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Superconductor-insulator transition driven by local dephasing

M. Cuoco(a,b) and J. Ranninger(a)

(a) Centre de Recherches sur les Tres Basses Temperatures associe a l’Universite Joseph Fourier,C.N.R.S., BP 166, 38042 Grenoble-Cedex 9, France and

(b)I.N.F.M. Udr di Salerno, Laboratorio Regionale SUPERMAT, Baronissi (Salerno), ItalyDipartimento di Fisica “E. R. Caianiello”, Universita di Salerno, I-84081 Baronissi (Salerno), Italy

We consider a system where localized bound electron pairs form an array of ”Andreev”-likescattering centers and are coupled to a fermionic subsystem of uncorrelated electrons. By meansof a path-integral approach, which describes the bound electron pairs within a coherent pseudospinrepresentation, we derive and analyze the effective action for the collective phase modes whicharise from the coupling between the two subsystems once the fermionic degrees of freedom areintegrated out. This effective action has features of a quantum phase model in the presence of aBerry phase term and exhibits a coupling to a field which describes at the same time the fluctuationsof density of the bound pairs and those of the amplitude of the fermion pairs. Due to the competitionbetween the local and the hopping induced non-local phase dynamics it is possible, by tuning theexchange coupling or the density of the bound pairs, to trigger a transition from a phase orderedsuperconducting to a phase disordered insulating state. We discuss the different mechanisms whichcontrol this occurrence and the eventual destruction of phase coherence both in the weak and strongcoupling limit.

PACS numbers: 74.20.-z, 74.25.-q, 74.20.Mn

I. INTRODUCTION

The problem of interacting Cooper pairs and/orbosons, together with the possibility of a quantumcontrol of long range phase coherence in low dimen-sional systems has received considerable interest since theexperiments1 on homogeneous lead and bismuth films,which exhibit a transition from a superconducting to aninsulating phase as a function of thickness.

It is known, since the work of Abrahams et al.2, thatno true metallic behavior can be expected to be observedfor 2D non-interacting electrons at T=0, because all thestates will be localized by an arbitrarily small amount ofdisorder. When one includes Coulomb interaction, thesituation is less clear, but the common belief is that ametallic phase should still not appear at T=0 in the pres-ence of disorder - though no rigorous proof is available3.Yet, in the presence of attractive interactions, one ex-pects a superconducting state both at T=0 and finitetemperature, even in presence of a finite amount of dis-order, due to its relative ineffectiveness for a transitionof Kosterlitz-Thouless type (the degree of relevance beinggiven by the Harris criterion4).

The existence of a superconducting state at T=0 in2D systems is then considered to be directly linked tothat of an insulating state, with no intermediate metallicphase present. One hence should be able to observe adirect superconductor-insulator transition (SIT) at zerotemperature in 2D as a function of disorder, interactionstrength, magnetic field or any other external parameterwhich can drive the system away from the superconduct-ing phase.

There are different theoretical scenarios which are com-monly discussed in connection with the salient featuresof such quantum phase transitions: i) dissipative mod-

els, considering a network of Josephson coupled super-conducting grains shunted by resistors5,6,7,8,9,10, wherethe competition between the Josephson intergrain cou-pling and the charging energy yields an increase of thephase fluctuations of the superconducting order param-eter and hence leads to a phase disordered state. Dis-sipation thereby plays the role of suppressing quantumphase fluctuations and thus competes with the chargingmechanism. ii) Bose-Hubbard models, where the super-conducting phase is due to the Bose-Einstein condensa-tion of charge-2e bosons and the insulating phase due to aproliferation of vortices and localization of pairs11,12,13,14.

While the SIT was long thought to be a paradigmof the above theories, recent experiments have thrownsome doubt about that. They revealed, what seems tobe a low-temperature metallic state which is intertwinedin such a SIT and thus requires a more appropriatetheoretical description. In magnetic field tuned experi-ments on Mo-Ge samples15, in granular superconductors(Ga films16 and Pb films17), and in Josephson junctionarrays18 a metallic phase has been observed. Moreover,the recent experiments of Kapitulnik et al.19, in which ametallic phase has been observed sandwiched betweenthe superconductor and the insulating phase, suggestthat perhaps two phase transitions accompany the lossof phase coherence in 2D superconductors : (i) a super-conductor to a “Bose metal” and (ii) a “Bose metal” toan insulator. A “Bose metal” in this context, is thoughtof just a gapless non-superfluid liquid with metallic liketransport20.

These experimental results have led to reconsider thewhole issue of the SIT. One recent proposal to handlesuch a new viewpoint of the SIT has been to reexam-ine the standard on-site charging model. By includ-ing nearest neighbor charging terms it was shown that

2

the resulting uniform Bose metal state lacks any traceof either phase or charge order21, due to a competi-tion between the order parameters which describe theonset of charge order and that of phase coherence. Adifferent point of view22 is, that in the quantum dis-ordered regime a cancellation happens between the ex-ponentially long quasi-particle scattering time and theexponentially small quasi-particle population, which ul-timately leads to a finite dc conductivity. Finally, intrin-sic as well as extrinsic sources of dissipation have beensuggested as potentially relevant for the occurrence ofa non-superfluid metallic phase in proximity to the su-perconducting phase19,23,24. In this context, the dissi-pation is a relevant perturbation25, which, depending onthe strength of the coupling to the dissipative source,can drive the system from a superconductor-to-insulator

transition to a superconductor-to-metal transition19,26.Several other ideas supported a scenario where the sys-tem could break into superconducting and insulating ”is-lands” weakly linked via percolating paths before goinginto a metallic phase, dominated by vortex dissipation26.This ”puddle” scenario matches with the view to describethe 2D superconductor-metal transition via supercon-ducting islands embedded in thin metal films27. Detailedanalysis of the puddle-like model considered strongly fluc-tuating superconducting grains embedded in a metallicmatrix28, predicting a metal-to-superconductor transition

with a metallic phase just above the transition domi-nated by Andreev reflections between the superconduct-ing grains. Finally, recent investigations directed the at-tention to new phases with Bose metallic features, wherethe dissipation is dynamically self-generated, as in thequantum phase glass model, where disorder in the distri-bution of the tunnelling amplitudes and quantum fluctu-ations destroy phase coherence20.

On similar lines as those dealt with in a great varietyof such different approaches discussed in the literature,our aim here is to investigate a system where the break-down of superconductivity situates itself in between thecase of a “bosonic” and a “fermionic” mechanism for su-perconductivity suppression. We focus on determiningthe possible ground states for interacting Cooper pairs,in close relation to the classical notions of superfluidityand localization of bosons and, on a more general basis,how the phase coherence can be tuned to a phase disor-dered state whose transport properties may be unconven-tional. In particular, we consider within the frameworkof a boson-fermion model (BFM), a system where lo-calized bound electron pairs (hard-core bosons) form anarray of ”Andreev”-like scattering centers coupled to afermionic subsystem of itinerant uncorrelated electrons(quantum pair-exchange). This scenario goes beyondthat of pure phase models widely discussed in the lit-erature, since here one is dealing with bosonic degrees offreedom (localized bound pairs) as well as fermionic ones(itinerant electrons) for which, due to the emergence ofpair correlations, one is dealing with both amplitude aswell as phase modes. The possibility to tune from short

to long-range phase coherence arises from the followingcompeting effects:

(i) on the one hand, the short range interaction be-tween bosons and fermion pairs (holes) induces a localphase locking in a configuration with a quantum super-position between bosons and electron pairs, leaving thecommon phase undetermined.

(ii) on the other hand, the itinerancy of the electronstends to lock and rigidly extend these initially arbitrarylocal phases. As a result, phase coherence develops overlonger distances by suppressing the quantum fluctuationsof the local phase, which will involve the dynamics ofamplitude fluctuations.Out of this competition one can recover either a super-fluid state in the regime of a small scattering rate or aphase disordered state in the limit where the pair ex-change dominates and the local quantum phase fluctua-tions do not allow for long-range phase coherence. As weshall show, the physics described by this scenario stronglydepends on the concentration of fermions and bosons,on the coupling strength, as well as on temperature andmagnetic fields. The purpose of this study is to analyzehow a transition from a superconductor to an insulatormay occur and discuss the features which can give riseto unconventional dynamics and, eventually, unconven-tional transport properties in the proximity of such atransition.

