Boson-fermion duality in four dimensions

# Boson-fermion duality in four dimensions

Takuya Furusawa    Yusuke Nishida Department of Physics, Tokyo Institute of Technology, Ookayama, Meguro, Tokyo 152-8551, Japan
October 2018
###### Abstract

Dualities provide deep insight into physics by relating two seemingly distinct theories. Here we propose and elaborate on a novel duality between bosonic and fermionic theories in four spacetime dimensions. Starting with a Euclidean lattice action consisting of bosonic and fermionic degrees of freedom and integrating out one of them alternatively, we derive a UV duality between a Wilson fermion with self-interactions and an XY model coupled to a compact U(1) gauge field. We find a continuous phase transition between topological and trivial insulators on the fermion side corresponding to Higgs and confinement phases on the boson side. The continuum limit of each lattice theory then leads to an IR duality between a free Dirac fermion and a scalar QED with the vacuum angle . The resulting bosonic theory proves to incorporate a scalar boson and dyons as low-energy degrees of freedom and it is their three-body composite that realizes the Dirac fermion of the fermionic theory.

## I Introduction

Particle statistics is one of the most fundamental outcomes from quantum mechanics, which classifies particles into either boson or fermion. Although bosons and fermions exhibit quite distinct behaviors, it is possible to transmute particle statistics in three spacetime dimensions (3D) by attaching one magnetic flux quantum to a particle Wilczek:1982a (); Wilczek:1982b (). In terms of quantum field theories, the statistical transmutation is implemented by coupling a matter field to a Chern-Simons gauge field Polyakov:1988 (), which has been successfully employed to account for the fractional quantum Hall effect Zhang:1989 (); Lopez:1991 (); Halperin:1993 ().

Recently, significant progress has been made regarding a web of dualities in 3D. Here a central position is occupied by the so-called boson-fermion duality, which relates a Wilson-Fisher boson coupled to a U(1) Chern-Simon gauge field at level one and a free massless Dirac fermion Karch:2016 (); Seiberg:2016 (). The boson-fermion duality can be thought of as the statistical transmutation in relativistic critical systems and generates other known dualities such as the particle-vortex dualities for bosons Peskin:1978 (); Dasgupta:1981 () and for fermions Son:2015 (); Wang:2016 (); Metlitski:2016 (); Mross:2016 () as well as new ones Karch:2016 (); Seiberg:2016 (). While the boson-fermion duality was originally proposed on the basis of the conjectured non-Abelian dualities Aharony:2016 (), it is now explicitly constructed on an array of coupled wires Mross:2017 () and on a Euclidean lattice Chen:2018a (). In particular, the latter approach has been extended to construct some of the non-Abelian dualities Chen:2018b (); Jian:arxiv ().

In spite of such great success in 3D, non-supersymmetric dualities in 4D have been far less explored Bi:arxiv (). This work is aimed at paving the way for an unseen web of dualities in 4D by constructing an analog of the boson-fermion duality. To this end, we employ the lattice construction approach developed in Ref. Chen:2018a () and extend it to 4D. Our starting point is a Euclidean lattice action consisting of the gauged XY model,

 SXY[θ,a]=−β∑x,μcos(Δμθx−ax,μ), (1)

and the gauged Wilson fermion Creutz:1983 (),

 SW[χ,a]=−∑x,μ(¯χx+μ1+γμ2eiax,μχx +¯χx1−γμ2e−iax,μχx+μ)+(4+M)∑x¯χxχx. (2)

Here is an angular variable, and are four-component Grassmann variables on a site , and is a compact U(1) gauge field on a link with . The gamma matrices obey , the lattice constant is set to be unity, and and are the dimensionless system parameters. Provided that is the forward difference, the action is invariant under the simultaneous gauge transformations of , , , and .

 Z[A]=∫π−πDa∫π−πDθ∫D¯χDχe−SXY[θ,a]−SW[χ,a+A], (3)

where the Wilson fermion is additionally coupled to an external gauge field . We note that an action for the dynamical gauge field is not included here corresponding to an infinite gauge coupling. Our strategy is such that the bosonic degrees of freedom is first integrated out to derive a purely fermionic theory and then the fermionic degrees of freedom to derive a purely bosonic theory. Because the resulting two lattice theories and their continuum limits necessarily share the same correlation functions, they constitute a novel boson-fermion duality in 4D, whose contents are to be elucidated below.

