The fact that on the vison line is
extended to in the bulk is explained in Appendix B. to
the following term in the bulk:

(7) |

Since this term is rotationally invariant, it will gap out vison lines along all directions. Thus with , and with the interaction term 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 is an integral multiple of . In the following paragraphs we will argue that when the confined bosonic state is a bosonic SPT state.

Ref. Vishwanath and Senthil (2013); Bi et al. (2015) pointed out that this bosonic SPT state can be described by a O(5) NLSM field theory with a topological term. Let us couple the 8 copies of He B to a five-component unit vector :

(8) |

where are five symmetric matrices in the flavor space that satisfy (e.g. a particular choice could be ). Under time-reversal transformation, . Following the calculation in Ref. Abanov and Wiegmann, 2000, we can show that for the He B phase with , after integrating out the fermions, the effective field theory for the vector contains a topological -term at :

(9) |

where is the volume of a four dimensional sphere with unit radius. Eq. (9) is precisely the field theory introduced in Ref. Vishwanath and Senthil, 2013; Bi et al., 2015 to describe the bosonic topological SC with time-reversal symmetry.

Using the field theory Eq. (9), we can demonstrate that
the boundary of this bosonic SPT state could be a
topological order, whose mutually semionic excitations
and are both Kramers’ doublet Vishwanath and Senthil (2013) (The so
called state)^{2}^{2}2The topological order at
the boundary has nothing to do with the bulk gauge
field that we will confine by proliferating the vison loops..
Ref. Fidkowski et al., 2013; Wang and Senthil, 2014; Metlitski et al., 2013 argued that
the boundary of 8 copies of He B is the (fermionized)
state. For the sake of completeness, we will repeat this argument.
Based on the field theory Eq. (9), the and
excitations at the boundary of the bosonic SPT phase
correspond to the vortex of boson field , and vortex of
respectively ^{3}^{3}3assuming tentatively an enlarged
symmetry for rotation of and respectively, 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 ^{4}^{4}4Here the time-reversal symmetry is a modified
time-reversal symmetry defined in Ref. Wang and Senthil (2014), which is
a product of ordinary time-reversal and a rotation of boson
field or . 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 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. Wang and Senthil, 2014 is a Kramers doublet. Also the
vortex of carries charge of , and vortex of
carries charge of , 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
state.

Combining all the results together, we conclude that the confined phase of 8 copies of He B is really the bosonic SPT phase with time-reversal symmetry. Furthermore, since this bosonic SPT state has classification, it implies that two copies of the bosonic state is trivial (which can be shown in our NLSM field theory by directly coupling two copies of Eq. 9 together Bi et al. (2015)), which then implies that 16 multiples of the He-B TSC is trivial under interaction. This conclusions is consistent with the well-known classification of DIII class fermionic SPT statesFidkowski et al. (2013); Wang and Senthil (2014); You and Xu (2014).

We can also give the 8 copies of He B phase various flavor symmetries, and we can construct many bosonic SPT phases with symmetry that contains as a normal subgroup by confining the bulk gauge field. Since all the free fermion SPT states in require the time-reversal symmetry, thus so far our approach does not allow us to construct bosonic SPT phases without .

*Construction of 2d bosonic SPT phases* —

Now let us look at examples. In the simplest fermionic SRE state is the topological superconductor (TSC) that does not require any symmetry, and the simplest bosonic SRE state is the so called “” state with chiral central charge at its boundary Kitaev (2006, ). In the following we will prove that if we couple copies of TSC to a gauge field, the gauge field can confine to a gapped bosonic state when and only when is a integral multiple of 16. And when , the confined phase is precisely the bosonic SRE state Plamadeala et al. (2013). First of all, when , the vison of the gauge field carries a Majorana fermion zero mode, which grants the vison a nonabelian statistics, thus when (and generally for odd integer ) the gauge field cannot enter its confined phase by condensing the vison. When is even, -copies of TSC is equivalent to an integer quantum Hall (IQH) state with Hall conductivity , thus a vison (half flux quantum) would carry charge , and has statistics angle under exchange. Thus the smallest that makes vison a boson is , and when , the gauge field can enter a confined phase by condensing the bosonic vison.