The outline of the paper is as follows. In Sect. II wewill introduce the BFM and describe its main qualitativefeatures and the phenomena which can be described byit. In Sect. III, we shall develop a path integral represen-tation of this model and derive an effective coarse grainedaction, which, after integrating out the fermionic degreesof freedom, is able to describe the low energy dynamicsof the phase and amplitude modes. In Sect. IV we willdiscuss the phase diagram based on the simplest approx-imation of such a coarse grained effective action in termsof a phase-only action and explore the transition froma superconductor to a phase disordered state as a func-tion of the coupling and the density of bosons. In Sect.V, we will study the physical features which arise fromthe intrinsic Berry phase term present in our effective ac-tion and which arises from the hard core nature of thebosons, represented by quantum pseudo spin- 1

2 variables.In section VI we compare the salient features of this BFMscenario with similar scenarios, such as the negative-UHubbard model, the Bose-Hubbard model and Joseph-son junction arrays. In the Discussion, section VII, wereview the main results obtained in this paper and indi-cate further developments planed for the near future.

II. THE BOSON-FERMION MODEL

The boson-fermion model (BFM) in recent years hasattracted considerable attention as a model capable ofcapturing basic physical properties in many body sys-tems with strong interaction, giving rise to the formation

3

of resonant pair states of bosonic nature inside a reser-voir of fermions. Such a scenario was initially proposedby one of us (JR) as an alternative to the scenario ofthe hypothetical and yet to be experimentally verifiedbipolaronic superconductivity. It was meant to describethe intermediary coupling regime between the adiabaticand anti-adiabatic limits in polaronic systems, where anexchange between localized bipolarons and pairs of un-correlated electrons can be assumed to take place (fora recent intuitive justification of such a scenario see forinstance ref.29).

As it has turned out, this boson-fermion scenariohas a much wider range of applicability than that forwhich it was initially proposed and seems to apply tovery different physical situation such as: hole pairingin semiconductors30, isospin singlet pairing in nuclearmatter31, d-wave hole and antiferromagnetic triplet pair-ing in the positive-U Hubbard model32 (and possibly alsoin the t-J model), entangled atoms in squeezed states inmolecular Bose Einstein condensation in traps33 and su-perfluidity in ultracold fermi gases induced by a Feshbachresonance34.

The BFM is reminiscent of an anisotropic Kondo lat-tice model in terms of a pseudospin- 1

2 , but characterizinglocalized electron pairs instead of localized impurity spinsas in the Kondo analogue. Its Hamiltonian is given by

H = (D − µ)∑

iσ

c+iσciσ + (∆B − 2µ)∑

i

(ρzi +1

2)

+∑

i6=j, σ

tij(c+iσcjσ +H.c.) + g

∑

i

(ρ+i τ

−i + ρ−i τ

+i

)

The pseudo-spin operators [ρ+i , ρ

−i , ρ

zi ] denote the local

bound electron pairs (bosons) and [τ+i = c+i↑c

+i↓, τ

−i =

ci↓ci↑, τzi = 1− τ+

i τ−i ] the itinerant pairs of uncorrelated

electrons. [c+iσ, ciσ] stand for the creation and annihila-tion operators of the itinerant electrons (fermions) andg is the strength of the boson ⇔ fermion pair exchangeinteraction. The hopping integral for the itinerant elec-trons, which is assumed to be different from zero onlyfor nearest neighbor sites, is given by t with a band halfwidth equal to D = zt, z denoting the coordination num-ber of the underlying lattice. The energy level of thebound electron pairs is denoted by ∆B . The number ofthe ensemble of bosons and fermions being conserved,ntot = nF↑ + nF↓ + 2nB, implies a common chemicalpotential µ for both subsystems. nB, nF↑,↓ indicate theoccupation number per site of the hard core-bosons andof the electrons with up and down spin states.

The exchange coupling between the bosons and thefermion pairs can be considered as an effective Andreev-like scattering leading to local states which are quantumsuperpositions of the form

|ψloc〉i =

∫dφi[cos(θi/2) cos(φi)ρ

+i + sin(φi)] ×

[cos(φi) + τ+i sin(θi/2) sin(φi)]|0〉. (1)

Such states evolve gradually out of the system of local-ized dephased bosons and essentially uncorrelated freefermions, which characterize the high temperature phaseof this model, when the temperature is decreased be-low a certain T ∗ ≃ g where resonant pairing (not boundpairs!) starts to be induced in the fermionic subsystem.These pair states have already features built in whichare reminiscent of those which characterize Cooper pair-ing of fermions as well as superfluidity of bosons. Thephases of the two coherent states, corresponding to thetwo subsystems, are the same and hence locked together,but are averaged over all angles as a consequence of theconserved particle number on any given site, ntot = 2.Roughly speaking, the ground state of the system is thengiven by a product state

∏i |ψloc〉i with cosθi = 1 and

which exhibits no phase correlations on any finite lengthscale. Let us next consider the effect of fermion hop-ping between adjacent sites. This will give rise to den-sity fluctuations on each of those individual sites and thushelp to stabilize an arbitrary but finite average value ofthe phases {φi} over a finite length and time scale. Inthis way the localized bosons and fermion pairs acquireitinerancy35,36,37 which eventually leads to a superfluidstate in both subsystems38, provided that the effect ofthe local correlations between the bosons and the fermionpairs can be sufficiently diminished, but remaining stillsufficiently strong to guarantee the formation of pairingin the fermionic subsystem. Achieving or not this situa-tion will depend on the relative importance of the localexchange coupling versus the fermion hopping rate (givenby the ratio g/t) and as well as on the concentration ofthe bosons, as shown by exact diagonalization study39 onfinite clusters of this BFM.

What we shall attempt in the present study is to de-scribe this physics in terms of an effective action for thephase and amplitude fields of the bosonic fields. In orderto achieve this we shall put the discussion on a level whichis more familiar, namely that one of Josephson junctionarrays and Bose-Hubbard models. For that purpose letus briefly sketch the analogy which exists (up to a certainpoint) between the BFM and those systems which havebeen widely discussed in the literature. A physically pos-sible realization of this BFM scenario can be imagined inform of a network of superconducting grains embeddedin a metallic environment and where the only mecha-nism of interaction between the grains and the fermionicbackground is that of Andreev reflections. Via such amechanism an electron (hole) is reflected on the grain asa hole (electron) leaving behind a surplus of two holes(electrons) in the fermionic subsystem and of two elec-trons (holes) in the grain. If the grains are such thatthey have a large charging energy, the fluctuations of thenumber of pairs on them are energetically unfavorableand hence are largely suppressed. We then have a sit-uation where the state of the grains switches essentiallybetween zero to double occupancy with respect to the av-erage occupation, any time an electron (hole) is reflectedat the interface of the grain. Thus, the quantum dynam-

4

ics of the single grain can be directly represented by apseudospin– 1

2 , in order to account for the doublets whichrepresent the two possible states of the grains.

For such a possible experimental setup, the effectivesites in the BFM have to be considered as defining a reg-ular array of grains and having the same periodicity asthe underlying lattice on which the fermions move witha hopping amplitude t. Moreover the size of the grainshas to be such that it is much smaller than the distancebetween them. A pictorial view of such an experimentalset up in given in Fig. 1. The analogy between the BFMand the array of superconducting grains which scatterpairs of fermions in a metallic matrix via Andreev likereflections, may ultimately serve as an experimental de-vice on which to test and analyze the theoretical issueswhich will be discussed in this paper.

FIG. 1: a) Schematic 1D representation of the BFM on a lat-tice (top and side view). The bosonic and fermionic particlesmove on two different arrays having the same periodicity: thefermions are indicated by circles and the bosons by squareson the respective arrays. b) The single site configurations forthe pseudospin and fermionic variables.

