Skyrmion Superfluidity in Two-Dimensional Interacting

Home , Thirring model

arXiv:1411.7885v1 [cond-mat.str-el] 28 Nov 2014 h oia tutr fteatcei kthdi Fig. in sketched is article the of structure a logical phase). of The superconducting existence of the version neu- prove tral (analog to phase us superfluid non-BCS allows parity-preserving This way. unified combines of that characteristics model lattice honeycomb (multi-layered) of physics the on insights 20]. get [18, to superconductors used mech- high-temperature be superconducting could non-BCS anisms These supercon- 19], of phase. 18, evidence ducting [8, smoking-gun BF the effect represents Meissner [16]. which states the Hall describe spin naturally and theory corre- [5] strongly systems some fermionic for lated theory effective a the is as theory candidate BF while used two-dimensional [15–17], parity-breaking superconductivity been non-BCS a have describe and skyrmions define to super- particular, of In physics the conductivity. characterize reveal which QFTs features topo- bosonic their Importantly, several of value 14]. the [13, and charge coeffi- term logical Hopf the the of of value front the in bosonic cient on fermionic, depending acquire statistics to anyonic quasipar- or model) (the the skyrmions in the present theory for ticles the allows in term [12] topological term) a (Hopf of addiction The 11]. phases [10, topological models protected symmetry sigma antifer- and [9] Heisenberg non-linear romagnets of physics the the describe contains class which (NLSM) Another QFT [6–8]. bosonic theories of QFTs BF topological and bosonic Chern-Simons by like superconductors described and quantum opportunely At fractional insulators are of topological states [1–5]. states, ground respectively Hall the theories time, same Thirring described the and be can Dirac fermions systems by For matter relativistic condensed self-interacting cor- matter. in emerging and of strongly phases free of topological description example, and the systems in related role fundamental a h olo hsti etri opoieanwfermionic new a provide to is letter this this of goal The Introduction.– 2 krinSprudt nToDmninlItrcigFer Interacting Two-Dimensional in Superfluidity Skyrmion nedsilnr hoeia cec eerhGop(iT Group Research Science Theoretical Interdisciplinary ASnmes 11.c 11.d 74.20.Mn 71.10.Fd, i 11.15.Yc, model numbers: lattice theor PACS the our of of implementation signature possible a a provides i in This to detected model. effects lattice superfluidity wit the effective effect for map Meissner al we effective is Finally, model an symmetry. the (skyrmi establishes that system show naturally we the whic which Moreover, of model, role. sigma quasiparticles Cooper-pairs non-linear the the of case, kind bi-layer new the a me superflui by is phase skyrmion theory superfluid effective parity-preserving fermionic non-BCS of a describing kind theory new a supports nti ril edsrb ut-aee oecm latti honeycomb multi-layered a describe we article this In 1 colo hsc n srnm,Uiest fLes Leeds Leeds, of University Astronomy, and Physics of School both unu edter QT plays (QFT) theory field Quantum krin n Fter nan in theory BF and skyrmions inoeioPalumbo Giandomenico Dtd eray1,2018) 13, February (Dated: ∗ 1 h krin eed ntenme flyr.For and fermions layers. (neutral) of like number behave the skyrmions of layer on double parity one statistics depends it the under skyrmions Interestingly, call kind the invariant (DSM). We new model is a skyrmion transformations. which to time-reversal equivalent model is and theory skyrmion 22], this of [20, that techniques particles. show fermionization we Dirac functional self-interacting using Thirring By supporting chiral-invariant [21] low-energy (2+1)-dimensional the model a in to rise, gives limit, which model Hubbard-like 1 corresponding the effective superfluidity and the fermionic model to among associated the structure properties logical physical describing the theories of field Sketch 1: FIG. E) IE,Wk-h,Siaa3109,Japan 351-0198, Saitama Wako-shi, RIKEN, HES), pcfial,tesse sdsrbdb fermionic a by described is system the Specifically, . n ar Cirio Mauro and oprlk Pairs Cooper-like n)hv ooi ttsisadreplace and statistics bosonic have ons) Skyrmions otrqiigabekn ftegauge the of breaking a requiring hout elqatmsystem. quantum real a n tclsymo uefliyta a be can that superfluidy skyrmion etical emdlo neatn emoswhich fermions interacting of model ce ette mn emoi observables fermionic among dentities oeuvln oaMxelB theory, Maxwell-BF a to equivalent so n ffntoa emoiain Such fermionization. functional of ans iy edrv h o-nryfield low-energy the derive We dity. DSM ecl obesymo oe.In model. skyrmion double call we h S J,Uie Kingdom United 9JT, LS2 , Fermionization emoi Observables Fermionic ∗ 2 Superfluidity o Energy Low Thirring

incSystems mionic Bosonization asv Photons Massive esnrEffect Meissner M 2 BF . 2 they can pair in order to form Cooper bosons. We focus on bi-layer systems where skyrmions behave as (neutral) bosons and represent the natural Cooper-like pairs in the v1 (fermionic) superfluid phase. In addiction, we show that v2 the system can also be described by a double(Maxwell)- BF (M2BF) theory which is a particular instance of a topologically massive gauge theory (TMGT). This equivalence can be shown either by integration of the scalar skyrmionic field or directly from the fermionic Thirring model by means of functional bosonization [23]. In the TMGT theory, effective photons acquire a mass as a consequence of topological interactions. This naturally leads FIG. 2: Tight binding for n = 2. Fermions hop along the to the London equations of superconductivity (fermion edges of two honeycomb lattice layers as described by the superfluidity) [18] which effectively combine Meissner ef- Hamiltonian in Eq. (1). For each spin species and each fect and infinite conductivity. We finally show how phys- layer the unit cell contains two fermions: a (orange) and b ical fermionic observables can probe the skyrmion super- (blue). Fermions in different layers but in unit cells with same fluid mechanism described by the model. in-layer coordinates interact through current-current interac- Lattice Model.– We consider n two-dimensional lay- tions (wavy line) as described by the Hamiltonian in Eq. (3). ers of spinful fermions stacked one on the top of each other (Fig. 2). Within each layer fermions are localized j free Hamiltonians by j =1,...,n so that H0 H0 . To on a honeycomb lattice. In the case n = 1 the fermion connect the layers we add current-current interactions→ to hopping is described by the following graphene-like (spin the free model s = , dependent) Hamiltonian ↑ ↓ H = n Hj + H , (3) s s s s s s j=1 0 I H = c (a †b v + a †b v + a †b ) 0 s,r r r+ 1 r r+ 2 r r (1) ± s s s s 2 +mc(ar†ar br†br) . g j P2 j5 2 P − with HI = 2 [( j Jµ) + ( j Jµ ) ], where µ = 0, 1, 2, where the spinor and the currents are, respectively, Here, the overall sign depends on the orientation of the j j Pj j T P jµ µ j5µ Ψ = a ↑ b ↑ a ↓ b ↓ , J = Ψ¯ γ Ψ and J = spin and ar and br are the fermion operators at posi- j j j Ψ¯ γ5γµΨ , and where the γs are Dirac gamma matrices. tion r Λ where Λ = n1v1 + n2v2 is the lattice of j j ∈ N √3 3 In the case of a single layer, we have that unit cells of the model (n1,n2 and v1 = ( 2 , 2 ) √3 3 ∈ and v2 = ( , ). This Hamiltonian describes hop- 2 2 2 2 2 HI =3g (a↑b↑ + a↓b↓) + (a↑b↓ a↓b↑) . (4) ping terms along− the links of the honeycomb lattice with | | | − | a real tunneling coefficient c and a staggered chemical The less compact, but similar, expression for the case potential (with energy scale mc2) and it can be exactly n = 2 can be found in the Supplemental Material [25]. solved. The spectrum becomes gapless at two indepen- Around each Fermi point P the low-energy effective dent points P in momentum space. The low energy physics is described by the following± partition function ± i SF physics around these points is effectively described by a ZF = Ψ¯ j Ψj e ~ , where SF = S0 + SI with standard massive Dirac Hamiltonian D D R n 3 ¯ 2 S0 = j=1 d xΨj (ci∂/ mc )Ψj 2 x y 2 2 n − n (5) † 3 g j 2 j5 2 H0 = d k Ψ (ckxα + ckyα + mc β)Ψ , SI = d x [( Jµ) + ( Jµ ) ] . ± ± P R2 j=1 j=1 X± Z h i (2) This modelR is nothingP but a (generalized)P chiral-invariant where the matrices α and β belong to an euclidean Clif- Thirring model [21]. In the following we will work in units ford algebra and where, for clarity, the energy scales c and such that c = ~ = 1 and without losing of generality we mc2 have been renormalized (for details, see appendix). will consider the physics only around one Fermi point. The spinors Ψ depend on the momentum space coor- Double skyrmion model and functional ± T dinate k as Ψ = a↑ (k±) b↑ (k±) a↓ (k±) b↓ (k±) fermionization.– We now introduce a double skyrmion where a are± the Fourier± transformed± ± fermion operators± model (DSM) which is a double O(3)-Hopf non-linear evaluated± at the Fermi points P respectively and where sigma model + ± k = (kx, ky) and k− = ( kx, ky). Note that it is possi- − 3 1 t µ t t ble to induce the same mass term in the above Hamilto- S(O(3) H)2 = d x (∂µm ∂ m )+ tnπH , − 2g2 nian by replacing the staggered chemical potential in (1) t= Z 0 with a standard Haldane term [24]. X± (6) We now consider the general case of n such layers (we will where n coincides with the number of layers, the fields 3 2 be mainly interested in the case n = 2) and label their m± R satisfy the non-linear constraint m± = 1 ∈ | | 3 and the two Hopf terms Ht are topological invariants which is in fact the original chiral-invariant Thirring [12]. Due to the different sign in front of the Hopf terms, model introduced in the previous section. As a final com- this theory describes independent skyrmions and anti- ment, we note that, alternatively, it is possible to show skyrmions which have opposite values of the topological that the this fermionic model can be bosonized to get the + 2 charges Q = Q− which assume only integer values (CP-CS) model [25]. T − T [25]. Each (anti-) skyrmion has a spin S given by [26] London action.– In this section we show that the ef-