The vison condensation can be formulated by the Chern-Simons
theory.Cheng and Gu (2014) Let us start from the Chern-Simons
description for -copies of TSC with even
(*i.e.* layers of IQH), and couple the fermion
currents () to the gauge field.
The Lagrangian density can be written as

(10) |

Here the gauge theory is described by the mutual Chern-Simon theory of two gauge fields and with the -matrix gauge theory, for instance see Ref. Kou et al. (2008); Xu and Sachdev (2009). can be considered as a Higgs field that Higgs the U(1) gauge structure of down to . The field couples to the fermion current with equal charge, and the field couples to the vison current in the gauge theory. The field can be treated as a Lagrangian multiplier and integrated out first, which leads to the constraint . This constraint can be solved by the following reparameterization , which is a standard representation of

(11) |

Substituting Eq. (11) into Eq. (10), we arrive at a bosonic theory in terms of the new set of gauge fields , as , where is the Cartan matrix of the Lie algebra(for even ). For , the -matrix reads

(12) |

which gives the Chern-Simons theory. We now extend by a block of trivial boson, given by the -matrix Cano et al. (2014), and define . One finds a transform , with , given by

(13) |

such that

(14) |

The last block describes a topological order.
The fermion excitations of this -matrix corresponds to the
original fermion in the TSC. The vison couples to
the last gauge field, *i.e.* it corresponds to the charge
vector , and is a boson ready to condense.
Thus after the vison condensation, the topological order
is destroyed and the original fermion is confined. The -matrix
is left with the upper block, which is exactly the
Cartan matrix of the Lie algebra. Since all the charge
vectors of the upper block are self-bosons, and they
are bosons relative to the vison, these charge vectors are
unaffected by the vison condensate. Thus we have shown by explicit
calculation that confining the fermions in 16-copies of
TSC leads to the bosonic SRE state.

Now let us investigate the TSC with a symmetry discussed in Ref. Gu and Levin, 2013. In this system the fermions with zero charge form a TSC, while fermions carrying charge form a TSC. This global symmetry is different from the gauge symmetry, since all the fermions in our system carry gauge charge. For one copy of the TSC coupled to the gauge field, the vison carries two independent Majorana fermion zero modes and , and the global symmetry acts . There is no nontrivial Hamiltonian for these two Majorana fermion modes that preserves the 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 TSC is formally equivalent to a quantum spin Hall (QSH) insulator: fermions that carry global charge and form and integer quantum Hall states respectively. Then after coupling to the gauge field, the vison would carry two complex localized fermion modes and , and a vison would carry charge of the global symmetry, which corresponds to , respectively (The symmetry in our system is just the subgroup of the U(1) symmetry of the integer quantum Hall state, and it is known that a vison, or a flux in a integer quantum Hall state carries charge, as was shown in Ref. Ran et al., 2008). Thus the condensate of the vison always spontaneously breaks the symmetry. This situation is very similar to the case discussed in Ref. Ran et al., 2008. The universality class of the confinement transition is the so-called XY transition, namely at the quantum critical point the symmetry order parameter has an anomalous dimension Isakov et al. (2011, 2012).

Eventually for four copies of this TSC, a vison carries four complex fermion modes , , , . The vison now can be a boson that does not carry any global charge, for example the state with and is a charge neutral boson. Thus condensing this vison would lead to a fully gapped nondegenerate bosonic state that preserves the global symmetry.

Now let us couple four copies of the TSC to a four-component unit vector :

(15) |

with , , , . The global symmetry acts as , and . After integrating out the fermions, the resulting theory is a O(4) NLSM with a topological -term at :

(16) |

where is the volume of a three dimensional sphere with unit radius, and this is precisely the field theory describing the bosonic SPT phase with symmetry, which was first studied in Ref. Levin and Gu, 2012. This field theory was studied in Ref. Xu and Senthil, 2013; Bi et al., 2015.

Finally we condense the vison in this system to confine the fermions. Similar to our previous -matrix calculation, we couple the four copies of TSC to the gauge field, as described by the Lagrangian density

(17) |

where the matrix is diagonal with the diagonal elements . In the theory, the global symmetry charge is given by the charge vector . Integrating out leads to the constraint , which can be solved by

(18) |

Substituting Eq. (18) into Eq. (17) yields a Chern-Simons theory with

(19) |

Correspondingly, the global charge is transformed to
, with the
transformation matrix taken from the first 4 rows of the
matrix in Eq. (18). In , the lower
block describes the topological order, which
contains the bosonic vison with neutral global charge (as
seen from ). As the vison condenses, the
topological order is removed, leaving the upper
block, *i.e.* the matrix, as the
-matrix describing a SRE bosonic state, with the global
charge (as taken from ).
Such a -matrix equipped with the symmetry
matches Lu and Vishwanath (2012) the Chern-Simons description of the
SPT state. Therefore after confining the fermions in four
copies of TSC, we obtain the bosonic SPT state
with global symmetry. This bosonic SPT state has
classification Chen et al. (2013); Levin and Gu (2012); Bi et al. (2015), which implies that 8
copies of the TSC with symmetry
is a trivial state, which is consistent with the well-known
classification of such TSC under
interaction Qi (2013); Yao and Ryu (2013); Ryu and Zhang (2012); Gu and Levin (2013); You and Xu (2014)