III. PATH INTEGRAL REPRESENTATION:

DERIVATION OF THE EFFECTIVE ACTION

A. Generalities

Let us now construct an effective action which de-scribes the BFM, with the aim to extract the dy-namical properties of the low energy degrees of free-dom of the phase and amplitude modes for the bosonsand fermion-pairs. We start by expressing the parti-tion function in terms of a coherent-state path inte-gral representation40, where the fermionic part is formu-lated by means of the usual Grassmann variables and thebosonic part is described by a pseudospin-coherent state

representation40,41:

Z =∏

i

∫DθiDφiDΨiDΨie

−A[Ψi,Ψi,θi,φi] (2)

where

A[Ψi,Ψi, θi, φi] =

∫dτ∑

i

[is(1 − cos θi) ∂τφi +

(∆B − 2µ) cos θi] +∑

〈ij〉

Ψj(τ)G−1ij Ψi(τ). (3)

τ denotes the imaginary Matsubara time variable anda Nambu spinor representation for the Grassmann vari-ables, related to that of the original fermionic operatorsby

Ψi =

(ci↑ci↓

)Ψi = (ci↑ ci↓) . (4)

The pseudospin is described by a bosonic fieldwhich in spherical coordinates is given by si =s(sin θi cosφi, sin θi sinφi, cos θi) (see Fig. 2). θi de-scribes the polar angle of the vector si with respect tothe north pole of the z axis, while φi is the azimuthal an-gle which defines the angular position of the basal planeprojection of this vector. The first term of the action A isthe Wess-Zumino term41,42, ensuring the correct quanti-zation of the quantum pseudospin variable. For any path,parameterized by φ(τ) and θ(τ), the contribution of thisterm is equal to i s times the surface area of the sphere be-tween this path and the north pole. For closed paths thishas exactly the form of the Berry phase43. The secondterm is linked to the density of the bosons nB(τ) throughthe cos θ(τ) dependence of the pseudospin. Finally, thelast contribution of the action contains the coupling be-tween the fermionic and bosonic subsystem through theGreen’s function Gij , determined by:

(K1 LL∗ K2

)Gij(τ − τ ′) = δ(τ − τ ′) (5)

where K1 = (−∂τ + µ)δij + tij , K2 = (−∂τ − µ)δij − tijand L = g sin θi(τ)e

i φi(τ)δij and L∗ being the conjugatefield of L. Integrating out the fermionic part, one obtainsthe action in terms of exclusively the bosonic fields:

A = −Tr lnG−1 +

∫dτ∑

i

[is(1 − cos θi(τ)) ∂τφi(τ) +

(∆B − 2µ) cos θi(τ)] (6)

where the trace has to be carried out over all internal aswell as space-time indices.

Up to this point no approximation has been made inthe derivation of the action which describes the couplingbetween the bosonic and fermionic degrees of freedom. Itis important to notice that the variation of the bosonicvariable θ(τ) describes both, the density fluctuations of

5

FIG. 2: Spherical representation of the pseudospin s includingthe Berry phase factor for one possible trajectory Γ. TheBerry phase term is exp[is

∮(1 − cos[θ]) ∂φ

∂τ] = eisA, where A

is the area of the surface enclosed in the trajectory Γ. Theblack part indicate the differential portion of the surface onthe sphere taken respect to the north pole.

FIG. 3: Representation of a possible path for the localpseudospin motion. The field l[θ(τ )] indicates the undula-tion of the pseudospin vector along the polar direction asit arises from the fluctuations of the average boson density(〈cos θ〉) and of the pairing amplitude (〈sin θ〉), while it pre-cesses around the z-axis due to the time evolution of the phasevariable φ(τ )).

the bosonic subsystem (via the projection of the pseu-dospin vector on the z axis, i.e. cos θ(τ), being the lon-gitudinal component) and the amplitude fluctuations ofthe fermionic pair field (via the transverse part as pro-jection of the pseudospin vector onto the basal plane, i.e.sin θ(τ)) (see Fig. 2)). The variable φ(τ) determines therotational degrees of freedom of the pseudospin vector,expressing its phase dynamics.

For extracting the relevant terms which control thelow energy dynamics of the coupled phase and amplitudemodes and performing an expansion, which is meaning-ful in terms of the phase variable, it is judicious to makethe following steps: i) gauge away the phase dependencefrom the term L which permits to separate the traceinto a part which does not depend on the phase of thebosonic field and another part that contains only spa-tial and time variations of φi(τ)

44, ii) rewrite the term Lafter the gauge transformation, as a sum of two pieces,one not dependent on time (which is linked to the aver-

age density of bosons) and another term containing thefluctuations with respect to its mean value.

The first operation is performed by applying to theoperators under the trace the rotation Ui = ei φi(τ)σ3/2,where σi denote the Pauli matrices. Hence,

Tr lnG−1 = Tr lnUG−1U−1 = Tr ln G−1 (7)

where

G−1ij =

[−∂τσ0 + [

i

2∂τφi(τ) − µ]σ3 + g sin θi(τ) σ1

]δij +

tijei [φi(τ)−φj(τ)]σ3/2. (8)

Since the bosonic density is fixed in average, we nowseparate the part which depends on the polar angle ina time independent contribution and its time dependentcorrection. That is, the term sin θi(τ) is decomposed intoits average value 〈sin θi〉 (which is determined by fixingthe density of bosons due to the spherical constraint) plusa time dependent contribution l[θ(τ)] which contains thefluctuation around its average value (Fig. 3). Thus thislocal field, due to the constraint, will describe both: thetime dependent variation of the density as well as of thepairing amplitude.

Next, let us write the Green’s function in the usualform as

G−1ij = G0

−1ij + Σij (9)

with G0−1ij = [−∂τσ0 − µσ3 + g σ1] δij and Σij = Tij +

Di +Ki, where

Di =i

2

∂φ(τ)

∂τσ3 (10)

Ki = g l[θ(τ)]σ1 (11)

Tij = tije−i(φi(τ)−φj(τ))

σ32 (12)

g = g〈sin[θ]〉. (13)

From this point onward we shall assume the averagedensity of bosons to be homogeneously distributed andthus given by nB = 1

2 (1 + 〈cos θ〉). This implies that thebare coupling g is renormalized to g as a consequenceof the spherical constraint of the pseudospin variable.Moreover, since 〈sin θ〉 =

√1 − 〈cos θ〉2 and the density

of bosons is fixed in average via a suitable choice of thechemical potential and of the on-site bosonic energy, onecan treat the exchange coupling as an external boson-density tunable parameter. The variation in the θ vari-able is then simply related to the variation of the bosonicdensity such that if θ varies in the range [0, π] then nBvaries in the interval [0, 1].

Before expanding the trace, let us write down explicitlythe expression of the zero order Green’s function, as itwill be frequently used in the following steps:

G0αβi (τ) =

1

β

∑

ωn

G0αβi (ωn) exp[−i ωn τ ] (14)

6

with

G0αβi (ωn) =

−iωn+µω2n+ω2

0

√ω2

0−µ2

ω2n+ω2

0√ω2

0−µ2

ω2n+ω2

0

−iωn−µω2n+ω2

0

(15)

and where we introduced ω0 = g.

B. Second order loop expansion

We now evaluate the contribution of the self energyΣij to the effective action. This is done in the usualway by making a loop expansion in the trace44. We shallconstruct that expansion up to second order in the timeand space derivatives of the phase variable and in theterms which contain both, the fluctuations of the densityand the amplitude. For that purpose we use the standardidentity:

Tr ln G−1 = Tr ln[G0

−1 + Σ]

= Tr lnG0−1 + Tr ln [1 +G0Σ] (16)

and then expand the second term of this expression up tosecond order in Σ, such as to keep all the contributionsup to quadratic order in the gradient of the phase. Thisgives

Tr ln G−1 ∼= Tr lnG0−1 + Tr[G0Σ] − 1

2Tr[G0Σ]2 . (17)

The first term of this expression is just a constant anddoes not contribute to the dynamics. In the second term

Tr[G0Σ] = Tr[G0T ] + Tr[G0D] + Tr[G0K] (18)

the only parts different from zero are Tr[G0iKi] andTr[G0iDi]. Tr[G0iTij ] gives no contribution once onemakes the trace over the site indices. Tr[G0iDi] intro-duces a contribution which is proportional to the chem-ical potential multiplied by the time derivative of thephase ∼ iµ∂τφ. We will see below that this contributiondescribes an effective off-set charge (in terms of the ter-minology of a similar Josephson junction array scenario)and which arises from the total fermionic and bosonicstatic density distribution via their dependence on thechemical potential. Tr[G0iKi] describes the lowest orderfluctuations of the bosonic (fermion pair) density due tothe presence of the spontaneous pair/hole creation out ofthe condensate. Its direct evaluation gives a contribution

A1 =

∫dτ1dτ2Trps[G0i(τ1, τ2)Ki(τ2, τ1)] (19)

where Trps represents the trace over the internal pseu-dospin index. With Ki(τ2, τ1) = Ki(τ1)δ(τ1 − τ2) andintegrating over the Matsubara times gives:

A1 =

∫dτT rps[G0i(0)Ki(τ)]

= 2gG012i (0)

∫dτl[θ(τ)]

= −g tanh[βω0]

ω0

∫dτl[θ(τ)], (20)

or in a compact form

A1 = E1

∫dτl[θ(τ)]

E1 = −g tanh[βω0]

ω0. (21)

Let us next come to the evaluation of the terms whichcontribute to the quadratic order in this loop expansionof the trace. The parts which are non zero in these termsare the following:

A2 = Tr [G0i Ki G0i Di]

A3 = Tr [G0i Ki G0i Ki]

A4 = Tr [G0i Di G0i Di]

A5 = Tr [G0i Tij G0j Tji] (22)

The term A2 contains processes where the fermionicbackground locally couples at different times to the fluc-tuations of the bosonic density and to the phase fluctua-tions:

A2 = Tr [G0 i Ki G0 i Di] =∫dτ1dτ2Trps[G0 i(τ1 − τ2)Ki(τ2)G0 i(τ2 − τ1)Di(τ1)].