2 fective theory described in Eq. (8) is equivalent to the S = (n/2)QT± . (7) London action which eﬀectively describes the physics of This shows that, depending on the number of layers, superconductivity. the statistics of the skyrmions can be either bosonic or In [18, 27] it is proven that (at low energy) a Maxwell the- fermionic. In particular, for a bi-layer systems (anti-) ory is equivalent to a CP model. We can use this to map the eﬀective theory in Eq. (8) to a (double Maxwell)-BF skyrmions behave like bosons and in our context take 2 the role Cooper-like pairs. theory (M BF) [8, 18] with action

Following [20], we now want to show that the partition 3 n λµν 1 µν SM2 BF = d x ǫ Bλ∂µAν 2 Fµν (A)F (A) function of this bosonic theory is equivalent to that one of − π − 4e 1 F (B)F µν (B) , the chiral-invariant Thirring theory by employing func- R 4e2 µν − (12) tional fermionization. Let us start by deﬁning the equiv- where F is the ﬁeld strength tensor and the two scales alent CP form [6] of the O(3) NLSM in Eq. (6) in which µν g and e are related as showed in [25]. We now follow [18] the Hopf terms are recast as Chern-Simons terms. We 0 which shows that this theory is equivalent to the London will refer to it as a double CP-Chern-Simons (CP-CS)2 partition function: model

3 1 µν 2 n 2 + + iS 2 i R d x[ 2 Fµν (A)F (A)+2e (∂µφ 2π Aµ) ] Z 2 = A A− z z−e (CP-CS) , (8) Zφ = A φe − 4e − . (CP-CS) D D D D D D Z Z (13) where We can see that the (2+2) degrees of freedom of the 3 n λµν t t massless fields A and B are mapped to the (3+1) degrees S(CP-CS)2 = t= d x [ 4π ǫ Aλ∂µAν 1 ± ±t t 2 (9) of freedom of a massive bosonic field A and a massless + 2 (i∂µ Aµ)z ] . Pg | R − | scalar field φ which, in this sense, represents a kind of T C Goldstone boson. The present mechanism, however, does Here, z± = z1± z2± with the fields z1±,z2± 2 ∈ not have any local order parameter like in ordinary BCS such that z = z±z±∗ + z±z±∗ = 1 and A = ± 1 1 2 2 µ± theory. The charge and currents associated with the field (i/2)z ∂| z |. We now proceed with the fermion- ±∗ µ ± A are −ization [25]. The fermions appear quite naturally. In fact, we begin by noticing that, by changing variables to δ δ 1 + 1 + ρ = δAL Jem = δAL , (14) A = (A + A−), B = (A A−), the difference of 0 µ 2 µ µ µ 2 µ − µ the two Chern-Simons terms transform into the BF term while the effective magnetic and electric fields inside the n d3xǫµνλB ∂ A leading to a (double CP)-BF theory π µ ν λ material are simply given by so that Z 2 = Z 2 . The BF term can now R CP -BF (CP-CS) be “linearized” by introducing n fermion species χ [18] i 1 iµν 1 0µν j E = ǫ F (A)µν Bmag = ǫ F (A)µν , (15) leading to the following intermediate partition function 2 2

+ where i = x, y. The effective physics described by the Z = A B z z− χ χ¯ D D D D D D massive field A, implies both a Meissner and infinite con- exp i d3x [¯χ (i∂/ m)χ √2A J jµ √2B J 5jµ) {R j j − j − µ χ − µ χ ductivity effects. In fact, the (effective) magnetic field + 1 (i∂ (A + tB ))zt 2 ] . g2 Pt= R | µ − µ µ | } intensity decays exponentially inside the material (Meiss- ± (10) P ner effect) due to the presence of superficial dissipation- For each value of the sign, the variable z± can be thought less screening currents. In particular we have that as specifying a coordinate system in a SU(2) algebra via ± iξ σj j the identification z± e j , where σ are the Pauli B = 0 (16) → mag matrices and ξj± are some fields [25]. Moreover, the gaussian integral over the fields A and B can be easily com- in the bulk of the material. As shown in [18] a zero puted. This cause the fields ξj± to effectively decouple voltage can be defined in the presence of steady currents. and changing the fermionic variables χj ΘΨj through These screening currents flow within a penetration a suitable phase Θ [25], we are left with → depth λ sg2 [25] from the boundary of the material. In this∝ sense, the system has infinite conductivity Z = Ψ Ψexp¯ i d3x Ψ¯ (i∂/ m)Ψ F j j j σ and follows the perfect conductivity relation E = σJem. D D 2 { 2 − (11) + g (J j )2 + g (J 5j )2 ] , R 2 µ P R2 µ } 4