## Ii Fermionic theory

### ii.1 Lattice action

The bosonic degrees of freedom, and , in Eq. (3) can be exactly integrated out in the same way as the 3D case Chen:2018a (). We first bring the partition function for the gauged XY model to its current representation,

 ∫π−πDθe−SXY[θ,a] =∑{j}(∏xδ∑μ∇μjx,μ,0)∏x,μIjx,μ(β)e−ijx,μax,μ, (4)

where is an integer variable on a link representing a current of boson and is the backward difference. On the other hand, the action for the gauged Wilson fermion can be expanded as

 e−SW[χ,a+A]=e−∑x(4+M)¯χxχx∏x,μ2∑p,q=01p!q! ×(h+x,μ[χ]eiax,μ+iAx,μ)p(h−x,μ[χ]e−iax,μ−iAx,μ)q, (5)

where and are the forward and backward hopping terms, respectively. They obey because being the matrices of rank two project out two linear combinations of the four Grassmann components.

By combining Eqs. (II.1) and (II.1), the integration over generates on each link, which equalizes the currents of boson and fermion indicating that they are bound together at the infinite gauge coupling. Therefore, the current conservation of boson, , is automatically satisfied under the integration over and in Eq. (3) because of the corresponding current conservation of fermion Chen:2018a (). After the summation over , we thus obtain

 Z[A] =∫D¯χDχe−∑x(4+M)¯χxχx∏x,μ2∑p,q=0Ip−q(β)p!q! ×(h+x,μ[χ]eiAx,μ)p(h−x,μ[χ]e−iAx,μ)q. (6)

Finally, by rescaling the fermion fields as and and reexponentiating the hopping terms, the partition function up to overall factors is provided by

 Z[A]∝∫D¯ψDψe−S′W[ψ,A]−Sint[ψ,A], (7)

where is the action for the Wilson fermion in Eq. (I) with its bare mass replaced by . Six self-interaction terms are now included in

 Sint[ψ,A] =∑x,μ∑p+q≥2Up,qp!q!(h+x,μ[ψ]eiAx,μ)p(h−x,μ[ψ]e−iAx,μ)q (8)

and the corresponding -body coupling constants read , , , and . The resulting fermionic theory in Eq. (7) is the self-interacting Wilson fermion, which constitutes the fermion side of our boson-fermion duality at the lattice level.

### ii.2 Phase diagram

The fermionic theory being obtained, we then elucidate its phase structure in the parameter space of with temporarily turned off. In particular, when , all the coupling constants in Eq. (II.1) vanish so that the Wilson fermion becomes free with its mass provided by . Therefore, the mass gap closes at , which is a critical point separating two gapped phases. One gapped phase on the side of is a trivial insulator because it is adiabatically connected to the atomic limit , while the other gapped phase on the side of is known to be a celebrated topological insulator Qi:2008 (). They cannot be adiabatically connected to each other as long as the time-reversal symmetry is respected Fu:2007 (); Moore:2007 (); Roy:2009 (). We also note that other critical points exist at , where doublers become massless, but are not considered here.

How the critical point separating the trivial and topological insulators extends toward can be determined perturbatively as long as . Because , , but , only the two-body coupling constants need to be kept to the leading order in . The Wilson fermion propagator is provided by , where the self-energy from a standard diagrammatic calculation (cf. the inset of Fig. 1) is found to be

 Σ(k)=1β∑μ(iγμsinkμ−coskμ)∫π−πd4l(2π)4 ×sin2lμ−coslμ[∑ν(1−coslν)+M′]∑νsin2lν+[∑ν(1−coslν)+M′]2+O(1/β2). (9)

The mass gap now closes at , which leads to as depicted in Fig. 1. Here the upper right side of the critical line corresponds to the trivial insulator with , while the lower left side to the topological insulator with .