(23)

Since G0i(τ1 − τ2) depends exclusively on the time dif-ferences, we introduce the new variables τ = τ1−τ2

2 and

η = τ1+τ22 , after which the integral over the Matsubara

time variables becomes:

A2 =1

2

∫dτdηT rps[G0i(τ)Ki(η − τ)G0i(−τ)Di(τ + η)].

(24)As we are interested in the gradient expansion in thebosonic phase and density, related to φ(τ) and θ(τ),we keep only the lowest order in their time derivatives.This is done by considering the expansion of the productKi(η − τ) Di(τ − η) around τ = 0 in order to separatethe parts which depend exclusively on the local variableof the Green function and those which are linked to thephase and the density fluctuations. The expansion thenreads as follows,

Ki(η − τ) Di(η + τ) ≃ Ki(η) Di(η) + τ{−∂Ki(η)

∂ηDi(η)

+∂Di(η)

∂ηKi(η)} +O(τ2) (25)

Due to the symmetry of the Green’s functions (even forτ → −τ), the linear contribution of the series expansionof Ki(η−τ) Di(τ−η) cancels in the effective action afterintegrating over the time. Hence, to lowest order oneobtains,

A2 =1

2

∫dτdηT rps[G0i(τ)Ki(η)G0i(−τ)Di(η)]

7

=g

2

∫dτ[G0

12i (τ)G0

11i (−τ) +G0

11i (τ)G0

12i (−τ)

− G022i (τ)G0

12i (−τ) −G0

12i (τ)G0

22i (−τ)

]×

∫dηi

2

∂φ(η)

∂ηl[θ(η)] (26)

or in a compact form,

A2 = E2

∫dη i

∂φ(η)

∂ηl[θ(η)] (27)

where the coefficient E2, after integrating over the Mat-subara time, is given by:

E2 =g2

4

µ (β ω0 − sinh [β ω0])

ω30(1 + cosh [β ω0])

. (28)

Let us next consider the term A3, which expressesthe coupling of the fermionic field to the fluctuations ofbosonic density at different times. In order to extract thelowest order gradient contributions, we follow the sameprocedure as that just used above, giving us:

A3 =1

2

∫dτ dηT rps[G0i(τ)Ki(η− τ)G0i(−τ)Ki(τ + η)]

(29)Performing again the expansion in time up to quadraticorder in the time derivatives of Ki(τ), we have:

Ki(η − τ) Ki(η + τ) ≃ Ki(η)2 − τ2 ∂Ki(η)

∂η

2

. (30)

and hence can rewrite A3 in the following way:

A3 =1

2

∫dτ dηT rps[G0i(τ)Ki(η)G0i(−τ)Ki(η)] +

1

2

∫dτ dη T rps[G0i(τ)K

′

i(η)G0i(−τ)K′

i(η)]. (31)

Carrying out the trace over the internal indices gives:

Trps[G0i(τ)Ki(η)G0i(−τ)Ki(η)] = [G022i (τ)G0

11i (−τ)

+2 G012i (τ)G0

12i (−τ) +G0

11i (τ)G0

22i (−τ)] (g2l[θ(η)]2)

(32)

so that

A3 = E3a

∫dη l[θ(η)]2 − E3b

∫dη∂ l[θ(η)]

∂η

2

. (33)

By evaluating the integrals over the Matsubara times weobtain:

E3a =g2

2

βω0g2 + µ2 sinh[βω0]

ω30(1 + cosh[β ω0])

E3b =g2

2[sech[(βω0/2)]2(−6βµ2ω0 + β3(−µ2ω3

0 + ω50)

24ω50

+6µ2 sinh[βω0])

24ω50

]. (34)

Applying the same procedure for the evaluation of A4

(which has the same functional form as A3 but involv-ing the coupling between the fermionic degrees of free-dom and the phase velocity at different time) one getsan analogous expression, providing we discard terms of

higher order in the derivatives than [∂φ(τ)∂τ ]2:

A4 ≃ 1

2

∫dτ dηT rps[G0i(τ)Di(η)G0( − τ)Di(η)]

=1

8C0

∫dη(

∂φ(η)

∂η)2. (35)

Here

C0 =

∫dτ{−[G0

11i (τ)G0

11i (−τ) − 2 G0

12i (τ)G0

12i (−τ) +

G022i (τ)G0

22i (−τ)]} , (36)

which after evaluating the integration over the Matsub-ara times gives,

C0 =sech[(βω0/2)]2(βµ2ω0 + g2 sinh[βω0])

2ω30

. (37)

Finally, we come to the evaluation of the last termA5 which involves the coupling between phase fluctua-tions on different sites and the process of single particlehopping. This contribution will yield terms which arequadratic in the time derivative of the phase (charging

like, in the terminology of a similar Josepson junctionarray scenario) and moreover will generate an effectivehopping induced inter-site phase coupling which turnsout to be similar to the Josepshon coupling in arrays ofsuperconducting grains.

Considering again the lowest order gradient expansioncontributions we have:

A5 = Tr [G0i Tij G0j Tji]

≃ 1

2

∫dτ dηT rps

{G0i(τ)Tij(η)G0j(−τ)Tji(η)−

τ2

[G0i(τ)

∂Tij(η)

∂ηG0j(−τ)

∂Tij(η)

∂η

]}

= P1 + P2 (38)

The terms P1,2 are given by:

P1 = −t2ij∫dτ [G0

12i (τ)G0

12i (−τ)]

∫dη cos[φi(η) − φj(η)]

P2 = −t2ij8

∫dτ τ2 [G11

0 i(τ)G110 j(−τ) +G22

0 i(τ)G220 j(−τ)]

∫dη[

∂φi(η)

∂η− ∂φj(η)

∂η]2. (39)

By carrying out the integration over the Matsubara timesone obtains for the hopping induced inter-site amplitude

8

and phase coupling:

P1 = EJ

∫dη cos[φi(η) − φj(η)]

EJ =t2ij g

2(−βω0sech[βω0/2]2 + 2 tanh[βω0/2]

)

8ω30

.

