Quantum Field Theory for the Three-Body Constrained Lattice Bose Gas -- Part I: Formal Developments
QQuantum Field Theory for the Three-Body Constrained Lattice Bose GasPart I: Formal Developments
S. Diehl,
1, 2
M. A. Baranov,
1, 2, 3
A. J. Daley,
1, 2 and P. Zoller
1, 2 Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria RRC “Kurchatov Institute”, Kurchatov Square 1, 123182 Moscow, Russia
We develop a quantum field theoretical framework to analytically study the three-body con-strained Bose-Hubbard model beyond mean field and non-interacting spin wave approximations. Itis based on an exact mapping of the constrained model to a theory with two coupled bosonic degreesof freedom with polynomial interactions, which have a natural interpretation as single particles andtwo-particle states. The procedure can be seen as a proper quantization of the Gutzwiller meanfield theory. The theory is conveniently evaluated in the framework of the quantum effective ac-tion, for which the usual symmetry principles are now supplemented with a “constraint principle”operative on short distances. We test the theory via investigation of scattering properties of fewparticles in the limit of vanishing density, and we address the complementary problem in the limitof maximum filling, where the low lying excitations are holes and di-holes on top of the constraintinduced insulator. This is the first of a sequence of two papers. The application of the formalism tothe many-body problem, which can be realized with atoms in optical lattices with strong three-bodyloss, is performed in a related work [14].
PACS numbers: 03.70.+k,11.15.Me,67.85.-d,67.85.Hj
I. INTRODUCTION
Lattice theories with constrained bosons have provento be a powerful description of various spin models andstrongly correlated systems in condensed matter physics[1]. On the other hand, such theories with constrainedlattice bosons have recently arisen naturally in effectivemodels for experiments with cold atoms in optical lat-tices. In the presence of large two-body and three-bodyloss processes, bosons in an optical lattice are describedon short timescales by a model with two-body and three-body constraints respectively [2, 3]. The behavior of theBose gas is changed drastically; for example, in the caseof the three-body constraint, the creation of an attrac-tive Bose gas with atomic (ASF) and dimer superfluid(DSF) phases is possible [3]. While the possibility of suchan ASF-DSF transition has been predicted earlier in thecontext of continuum attractive Bose gases near Feshbachresonances [4, 5], the constrained lattice system offers anintrinsic stabilization mechanism to observe such a phe-nomenology in experiments. This serves as one motiva-tion to study such models theoretically in more detail, inparticular exploring the consequences of the presence ofthe constraint. We also note that by the same dissipativeblockade mechanism, constrained models with fermionsmay be created [6, 7].Here our goal is to describe the physics of a constrainedboson system beyond a mean-field plus spin wave ap-proach (see e.g. [1]). In order to do this, we find an ex-act mapping of the original constrained bosonic Hubbardmodel to a theory of two coupled unconstrained bosonicdegrees of freedom which interact polynomially. The re-sulting theory is conveniently analyzed in the frameworkof the quantum effective action, which makes it possibleto study both thermodynamical and dynamical proper- ties of the system via various many-body techniques. Asa consequence, we can demonstrate several remarkablefeatures of the three-body constrained attractive Boselattice gas, which are uniquely tied to this constraint andnot treated properly within a simple mean field plus spinwave approach. In particular, we show the emergence ofan Ising quantum critical point on the phase transitionline between atomic to dimer superfluid phases, whichgenerically is preempted by the Coleman-Weinberg mech-anism [8] where quantum fluctuations drive the phasetransition first order, rendering the correlation length fi-nite [4, 5, 9–11]. We also show the presence of a bicriticalpoint [12] in the strongly correlated regime at unit fillingof bosons, which is characterized by energetically degen-erate orders. In our case this corresponds to the coex-istence of superfluidity and a charge density wave. Fur-thermore, quantitative effects of quantum fluctuations onthe position of the phase boundary can be investigatedsystematically.The formalism we develop here has much broader ap-plications than the bosonic lattice gas with a three-bodyconstraint. In particular, it could be used to treat sys-tems with effective constraints arising from large interac-tion parameters, and it is also applicable to constrainedfermionic models, which could arise due to strong lossfeatures in 3-component fermion gases [7, 13]. As a re-sult, we focus in this paper on presenting the quantumfield theoretical construction in detail, and give bench-mark calculations for this method. Its application to themany-body attractive lattice Bose gas with three-bodyconstraint is left to [14], where we discuss in detail theresults outlined above. The results of the present and therelated paper [14] are summarized in [15], where we alsoindicate how to probe our findings in experiments withultracold atoms. a r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug This paper is organized as follows. We begin by re-viewing the microscopic derivation of the three-body con-strained model via a dissipative blockade mechanism inSec. II. In Sec. III we give an overview of our formalismand a comparison to existing methods for treating con-strained models. Sec. IV contains the central result ofthis paper, the mapping of the constrained model to acoupled boson theory already anticipated above. In Sec.V we apply the formalism to the two limits where truemany-body effects are absent, n = 0 and maximum filling n = 2. There our boson model reduces to Feshbach-typemodels, which we analyze in terms of Dyson-Schwingerequations [16]. We perform the above mentioned bench-mark calculations at n = 0. Furthermore, we investi-gate hole/di-hole scattering and bound state formationat n = 2, and present a fourth order perturbative cal-culation for the dimer-dimer interaction. This will beneeded as an input for the many-body theory in [14].Our conclusions are drawn in Sec. VI. II. DERIVATION OF THE CONSTRAINEDMICROSCOPIC MODEL