Fermionization rules and physical observables.– In this way, the effective Meissner effect in Eq. (16) can The aim of this section is to map the effective super- now be written as follows fluidity physics which describes the model to fermionic J 5j =0 . (19) observables. To this end, we introduce a minimal cou- h j 0 iF pling interaction with two external fields Aext and Bext The validity of suchP a prediction is confirmed by the to the fermionic Lagrangian density in Eq. (11) via a skyrmionic interpretation of the model. In fact, Eq. j µ 5j µ minimal coupling J A + J B so that ZF (19) can be derived from an alternative dual point of j µ ext j µ ext → ZF (Aext,Bext). We can then track these new terms as we view. We first notice that the expectation value of the P P µ 1 µνλ follow back all the steps that lead us from Eq. (10) to Eq. skyrmion currents [25] JS± = 2π ǫ ∂ν Aλ± can be writ- (11). The additional terms only cause a shift in the Dirac 1 j 5j ten as JS± (CP-CS)2 = ( J F J F ) by 5 h i ± n√2 h j i ∓ h j i operator i∂/ m i∂/ m + A/ext + γ B/ ext which leads using the fermionization rules. Now, by definition, the − → − P P to the equivalence ZF (Aext,Bext)= ZCP2BF (Aext,Bext), topological charges are the spatial integral of the 0th 2 0 where this last partition function has an additional term component of the current Q± = d xJ ± so that n µνλ ext ext 1 ext ext T S ǫ ( Bµ ∂ν Aλ + Aµ ∂ν Bλ + Bµ ∂ν Aλ ) in √2π √2 n + n 2 0t − 0 = (Q + Q−)= dR x J 2 the action [25]. By taking derivatives of the partition √2 T T √2 t= S (CP-CS) R − 2 5j − ±h i functions with respect to the external fields [25], this al- = d x J F , h j 0 i R P lows us to prove the following “fermionization” rules (20) R P j n consistently with Eq. (19). J = ǫ ∂ B 2 j µ F √2π µνλ ν λ CP BF h 5ji nh i (17) At the same time, as mentioned above, the system sup- J = ǫ ∂ A 2 , hPj µ iF − √2π h µνλ ν λiCP BF ports steady state currents within a penetration depth λ g2 distance from the boundary. By tuning the pa- where theP expectation values F and CP2BF are calcu- ∝ lated with respect to the groundhi statehi of the fermionic rameter g to allow the fermionization rules to hold, the and bosonic theory respectively. Drude relation E = σJem maps to the fermionic contraint J5j = σ Jj where σ . Similarly, we can we can map observables of the h j i h j i → ∞ London theory to those of a double(Maxwell)-BF Conclusions.– In this article we proposed a fermionic µνλ A tight-bindingP P model which naturally supports the by adding and tracking source terms Fµν (A)ǫ Jλ µνλ B two main ingredients of fermionic superconductivity: and Fµν (B)ǫ Jλ to the latter theory [25] so that A B Cooper-like pair formation and Meissner effect. In order ZM2BF ZM2BF(J ,J ). This leads to the equivalence →A B A B to prove these effects, we employed functional fermion- ZM2BF(J ,J ) = Zφ(J ,J ) (see [25] for its explicit expression). We can now use these correspondences to ization to show the equivalence between the effective relate the current (ρ, Jem) and the fields (Bmag, E) de- fermionic theory describing the lattice system (a chiral- fined in Eqs. (14) and (15) to fermionic observables. By invariant Thirring model) and a double skyrmion model. A B inspecting Eq. (14) and the expression of Zφ(J ,J ), we This model supports skyrmions with bosonic statistics 2π δ i 2π δ (Cooper-like pairs) in the bi-layer case and it is for- first notice that ρ = n JLB and Jem = n δJLB which leads 0 i mally equivalent to a double Maxwell-BF theory which 2π δZM2BF i 2π δZM2BF to ρ φ = n δJB and Jem φ = n δJB (where we h i 0 h i i describes an effective Meissner effect. Moreover, we pro- implicitly impose J B = 0, after the derivative has been vide fermionic observables associated to the superfluid taken). Finally, from the equivalences between London, phase to detect a signature of such effects in a possible (double Maxwell)-BF and (double CP)-BF theories we implementation of the lattice model in a real (or simu- find the following correspondence lated [28–30]) quantum system. A straightforward gener-

0µν 2√2π j alization of our model to the (charged) superconducting ρ = ǫ F (B) 2 = J φ µν M BF n j 0 F case can be obtained once neutral fermions are replaced h ii h iµν i 2√2π h j i J = ǫ F (B) 2 = J . h emiφ h µν iM BF n hPj i iF with charged ones and an external electromagnetic field (18) coupled with them is taken into account. Finally, an P Similarly, the electric and magnetic fields are simply open question related to this work concerns the possible i 1 δZφ 1 δZφ given by E = 2 δJA and Bmag = 2 δJA , which implies existence of Abrikosov-like vortices and the presence of h i i h i 0 i √2π 5j √2π 5j Majorana states localized at their cores [31–33]. We leave E φ = n j Ji F and Bmag φ = n j J0 F afterh i using theh fermionizationi h rules.i These findingh arei the study of these important aspects to future works. summarized inP the following table P Acknowledgments.– G. P. thanks Jiannis K. Pachos for comments and useful suggestions and Franco Nori for the hospitality in his group at RIKEN where this Electromagnetic Quantities Fermionic Observables j work was completed. M.C. thanks Jiannis K. Pachos ρ φ j J0 F and Franco Nori for comments and for the financial (and h i h j i Jem φ j J F moral!) support in Leeds and Wako-shi where this work h i hP 5ji E φ j J F was started and completed respectively. G. P. is sup- h i h P 5ji Bmag φ J F ported by a EPSRC Grant. M.C. is supported by the h i hPj 0 i P 5

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This supplementary material gives the details of the derivations shown in the main text.

TIGHT-BINDING MODEL

In this section we show that the interacting tight binding model described in the main text is equivalent, at low energy, to a (chiral-invariant) Thirring model. Let us start with the free model: a graphene-like tight binding model with a staggered chemical potential term

s 2 2 H = c (a†b + a†b + a†b )+ mc a†a mc b†b , (21) 0 ± r r+v1 r r+v2 r r r r − r r " r r r # X X X where the sign depends on the spin variable, c and mc2 are an energy scales and v = ( √3 , 3 ) and v = ( √3 , 3 ). The 1 2 2 2 − 2 2 Brillouin zone is deﬁned as BZ = p : p p + p p with p ,p [0, 1) and p = 2 ( 3 , √3 ), p = 2 ( 3 , √3 ), so { 1 1 2 2} 1 2 ∈ 1 3√3 2 2 2 3√3 − 2 2 that p = 2 ( 3 (p p ), √3 (p + p )) which leads to deﬁne p = (p ,p ), with p = 1 (p p ) and p = 1 (p + p ). 3√3 2 1 2 2 1 2 x y x √3 1 2 y 3 1 2 − 2πipr 2πipr − By performing a Fourier transform ar = p e ap, br = p e bp we can write

P P 2 2 mc f(p) ap H0 = d p ap† bp† 2 , (22) ± f ∗(p) mc bp Z − 2πip 2πip where f(p)= c(1 + e− 1 + e− 2 ). By solving the equation f(p) = 0 we ﬁnd two points in the Brillouin zone for 2π 4π 4π 2π which the kinematic energy is zero. These two Fermi points are: P+ = ( , ) and P = ( , ). We now want 3 3 − 3 3 to expand the kinematic term around these two points. In particular we have: ∂f = √3 + i , ∂f = √3 + i , ∂p1 2 2 ∂p2 2 2 + − + ∂f = √3 + i , ∂f = √3 + i . By writing P = P + k p + k p for small k and k we have, at ﬁrst order: ∂p1 2 2 ∂p2 2 2 1 1 2 2 1 2 − − − ± √3 i √3 i 3 3 √3 i √3 i 3 3 f+ = ( 2 + 2 )k1 + ( 2 + 2 )k2 = 2 kx + 2 iky and similarly f = ( 2 + 2 )k1 + ( 2 + 2 )k2 = 2 kx + 2 iky with k = 1−(k k ) and k = 1 (k + k −). This allows us to write the− following matrices associated− with the Hamiltonian x √3 1 − 2 y 3 1 2 kernels around the two Fermi points as