### ii.3 Continuum action

The continuous phase transition between the trivial and topological insulators allows us to take the continuum limit of the fermionic theory so far defined on the lattice. Because all the self-interactions in Eq. (II.1) are irrelevant about the Gaussian fixed point corresponding to a free massless Dirac fermion, the fermionic action in Eq. (7) is reduced to

 SF[ψ,A]=∫d4x¯ψ[γμ(∂μ−iAμ)+m]ψ+SM[A]. (10)

Here the continuum limit is taken with the physical mass of kept constant and with is the Maxwell action for the external gauge field , which is generated by integrating out high-energy degrees of freedom. We note that summations over repeated indices are implicitly assumed only in continuum actions.

Because the resulting free Dirac fermion in Eq. (10) is massive when , it can be further integrated out leading to

 −lnZ[A]→SM[A]+iΘ32π2∫d4xϵμνστFμν[A]Fστ[A], (11)

with the vacuum angle provided by  Qi:2008 (); Hosur:2010 (). Therefore, the low-energy effective action is found to be the usual electrodynamics with for the trivial insulator , but the axion electrodynamics with for the topological insulator  Wilczek:1987 (); Qi:2008 (), as indicated in Fig. 1. This completes the fermion side of our boson-fermion duality in the continuum limit.

## Iii Bosonic theory

### iii.1 Lattice action

Turning to the boson side of our boson-fermion duality, the fermionic degrees of freedom, and , in Eq. (3) can be formally integrated out leading to

 Z[A]=∫π−πDa∫π−πDθe−SXY[θ,a]−SG[a+A], (12)

where is from the Wilson fermion determinant. Because the action for the dynamical gauge field is now generated, the resulting bosonic theory in Eq. (12) can be thought of as a variant of the Abelian Higgs model, which proves to be dual to the self-interacting Wilson fermion in Eq. (7) at the lattice level. They share the same partition function and thus must share the same phase diagram depicted in Fig. 1, consisting of the two gapped phases separated by the critical line along on the side of .

In order to understand the two gapped phases in terms of the bosonic theory, we recall that phases of the Abelian Higgs model are typically classified into Coulomb, Higgs, and confinement phases Fradkin:1979 (); Borgs:1987 (). Although the Coulomb phase is ineligible for our phase diagram because the gauge field therein is massless, it acquires a mass gap in the other two phases Osterwalder:1978 (). It is then reasonable to identify the gapped phase at as the Higgs phase because XY spins tend to order at lower temperature, while the confinement phase is left for the other gapped phase at . The latter identification is also reasonable because it is adiabatically connected to the limit , where the action for the dynamical gauge field is reduced to the simple plaquette action, , with an infinite gauge coupling.

The above understanding and the correspondence to the fermion side are summarized in Table 1, which is to be further confirmed below by reproducing the low-energy effective action in Eq. (11) from the boson side. We note that no order parameter can distinguish the Higgs and confinement phases with the fundamental charge Fradkin:1979 (), which is indeed consistent with the fact that no order parameter can distinguish the topological and trivial insulators on the fermion side. Unlike in the typical Abelian Higgs model, our Higgs and confinement phases cannot be adiabatically connected to each other as long as the time-reversal symmetry is respected because of the distinct vacuum angles .

### iii.2 Continuum action

The continuous phase transition between the Higgs and confinement phases is controlled by on the side of , where it is possible in principle to take the continuum limit of the bosonic theory so far defined on the lattice. Although it is intractable in practice because the system is strongly coupled about the critical point, we proceed by assuming that the continuum limit may be considered separately for and in Eq. (12). First, regarding the gauged XY model, it is convenient to think of it as a limiting case of the gauged complex scalar model, , with . Its continuum limit is then provided by the Ginzburg-Landau action, , where the continuous phase transition is controlled by so that with defined as the critical point corresponds to .

On the other hand, regarding the pure gauge action of , we recall that it is generated by integrating out and in Eq. (3). Because the mass of the bare Wilson fermion is negative on the side of , is reduced to the Maxwell action with the vacuum angle , , in the continuum limit [cf. Eq. (11)]. Therefore, the above consideration leads us to propose that the bosonic action in Eq. (12) turns into the form of

 SB[ϕ,a,A]=∫d4x[|(∂μ−iaμ)ϕ|2+r|ϕ|2+λ|ϕ|4] +SM[a+A]+i32π∫d4xϵμνστFμν[a+A]Fστ[a+A]. (13)

This is none other than the scalar QED with the vacuum angle , which proves to be dual to the free Dirac fermion in Eq. (10) with corresponding to . We note that the dynamical gauge field needs to be understood as compact allowing for magnetic monopoles and thus the confinement phase.