In this section, we review how the three-body hardcoreconstraint for the Bose-Hubbard model emerges fromstrong three-body loss [3].We consider bosons in the lowest Bloch band of anoptical lattice, which are described by the Bose-HubbardHamiltonian ( (cid:126) ≡ H BH = − J (cid:88) (cid:104) i,j (cid:105) a † i a j − µ (cid:88) i a † i a i + U (cid:88) i a † i a i , (1)where a i ( a † i ) is the bosonic annihilation (creation) opera-tor at site i . J is the hopping rate, µ the chemical poten-tial, and U the onsite interaction. The convention (cid:104) i, j (cid:105) first sums over all sites i , and then over the neighbour-hood of each i spanned by sites j , (cid:80) (cid:104) i,j (cid:105) = (cid:80) i (cid:80) (cid:104) j | i (cid:105) .This model is valid in the limit where J, U n (cid:28) ω , where ω is the separation between Bloch bands and n is themean density.Three-body loss in this system is due to inelastic colli-sions of three atoms, two of which form a deeply boundmolecule. Together with the third atom, molecule for-mation is compatible with energy and momentum con-servation, unlike the case of two-particle collisions. Sincethe binding energy of the molecule typically strongly ex-ceeds optical lattice depths, the resulting kinetic energyof the products couples them to the continuum of un-bound states, thus leading to their escape from the lat-tice. This picture allows us to write down a zero tem-perature master equation in the Markov approximation,which in the simplest approximation neglects loss arisingfrom particles on neighbouring lattice sites. This master equation is given by˙ ρ = − i (cid:16) H eff ρ − ρH † eff (cid:17) + γ (cid:88) i a i ρa † i , (2) H eff = H − i γ (cid:88) i a † i a i , where the decay rate γ can be roughly estimated fromthe experimentally measured continuum loss rate via theusual Wannier construction [3]. We have absorbed theSchr¨odinger type terms of the dissipative evolution intoan effective Hamiltonian with imaginary, and thus decay,term. The remaining recyling term couples sectors inthe density matrix with particle number N, N − , ... andensures a norm conserving time evolution.We are now interested in the limit of strong loss, γ (cid:29) J, U , which suggests a perturbative expansion in1 /γ . The scale γ only couples states with three andmore particles per site. We therefore define a projector P onto the subspace with at most 2 atoms per site, andits complement as Q = 1 − P . To second order in pertur-bation theory we then find the projected Hamiltonian H P, eff ≈ P H BH P + P H BH Q ( QH BH Q ) − QH BH P = P H BH P − iΓ2 (cid:88) i P c † i c i P, (3)and the corresponding Master Equation is˙ ρ P = − i (cid:16) H P, eff ρ P − ρ P H † P, eff (cid:17) + Γ (cid:88) i P c i ρ P c † i P, (4)with ρ P = P ρP . The effective decay rate and jumpoperators are given byΓ ≈ J γ , c i = a i (cid:88) (cid:104) j | i (cid:105) a j / √ . (5)The respective terms in H P, eff have simple interpreta-tions: The leading term describes the coherent dynamicsof lattice bosons, but with the constraint of not popu-lating a single site with more than two particles. Thisprojected part H = P H BH P can thus be written as a3-body constrained Bose-Hubbard Hamiltonian H = − J (cid:88) (cid:104) i,j (cid:105) a † i a j − µ (cid:88) i a † i a i + U (cid:88) i a † i a i ,a † i ≡ . (6)The leading correction is imaginary and describes parti-cle number loss. The decay rate Γ ∼ /γ is, however,very small in the considered limit. Consequently, overtimescales τ = 1 / Γ, one realizes indeed the physics of theBose-Hubbard model with 3-body hardcore constraint.In [3], we have shown that e.g. in atomic Cesium systemsclose to the zero crossing of the scattering length, the lossrate γ is the dominant energy scale. There, we have alsoanalyzed the many-body dissipative dynamics of Eq. (2)in the regime described by Eq. (3) with exact DMRGmethods in one spatial dimension, including the specifi-cation of a scheme with which the ground state of P HP can be reached. This motivates a more detailed analyti-cal investigation of the zero temperature phase diagramof Eq. (6). As mentioned in the introduction, such a sce-nario with dominant three-body loss also arises naturallyin three-component fermion systems close to a Feshbachresonances [13], where it arises due to the proximity torapidly decaying Efimov state.
III. OVERVIEW
In this section, we provide an overview and discussionof our construction and compare it with existing theoret-ical approaches.The starting point is a truncation of the onsite bosonicHilbert space to three states corresponding to zero, sin-gle and double occupancy. Following Altman and Auer-bach [24], we introduce three operators creating thesestates out of an auxiliary vacuum state. The operatorsare not independent but obey a holonomic constraint,such that one of the degrees of freedom can be elim-inated. We propose a resolution of this constraint ina fashion that (i) produces polynomial interactions be-tween the two remaining operators and (ii) allows us tointerpret them as standard bosonic degrees of freedom.The Hamiltonian written in terms of these bosonic oper-ators is an involution on the physical Hilbert space (as-sociated to onsite occupation ≤ > n (cid:38) n (cid:46) U . Wemay view this result as a built-in Hubbard-Stratonovichtransformation on the level of the Hamiltonian, whichimportantly respects the constraint. At this point, wenote an important difference of our approach to the oneby Dyson and Maleev [17], which map a spin model to amodel with a single bosonic degree of freedom. The gen-eralization of a Hubbard-Stratonovich transformation tosuch a model would be problematic.Other related approaches to constrained models havebeen put forward in the context of slave boson theories forthe description of strongly correlated fermions [18, 19], orin the Schwinger-boson or Holstein-Primakoff approachto spin models [1]. While these formulations are in princi-ple exact, the practical implementation of the constraintis done on an approximate level only [19], or in a fashionthat makes it difficult to assess coupling constants of aneffective low energy theory or thermodynamic quantities[18]. Our approach is tailored to make these accessible.Furthermore, we stress that our exact implementation ofthe constraint differs from the conventional approximatetreatment via expansion of the square root as done e.g.for the case of large spin S in the Schwinger-boson orHolstein-Primakoff approaches [1]. We provide concretebenchmarks for our procedure through the analysis of thevacuum problem of a few scattering particles. While ourapproach yields the correct non-perturbative Schr¨odingerequation for two-particle scattering and the right coeffi-cient for an induced nearest-neighbour dimer-dimer inter-action, the square root expansion would produce wrongprefactors.Clearly, going beyond the mean field approach inconstrained bosonic models is nowadays possible mak-ing use of powerful numerical techniques such as Quan-tum Monte Carlo [20–22] or variational simulations [23].However, often a more analytical understanding of thephysics leading to a certain quantitative effect is desir-able. Furthermore, as we have mentioned above, our for-malism can be extended to fermion systems, which is notstraightforward with numerically exact techniques. IV. MAPPING THE CONSTRAINED MODELTO AN INTERACTING BOSON THEORY