¯ 3 H+ = 2 c(σxkx + σyky)+ mσz − 3 (23) H¯ = c(σxkx σyky)+ mσz , − − 2 − where σx,y,z are the Pauli matrices and conclude that the low-energy physics is described by the following Hamiltonian

s 2 H = d k Ψ′†H¯ Ψ′ , (24) 0 ± s s R T H¯ 0 where Ψ = a+(k) b+(k) a (k) b (k) and H¯ = + . s′ s s s− s− 0 H¯ − This spinor does not have a form suited for our purposes. In fact, in a later stage, we are interested to add non-trivial interactions among all the fermions that compose tha spinor (we are interested in chiral interactions involving a γ5 matrix so that we cannot reduce to a 2-spinor). Unfortunately, the spinor presented above involves degrees of freedom evaluated at diﬀerent Fermi point so that interactions are not readily available. The solution is to simply mix the spin quantum number with the Fermi point label to obtain a 4-spinor at each of the Fermi points. This allows us to describe the low-energy limit of the model as ¯ ¯ 2 H+ 0 H 0 H = d k Ψ+† Ψ+ +Ψ† − Ψ , (25) 0 H¯+ − 0 H¯ − Z − − − T where Ψ = a↑ (k) b↑ (k) a↓ (k) b↓ (k) . We can rewrite the previous expression as ± ± ± ± ± 2 x y x y H = d k tΨ+† (kxα + kyα + mβ)Ψ+ + tΨ† ( kxα + kyα + mβ)Ψ , (26) − − − Z h i 7

3 x,y σx,y 0 σz 0 where we rescaled 2 t t, α = and β = . We can now change variables kx kx − → 0 σx,y 0 σz →− in the second integral to get − −

2 x y x y H = d k tΨ+† (kxα + kyα + mβ)Ψ+ + tΨ† (kxα + kyα + mβ)Ψ , (27) − − Z h i where we redeﬁned Ψ (kx, ky) Ψ ( kx, ky). We stress that the low− energy→ description− − of the model is given by a sum of two such action corresponding to the physics around the two Fermi points. Since we do not have any interaction coupling the degrees of freedom around the Fermi points we omit the momentum space label with the agreement that all the following quantities can be evaluated at either Fermi point. We are now ready to introduce± the last quantum number. We consider two identical layers (labelled by j =1,...,n) each one with the structure presented above. This allows us to opportunely introduce inter-layers interaction terms as follows. The full Hamiltonian of the system is

j H = H0 + HI , (28) j X where j j j H0 = H0↑ + H0↓ , (29) with j j 2 j j m f(p) ap↑ H0↑ = d p ap↑† bp↑† j f ∗(p) m bp↑ − j (30) j R 2 j m f(p) ap↓ H0↓ = d p ap↓† bp↓† j , − f ∗(p) m b ↓ − p R and where we introduced interaction terms g2 H = [( J j )2 + ( J j5)2] , (31) I 2 µ µ j j X X jµ µ j5µ 5 µ j j j j T where J = Ψj†γ Ψj and J = Ψj†γ γ Ψj with Ψj = a ↑ b ↑ a ↓ b ↓ . The gamma matrices are defined as 0 x x y y 3 γ = β = σz σz, γ = βα = iσy I, γ = βα = iσx I. There is also a further gamma matrix γ = iσz σx such ⊗ ⊗ 5 0 ⊗1 2 3 ⊗ that we can define the ”fifth” gamma matrix γ = iγ γ γ γ = σz σy. Using the results reported so far, the action describing the low ene− rgy⊗ of this model is (at each Fermi point)

S = S0 + SI , (32) with 3 2 2 S0 = d xΨ¯ 1(ic∂/ mc )Ψ1 + Ψ¯ 2(ic∂/ mc )Ψ2 2 − − (33) S = d3x g [( J j )2 + ( J j5)2] . I R 2 j µ j µ This shows that, at low energy, the modelR is describedP by a chiral-invP ariant Thirring model. In particular, in the case of a bi-layer (n = 2), we have that

2 j j j j 2 j j j j 2 1 2 1 2 2 1 2 1 2 2 HI = 3g j (a ↑b ↑ + a ↓b ↓) + (a ↑b ↓ a ↓b ↑) + (a ↑a ↓ + a ↓a ↑) + (a ↑a ↑ a ↓a ↓) 1 2| 1 2 2 |1 2 | 1 2− 2 |1 2| 1 2 1 2 | |1 2 2 − | + (b ↑b ↓ + b ↓b ↑) + (b ↑b ↑ b ↓b ↓) + (a ↑b ↓ + b ↑a ↓ a ↓b ↑ b ↓a ↑) + | 1P2 1 2 | 1 |2 1 −2 2 | 2 |1 2 1 1 −2 1 −2 | (34) (a ↑b ↑ + a ↓b ↓ + b ↑a ↑ + b ↓a ↓) + (a ↓b ↑ a ↑b ↓)†(a ↓b ↑ a ↑b ↓)+ | 2 1 2 1 1 2 1 2 | − − (b ↑a ↑ + b ↓a ↓)†(b ↑a ↑ + b ↓a ↓) + h.c. . DOUBLE SKYRMION MODEL AND FUNCTIONAL FERMIONIZATION

In this section, we show that a double O(3)-Hopf non-linear sigma model (being equivalent to a double CP-CS theory) is mapped in a chiral-invariant Thirring model by generalizing the fermionization techniques introduced in [20, 22]. Moreover, we derive fermionization rules which map observables of the fermionic model to those ones of the bosonic (double CP-CS) model. 8

Double skyrmion model and double CP-CS theory

The double skyrmion model is deﬁned by the following partition function

+ 3 1 + µ + + 1 µ Z(O(3) H)2 = m m− exp i d x (∂µm ∂ m )+ nπH + (∂µm−∂ m−) nπH− , (35) − D D 2g2 2g2 − Z Z 0 0 with the constraint m2 = 1 and H is the Hopf invariant deﬁned as [12]

µνλ ǫ 3 1 1 1 H± = d x tr (U − ∂µU )(U − ∂ν U )(U − ∂λU ) , (36) 24π2 ± ± ± ± ± ± Z 1 where i mi±σi = U − σ3U . On the other hand, a± double± CP-CS model is defined by the partition function P + + 3 n λµν + + n λµν Z 2 = A A− z z− exp i d x ǫ A ∂ A ǫ A−∂ A− (CP-CS) D D D D { 4π λ µ ν − 4π λ µ ν 1 3 + + 2 1 3 2 (37) + 2 d x (i∂ A )z + 2 d x (i∂ A−)z− , R g | µ − µ | R g | µ − µ | } i T R C R 2 where z± = z1± z2± with the fields z1±,z2± such that z± = z1±z1±∗ + z2±z2±∗ = 1. The low-energy equivalence of these two models∈ comes from using| | a saddle point approximation to integrate the fields + A and A−. This results in the following identities [6] 1 2 1 µ 2 (∂µ Aµ±)z± = 2 (∂µm±∂ m±) g0 2g0 | − | i A± = z±∗∂ z±  µ − 2 µ (38)  m± = zα±∗σαβzβ±  1 3 λµν 4π d xǫ Aλ±∂µAν± = πH± .  The spin associated with the double skyrmionR model is given by [26] 2 (Q±) S = n T , (39) 2 2 0 0 µ where QT± = d xJS± is the topological charge with JS± the 0th components of the two skyrmion currents JS± = 1 µνλ abc 1 µνλ 8π ǫ ǫ ma±∂ν mb±∂λmc± = 2π ǫ ∂ν Aλ [6]. Due to the different sign in front of the Hopf terms, this theory describes R + independent skyrmions and anti-skyrmions which have opposite values of the topological charges QT = QT− which assume only integer values. Skyrmions have fermionic or bosonic statistics depending on the value of −n. For odd n, skyrmions are fermions so that the emergence of fermionic superfluidity must be related to a condensation into bosons. Insted, when n is even, skyrmions behave like bosons and already simulate the Cooper-like pairs properties.