(40)

Similarly, by calculating the coefficient of the term P2,one obtains the strength of a mutual capacitance term(in the terminology of a similar Josephson junction arrayscenario) for neighboring effective sites:

P2 =1

8C1

∫dη[

∂φi(η)

∂η− ∂φj(η)

∂η]2

C1 = −t2ij∫ β/2

−β/2

dτ τ2 [G11i (τ)G11

i (−τ) +G22i (τ)G22

i (−τ)]

= −t2ij[exp[βω0](−6βg2ω0 + β3ω3

0(µ2 + ω2

0)

12(1 + exp[βω0])2ω50

+6g2 sinh[βω0])

12(1 + exp[βω0])2ω50

]. (41)

The final effective action is then given by the sum ofall the terms evaluated above which are grouped togetherin form of three different contributions:

S =

∫dτ [Sφ + Sφ−θ + Sθ] . (42)

Sφ, Sθ, Sφ−θ are the contributions arising from exclu-sively (i) the phase dynamics, (ii) the fluctuations of thebosonic density and of the amplitude l[θ] and (iii) thecoupling between them. They are given by:

Sφ =∑

i

1

8(C0 + zC1) (

∂φi(τ)

∂τ)2

−∑

〈i,j〉

[1

8C1∂φi(τ)

∂τ

∂φj(τ)

∂τ+ EJ cos[φi(τ) − φj(τ)]

]

+ i∑

i

µ∂φ(τ)

∂τ

Sθ =∑

i

[E1 l[θi(τ)] + E3al[θi(τ)]

2 − E3b[∂ l[θi(τ)]

∂τ]2]

Sφ−θ =∑

i

[i s

∂φi(τ)

∂τ(1 − cos[θi(τ)])

+ i E2∂φi(τ)

∂τl[θi(τ)]

]. (43)

We expand the Berry phase contribution (the first termin Sφ−θ) up to second order in l[θ(τ)] and subsequentlyredefine this field as l[θ(τ)] = a+ bl[θ(τ)], with the timeindependent constants a, b chosen in such a way as toeliminate any terms linear in l in the the action S. Wethen find two contributions to the action which are linearin ∂φ(τ)/∂τ : One which is time independent and which

can be absorbed into the chemical potential and anotherone which is quadratic in l[θ(τ)]. With this, Sφ−θ can berewritten as:

Sφ−θ =∑

i

i∂φi(τ)

∂τqi(τ) , (44)

where qi(τ) is quadratic in l[θ(τ)].The effective action thus constructed for the BFM

(Eqs. (42,43)) is similar to that of Josephson junctionarrays with nearest neighbor, as well as, self capacitance,Josephson coupling and off-set charge terms. The actionfor the BFM goes however beyond that for such Joseph-son junction arrays in the following respects. We haveextra terms which control the dynamics of the amplitudemodes, given by Sθ and an intrinsic Berry phase termwhich gives rise to a direct phase - amplitude coupling(Eq. 44), where the dynamical amplitude fluctuationswould correspond to a time dependent offset charge termin an analogous Josephson junction array picture. Fi-nally, the Berry phase term, being an intrinsic topologi-cal term, may give rise to a Magnus force on a vortex, asit will be discussed in section V.

IV. THE SUPERCONDUCTOR - INSULATOR

PHASE BOUNDARY

In a first approach to analyze the stability region oflong-range phase superconducting coherence, we exam-ine the effective action (eqs. 42,43), restricting ourselvesto the phase only part of it. This means a study of theSIT driven by a competition between the phase coherenceinduced by pair hopping and the disrupting effect of localboson density (or equivalently pair field amplitude) fluc-tuations in the presence of a source term for the bosonswhich controls the global boson density via an effectivechemical potential. Within such an approximation ourstudy is equivalent to that of Josephson junction arrays,except that the effective coupling constants entering insuch an action depend in a highly non trivial way on theparameters which characterize the original BFM Hamil-tonian. This, as we shall see, will lead to novel featuresconcerning the phase diagram with a SIT for the BFMwhen we examine it in terms of the boson-fermion ex-change coupling g/t and the boson concentration nB.

We shall determine the phase diagram by means of theso-called coarse graining approximation which has beensuccessfully applied for this kind of problem and whichpermits to capture the relevant qualitative and quanti-tative features of such a Josephson junction array likeaction45,46. It is known that, as a consequence of theuncertainty relation between the phase φi and the pairnumber operator Qi = i ∂

∂φi, the system can switch from

a phase ordered to a disordered state. An essential partof this study will concern how the relevant parametersof the BFM Hamiltonian influence the equivalent am-plitudes of the Josephson coupling, the capacitance and

9

the off-set charge terms. Thus, at zero temperature, byfixing the bosonic density distribution, we find that thevariation of the ratio between the Josephson couplingenergy EJ and the charging energy EC = C−1

0 /2, whichcontrols the phase-density interplay, increases from zero,goes through a maximum and then decreases to zero withincreasing g/t . Or else, if one fixes the coupling g/t, thenby varying the density of the bosonic distribution oneis able to control the effective coupling which appearsin EJ and Cij bt varying the average bosonic density

g = 2g√nB(nB − 1)). It is thus immediately evident

that in the BFM scenario there is a non-trivial interplaybetween the renormalization of the Josephson couplingand the charging effect. If one goes to the limit of empty(nB = 0) and full bosonic occupation (nB = 1), theeffective coupling g → 0, and the critical temperatureconsequently reduces to zero.

The general form of the action we then have to examineis given by:

Sphase =

∫ β

0

∑

i

i

2

∂φi(τ)

∂τqi −

EJ∑

ij

αij cos[φi(τ) − φj(τ)] +

∑

i,j

1

8

∂φi(τ)

∂τCij

∂φj(τ)

∂τ. (45)

The first term describes the effect of a static off-set charge

(qi) The second term contains the physics of the pair hop-ping processes, an analog to the Josephson tunnellingprocesses, and has a coupling strength EJ with αij = 1if (i, j) are nearest neighbors and zero otherwise. Fi-nally, the third term describes the charging term aris-ing from the local exchange of boson and fermion pairsand from quasi-particle hopping between nearest neigh-bor sites. The strength of this charging type interactionis given by Cij = (C0+zC1)δi,j−C1

∑p δi,i+p which rep-

resents an effective general capacitance matrix, with thevector p running over the nearest neighbors. C0 denotesthe self-capacitance and C1 the mutual one (z being thecoordination number).

As mentioned before, to extract the phase diagram wemake use of the coarse graining approximation45. Themain idea of this approach is to introduce a Hubbard-Stratonovich auxiliary field which is conjugate to the av-erage of 〈eiφi〉 and which plays the role of an order pa-rameter for the transition from a superconducting to aninsulating state. Since the phase transition has a contin-uous character, one can expand the action in powers ofthe auxiliary field and determine the occurrence of phasecoherence by looking at the coefficients of the quadratic-term in the limit of long-wavelengths and zero frequency.

We briefly sketch the main steps of such an approxi-mation and adapt it to the present scenario of the BFM.The partition function for Sphase is given by the sum ofall the possible paths of the phase variables in the imag-

inary time and in the real space:

Z =

∫ ∏

i

Dφi exp

{∫ β

0

∑

i

− i

2

∂φi(τ)

∂τqi(τ)

+ EJ∑

ij

αij cos[φi(τ) − φj(τ)]

− 1

8

∑

ij

∂φi(τ)

∂τCij

∂φj(τ)

∂τ

To perform the Hubbard-Stratonovich transforma-tion, one rewrites the Josephson coupling term asEJ2

∑ij exp[iφi]αij exp[iφj ] and then, by using the usual

Gaussian identity, introduces an auxiliary field ψi(τ).The partition function Z then becomes:

Z =

∫ ∏

i

Dψ∗iDψiDφi exp

∫ β

0

dτ [(−2

EJ)∑

ij

ψ∗i α

−1ij ψj

exp

{∫ β

0

dτ

[∑

i

i

2

∂φi(τ)

∂τqi(τ) −

∑

i

(ψieiφi − ψ∗e−iφi)

−∑

i,j

1

8

∑

ij

∂φi(τ)

∂τCij

∂φj(τ)

∂τ

. (46)

Hence, starting from the effective action for the aux-iliary field, one can perform an expansion up to secondorder in ψ and thus derive the usual Ginzburg Landautype free energy functional which permits to determinethe boundary line between the superconducting and theinsulating state. The corresponding effective action forthese auxiliary fields ψ, ψ∗

Sψ = ln

[∫ ∏

i

Dφi exp

(∫ β

0

dτ

{∑

i

i

2

∂φi(τ)

∂τqi(τ)

−∑

i

(ψi(τ)eiφi(τ) − ψ∗

i (τ)e−iφi(τ))

−∑

i,j

1

8

∂φi(τ)

∂τCij

∂φj(τ)

∂τ

(47)

is then expanded to second order, giving:

Sψ =

∫ β

0

dτdτ′

χij(τ, τ′

)ψ∗i (τ)ψj(τ) +O(ψ4), (48)

where

χij(τ, τ′

) = 〈ei[φi(τ)−φj(τ)

′]〉0 (49)

denotes the two-time phase correlator, which is equal tothe second derivative with respect to the auxiliary fieldψ and its conjugate at different time and space positions,evaluated around their zero values. By performing thefunctional derivation, one gets the following expressionfor it:

10

χij(τ, τ′

) =

∫ ∏iDφie

i[φi(τ)−φj(τ)

′]exp[−S0]∫ ∏

iDφi exp[−S0](50)

with S0 being the part of the action which contains ex-clusively the charging contributions, i.e. :

S0 = exp

[∫ β

0

dτ∑

i

i

2

∂φi(τ)

∂τqi(τ) (51)

−∑

i,j

1

8

∂φi(τ)

∂τCij

∂φj(τ)

∂τ

.