In this section we derive the mapping of the bosonicmodel with three-body hardcore constraint Eq. (6) toan unconstrained model with two bosonic degrees of free-dom, coupled via polynomial interactions. This construc-tion depends on the filling of the lattice. We concentrateon the “vacuum limit” of zero density and temperaturefirst, where one deals with a few particles and many-bodyeffects, such as spontaneous symmetry breaking, are ab-sent. The generalization to arbitrary filling 0 ≤ n ≤ A. Reduction to three on-site states
We start from the 3-body constrained Bose-HubbardHamiltonian Eq. (6). The constraint makes it possibleto restrict the Hilbert space on each site to three stateswith occupation 0,1,2. We introduce three operators thatcreate ”particles” in these states. Such a slave-boson typeprocedure has been proposed previously by Altman et al. [24] in the context of the repulsive Bose-Hubbard model,where it constitutes an approximation. Here this step isexact: | α (cid:105) = t † α,i | vac (cid:105) = 1 √ α ! a † αi | vac (cid:105) , α = 0 , , . (7)The operators t α,i are so far only defined by their actionof the “vacuum” state | vac (cid:105) and we discuss their commu-ation relations below. They obey a holonomic constraint, (cid:88) α =0 t † α,i t α,i = . (8) In the space spanned by | (cid:105) , | (cid:105) , | (cid:105) we may express thecreation operator as a † i = √ t † ,i t ,i + t † ,i t ,i . (9)Hence we can express the original Hamiltonian equiv-alently in terms of the new operators, defining non-hermitian kinetic operators K (10) i = t † ,i t ,i , K (21) i = t † ,i t ,i , (10)and hermitian potential energy operatorsˆ n ,i = t † ,i t ,i , ˆ n ,i = t † ,i t ,i , (11)we obtain H kin = − J (cid:88) (cid:104) i,j (cid:105) (cid:2) K (10) i K (10) † j + 2 K (21) † i K (21) j (12)+ √ K (21) i K (10) † j + K (10) i K (21) † j ) (cid:3) ,H pot = − µ (cid:88) i ˆ n i + U (cid:88) i ˆ n ,i , ˆ n i = ˆ n ,i + 2ˆ n ,i . We observe that the local terms become simple(quadratic) in this representation, while the nonlocalterms are fourth order in the operators, giving rise to“kinematic” interactions. The onsite interaction parttherefore can be treated exactly, while the complexity ofthe problem is now encoded in the hopping term. This isreminiscent of the conventional Gutzwiller (mean field)approach to the Bose-Hubbard model. Indeed, imple-menting a Gross-Pitaevski-type mean field theory for theabove Hamiltonian by formally replacing the operatorsby complex valued amplitudes t α,i → f α,i , the aboveHamiltonian operator reduces to the Gutzwiller energyexpression E GW = (cid:104) ψ | H | ψ (cid:105) for the case that the wavefunction | ψ (cid:105) = (cid:81) i | ψ (cid:105) i is truncated to the three low-est Fock states on each site, | ψ (cid:105) i = (cid:80) α =0 f α,i | α (cid:105) i . Inthis case, the holonomic constraint reduces to the nor-malization condition for the wave function, i (cid:104) ψ | ψ (cid:105) i = (cid:80) α f ∗ α,i f α,i = 1 ∀ i .It may be tempting to try to develop a many-bodytheory including the description of spontaneous symme-try breaking at finite density for the above Hamiltoniandirectly in terms of the t α operators in the sense of aBogoliubov-type theory on top of the Gross-Pitaevskimean field, via a replacement of the type t α,i = f α,i + δt α,i familiar from low density continuum theories. Such aprocedure, however, leads to severe consistency problemswhen encompassing the full range of densities allowed bythe three-body hardcore constraint, 0 ≤ n ≤
2. There-fore, we first focus on the “vacuum limit” n = 0 whereno spontaneous symmetry breaking is present, and wherethe Hamiltonian (12) describes the physics of a few scat-tering particles. The generalization to arbitrary densityis performed in Sec. IV C. B. Implementation of the Holonomic Constraint:Interacting Boson Theory
The holonomic constraint is now used to eliminate oneof the operators. In the limit n = 0, all the amplituderesides in the zero-fold occupied state, and the mean fieldvacuum is described by | Ω (cid:105) = (cid:81) i t † ,i | vac (cid:105) . We thus elim-inate the operators t ,i . The remaining two operatorsdescribe excitations on top of this mean field vacuum,and will have a natural interpretation in terms of atomsand dimers. At this point, it is transparent that ourconstruction builds on the proper choice of the qualita-tive features of the physical vacuum. This prerequisitesa certain understanding of the physics, and introducesa bias in our construction. The subsequent construc-tion is however exact, and can be used to quantitatively calculate properties of the system. The situation is actu-ally similar to the treatment of interacting Bose gases inthe continuum, where a condensation of the particles inthe zero mode is assumed to expand in the fluctuationsaround this mean field.The holonomic constraint can, in principle, be imple-mented by the replacements t † ,i t ,i → t † ,i e i ϕ i (cid:112) − ˆ n ,i − ˆ n ,i , (13) t † ,i t ,i → (cid:112) − ˆ n ,i − ˆ n ,i e − i ϕ i t ,i . The holonomic constraint is local and only restrictsthe amplitude of t ,i , therefore we introduce a phase-amplitude representation | t ,i | exp i ϕ i . We first discussthe role of this phase and then turn to the more interest-ing question of the amplitude.Inserting the replacement into Eq. (12), we observethat a local redefinition of t and t as t ,i → ˜ t ,i = t ,i e i ϕ i , t ,i → ˜ t ,i = t ,i e i ϕ i (14)for all i completely removes the phase from the Hamilto-nian. Therefore, we can work in this rotated frame fromthe outset, and simply consider the two complex valuedoperators ˜ t ,i . We will drop the tilde in the following.Now we study the amplitude. Obviously the squareroots are impracticable for any field theory calculationwhere one has to work with polynomials in the field op-erators. However, on our subspace, the matrix elementsof both (1 − ˆ n ,j − ˆ n ,j ) / and (1 − ˆ n ,j − ˆ n ,j ) are thesame: either 1 or 0. Consequently, on the subspace wemay replace t † ,i t ,i → t † ,i X i , t † ,i t ,i → X i t ,i , (15) X i = 1 − ˆ n ,i − ˆ n ,i . Note that the second expression could also be replacedby (1 − ˆ n ,i − ˆ n ,i ) / t ,i → t ,i , since the action of thisoperator is nonzero only if originally there is an atom onthe i -th site, but when this atom is annihilated by t ,i with an empty state left, the action of the square root issimply unity. However, for hermeticity issues we preferto work with the variant in Eq. (15). Formally, we may justify the replacement of the squareroot by a polynomial by the following formula: Considera linear operator ˜ X with the property ˜ X = ˜ X . Then fora function f of this operator one has, using the Taylorrepresentation, f ( ˜ X ) = ∞ (cid:88) n =0 f ( n ) (0) n ! ˜ X n = f (0) + ˜ X ∞ (cid:88) n =1 f ( n ) (0) n ! 