Functional fermionization

Following [20], we now want to show that a theory described by a double CP-Chern-Simons is equivalent to a chiral-invariant Thirring model. Our starting point is the partition function + + 3 n λµν + + n λµν Z 2 = A A− z z− exp i d x ǫ A ∂ A ǫ A−∂ A− (CP-CS) D D D D { 4π λ µ ν − 4π λ µ ν 1 3 + + 2 1 3 2 (40) + 2 d x (i∂ A )z + 2 d x (i∂ A−)z− , R g | µ − µ | R g | µ − µ | } R R i z1± C 2 where z± = where z1±,z2± such that z± = z1±z1±∗ + z2±z2±∗ = 1. We now perform the following change z2± ∈ | | of variables A+ = A + B µ µ µ (41) A− = A B , µ µ − µ µνλ + + so that, after an integration by parts on a manifold without boundary we get ǫ (Aµ ∂ν Aλ Aµ−∂ν Aλ−) = µνλ − 4ǫ Bµ∂ν Aλ and an equivalent partition function

+ 3 n µνλ Z(CP)2 BF = A B z z− exp i d x π ǫ Bµ∂ν Aλ − D D D D 1 + 2 1 2 (42) + 2 (i∂ (A + B ))z + 2 (i∂ (A B ))z− . R g | µ − µ µ R | g | µ − µ − µ | io 9

Now, we can introduce n species of fermions in order to “linearize” the BF term [18]

n 3 µνλ n jµ 5jµ i R d xǫ Bµ∂ν Aλ i P (χ ¯j (i∂/ m)χj √2AµJ √2BµJ ) e π = χ χe¯ j − − χ − χ , (43) D D Z n 3 µνλ n jµ 5jµ i R d xǫ Bµ∂ν A i P (χ ¯j (i∂/ m)χj AµJ BµJ ) jµ µ where we used the identity e 2π λ = χ χe¯ j − − χ − χ and where J =χ ¯ γ χ D D χ j j and J 5jµ =χ ¯ γ5γµχ . In this way the partition function becomes χ j j R

+ 3 jµ 5jµ Z = A B z z− χ χ¯ exp i d x j (¯χj (i∂/ m)χj √2AµJχ √2BµJχ ) D D D D D D { − − − (44) 1 + 2 1 h 2 R+ 2 (i∂µ (Aµ + Bµ))z + R2 (i∂µ P(Aµ Bµ))z− . g | − | g | − − | } i We now consider the following change of variables and notation

± j z1± z2±∗ iξ σ z± Z˜± − e j , (45) → ≡ z± z±∗ ≡ 2 1 which allows us to write (omitting the labels) the CP terms as ± 1 [(i∂ (A + B )]z 2 Tr [∂ (A + B )σ ]Z˜ 2 , (46) µ − µ µ | → 2 | µ − µ µ 3 | where

2 [i∂ (A + B )σ ]Z˜ Z˜†[ i∂~ (A + B )σ ] [i∂~ (A + B )σ ]Z.˜ (47) | µ − µ µ 3 | ≡ − µ − µ µ 3 · µ − µ µ 3 The z-dependent terms can now be rewritten as

1 2 1 j ˜ 2 2 Tr [i∂µ (Aµ + Bµ)σ3]Z = 2 Tr [ ∂µξj σ (Aµ + Bµ)σ3]Z | − | 1 | − j −µ j | 2 µ µ j j = 2 Tr(∂µξj σ )(∂ ξj σ ) + (Aµ + Bµ) + ∂µξj (A + B )(σ σ3 + σ3σ ) 1 2 2 ξi µ µ j j (48) = 2 Tr[(∂µξj ) + (Aµ + Bµ) + Jµ (A + B )(σ σ3 + σ3σ )] 2 2 ξ3 µ µ = [(∂µξj ) + (Aµ + Bµ) + Jµ (A + B )] ,

ξi i where Jµ =2∂µξ . With this transformation the action becomes

+ 3 jµ 5jµ Z = A B ξ ξ− χ χ¯ exp i d x (¯χ (i∂/ m)χ √2A J √2B J ) D D D D D D j j − j − µ χ − µ χ 1 + 2 2n ξ+3 h µ µ 1 2 2 ξ−3 µ µ R+ 2 (∂µξj ) + (Aµ + Bµ) + RJµ (AP+ B ) + 2 (∂µξj−) + (Aµ Bµ) + Jµ (A B ) g g − − (49) + 3 jµ 5jµ = An B ξ ξ− χ χ¯ exp i d x (¯χ (i∂/ m)χ √2A J √2B J ) oio D D D D D D j j − j − µ χ − µ χ 1 + 2 2 n2 2 h µ ξ+3 ξ−3 µ ξ+3 ξ−3 +R 2 (∂µξ ) + (∂µξ−) + A +R B +PA (J + J )+ B (J J ) . g j j µ µ µ µ µ − µ n oio We now change variables as

A + ξj = ξj + ξj− B + (50) ξ = ξ ξ− . j j − j so that

+ 3 jµ 5jµ Z = A B ξ ξ− χ χ¯ exp i d x (¯χ (i∂/ m)χ √2A J √2B J ) D D D D D D j j − j − µ χ − µ χ 1 1 A 2 1 B 2 n 2 h2 µ ξA3 µ ξB 3 +R 2 (∂µξj ) + (∂µξj ) + ARµ + BµP+ A Jµ + B Jµ g 2 2 (51) + 3 1 1 A 2 1 B 2 = An B ξ ξ− χ χ¯ exp i d x (¯χ (∂/ m)χ + 2 oio(∂ ξ ) + (∂ ξ ) D D D D D D j j − j g 2 µ j 2 µ j 1 2 2 µ n jχ h1 ξA3 µ 5jχ 1 ξB 3 R 2 (A + B )+ A ( √2 JR + P2 J ) +B ( √2 J + 2 J ) , − g µ µ − j µ g µ − j µ g µ io where P P

ξA3 ξ+3 ξ−3 A Jµ = Jµ + Jµ = 2∂µξj B + − (52) J ξ 3 = J ξ 3 J ξ 3 = 2∂ ξB . ( µ µ − µ µ j 10

3 1 µ µ 3 g2 µ i R d x( 2 AµA +JµA ) i R d x JµJ Thanks to the general gaussian integral identity Ae− − 2g = e− 2 we get D

3 1 R µ ˜A µ 3 g2 A Aµ i R d x( 2 AµA +J µA ) i R d x J˜ J˜ Ae g = e− 4 µ , (53) D Z and