The resulting bosonic theory in Eq. (III.2) must be consistent with the low-energy effective action obtained from the fermion side. First, in the Higgs phase with , the scalar field condenses and thus generates a mass term written in the unitary gauge. Then, by integrating out the massive gauge field with higher derivative terms neglected, Eq. (11) is reproduced with . On the other hand, in the confinement phase with , the scalar field is massive and is thus integrated out leaving behind . When is absent corresponding to the limit of , no effective action for is left after the integration over . Then, by turning on adiabatically within the confinement phase, the Maxwell action for may be generated with protected by the time-reversal symmetry so as to reproduce Eq. (11). Therefore, the topological and trivial insulators on the fermion side are consistently realized by the Higgs and confinement phases on the boson side, as shown in Table 1.

### iii.3 Dyons

In order to gain further insight into the confinement phase, we need to handle the compact gauge field, which is conveniently formulated on a Euclidean lattice again. In the Villain representation, the pure gauge action resulting from Eq. (III.2) at reads

 S[a,A]=14e2∑x,μ,ν(Fx,μν[a+A])2 +iΞ32π2∑x,μ,ν,σ,τϵμνστFx,μν[a+A]Fx,στ[a+A], (14)

where in our case and is the field strength tensor with an integer variable on a plaquette Cardy:1982a (); Cardy:1982b (). The monopole current is provided by and satisfies the conservation law, , automatically. The presence of such magnetic monopoles can be made explicit by bringing Eq. (III.3) to its dual representation Peskin:1978 (); Herbut:2007 (),

 S′[ζ,b] =−β′∑x,μcos(∇μζx−bx,μ)+14e′2∑x,μ,ν(fx,μν[b])2 +iΞ′32π2∑x,μ,ν,σ,τϵμνστfx,μν[b]fx,στ[b], (15)

where is an angular variable and is a noncompact U(1) gauge field on a link with its field strength tensor provided by . The dimensionless system parameters are found to be , , and , as detailed in Appendix.

The above two lattice actions are related via the electromagnetic duality and share the same partition function up to overall factors. Corresponding to the Coulomb and confinement phases of the pure gauge theory in Eq. (III.3), the noncompact Abelian Higgs model in Eq. (III.3) is known to exhibit the Coulomb and Higgs phases, at least when  Borgs:1987 (). In particular, in the Higgs phase with a suitable gauge fixing, the scalar field condenses and thus generates a mass gap for the gauge field Kennedy:1985 (); Kennedy:1986 (). Because is introduced as a Hubbard-Stratonovich field coupled to [cf. Eqs. (18) and (21) in Appendix], the electrically charged under actually carries both electric and magnetic charges of and , respectively, under and . Therefore, it is a magnetic monopole when but becomes a dyon in our case with because of the Witten effect Witten:1979 (), whose condensation leads to the confinement of electric charges.

We thus find that the bosonic theory in Eq. (III.2) incorporates a dyon as an elementary degree of freedom relevant to low-energy physics. It is a boson carrying the electric and magnetic charges of under and under in units of . Its time-reversal partner denoted by is also a boson and carries , while a pair of carrying is transmuted into a fermion because of the charge-monopole statistical interaction Goldhaber:1976 (); Metlitski:2013 (); Wang:2014 (); Metlitski:2015 (). Then, combined with the scalar boson carrying , a three-body composite of remains a fermion but carries only just as the Dirac fermion in Eq. (10). Therefore, as compared in Table 2, the Dirac fermion on the fermion side is consistently realized by the three-body composite of the scalar boson and dyons on the boson side. All the correlation functions with respect to them should match each other between the bosonic and fermionic theories, which constitutes our boson-fermion duality in the continuum limit.