1= f (0)(1 − ˜ X ) + ˜ X ∞ (cid:88) n =0 f ( n ) (0) n ! 1= f (0)(1 − ˜ X ) + f (1) ˜ X. (16)In our case, we have ˜ X = 1 − X i = ˆ n ,i + ˆ n ,i for all i , and f ( ˜ X ) = (cid:112) − ˜ X . Indeed, ˜ X = ˜ X . Seen as a function of X i = 1 − ˜ X , we have f ( X i ) = √ X i = f (0) X i + f (1)(1 − X i ). Since also X i = X i the latter result would havebeen obtained from the Taylor representation of f ( X i )around 0. The auxiliary operator ˜ X is introduced tocircumvent an expansion of the square root around 0,but leads to the same result. No approximation has beenused here.Having implemented the constraint, we are now goingto show that the remaining operators t , t can be treatedas standard bosonic operators . Consequently the Hamil-tonian (or the corresponding action) with the above re-placements will lend itself for a treatment with well es-tablished field theoretic methods. To show that we mayinterpret t , t as bosonic operators, we assume a bosonicHilbert space for atoms t and dimers t at each site i ,which reads H i = {| n i (cid:105)| m i (cid:105)} , n i , m i = 0 , , ... . The com-plete Hilbert space is H = (cid:81) i H i . We divide the onsiteHilbert spaces into a physical subspace P i and an unphys-ical one U i , H i = P i ⊕ U i ; the subspaces are orthogonalby construction. The physical subspace is spanned bythe combinations P i = {| i (cid:105)| i (cid:105) , | i (cid:105)| i (cid:105) , | i (cid:105)| i (cid:105)} . (17)In our construction of introducing atom and dimer oper-ators, the state with two atoms on one site is representedas one dimer | i (cid:105)| i (cid:105) ; | i (cid:105)| i (cid:105) instead is already part of U i .A first observation is that standard bosonic operatorshave the same action on the physical subspace as the orig-inal operators t , t defined via Eq. (7), since the bosonic √ n -enhancement ( b † | n (cid:105) = √ n + 1 | n + 1 (cid:105) for bosonic op-erators b † ) is either 0 or 1 on the physical subspace. Theassumption of t , t being bosons is thus consistent onthe physical subspace.Next we consider the Hamiltonian Eq. (12) with theconstraint implemented via Eq. (15), with the goal toshow that the time evolution generated by this Hamilto-nian does not couple the two subspaces. It reads H kin ≡ (cid:88) (cid:104) i,j (cid:105) H i,j = − J (cid:88) (cid:104) i,j (cid:105) (cid:2) t † ,i X i X j t ,j + 2 t † ,i t ,j t † ,j t ,i + √ t † ,i t ,i X j t ,j + t † ,i X i t † ,j t ,j ) (cid:3) ,H pot = (cid:88) i ( U − µ )ˆ n ,i − µ ˆ n ,i , (18)In the kinetic term, the first expression describes the con-ditional hopping of single atoms, the second representsthe exchange of a dimer and an atom on neighbouringsites, and the last one describes the conditional bilocalsplitting and recombination of a dimer into atoms. Itcan be easily shown that H maps physical on physicalstates, and unphysical on unphysical ones, while thereare no transitions between the subspaces generated by(18). For that purpose it is sufficient to check that H i,j | n i (cid:105)| m i (cid:105)| n j (cid:105)| m j (cid:105) (19)is in P i P j if the initial state is in P i P j (a simple 9 dimen-sional space). Since this means that all matrix elements (cid:104) u | H ij | p (cid:105) = 0, this is sufficient to conclude that startingin U the mapping will be into U , since for the hermitian H one has (cid:104) p | H ij | u (cid:105) = (cid:104) u | H ij | p (cid:105) ∗ = 0. In consequence H can be written in the form H = H P ⊗ U + P ⊗ H U . (20)In other words, P = (cid:81) i P i and U = (cid:81) i U i are invariantsunder application of H , and thus also repeated applica-tion does not lead out of the subspaces. Therefore, wealso haveexp( − βH ) = exp( − βH P ) ⊗ U + P ⊗ exp( − βH U ) . (21)The partition sum is the given by Z = Tr exp( − βH ) (22)= (cid:88) { p,u } ( (cid:104) p | , (cid:104) u | ) (cid:18) exp( − βH P ) 00 exp( − βH U ) (cid:19) (cid:18) | p (cid:105)| u (cid:105) (cid:19) = (cid:88) { p } (cid:104) p | exp( − βH P ) | p (cid:105) + (cid:88) { u } (cid:104) u | exp( − βH U ) | u (cid:105) . Thus we get contributions from both the physical andthe unphysical part of the Hilbert space. However, thekey point is that they do not mix; thus the answers foundfor the physical part will be correct, and we only need tofind the criterion to discriminate the physical from theunphysical part of the partition sum.This issue is addressed in the last step of the construc-tion. Indeed, such a setting is provided by using the effective action to encode the physical information of thetheory, as we will now outline. First we represent the par-tition function as a Euclidean functional integral, which is straightforward as we simply have to quantize a theorywith two coupled bosonic degrees of freedom [25]: Z = (cid:90) D t D t exp − S [ t , t ] , (23) S = (cid:90) dτ (cid:16) (cid:88) i t † ,i ∂ τ t ,i + t † ,i ∂ τ t ,i + H [ t , t ] (cid:17) , where H is the Hamiltonian above, however to be inter-preted in the Heisenberg picture with (imaginary) timedependent fields, and the fields are now classical fluctu-ating variables. S is the classical Euclidean action. Inthe next step we introduce a source term in the partitionfunction, Z [ j , j ] = (cid:90) D t D t exp − S [ t , t ] (24)+ (cid:90) dτ (cid:88) i ( j † ,i t ,i + j † ,i t ,i + c.c. ) ,Z = Z [ j = j = 0] . The source terms introduce linear terms in t , , whichmix the physical and the unphysical sectors. Since j , is only used in a pivotal sense to generate the correla-tion functions upon functional differentiation, and set tozero at the end of the calculation, this does not pose anyconceptual problems. The situation is analogous to theeffect of the source term on the symmetries of the theory,which are broken explicitly for nonzero sources.The effective action is defined as the Legendre trans-form of the free energy W [ j ] = log Z [ j ] (we introduce theshorthands ˆ χ = ( t , t † , t , t † ) , j = ( j , j † , j , j † ) [26]:Γ[ χ ] = − W [ j ] + (cid:90) j T χ, χ ≡ δW [ j ] δj , (25)where χ = (cid:104) ˆ χ (cid:105) is the field expectation value or the “clas-sical” field. By the Legendre transform, the active vari-able is changed from j to χ . The effective action has thefollowing representation in terms of a functional integral,exp − Γ[ χ ] = (cid:90) D δχ exp − S [ χ + δχ ] + (cid:90) j T δχ,j = δ Γ[ χ ] δχ , (26)where δχ ≡ ˆ χ − χ . The last identity is the full quan-tum equation of motion, and the equilibrium situation weare interested in is specified by j = 0 where no mixingbetween the physical and the