3 1 µ ˜B µ 3 g2 B Bµ i R d x( 2 BµB +J µB ) i R d x J˜ J˜ Be g = e− 4 µ . (54) D Z A B By identifying J˜A = √2 J jχ + 1 J ξ 3 and J˜B = √2 J 5jχ + 1 J ξ 3, we can integrate over A and B to get µ − j µ g2 µ µ − j µ g2 µ P P + 3 1 1 A 2 1 B 2 Z = ξ ξ− χ χ¯ exp i d x (¯χ (i∂/ m)χ + 2 (∂ ξ ) + (∂ ξ ) D D D D { j j − j g 2 µ j 2 µ j g2 1 3A jχ 2 h g2 1 3B j5χ 2 R ( 2 Jµ √2 j JµR )) P (( 2 Jµ √2 j Jµ )) − 4 g − − 4 g − (55) + 3 1 1 i A 2 1 B 2 = ξ ξ− χ χ¯ expP i d x (¯χj (i∂/ m)χjP+ 2 (∂µξ ) + (∂µξ ) D D D D { j − g 2 j 2 j g2 χj 2 5χj 2 h 1 3A 3B 1 3A jχ 3B 5jχ R ( J ) + ( J R ) P( 2 ) J + J + J ( J )+ J ( J ) . 2 j µ j µ − 4g µ µ √2 µ j µ µ j µ P P P P i√2(ξ3A+γ5ξ3B ) We now change spinor variables χ = ΘΨ, with Θ = e so that the kinematic partχ ¯j (i∂/ m)χj of the fermionic action gives us a piece −

√2(Ψ¯ γµ∂ ξ3AΨ + Ψ¯ γµ∂ ξ3Aγ5Ψ )= 1 J 3A( J jΨ)+ J 3B ( J 5jΨ) , (56) − j µ j j µ j − √2 µ j µ µ µ P P which simpliﬁes the expression for the partition function to

+ ¯ 3 ¯ 1 3 1 A 2 1 B 2 Z = ξ ξ− Ψ Ψexp i d x j (Ψj (i∂/ m)Ψj + 2 d x (∂µξj ) + (∂µξj ) D D D D − g 2 2 (57) g2 Ψj 2 g2 n 5Ψj 2h 1 2 3A 1 2 3B R+ ( J ) + ( JR ) P( 2 ) J ( 2 ) J R , 2 j µ 2 j µ − 4g µ − 4g µ P P io where we used the fact that

J jχ = J jΨ µ µ (58) J j5χ = J j5Ψ . µ µ We then see that the ﬁelds ξ do not interact with the fermion so that we can integrate them out to get

2 2 Z = Ψ Ψexp¯ i d3x (Ψ¯ (i∂/ m)Ψ + g ( J Ψj )2 + g ( J 5Ψj )2 , (59) F D D { j j − j 2 j µ 2 j µ R R hP P P i which is in fact the original chiral-invariant Thirring model we started from.

Fermionization rules

In this subsection we map observables for the Thirring model and observables for the bosonic double CP-CS theory. To this end, we begin by introducing external ﬁelds in the fermionic theory of Eq. (59) via a minimal coupling

Ψ 5Ψ ¯ 3 ¯ g2 Ψj 2 g2 5Ψj 2 ZF (J ,J ) = Ψ Ψexp i d x j (Ψj (i∂/ m)Ψj ( j Jµ ) + ( j Jµ ) D D { − 2 2 (60) j µ 5hj µ R+ j JµAext +R j Jµ PBext , P P P P i where we omitted the label Ψ in the currents in light of the identities in Eq. (58). We now notice that all the steps done in the fermionization process described in the previous subsection can be reversed. We then replace the Thirring action in Eq. (59) with Eq.(60) and follow all the fermionization steps back. The newly introduced terms depending on the external ﬁelds can be carried over until Eq. (44) by simply replacing

i∂/ m i∂/ m + A/ + γ5B/ . (61) − → − ext ext 11

We can then change variables to

A¯µ = Aµ 1 Aµ − √2 ext (62) B¯µ = Bµ 1 Bµ . ( − √2 ext before integrating out the fermions. This allows us to use Eq. (43) as it is and to get, in place of Eq. (40)

Ψ 5Ψ ¯ ¯ + 3 n ¯µ ν ¯λ ZCP2 BF (J ,J ) = A B z z− exp i d x π ǫµνλB ∂ A − +D1 (Di∂ D (A¯D + B¯ {+ 1 Aext + 1 Bext)z+ 2 R g2 | µ − µ µ R√2 µ √2 µ | 1 ¯ ¯ 1 ext 1 ext 2 + 2 (i∂ (A B + A B ))z− g | µ − µ − µ √2 µ − √2 µ | } = A B z+ z exp i d3x n ǫ (Bµ 1 Biµ )∂ν (Aλ 1 Aλ ) − π µνλ √2 ext √2 ext D D D D { − − (63) 1 + 2 h1 2 R+ g2 (i∂µ (Aµ + Bµ))zR + g2 (i∂µ (Aµ Bµ))z− | − + | 3 n | −µ ν λ− | } = A B z z− exp i d x ǫµνλB ∂ A i D D D D { π 1 + 2 1 2 + 2 (i∂ (A + B ))z + 2 (i∂ (A B ))z− R g | µ − µ µ R | g | µ − µ − µ | + n ǫ ( Bµ ∂ν Aλ + Aµ ∂ B )+ n ǫ Bµ ∂ν Aλ o . √2π µνλ − ext ext ν λ 2π µνλ ext ext i This immediately allows us to prove the following fermionization rules

j n ν λ J ǫµνλ∂ B j µ ↔ √2π (64) J 5j n ǫ ∂ν Aλ , ( Pj µ ↔ − √2π µνλ which hold in the following sense P

j n ν λ J = ǫ ∂ B 2 j µ F √2π µνλ CP BF h 5ji n h ν iλ − (65) j Jµ F = ǫµνλ ∂ A CP2 BF , ( P h i − √2π h i − P where the expectation values F and CP2 BF are calculated with respect to the ground state of the fermionic hi hi − Ψ 5Ψ and bosonic theory respectively. The proof for this is a simple consequence of the duality ZF (J ,J ) = Ψ 5Ψ ZCP2 BF (J ,J ). In fact −

δZF δZCP2−BF n ν λ Jµ F = µ = µ = ǫµνλ ∂ B CP2 BF h i δAext δAext √2π h i − (A,B)ext=0 (A,B)ext =0 δZ (66) 5 δZF CP2−BF n ν λ Jµ F = µ = µ = ǫµνλ ∂ A CP2 BF . h i δBext δBext − √2π h i − (A,B)ext =0 (A,B)ext =0

SUPERFLUIDITY PHYSICS

In this section we show that the low energy physics of the model is described by a London action which imply a Meissner effect for the effective magnetic field and dissipationless currents. We then associate effective superfluidity effects to fermionic identities between observables for the original tight binding model.

London Action

In this subsection we describe the low-energy physics of the model with a London action. In [18], it is proven that (at low energy) a Maxwell theory is equivalent to a CP model, namely

i 3 2 R d x (i∂µ Aµ)z i 3 µν g2 R d xF (A)µν F (A) A z z†δ(z†z 1)e 0 | − | = Ae− 4e2 , (67) D D D − D Z Z where z = (z ,z ) with z ,z C and z 2 + z 2 = 1 and e2 = 24π M where M is given by the consistency condition 1 2 1 2 ∈ | 1| | 2| | | d3k 1 1= ig2 , (68) 0 (2π)3 k2 M 2 Z − 12 which renormalizes the coupling strength in relation to a momentum cut-oﬀ k =Λ as | | Λ sin θd3k k2 4π Λ k2 4π Λ 1= ig2 = ig2 = ig2 (Λ m arctan ) , (69) 0 (2π)3 k2 M 2 0 (2π)3 k2 M 2 0 (2π)3 − M Z0 − Z0 − which, for Λ = 3 3 s M with s 1 (low kinetic energy limit) allows us to write 2 ≪ q 3 1 = ig2 4π (Λ M( Λ + 1 Λ M)) , (70) 0 (2π)3 − M 3 M 3 which leads to (2π)2 M = 2 , (71) | | sg0 or, equivalently