According to the scheme outlined above, we now developthe partition function Z up to second order in the auxil-iary fields, thus putting it into a familiar form:

Z =

∫ ∏

i

Dψ∗iDψie

−Fψ (52)

with

Fψ =

∫ β

0

dτdτ′∑

ij

ψ∗i (τ)

[αijδ(τ − τ

′

) − χij(τ, τ′

)]ψi(τ

′

).

(53)The determination of the conditions for the boundary linebetween the superconducting and the insulating phasethen reduces to the explicit evaluation of χij(τ, τ

′

). Theresult for that has been first obtained in Ref. [48], andwe sketch below the main steps of this derivation.

In determining χij(τ, τ′

) it has turned out to be es-sential to treat the phase variable φi in a compact form.In order to separate the imaginary time evolution of thephase into a periodic part φi(τ) and an non-periodic part,one introduces the following parameterization:

φi(τ) = φi(τ) +2πniτ

β(54)

with φi(0) = φi(β) and ni being an integer which countshow many times the phase winds over an angle which isa multiple of 2π.

The use of such a relation allows to express the sumover all φi as an integration over φi plus a sum over all thepossible integer values of the winding number ni. Afterperforming a number of suitable algebraic operations46,one ends up with the following expression for the two-time phase correlator:

χij(τ) = δijexp[−2C−1

ii |τ |]∑

{ni}exp

[−∑ij 2βC−1

ij NiNj

] ×

∑

{ni}

exp

−∑

ij

2βC−1ij NiNj −

∑

k

4C−1ik Nkτ

,(55)

where Ni = qi/2 + ni. This two-time phase correlatoris local in space and its time dependence follows an ex-ponential behavior, if one assumes that the static off-

set charge qi has a distribution which is homogeneous

in space. In the general case, of a dynamic offset charge

qi(τ), this time dependence will be modified and can leadto qualitative changes in the nature of the SIT.

Having obtained the expression for the local two-timephase correlator in the time and space representation,we can now express the effective Ginzburg-Landau freeenergy functional in the Fourier space in the followingway:

Fψ =1

βL

∑

n,k

ψ∗k(ωn)

[α−1k − χk(ωn)]ψk(ωn)

]. (56)

Expanding the inverse matrix α−1ij in the form α−1

k =

(1/z)+ k2(a2/z2) + ..., this free-energy functional finallyis written as:

Fψ =1

βL

∑

k,n

[2

zEJ− χ0 + ak2 + bω2

n + ...

]|ψk(ωn)|2.

(57)

The transition line is then given by the condition thatthe coefficient of the quadratic term vanishes in the limitof vanishing k and ω, that is:

1 − zEJ2χ0(0) = 0 . (58)

For a quantitative analysis, one has to determine theexplicit zero frequency limit of the two-time phase cor-relator. As mentioned above, the inclusion of the timedependent offset-charge coming from the fluctuations ofthe bosonic density can modify the low frequency behav-ior of the local phase correlations.

In order to get a first insight into the underlyingphysics at play here we evaluate the two time phase cor-relator in the limit of a purely local capacitance (the socalled self-charging limit), by keeping only the on-sitepart in the original structure of the capacitance matrix.Under those conditions the evaluation of the two timephase correlator at finite frequency gives:

χii(ωn) =1

Z0

∑

{ni}

F [ni]

(1

C−1ii

−[iωn − 4C−1

ii ni])

(59)

with F [ni] = exp[−∑i 2βC−1ii n

2i ] and Z0 =

∑{ni}

F [ni].

With these expressions we finally can cast Eq. 58 inthe form

1 − zEJ4EC

∑n

exp[−4βECn2]

1−4n2∑m exp[−4ECβm2]

= 0 , (60)

expressed in terms of the Josephson coupling EJ andthe charging energy EC = C−1

ii /2, as given by theEqs.(37,40,41).

Given this defining equation for the boundary betweenthe superconducting and the insulating phase, we want

11

to see now how the intrinsic dependence of the Joseph-son and charging energy on the exchange coupling andthe density of bosons, manifests itself in the competitionbetween phase and boson density degrees of freedom. Inthe loop expansion given in the previous section for theBFM, we have obtained the amplitude of the intersitephase coupling and the charging effect as a function oftemperature, the ratio g/t and the bosonic density. Itis the non trivial dependence of those coupling constantson the parameters of the original microscopic Hamilto-nian of the BFM which leads to the interesting featuresin the competition between the phase and boson densityfluctuations. Thus, upon increasing the pair exchangecoupling it is expected that the Josephson coupling is re-duced while the charging energy is increased due to thelocal transfer between fermions pairs and bosons as wellas due to the single particle hopping of fermions betweennearest neighbor sites.

We determine the critical line for two different cases inorder to highlight the role played by, on the one hand,the coupling and, on the other hand, by the effectiveboson density. In Fig. 4, we illustrate the transitionline Tφ, separating a phase coherent state from a phasedisordered one, as a function of the coupling strengthand the effective boson density nB. The evolution ofthe transition line is non monotonic as a function of g/tand goes through a maximum at gmax ∼ t. The quantumcritical point, where the SIT occurs, is given by gcrit ∼ 2t.The critical behavior close to the transition is that of anXY model in d+1 dimensions.

More interesting still is the behavior of Tφ as a functionof the nB. The variation of Tφ is qualitatively differentfor the different parameter regimes: a) the weak couplingcase for g < gmax, b) the intermediate one with gmax <g < gcrit and c) the strong coupling limit for g > gcrit(see Fig. 5). Going from the limit a to c we find that

0.0 0.5 1.0 1.5 2.0 2.5 3.0g/t

0.0

0.1

0.2

0.3

0.4

0.5

T/t

SCI

FIG. 4: Phase diagram as a function of g/t describing theboundary between an insulating(I) and superconducting(SC)phase with long range phase coherence for the case of fullysymmetric limit (∆B = 0, nF↑ = nF↓ = 2nB = 1).

0.00 0.20 0.40 0.60 0.80 1.00 nB

0.00

0.10

0.20

0.30

0.40

T/t

SC

I

FIG. 5: Phase boundary lines between the superconduct-ing(SC) and insulating(I) state at different values of the cou-pling and upon varying the average density of bosons. Solid,dotted and dashed line stand for values of the coupling con-stants for which g < gmax, gcrit > g > gmax and g > gcrit,respectively.

with increasing nB the critical line decreases to zero, goesthrough a maximum starting at a finite T φ(nB = 1) andfinally shows a SIT at a critical density.

V. ROLE OF BERRY PHASE TERM: AN

INTRINSIC MAGNUS FORCE ON VORTICES

In the preceding section we have analyzed the basicphysics resulting from the effective action by neglecting,(i) the influence of any feedback between the density andamplitude fluctuations (included in the field l[θi]) on thephase dynamics, and (ii) the contribution from the Berryphase term, responsible for the correct quantization ofthe pseudospin variable, which is given by the integralover all the possible paths of is(1 − cos[θi])∂τφi. In thissection, we discuss the consequences of the presence ofsuch a Berry phase term in the case where the phaseaction has a vortex solution. The existence of a vortexsolution is assured for 2D systems where the phase cor-relations are described by a XY type dynamics. We shallshow here that the Berry phase term will produce an in-trinsic Magnus Force on the vortex which is analogous tothe Lorentz force for a charged particle, whose effectivemagnetic field depends on the spatial distribution of theθ field. Using the correspondence with the bosonic pseu-dospin variable, this implies a relation with the spatialdensity distribution of the bosons. The Magnus force hasbeen widely discussed in the context of normal BCS typesuperconductors where it has been shown that it arisesfrom the Berry phase caused by the adiabatic motion ofa vortex along a closed loop coming back to its startingposition48. The adiabatic vortex motion on a loop in thesuperconducting state turns out to be affected by an ef-fective magnetic field generated by the supercurrent aris-ing from the gradient of the phase which encircles such a

12

vortex. In the BFM scenario discussed here, we find thaton top of the usual contribution due to the superfluidelectrons, there is an intrinsic Berry phase term whichwill generate such a Magnus force and which has an in-tensity proportional to 2nB − 1, where nB is the bosonicdensity. This Magnus force, arising solely from the quan-tum nature of the pseudospin variable, is clearly indepen-dent on any superconducting state, and does not requirea coherent superfluid current induced by the presence ofthe vortex itself. Moreover, since nB can be controlledexternally, one has the possibility to tune the strengthof the effective magnetic field acting on the vortex, andhence to alter its dynamical properties. This will resultin a possible measurable effect on transport coefficients,such as the Hall coefficient, resistance, etc.