unphysical sector occurs.When fluctuations are unimportant, the integration overthe δχ can be dropped and the above equation reducesto Γ[ χ ] = S [ χ ], i.e. the quantum effective action reducesto the classical one.The effective action expresses the theory in terms of thefields χ . The vertex expansion generates the one-particleirreducible (1PI) correlation functions,Γ[ χ ] = (cid:88) l l ! (cid:90) x ,....,x l Γ ( l ) i ,...,i l ( x , ..., x l ) χ i ( x ) · ... · χ i l ( x l ) . (27)Usually, the coupling coefficients of the expansion areonly restricted by the symmetries of the theory – the ef-fective action is the most general polynomial in the fields χ which is compatible with the latter. Thus, formulatingthe theory in terms of physical objects – the fields χ –offers the advantage of directly leveraging the power ofsymmetry considerations from the microscopic (or clas-sical) to the full quantum level.In complete analogy, we can make use of the restric-tions present in the microscopic Hamiltonian when com-puting the quantum effective action. Since, as we haveshown above, no couplings mapping from U → P are gen-erated, we may write down the most general form for theeffective action for the physical sector of the theory bydirectly excluding couplings which would violate this con-straint. In practice, this concerns processes which changethe on-site occupation number. For example, a processwhich involves creation of a dimer on site i must be ac-companied by an appropriate constraint that the site beempty prior to the process. Thus, the operator t † mustalways appear in the combination t † X i . Furthermore re-quiring hermeticity of the terms appearing in the effectiveaction, we conclude that the effective nearest-neighbourdimer hopping term is of the form J eff t † ,i X i X j t ,j . Inthe practical calculation, we may restrict ourselves tothe computation of the coefficient which is simplest toextract – obviously the quadratic one.In sum, we have obtained the following result: theusual symmetry constraints on the quantum effectiveaction are now supplemented by a further fundamen-tal principle, namely the restrictions present in the mi-croscopic theory which originate from the hardcore con-straint. C. Arbitrary density
The construction presented in the last section focusedon the zero density limit, describing the scattering of fewparticles in the absence of many-body effects. At a finitedensity, the low temperature physics of bosons is char-acterized by the spontaneous breaking of global phaserotation symmetry U (1). The ground state exhibits acondensate mean field, which has to be incorporated inthe theoretical description of the system. One customaryapproach is to quantize a theory with degrees of freedomˆ b = s + δb via the path integral, where δb is the fluctu-ation around the classical field s . However, in our casethis procedure does not work, since possible values of themean field lie on the compact interval [0 , et al. [27], who implement the procedure onthe mean field plus spin wave level. We will see that ourtreatment of the constraint can be applied also in thiscase, such that we arrive at an exact formulation of theproblem in the presence of spontaneous symmetry break-ing. The procedure consists in first introducing the meanfield via a unitary rotation in the space of operators (Eq. FIG. 1: Rotation to a new ground state: The vacuumstates n = 0(2) are described by the red mean field vec-tor pointing in positive (negative) z direction. The fluctu-ations (black arrows) form a coordinate system in which thedirection collinear to the mean field vector is eliminated viathe implementation of the constraint. All mean field vec-tors not in the z direction describe a homogeneous superfluidground state. A situation analogous to the one for the vacuais achieved via an appropriate rotation of the coordinates forthe fluctuations. (29) below), and then quantizing this theory of operatorswhich are free of expectation values via the path integral.In this way, we can obtain a picture which is fully con-sistent with general features of many-body theories with U (1) symmetry, in particular, we can derive Goldstone’stheorem within our framework [14].Here we concentrate on homogeneous ground states,noting that the implementation of spatially dependentorder parameters – such as a charge density wave – isstraightforward. One then simply has to use rotationmatrices which vary from site to site. Such a situationwill be encountered in [14].Our treatment of the vacuum problem at n = 0 startedfrom the idea that in this case, all the amplitude resides inthe zero-fold occupied state, and that fluctuations aroundthis state have to be considered. The excitations on top ofthis “mean field” vacuum, defined as | Ω (cid:105) = (cid:81) i t † ,i | vac (cid:105) ,then turned out to be single atoms and dimers, respec-tively, as expected intuitively, and we have formulatedthe corresponding quantum field theory to describe theirscattering properties.In the many-body problem, we proceed in completeanalogy by first introducing a mean field vacuum. Ageneral homogeneous mean field vacuum may be writtenas | Ω (cid:105) = (cid:89) i (cid:0) (cid:88) α r α exp( iαφ ) | α (cid:105) i (cid:1) (28)= (cid:89) i (cid:0) (cid:88) α r α exp( iαφ ) t † α,i (cid:1) | vac (cid:105) ! = (cid:89) i b † ,i | vac (cid:105) . The introduction of b † ,i as the new vacuum creation op-erator implies the need for a redefinition of the remain-ing two degrees of freedom. Such a transformation isperformed via a two-parameter unitary rotation, whoserotation angles are chosen such that the new operatorsfluctuate around the new vacuum state and do not fea-ture expectation values (cf. Fig. 1), b † α,i = ( R θ R χ ) αβ t † β,i (29)with the explicit form of the rotation matrices R θ = cos θ/ θ/ φ − sin θ/ − φ θ/ , (30) R χ = χ/ − sin χ/ i φ χ/ − i φ cos χ/ . A finite θ ( χ ) corresponds to a finite amplitude in | (cid:105) ( | (cid:105) ).The precise relation is r = cos θ/ , r = sin θ/ χ/ , r = sin θ/ χ/ . (31)The strategy is to first rotate to the new mean field stateby inverting the unitary matrix in Eq. (29) and sub-sequently implement the constraint. Analogous to theprocedure in the physical vacuum, we may now eliminatethe operator b which is chosen to include the expectationvalue. We note that the local rotating frame transforma-tion (14) can be applied also here, showing that the phaseof b is irrelevant for the Hamiltonian. Consequently wecan implement the constraint by the formal replacement b ,i → X i ≡ − b † ,i b ,i − b † ,i b ,i , b † ,i b ,i → X i . (32)The second expression is simply a rearrangement of theholonomic constraint. The resulting bosonic Hamilto-nian, which is then quantized by means of a functionalintegral, is rather complex, and its explicit form