2 2 (2π) e = 24π 2 . (72) sg0 By using this mapping we can map our double (CP-CS) to a double (CS-Maxwell) theory

3 n λµν + + n λµν 1 + µν + 1 µν S 2 = d x ǫ A ∂ A ǫ A−∂ A− 2 F (A )F (A ) 2 F (A−)F (A−) . (73) M-CS 4π λ µ ν − 4π λ µ ν − 4e µν − 4e µν + By defining newR fields A and B as A = A + B and A− = A B , we get the dual theory with action µ µ µ µ µ − µ 3 n λµν + 1 µν 1 µν SM2 BF = d x ǫ Bλ∂µAν 2 Fµν (A)F (A) 2 Fµν (B)F (B) , (74) − π − 4e − 4e so that our theory is defined byR

3 n λµν 1 µν 1 µν i R d x[ π ǫ Bλ∂µAν 2 Fµν (A)F (A) 2 Fµν (B)F (B)] ZM2 BF = A Be − 4e − 4e , (75) − D D We now follow [18] and replaceR the (2+2) degrees of freedom associated with the fields A and B with the (3+1) degrees of freedom associated with a massive bosonic field B and a massless scalar field φ. In this sense, the field φ can be thought as a Goldstone boson associated with the breaking of the U(1) symmetry for the field A. Differently from a conventional BCS theory this happens without a local order parameter. The emergence of a mass for the boson field A already is a signature of the physics associated with the Meissner effect. We now introduce an antisymmetric tensor field Zµν through

Zδ(Z F (B))=1 . (76) D µν − µν Z The delta function can be represented as

1 3 µαβ δ(T )= L ei R d xLµǫ Tαβ . (77) µν 2π D µν Z By reabsorbing constant factors we get

Z = A B Z Lexp i d3x n ǫλµν A Z 1 Z Zµν 1 F (A)F µν (A) 2π λ µν 4e2 µν 4e2 µν (78) +DL ǫDµαβD(Z D F (B)) −. − − R µ αβ − µν R By using again the representation of the delta function given above, we now perform the integration over B to

Z = A Z Lδ(ǫ ∂αLβ)exp i d3x n ǫλµν A Z 1 Z Zµν µαβ 2π λ µν 4e2 µν (79) D1 DF D(A)F µν (A)+ L ǫµαβZ .− − R− 4e2 µν µ Rαβ α β The constraint ǫµαβ∂ L = 0 can be implemented by imposing Lµ = ∂µφ where φ is a scalar ﬁeld. In this way we get

1 1 n Z = A Z φexp i d3x F (A)F µν (A) Z Zµν +(∂ φ A )ǫµαβZ . (80) D D D −4e2 µν − 4e2 µν µ − 2π µ αβ Z Z io 13

The integration over Z is a gaussian integral which leads to

3 1 µν 2 n 2 i R d x[ Fµν (A)F (A)+2e (∂µφ Aµ) ] Z = A φe − 4e2 − 2π . (81) φ D D Z From this we can see that, indeed, the field A acquires a mass which breaks the gauge symmetry of the original model. The charge and currents associated with the field are, by definition,

δ ρ = Lφ δA0 δ (82) Lφ ( Jem = δA , where φ is the Lagrangian associated with the partition function Zφ. We now observe that πφ, the momentum conjugateL to the variable φ is

δ πφ = L δ∂0φ δ = Ln δ(∂0φ 2π A0) 2π δ− (83) = L n δ(A0) 2π = n ρ . This shows that the charge density ρ is the canonical momentum conjugate to the ﬁeld φ. The Hamilton equations δ of motion are ∂0φ = δρH and this gives ∂φ V = , (84) ∂t where V is the voltage. So, the presence of steady state currents implies V =0 . (85) Since we have a situation with time independent currents and zero potential energy we can use the Drude formula J = σE , (86) to describe this model with σ = . ∞

Observables

We now want to associate identities between fermionic physical observables to two key superconducting features: Meissner effect and infinite conductance. µνλ A µνλ B To this end, let us add source terms Fµν (A)ǫ Jλ and Fµν (B)ǫ Jλ to the theory described by Eq. (75) A B 3 n λµν 1 µν Z 2 (J ,J ) = A Bexp i d x ǫ B ∂ A F (A)F (A) M BF D D π λ µ ν − 4e2 µν (87) 1 F (B)F µν (B)+ F (A)ǫµνλJ A + F (B)ǫµνλJ B , R− 4e2 µν R µν λ µν λ These terms can be thought as a non-minimal coupling of a current J to the field A, B. Alternatively, we can think µ ν λ of it as a minimal coupling 2AµJ˜ between the field A and a current J˜ = ǫµνλ∂ J . We now track the source term− while following the steps that brought us from Eq. (75)to Eq. (81). In particular, this µνλ A λµν amounts in adding the term Fµν (A)ǫ Jλ to each action and replacing, from Eq. (78) on, the term ǫ AλZµν λµν B → ǫ (Aλ + Jλ )Zµν which then leads to the generalization of Eq. (81)

3 1 µν 2 n B 2 µνλ A A B i R d x[ Fµν (A)F (A)+2e (∂µφ Aµ+Jµ ) +Fµν (A)ǫ Jλ ] Z (J ,J )= A φe − 4e2 − 2π . (88) φ D D Z Now, this is our final effective theory describing an effective electromagnetic potential inside our material. It has all the key features of superfluidity: the boson acquires a mass due to the interaction with the scalar field φ which acts analogously to a Goldstone boson. This effect describes the Meissner effect. On the other hand, in the steady state, we showed above that this system describes infinite conductivity compatible with the physics of a perfect conductor. The effective magnetic and electric fields inside the material are given by Ei = 1 ǫjµν F (A) 2 µν (89) B = 1 ǫ0µν F (A) . mag 2 µν 14

To establish a correspondence between these effects and fermionic observables, we want to find the fermionic version of both the electromagnetic charges and currents (ρ, Jem) defined in Eq. (82) and the effective electromagnetic fields (Bmag, E) defined in Eq. (89). From the definition in Eq. (82) and from the expression of the action associated with Zφ in Eq. (88) we find that

δ 2π δ φ ρ = L = LB δA0 n J 0 (90) i δ 2π δ φ J = Lk = LB , ( δA n δJi where i =1, 2. This leads immediately to

A B 2π δZφ(J ,J ) ρ φ = n δJB h i 0 JA,JB =0  A B (91)  i 2π δZφ(J ,J )  J φ = n δJB .  h i i JA,JB =0   But, from the duality relation Zφ = ZM2BF proved above we also have

A B 2π δZM2BF (J ,J ) µν0 2 ρ φ = n δJB = ǫ Fµν (B) M BF h i 0 h i JA,JB =0  A B (92) i 2π δZM2BF (J ,J ) µνi  2  J φ = n δJB = ǫ Fµν (B) M BF .  h i i h i JA,JB =0   This relation, together with the equivalence between (double-Maxw ell)-BF and (double CP)-BF theories and with the fermionization rules given in Eq. (65) implies

ρ = √2π J j φ n j 0 F (93) h ii √2π h j i ( Jem φ = Ji F , h i n Pj h i j P where, we remark that, in the symbol Ji , the label j corresponds to the layer index, while i corresponds to a spatial component (i =1, 2) of the current. On the other hand, the expectation values for the eﬀective electric and magnetic ﬁelds can be simply written as

A B i 1 δZφ(J ,J ) E = 2 δJA h i i JA,JB =0  A B (94)  1 δZφ(J ,J )  Bmag = 2 δJA ,  h i 0 JA,JB =0   which using the duality, leads to 