To be more explicit and following the procedures usedin different approaches treating with the Magnus forceproblem, let us assume that one has a vortex centeredat the position R. Let us furthermore consider thatwe are in the continuum and at zero temperature sothat the Berry phase term in the Lagrangian now readsLB =

∫d2rcos[θ(r, t)]∂tφ(r, t). The contributions linear

in ∂tφ(r, t) and having time independent coefficients donot contribute to the dynamics. Moreover, since we con-sider that the phase φ and the variable θ are linked to avortex solution centering at R, it is judicious to introducea new relative variable r −R, which defines the intrinsicposition dependence of the variables with respect to thevortex center.

Starting from the above given expression for the La-grangian LB and making a simple change in the timederivative we can rewrite it in the following form:

LB =

∫d2rcos[θ(r −R, t)]∂tφ(r −R, t) (61)

=dR

dt

∫d2rcos[θ(r −R, t)]∇rφ(r −R, t). (62)

At this point, one recognizes that the term∫d2rcos[θ(r − R, t)]∇rφ(r − R, t) plays the role of an

effective vector potential AB and dRdt represents the ve-

locity of the vortex. This Lagrangian is thus equivalentto that of a charged particle in presence of an effectivemagnetic field in the z-direction given by B = ∇×A. B

hence creates a Magnus force FM = −(dRdt × z)2πcos[θ0]which is transverse and proportional to the vortex ve-locity and whose strength is linked to the magnitude ofthis effective magnetic field Bz = −2πcos[θ0]. Its magni-tude is given by the asymptotic value of the backgroundbosonic density at large distance cos[θ0] for r → ∞.

In case of a 2D superconductor described by the BFMscenario, the usual effective magnetic field −2πρs aris-ing from the supercurrent of the electrons circling thevortex has thus to be supplemented by the above men-tioned contribution B (intrinsic to such a scenario) andpermits in principle to modulate the total Magnus forceupon changing the density of the bosons. Clearly, a fulldescription of such an eventuality, which is beyond thescope of the present analysis, has to be studied in more

detail and has to include the derivation of the effectiveaction in presence of magnetic field such as to determinethe superfluid density of the electrons ρs.

At this stage we simply want to point out that the signof the magnetic field arising from the Berry phase termin the action of the BFM changes if one tunes the bosonicdensity between zero and unity. In terms of the variablecos[θ] this implies a variation in the range [−1, 1]. Inother words, the sign of the intrinsic Berry phase inducedMagnus force will change at nB = 1/2 when going fromthe limit of small density of bosons to high density. Thisextra contribution to the Magnus force has to be addedof course to the conventional one known for standardsuperconductors.

VI. COMPARISON WITH NEGATIVE-U AND

BOSE-HUBBARD MODELS

A frequently asked question concerns the qualitativedifferences which exist between the BFM and similar sce-narios such as the negative-U Hubbard model and purebosonic systems such as the Bose-Hubbard model andJosephson junction arrays.

Let us start with a comparison of the BFM and thenegative-U Hubbard model which has been studied in agreat variety of different approaches and discussed espe-cially in connection with the BCS-BEC crossover. Themain issue of such a comparison is the occurrence or notof a quantum critical point in the negative-U Hubbardscenario at a finite value of the coupling U which intro-duces the pairing among the electrons. As we have shownin this paper, in the BFM there exists a critical value forthe exchange coupling g/t responsible of the pairing inthe fermionic subsystem above which the system under-goes a SIT. We have seen that within a path integralformalism, one can extract an effective action describingsuch a transition as being primarily due to the interplaybetween the local quantum pair exchange (between a bo-son and a fermion pair) and the non-local inter-site pairhopping leading to a phase dynamics originating from theitinerancy of the fermions. Due to the non-trivial depen-dence of the strength of these two competing mechanismson the microscopic parameters of the BFM, there occurs aSIT for a finite density as well as finite exchange couplingg/t. For the negative-U Hubbard scenario this is not thecase and the SIT does not occur at any finite value of theratio U/t49. This model merely describes a continuouscross-over between a BCS superconductor and a BEC oftightly bound electron pairs as U is increased from 0 to∞.

In order to better understand this difference betweenthe BFM and the negative-U Hubbard model, let us con-sider two scenarios in equivalent situations, i.e., the halffilled band case for the negative-U Hubbard model andthe fully symmetric case for the BFM (with the bosoniclevel lying in the middle of the fermionic band such thatboth, the fermionic band as well as the bosonic level are

13

half occupied). We then address the question how, in thestrong coupling limit, the pair hopping is generated outof the basic configurations of states and processes whichcontribute in this regime.

For the case of the negative-U Hubbard model withfermions interacting via a local attractive potential, theground state wave function in the limit of U → ∞ andat half-filling is highly degenerate and is composed of allthe possible configurations comprising equal distributionof zero and doubly occupied site. In this limit, one canperform a mapping of this model on the hard-core bosonmodel described by the Hamiltonian:

HU = −∑

〈ij〉

2t2ijUb+i bj +

∑

〈ij〉

2t2ijUninj − µ

∑

i

ni. (63)

b+i (bi) and ni = b+i bi stand for the creation (destruc-tion) operators of hard-core bosonic particles (tightlybound electron pairs) and for their density operator, re-spectively. Due to the presence of a coupling, which isisotropic both in the boson hopping and in the chargeinteracting channel, one has a superconducting state forany finite value of the ratio U/t, and possibly a super-solid phase due to the symmetry of the charging interac-tions characterized by a coexistence of diagonal and off-diagonal long range order. This implies that the quantumcritical point is strictly pushed to U → ∞.

In the BFM the phase space in the large g/t limit iscompletely different. In the fully symmetric case of thismodel (corresponding to a total density equal twice thenumber of sites (ntot = 2)), the ground state configura-tion is given by a wave function where all the bosons arestrongly coupled with pairs of fermions, thus resulting ina product state of local bonding states:

|ψ0〉 =∏

i

1√2(ρ+i + τ+

i )|0〉. (64)

This wave function can be viewed as a ferromagneticIsing-type state in the sense that it is made up of a ferro-type bonding order and is not degenerate with respectto all the other possible local configurations. Moreover,by construction it contains bonding-bonding correlationson a long range (bonding solid), but has zero phase cor-relation length. The latter can be shown by evaluatingthe static correlation function for the bosons or, equiv-alently, the fermion pairs which gives 〈ψ0|ρ+

i ρj |ψ0〉 = 0and 〈ψ0|τ+

i τj |ψ0〉 = 0 for any distance |i− j|.Next, let us consider the low energy configurations

which are mixed into such a g/t = ∞ ground state whenthe kinetic energy operator Ht =

∑i6=j, σ tij(c

+iσcjσ +

H.c.) is switched on. Applying Ht to |ψ0〉, there willoccur states which are separated by an energy gap withrespect to |ψ0〉. The low energy states to be consid-ered as the relevant quantum mixed configurations areof non-bonding nature, such as C+

iσ|0〉 = c+iσ|0〉 andS+iσ|0〉 = ρ+

i c+iσ|0〉. In order to construct those states, let

us to begin with consider a local excitation on two ad-jacent sites i, j given by S+

iσC+j−σ|0〉 in the background

of bonding states. This configuration can be seen as aferro-type order interrupted by two domain walls of non-bonding/bonding nature. Now let us consider the dy-namics of such objects and under which conditions theycan be rendered itinerant in a way which leaves the num-ber of bonding configurations unchanged. The problemis thus analogous to that of domain wall dynamics in anIsing type systems. It so happens (a detailed discussionof this is beyond the scope of the present study and willbe given at a later stage elsewhere), that the local degen-eracy of the non-bonding states will be removed by theaction of the kinetic term. In this way it induces a globallowering of their excitation energy due to the dispersionof each domain wall which is of the form

EDW (k) =1

2[ǫk + 6g ±

√(ǫk − 2g)2 + 4t2] (65)

and where ε(k) = −2ztcosk. Only after the condensationof the domain wall like excitations in the presence of thebonding state background one can meet the conditionsfor setting up long range boson and fermion-pair phasecorrelations .