and anal-ysis are discussed in [14]. However, it exhibits a simplestructure, H = E GW + H SW + H int . (33) The phase relation between the amplitudes f α = r α e i θ α , θ α = αφ is a consequence of spontaneous symmetry breaking in Fockspace. φ is the spontaneously chosen phase of the condensate. E GW is the Gutzwiller mean field energy and H SW de-scribes the quadratic spin wave theory . The correctionsto the mean field phase diagram, as well as nontrivial ef-fects in the deep infrared physics which we analyze in[14] are not captured at this quadratic level. They areall encoded in the interaction part H int .At this point, let us compare our findings to the workof Huber et al. [27]. We have verified explicitly that thequadratic part of the Hamiltonian coincides with the spinwave or Bogoliubov theory obtained in that work, thoughthe authors use a different prescription for the resolutionof the constraint for a single b operator. The reason isthat whenever the operator b appears in multiplicativecombination with another operator b , b , the replace-ment prescribed by the first expression of Eq. (32) givesrise to at least cubic terms neglected in the spin wave ap-proximation. In contrast, when the combination in thesecond expression of Eq. (32) appears, one simply has torearrange the constraint and no difference between thetwo approaches appears.The many-body problem is completely specified onlyupon indicating conditions which determine the two ro-tation angles θ, χ . In [14] we show that they are fixed bygap equations which emerge as a consequence of Gold-stone’s theorem. Furthermore, the chemical potentialis fixed via the equation of state when calculations atfixed filling are intended. These conditions are exact butimplicit, and can be resolved approximately only. Thesimplest approximation reproduces the classical or meanfield solution as anticipated above, but it is possible togo beyond this simple scheme with our setting [14]. D. Relation to a Spin-1 Model
From the availability of only three onsite states, it isclear that there exists a mapping of the 3-body con-strained Bose-Hubbard model to a spin-1 model. Nev-ertheless, it seems advantageous to us to work with theabove construction of mapping to a coupled boson theory.The reason is twofold.First, the practical analytical analysis of spin modelsbeyond the standard mean field plus non-interacting spinwave approximation is technically very hard. Indeed,usually one resorts to introducing an artificial smallnessparameter (inverse number of field components 1 /N , in-verse total spin 1 /S ), such that a Gaussian theory be-comes exact in the limit N, S → ∞ , and organizes anexpansion scheme around this noninteracting fixed point(1 /N, /S expansions). This is then followed by a con-tinuation of the results to the physical system of inter-est, where N and S are typically small. However, in A linear contribution, as naively expected in the expansion aboutthe condensate, does not occur due to Goldstone’s theorem, seeRef. [14]. our model S = 1, and most of the effects which we dis-cuss here and in [14] – ranging from the non-perturbativeformation of the dimer bound state in vacuum over cor-rections to the phase boundary to the true nature of thephase transition – are not accessible to leading order inthe abovementioned schemes, since they are all rooted inthe intrinsic non-linearities of the theory.Second, typically the direct mapping of a bosonic the-ory with hardcore constraint yields a rather complicatedeffective microscopic spin Hamiltonian, which furthercomplicates an analysis in terms of spin degrees of free-dom. For example, in our case the corresponding spinmodel would feature cubic and quartic bilocal spin in-teractions ∼ s zi s + i s − j + h . c ., ∼ s zi s + i s − j s zj + h . c . with in-teraction constants of the same order as the quadraticterms [27]. These terms break the rotation symmetriestypically present in generic Heisenberg models for mag-nets. Physically, the appearance of such terms has tobe expected, since none of these rotation symmetries arepresent in the hardcore boson model. Clearly, mappingthe constrained model to a theory of unconstrained cou-pled boson degrees of freedom, which find a natural in-terpretation in terms of single particle and bound statedegrees of freedom, is closer to the physics of the bosontheory with hardcore constraint.Reversely however, we emphasize that the class of spinmodels which can be readily encompassed within our for-malism is made up of Heisenberg XX models in externalfields, with possible extensions to XXZ models with smallanisotropy. The power of conventional field theory tech-niques can thus be applied to such models. V. FLUCTUATIONS IN THE VACUUMPROBLEM
In this section, we analyze the quantum field theoryderived above in the limits n = 0 ,
2. This provides auseful starting point for the treatment of the many-bodyproblem addressed in [14]. First we focus on the two-body problem, for which we present the exact solutionwithin our framework. The Schr¨odinger equation for two-particle scattering is correctly reproduced. Clearly, fortwo particles, the physical three-body constraint cannot play a role. However, a mathematically wrong or only ap-proximate implementation of the constraint will producea wrong scattering equation as we see explicitly. Second,we use the language of Feynman diagrams to explicitlycalculate the two-dimer interaction strength up to fourthorder in the perturbative regime J/ | U | (cid:28)
1. This pro-vides another benchmark for our formalism. Third, weconsider hole scattering in the limit n = 2 and discussthe formation of a di-hole bound state, which exhibitsproperties different from the dimer bound state at n = 0.Below we introduce the formalism used to do concretecalculations, the Dyson-Schwinger equations, demon-strating how to use the powerful methods of quantumfield theory in our problem. These equations providean exact hierarchy of relations between correlation func-tions. We find that in the vacuum limit, where the systemof equations describes few-particle scattering, the equa-tions for the dimer self energy and the splitting vertexare one-loop. Furthermore, the atom self energy is notrenormalized. These three generalized couplings form aclosed system of equations, decoupling from higher in-teraction vertices. We find the exact solution for theseequations. These ingredients lead to the exact solutionof the two-body problem, which manifests itself in theemergence of the non-perturbative Schr¨odinger equationfor the bound state.In order to compute higher interaction vertices, weneed to take higher loop diagrams into account, and thesystem of equations for these vertices is not closed. How-ever, we can establish a perturbative expansion of theequations in the limit J/ | U | →