A B i 1 δZM2BF (J ,J ) µνi 2 E = 2 δJA = ǫ Fµν (A) M BF h i i h i JA,JB =0 A B (95) 1 δZM2BF (J ,J ) µν0 2 Bmag = 2 δJA = = ǫ Fµν (A) M BF . h i 0 h i JA,JB =0

Now, thanks to the fermionization rules, we have B = √2π J 5j mag φ n j 0 F (96) h i i − √2π h 5j i ( E φ = Ji F . h i − n Pj h i All these ﬁndings can be nicely summarized in the following table.P Electromagnetic Quantities Fermionic Oservables ρ J j h iφ j h 0 iF J Jj h emiφ Pj h iF B J 5j h magiφ Pj h 0 iF E J5j h iφ Pj h iF P 15

We now use these correspondences to map the superfluidity effects to identities among fermionic observables. The (effective) Meissner effect is characterized by the expulsion of the (effective) magnetic field from the sample and, in our case, it reads

Bmag =0 , (97)

1 within a penetration depth from the boundary given by λ e2 [18], which, by using Eq. (72) can be written in terms of the interaction strength g as λ g2. From the table above∝ we see that this implies ∝ J 5j =0 . (98) h 0 iF j X which is an identity that has to be satisﬁed in the original fermionic model. On the other hand, an inﬁnite conductivity is represented by a Drude formula J = σE with σ = and its correspon- dent fermionic identity is given by ∞

Jj = σ J5j . (99) h iF h iF j j X X We note that superconducting currents only flow within a distance λ g2 from the boundary of the sample. This means that the superfluidity of our model has a tunable penetration depth∝ (depending on the interaction strength g). On can use this feature to insure that the identity in Eq. (99) is valid inside the bulk of the material where the fermionization rules hold. The observable identity in Eq. (98) is consistent with the skyrmionic interpretation of the effective superfludity proposed in this article. In fact, the skyrmion currents for the theory in Eq. (37) can be written as [6]

µ 1 µνλ J ± = ǫ ∂ A± . (100) S 2π ν λ Now, by using the fermionization rules and the change of variables in Eq.(41) we have

µ+ 1 µ µ5 J 2 = ( J J ) S (CP-CS) n√2 j j F j F h µ i 1 h iµ − h µi5 (101) J − 2 = ( J + J ) , ( h S i(CP-CS) − n√P2 j h j iF h j iF which connects the skyrmionic currents (which involves bosonsP for n even) to the fermionic observables. Now, from Eq. (35) we can see that the two skyrmions have opposite topological charge QT

+ Q = Q− , (102) T − T where

2 0 QT± = d xJS± . (103) Z But, this gives us another way to prove Eq. (98), namely

Eq. 102 n + 0 = (Q + Q−) − √2 T T Eq. 103 n 2 +0 0 = d x J 2 + J − 2 (104) − √2 h S i(CP-CS) h S i(CP-CS) Eq. 101 2 5j = d xR J F , j h 0 i which is consistent with Eq. (19) which wasR derivedP a consequence of the eﬀective Meissner eﬀect.

BOSONIZATION

We now want to obtain the mapping between the chiral-invariant Thirring model and the (double Maxwell)-BF theory through a generalization of the bosonization techniques introduced in [23]. Following what done before, we stress that we are going to treat each Fermi point independently. We then omit the momentum space label with ± 16 the agreement that all the following is valid around each Fermi point. Our starting point is the fermionic model given in Eq. (32) which we rewrite here

S = S0 + SI , (105) with 3 2 2 S0 = d xΨ¯ 1(ic∂/ mc )Ψ1 + Ψ¯ 2(ic∂/ mc )Ψ2 2 − − (106) S = d3x g [( J j )2 + ( J j5)2] . I R 2 j µ j µ We now use the Hubbard-Stratonovich transformationR P to writeP

2 iSI ig j 2 j5 2 e = exp 2 [( j Jµ) + ( j Jµ ) ] 3 1 µ j µ 1 µ j5 µ (107) i R d x( aµa +P J a + bµb +P J b ) = a be 2g2 j µ 2g2 j µ . D D P P The full action of the model can thenR be written as

3 2 2 1 µ 1 µ S = d x Ψ1†(ic∂/ mc )Ψ1 +Ψ2†(ic∂/ mc )Ψ2 + 2 aµa + 2 bµb − − 2g 2g (108) (1) µ (1)5 µ (2) µ (2)5 µ R+Jµ ha + Jµ b ++Jµ a + Jµ b , i (1,2) µ (1,2)5 5 µ where J =Ψ1†,2γ Ψ1,2 and J =Ψ1†,2γ γ Ψ1,2. By choosing c = 1, we can write the partition function of the model as

Z = a beiS , (109) F D D Z where

1 2 S = SΨ + SΨ + SF , (110) with

1 3 5 S = d x Ψ†(i∂/ + a/ + γ /b m)Ψ Ψ 1 − 1 R h i 2 3 5 (111) S = d x Ψ†(i∂/ + a/ + γ /b m)Ψ Ψ 2 − 2 3 h 1 µ 1 µ i SF = R d x 2g2 aµa + 2g2 bµb . h i 1 R 1 2 1 Dimensional analysis shows that [Ψ] = L− ,[m]= L− ,[g ]= L,[a] = [b]= L− . To begin we can integrate out the fermionic degrees of freedom to get

i(S1 +S2 ) iS Ψ Ψ† Ψ Ψ† e Ψ Ψ = e eff , (112) D 1D 1D 2D 2 where R / 5/ n m µνλ ∂ Seﬀ = in log det(∂ + a/ + γ b m)= 2π m ǫ bµ∂ν aλ + O( mc ) . (113) − − | | We can now rewrite the low-energy limit of S as R n 1 1 S = d3x ǫλµν b ∂ a + a aµ + b bµ . (114) 2π λ µ ν 2g2 µ 2g2 µ Z By changing variables as a = 1 (t + s ) and b = 1 (t s ) we get µ 2 µ µ µ 2 µ − µ n n 1 1 S = d3x ǫλµν t ∂ t ǫλµν s ∂ s + s sµ + t tµ . (115) 8π λ µ ν − 8π λ µ ν 4g2 µ 4g2 µ Z We now deﬁne the following interpolating action

3 1 µ λµν + λµν + + SI = d x( 4g2 sµs + q1ǫ sλ∂µAν q2ǫ Aλ ∂µAν ) 3 1 µ λµν − λµν (116) + d x( t t + q ǫ t ∂ A + q ǫ A−∂ A ) , R 4g2 µ 1 λ µ ν− 2 λ µ ν− R 17 having

2 2 1 µ 1 µ q1 λµν q1 λµν + iSI i( 2 sµs + 2 tµt ǫ sλ∂µsν + ǫ tλ∂µtν ) A A−e = e 4g 4g − 4q2 4q2 , (117) D D Z q2 which is our original theory for 1 = n while 4q2 8π

3 + + − − 2 2 + +µν 2 2 − −µν iSI i R d x(q2A ∂µA q2A ∂µA +(2q1g ) (F F )+(2q1g ) (F F )) s te = e λ ν − λ ν µν µν . (118) D D Z + µ µ µ µ By defining new fields A and B as A = A + B and A− = A B , we get the dual theory with action µ µ − n 1 1 S = d3x ǫλµν B ∂ A + F (A) F (A)µν + F (B) F (B)µν , (119) π λ µ ν 4e2 µν 4e2 µν Z 2 2 2πq2 π q1 (where e = 2 = 2 2 ), which is nothing but the (double Maxwell)-BF theory. Then we have found the wanted 4nq1g n g low-energy correspondence between different theories in the low-energy limit by using functional bosonization

ZM 2 BF ZF . (120) − ≈