## Iv Summary

We have proposed and elaborated on the novel boson-fermion duality in 4D. Starting with the Euclidean lattice action in Eq. (3) and integrating out the bosonic or fermionic degrees of freedom alternatively, we derived the UV duality between the self-interacting Wilson fermion in Eq. (7) and the variant Abelian Higgs model in Eq. (12). The continuum limit of each lattice theory then led to the IR duality between Eqs. (10) and (III.2), which reads

 free Dirac fermion ↔ scalar QED with vacuum angle π.

The resulting bosonic theory incorporates a scalar boson and dyons as low-energy degrees of freedom and they condense in the Higgs and confinement phases, respectively. These two gapped phases with the distinct vacuum angles in Eq. (11) are separated by a new type of continuous phase transition, i.e., Higgs-confinement criticality, which is dual to the topological phase transition on the fermion side. It is the three-body composite of the scalar boson and dyons on the boson side that realizes the Dirac fermion of the fermionic theory.

Regarding future works, it is important to further confirm our boson-fermion duality with complementary approaches such as the operator-based coupled-wire construction Mross:2016 (); Mross:2017 () as well as numerical simulations being successful in 3D Senthil:arxiv (). The extension to non-Abelian dualities is currently underway. Last but not least, whether our boson-fermion duality generates a web of dualities in 4D is indeed worth elucidating.

###### Acknowledgements.
The authors thank Yohei Fuji, Akira Furusaki, Shigeki Onoda, Shinsei Ryu, Yuya Tanizaki, and Keisuke Totsuka for valuable discussions. This work was supported by JSPS KAKENHI Grants No. JP15K17727 and No. JP15H05855.

## Appendix A Derivation of Eq. (iii.3) from Eq. (iii.3)

Here we present how to derive Eq. (III.3) from Eq. (III.3) via the electromagnetic duality based on Refs. Peskin:1978 (); Herbut:2007 (). Starting with the partition function for the pure gauge theory in Eq. (III.3),

 Z=∑{n}∫π−πDaexp[−14π∑x∑μ<ν∑σ<τFx,μν[a+A](2πe2δμσδντ+iΞ2πϵμνστ)Fx,στ[a+A]], (16)

 Z∝∑{n}∫π−πDa∫∞−∞Dgexp[−14π∑x∑μ<ν∑σ<τgx,μν(2πe2δμσδντ+iΞ2πϵμνστ)−1gx,στ−i2π∑x∑μ<νFx,μν[a+A]gx,μν]. (17)

This identity is readily confirmed with the equation of motion,

 gx,μν∣∣∣EoM=−i∑σ<τ(2πe2δμσδντ+iΞ2πϵμνστ)Fx,στ[a+A], (18)

and the matrix inversion under and reads

 (2πe2δμσδντ+iΞ2πϵμνστ)−1 =2π(2π/e)2+(Ξe/2π)2δμσδντ−iΞ2πe2(2π/e)2+(Ξe/2π)2ϵμνστ (19) ≡2πe′2δμσδντ+iΞ′2πϵμνστ. (20)

Then, with the understanding of for , the integration over in Eq. (17) generates a Kronecker delta imposing on each link, which is automatically satisfied by changing the variable from to so that

 gx,μν=12∑σ,τϵμνστfx,στ[b]=∑σ,τϵμνστ∇σbx,τ. (21)

Therefore, the partition function is now provided by

 Z=∫∞−∞Dbz[b]exp[−14π∑x∑μ<ν∑σ<τfx,μν[b](2πe′2δμσδντ+iΞ′2πϵμνστ)fx,στ[b]] (22)

with

 (23)

Here the variable is changed again from to , which must be constrained by on each site as ensured by the integration over . Finally, the Jacobi-Anger expansion with the asymptotic form of the modified Bessel function,

 eβcosθ=∑N∈ZIN(β)eiNθ→eβ√2πβ∑N∈Ze−N22β+iNθ(β→+∞), (24)

 z[b] =limβ′→+∞∑{J}∫π−πDζexp[−12β′∑x,μ(Jx,μ)2+i∑x,μJx,μ∇μζx−i∑x,μJx,μbx,μ] (25) ∝limβ′→+∞∫π−πDζexp[β′∑x,μcos(∇μζx−bx,μ)]. (26)

By combining Eqs. (22) and (26), we arrive at the partition function for the noncompact Abelian Higgs model in Eq. (III.3).

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