0. In [14], we establishthe relation of the resulting effective theory to a spin-1/2model in this limit.
A. Dual Feshbach Model
Before embarking the calculations, let us briefly dis-cuss the microscopic model emerging from our quantiza-tion prescription. We start with the microscopic action,obtained from the Hamiltonian Eq. (18) using Eq. (23).The complex fields t † , , t , are now fluctuating classicalvariables and we may permute them at will, S [ t , t ] = (cid:90) dτ (cid:16) (cid:88) i (cid:104) t † ,i ( τ )( ∂ τ − µ + U ) t ,i ( τ ) + t † ,i ( τ )( ∂ τ − µ ) t ,i ( τ ) (cid:105) (34) − J (cid:88) (cid:104) i,j (cid:105) (cid:104) t † ,i ( τ ) t ,j ( τ ) X i ( τ ) X j ( τ ) + 2 t † ,i ( τ ) t ,j ( τ ) t † ,j ( τ ) t ,i ( τ )+ √ (cid:2) t † ,i ( τ ) t ,i ( τ ) t ,j ( τ ) + t ,i ( τ ) t † ,i ( τ ) t † ,j ( τ ) (cid:3) X j ( τ ) (cid:105)(cid:17) . This action has the form of a Feshbach resonance model(see [29] for the fermion case, and [4, 5] for the bosonic case) on the lattice, with non-local interaction parame-0ters: The dimer degree of freedom couples to the chem-ical potential with double strength, taking care of thedouble atom number in the dimer. The role of the ”de-tuning” of the dimer state from the atoms is played bythe onsite interaction. We note an interesting duality tothe standard Feshbach model in the continuum for atomand dimer degrees of freedom obtained from a Hubbard-Stratonovich decoupling of an attractive (local) two-bodyinteraction u : Such a procedure generates a detuning ν ∼ /u , while in our case the detuning ν ∼ U . Physi-cally, this means that the bound state formation in ourlattice scenario is a weak coupling phenomenon, whilebeing a strong coupling (resonant) effect in the contin-uum. Further note that the usual lattice constructionusing the single band approximation is delicate for sys-tems close to Feshbach resonances [30], and a realizationof lattice Feshbach models in the resonant case is there-fore not a straightforward task, while it is realized in anatural way here.The physically most important coupling is the Fesh-bach or splitting vertex. It describes the formation of a dimer out of two atoms and its reverse. In contrast tothe conventional continuum Feshbach model, the split-ting vertex in our model is bi-local, which in momentumspace induces a form factor but does not lead to com-plications and still allows for an exact solution of thetwo-body problem. The term in the third line describesa non-local interaction between atoms and dimers.In the loop calculations we prefer to work in frequencyand momentum space. With the definitions t α,i ( τ ) = (cid:90) q e i qx i t α,q , t † α,i ( τ ) = (cid:90) q e − i qx i t † α,q , (35) x i = ( τ, x i ) , q = ( ω, q ) , (cid:90) q = (cid:90) dω π (cid:88) q ,(cid:15) q = J d (cid:88) λ =1 cos( qe λ ) , ∆ X q,k = +( t † ,q t ,k + t † ,q t ,k )the Fourier transformed action reads S [ t , t ] = (cid:90) q (cid:104) t † ,q (i ω − µ − (cid:15) q ) t ,q + t † ,q (i ω − µ + U ) t ,q (cid:105) (36) −√ (cid:90) q ,q ,q δ ( q − q − q )( (cid:15) q + (cid:15) q )( t † ,q t ,q t ,q + h.c.) − (cid:90) q ,...,q δ ( q − q + q − q )[ (cid:15) q − q + (cid:15) q − q ] t † ,q t ,q t † ,q t ,q + √ (cid:90) q ,...,q δ ( q − q − q + q − q )[ (cid:15) q − q + (cid:15) q − q ]( t † ,q t ,q t ,q ∆ X q ,q + h.c.)+2 (cid:90) q ,...,q δ ( q − q + q − q )[ (cid:15) q + (cid:15) q ] t † ,q t ,q ∆ X q ,q − (cid:90) q ,...,q δ ( q − q + q − q + q − q )[ (cid:15) q + q − q + (cid:15) q + q − q ] t † ,q t ,q ∆ X q ,q ∆ X q ,q . B. Dyson-Schwinger Equations
Dyson-Schwinger equations (DSEs) [16] are a directconsequence of the shift invariance of the functional in-tegral:0 = 1 Z [ j ] (cid:90) D ( δ ˆ χ ) δδ ˆ χ exp − S [ ˆ χ ] + j T δ ˆ χ (37)= 1 Z [ j ] (cid:90) D ( δ ˆ χ ) (cid:16) − δSδ ˆ χ + j (cid:17) T exp − S [ ˆ χ ] + j T δ ˆ χ. Switching to the effective action, i.e. requiring j = δ Γ /δχ , the above equation turns into δ Γ δχ = (cid:68) δSδ ˆ χ (cid:69)(cid:12)(cid:12)(cid:12) j = δ Γ /δχ . (38)This is the DSE for the one-point function. To reveal thestructure of the DSEs for higher N -point functions, weconsider a general classical action with M vertices. Wewrite the classical action in a vertex expansion, S [ ˆ χ ] = S [ χ ] + M (cid:88) N =1 N ! S ( N ) α ...α N δ ˆ χ α · ... · δ ˆ χ α N . (39)1Here α i is a multi-index collecting field type as well asspace and timelike (or momentum and frequency) indices. S and S ( N ) still depend on the classical field χ . Plug-ging the vertex expansion into Eq. (38) relates the fieldderivative of the effective action to 1PI Green functionsup to order M . We can turn the DSE into a manifestlyclosed equation, i.e. an equation which is expressed solelyin terms of the effective action and its functional deriva-tives. It reads δ Γ δχ β = S (1) β + 12! S (3) α α β G α α (40)+ M (cid:88) N =4 N − S ( N ) α ...α N − β (cid:104) N − (cid:89) i =3 G α i κ i δδχ κ i (cid:105) G α α . For N = 4 the derivative operator in the squared brack-ets is just the unit matrix. The full propagator is denotedby G and we have the relation G αβ = (Γ (2) ) − αβ . The fullpropagator as well as the classical vertices are functionsof the classical field, S ( N ) = S ( N ) [ χ ] , G = G [ χ ], suchthat the DSE for the N -point correlation function can beobtained by taking N − δ Γ /δχ β de-pends on correlation functions up to order M −