Extra symmetries can be added to the four copies of TSC discussed above, and other bosonic TSC can be constructed in the same way. Construction of 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 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 Lu and Vishwanath (2012) and semiclassical NLSM field theory Bi et al. (2015) 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 symmetry in , 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.

The authors are supported by the the David and Lucile Packard Foundation and NSF Grant No. DMR-1151208.

## Appendix A Appendix A. Construction of Bosonic SPT

In this appendix, we construct the Haldane phase using four copies of Kitaev’s chains with the time-reversal symmetry. Let us start from the fermionic SPT phase composed of four copies of Kitaev’s chains coupled to a fluctuating three-component unit vector :

(20) |

with , , . The time reversal symmetry acts as and followed by the complex conjugation (denoted ). Note that the time reversal operator behaves as on the Majorana fermions . After integrating out the fermions, the resulting theory is a O(3) NLSM with a topological -term at :

(21) |

where is the volume of a two dimensional sphere
with unit radius, and this is precisely the field theory
describing the bosonic SPT phase with symmetry,
*i.e.* the Haldane phase of spin
chain Haldane (1983a, b).

Then we can couple the fermions to a gauge field, namely we impose the following gauge constraint on every site: . 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 asHermele (2007)

(22) |

Under time-reversal transformation, .

Two three-component vector operators can be conveniently constructed with these edge Majorana operators ():

(23) |

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)SU(2), generated by and respectively. For the fermions in , the SU(2) rotation corresponds to a left rotation with SU(2), while the SU(2) rotation corresponds to a right rotation with SU(2).

Under the constraint , which
is equivalent to the requirement of gauge neutrality, *i.e.*
. Therefore under the gauge constraint, the physical
state of the boundary is only two fold degenerate, and these
states are invariant under SU(2). This means that
we are free to combine time-reversal symmetry with a
SU(2) transformation. For example, we can define a
new time-reversal transformation , this new time-reversal
transformation satisfies , 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.

## Appendix B Appendix B. Vison Loops in He B TSC

In this appendix, we derive the effective theory along the vison loop in the He B TSC. Let us start with Eq. (4), and first consider a straight vison line along the -direction. The vison line can be considered as a thin hollow cylinder through the bulk of the TSC with a flux (-flux) threading through the hole of the tube. For this configuration, it could be convenient to use the cylindrical coordinate defined as . Applying the coordinate transform to Eq. (4), the Schördinger equation reads

(24) |

where is the spin connection that corresponds to threading the -flux (as ). The low-energy fermion modes around the vison line are given by the following ansatz in the asymptotic limit,

(25) |

Substitute Eq. (25) to Eq. (24), one can see must satisfy in order to obtain the low-energy modes (whose energy as the -direction momentum ). The matrix has two eigenvectors of the eigenvalue:

(26) |

corresponding to the two counter-propagating Majorana modes along the vison line. It is straight forward to see that the matrix represented on the basis becomes the matrix , so the effective Hamiltonian should be as shown in Eq. (5).

In general, any operator (as a matrix) defined in the bulk can be thus projected to the subspace of the fermion modes along the vison line, as the corresponding matrix by ()

(27) |

In Tab. 1, we conclude the projection of all Hermitian matrices (16 complete basis) to the 2-dimensional subspace of counter propagating Majorana modes along the vison line. This establishes the correspondence between the operators in the bulk and that on the vison line. One can see in the bulk would correspond to on the vison line. So the action of the time-reversal symmetry is reduced to on the vison line.

Given the effective Hamiltonian Eq. (5) and the above symmetry on the vison line, it seems that if we make even copies of the system, the vison line can be gapped out by a bilinear mass term of the form (with ) which does not breaks the time-reversal symmetry. However, this is only true for our analysis of the straight vison line along the -direction. Because according to Tab. 1, the mass term would extend to the bulk as , which can not gap out the vison lines along any other directions, as commutes with both and . Therefore it is impossible to fully gap out the vison loop by any fermion bilinear term.

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