This simple sketch of the nature of the excitations inthe BFM implies that the onset of phase coherence can-not be activated for an arbitrarily small hopping am-plitude since one has to overcome the energy gap be-tween the ground state and the manifold of non-bondingstates, which is of the order of ∼ 2g. As we can see fromthe expression for the dispersion of the domain walls,this is achieved when EDW (k) becomes zero for k = 0,

i.e., when g/t = 14 (2 +

√2 + 4z2). For z=2 this gives

g/t ∼ 2.0, which determines when those defects-like ex-citations become gapless and thus induce a proliferationwithin the background of the bonding states. It is worthpointing out that this value for the critical exchange cou-pling matches rather well the value obtained from ourfunctional integral approach discussed in this paper (seeFig 3).

The picture which hence emerges for the BFM is verysimilar to that of a XY model in a transverse field. There,the hopping term is responsible of the XY dynamics,while the local boson-fermion pair exchange provides therole of the effective transverse field. This itself is alreadya strong indication that this model is of different natureto that of the negative-U Hubbard scenario, for whichwe know that it is, on the contrary, akin to an isotropicHeisenberg model.

A further essential difference between the BFM and thenegative-U Hubbard model can be seen from their respec-tive path-integral formulations. When the path integralrepresentation is considered for this model50, the firststep is to make use of a Hubbard-Stratonovich transfor-mation to rewrite the quartic interaction term in a bilin-ear form, where the fermionic operators are now coupledto random auxiliary fields. Limiting oneself to purelysuperconducting order, the usual procedure for manipu-lating such a bilinear action is to separate the complexauxiliary field into its modulus and a pure phase part

14

before performing an expansion around the saddle pointsolution for the amplitude of this auxiliary field. Thisway of proceeding allows then to extract an effective ac-tion for the slow phase dynamics. This is distinctly dif-ferent from the functional integral representation for theBFM which we have presented above and where fromthe very beginning and throughout such a procedure thefermionic pair fields are coupled to physically real bosonicmodes which have their own proper dynamics. As wehave seen in section III, the bosonic part of the actionis treated within a coherent pseudo-spin representationwhere the dynamics of the pseudospin is parameterizedby the time dependence of the spherical variables. Therole of the dynamics within the spherical representationof the bosonic field and of the Berry phase term (whichis a consequence of the quantum interference in the localpseudospin space) is a distinct feature of this BFM andpresents specific differences with respect to the negative-U model. Such differences imply in particular that in theBFM case one has feedback effects between the ampli-tude and the phase fluctuations which are totally absentin an equivalent description of the negative-U Hubbardmodel. Such features are important because of intrinsicdissipation effects that eventually can change the natureof the transition and possibly are relevant for the emer-gence of a ”bosonic metal” in proximity of the SIT, afeature which seems to be outside the framework of thenegative-U Hubbard model.

Let us conclude this comparison of the BFM with sim-ilar models with a brief discussion on the Bose-Hubbardmodel and Josephson junction arrays. The main aspectwhich emerges as a common denominator in all thosemodels, at least as far as the phenomenon of the SITis concerned, is that the mechanism which is responsi-ble for the degrading of the phase coherence is analogousand originates from the competition between the phaseand charge degrees of freedom. In the Bose-Hubbardmodel this manifests itself as a competition between theboson hopping and their charge repulsion, while in theJosephson junction array scenario it appears as an in-terplay between the Josephson tunnelling amplitude andthe charging energy.

In spite of this, at first sight, similarity between theBFM and those scenarios, we would like to stress that ina system where the dynamics is described by the BFMscenario, the interplay between the phase and amplitudefluctuations is intrinsically related to the coupling be-tween the fermionic and bosonic degrees of freedom. Thisintroduces amplitudes of the different processes at workin form of a non-trivial dependence on the microscopicparameters of the starting Hamiltonian. Furthermore,as we have seen in the above discussion of the effectiveaction of the BFM, this model contains features whichgo beyond those which characterize the pure Josephson-type dynamics and which arise from the peculiar feedbackbetween the fluctuations of the bosonic density (or am-plitude pair field) and fluctuations of the phase. Last notleast, the appearance of an intrinsic Berry phase-term in

the BFM scenario together with the effect of dissipationdue to the fermionic dynamics, can give rise to an uncon-ventional phenomenology when topological phases play arole, and especially in the presence of vortices.

VII. CONCLUSION

In this paper we examined the nature of a superconduc-tor - insulator transition in a system of localized bosonsand itinerant fermions coupled together via a pair ex-change term. An effective action was derived from sucha microscopic model, which, after integrating out thefermionic fields, could be phrased in terms of amplitudeand phase fluctuations of the bosons. In order to makethe presentation more familiar we discussed the action ina terminology frequently employed in connection with thestudy of Josephson junction arrays. We stress that oursystem does not necessarily imply any charged fermionsand bosons.

Considering the phase-only part of the effective actionit is fully equivalent to the quantum phase model for theJosephson junction arrays, discussed in terms of: (i) aJosephson coupling term, (ii) a charging or capacitance

term and (iii) an offset charge term. Equivalent to thatin our scenario is: (i) a boson hopping term, (ii) a termwhich takes into account the reduction in hopping am-plitude due to a fluctuating local boson density arisingfrom the intrinsic on-site exchange coupling between thebosons and the fermions and (iii) a chemical potentialterm controlling the bosonic concentration. New in thepresent study within already the lowest (phase-only) ap-proach to the boson-fermion system is the intricacy ofthe dependence of the effective Josephson coupling, thecapacitance term and the off-set charge term on the pa-rameters of the initial Hamiltonian, i.e., the exchangecoupling and the density of bosons (or else the total par-ticle density). It turns out that within the lowest (phase-only) contribute a superconductor to insulator transitioncan not only be triggered by a change in the exchangecoupling but also by a variation of the boson density.The latter presents evidently a certain interest from theexperimental point of view and can possibly be tested insuch transition occurring in optical lattices for ultracoldfermi gases with Feshbach resonance pairing.

Apart from the phase-only part of the effective ac-tion we established the existence of an intrinsic Berryphase term, which arises from the hard core nature ofthe bosons and gives rise to an additional Magnus forcewhen the system is such that topological ground stateconfigurations like vortices are stabilized. This again isof potential interest for experiments since in principle onecan change the sign of the Magnus force upon changingthe density of bosons and thus deviating the motion of avortex from one direction into the opposite one. Thesevery preliminary results will be dealt with in greater de-tail in some future studies.

A further property of the Berry phase term is that it

15

gives rise to a bilinear coupling term between the phaseand amplitude fluctuations, and is hence much more di-rect and relevant than similar terms in, for instance, sce-narios based on the negative-U Hubbard model wherethey occur only at a much higher order in a correspond-ing loop expansion of the trace in the effective action.This again merits to be investigated in some detail withthe aim to study the dissipation introduced by such am-plitude phase coupling and its effect on the nature ofthe transition, in view of exploring the possibility of anintermediary bosonic metallic ground state.

Finally, we have dealt with a frequently asked questionconcerning the differences between the negative-U Hub-bard scenario - mainly studied in connection with theBCS-BEC cross-over and the presently studied boson-fermion model. Far from being able to give a completeaccount for the major differences we found mainly twoaspects which distinguish the physics of these models ona qualitative and robust level. The one is that in thenegative-U Hubbard model a superconductor - insulator

transition can not take place at any finite coupling U, norcan such a transition be triggered by the change in par-ticle concentration. The second point is that the groundstates in the strong coupling limit of the two models arequite different: a highly degenerate ground state for thenegative-U Hubbard model with excitations being con-trolled by an isotropic Heisenberg model when the hop-ping term is switched on. Contrary to that, for the bosonfermion model the ground state is non-degenerate (cor-responding to a ferro pseudo-magnetic Ising type systemof singlets formed by bosons and fermion pairs), which,after switching on the fermion hopping term, gives riseto propagating domain like structures. The topologicalstructures appearing in the ground state might have mea-surable consequences in the transport properties near thesuperconductor-insulator transition.

This and the other preliminary studies mentionedabove will require a detailed analysis and will be dis-cussed in future.

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