1. Thus,the DSE for the N -point function features vertices upto order M + N −
2. Furthermore the self-consistencyequations for the correlation functions for a theory withclassical vertices up to order M features M − M vertex has M − C. Exact Solution of the Two-Body Problem
The scattering problem is described by two coupled in-tegral equations for the exact dimer self-energy and forthe exact Feshbach vertex, cf. Fig. 2 and App. A forthe derivation. Inserting the frequency and momentumconfigurations appropriate for two-body scattering as de-picted in Fig. 2 and integrating out the frequencies, weobtain G − d ( E, k ) = G (0) − d ( E, k ) + Σ( E, k ) , Σ( E, k ) = − √ (cid:90) d d q (2 π ) d Γ k ( q )Γ (0) k ( q ) E + Γ (0) k ( q ) , Γ k ( p ) = Γ (0) k ( p ) + (cid:90) d d q (2 π ) d Γ k ( q )(Γ (0) k ( p ) + Γ (0) k ( q )) E + Γ (0) k ( q ) , (41)with the definitionsΓ (0) k ( q ) = − √ (cid:15) q + (cid:15) q − k ) , E = √ ω − µ ) , (42)where ω is the Euclidean external frequency. Thedifference between the full ( G d ( E, k )) and the bare( G (0) d ( E, k )) Green’s function is the dimer self-energy, FIG. 2: Scattering equations in vacuum. The solid linesrepresent atom propagators. Full vertices are signalled withheavy blobs. (a) Equation for the dimer self-energy. The fullFeshbach vertex is needed for its solution. (b) Renormaliza-tion of the Feshbach vertex. External momentum configura-tions are chosen as needed in (a). Σ( E, k ) = G − d ( E, k ) − G (0) − d ( E, k ). Note carefully theappearance of both external ( p ) and loop ( q ) momentain the equation for the full Feshbach vertex. The secondequation can be solved independently of the first one.This can be done by choosing the following ansatz forthe full vertex,Γ k ( q ) = Γ (0) k ( q ) γ (0) ( E, k ) + γ (1) ( E, k ) , (43)where the unknown γ (0) is dimensionless while γ (1) car-ries dimension of energy. The two unknown functionsdepend only on the external center-of-mass momentum k and the energy variable E ; the dependence on the rela-tive momentum q only appears in the coefficient of γ (0) .Comparing coefficients this ansatz yields the followingsystem of coupled equations for the unknowns γ (0) = 1 + γ (0) (1 − EI ) + γ (1) I, (44) γ (1) = [ γ (1) − γ (0) E ](1 − EI ) , where we use the abbreviation I ( E, k ) = (cid:90) d d q (2 π ) d E + Γ (0) k ( q ) (45)and the simplifications (cid:90) d d q (2 π ) d Γ (0) k ( q ) E + Γ (0) k ( q ) = 1 − EI, (46) (cid:90) d d q (2 π ) d Γ (0) k ( q )Γ (0) k ( q ) E + Γ (0) k ( q ) = E ( EI − . The solution of the above equations is γ (0) = 1 , γ (1) = E − I − , (47)Γ k ( q ) = ( E + Γ (0) k ( q )) − I − . E, k ) = 1 √ I − − E ) = − (i ω − µ ) + (48) (cid:104) (cid:90) d d q (2 π ) d − (cid:15) q + (cid:15) q − k ) + i ω − µ (cid:105) − , such that the equation for the full inverse dimer Green’sfunction becomes, with G (0) − d ( E, k ) = i ω − µ + U , G − d ( E, k ) = U + (cid:104) (cid:90) d d q (2 π ) d − (cid:15) q + (cid:15) q − k ) + i ω − µ (cid:105) − . (49)The presence of a bound state is signalled by a pole in thedimer Green’s function at zero center-of-mass momentumand zero external frequency, G − d ( ω = 0; µ, k = 0) = 0.The chemical potential µ in the physical vacuum canbe interpreted as the binding energy [28] after an ap-propriate decomposition which ensures that the atom atrest has no kinetic energy, µ = µ b − Jz . This defini-tion separates true kinetic from true potential (binding)energy, and finite momentum excitations have positiveenergy, δ(cid:15) q = J (cid:80) λ (1 − cos qe λ ). Then, we have forthe binding energy E b = 2 µ b , and we note that theatoms are gapped out with half the binding energy, sincetheir Green’s function involves − µ b . Thus, the molec-ular degrees of freedom are the lowest excitations sincein contrast to the atoms they are massless (pole condi-tion). For vanishing binding energy we have µ b = 0, suchthat the situation is reversed: The atoms are the gaplessexcitations, while the molecules are gapped. Introduc-ing dimensionless and dimensionally invariant variables˜ U = U/ ( Jz ) , ˜ E b = E b / ( Jz ) the pole condition leads to1 | ˜ U | = (cid:90) d d q (2 π ) d − ˜ E b + 2 /d (cid:80) λ (1 − cos qe λ ) . (50)This is precisely the Schr¨odinger equation for the dimerbound state. We discuss the formation of the boundstate, since it will be interesting to confront these well-known results to the formation of di-hole bound states at n = 2 which shows different properties. The bound stateforms at the critical ˜ U d in d dimensions where E b = 0. Inthree dimensions, the integral evaluates to a finite value,while in two dimensions a logarithmic infrared divergencepushes ˜ U to zero:˜ U ≈ − , ˜ U = 0 . (51)This has to be compared to the mean field result at zerodensity, ˜ U mf = −
2, obtained from the classical inversedimer Green’s function G (0) − d ( ω = 0; µ, k = 0). Weobserve a substantial downshift in the critical interac-tion strength. Close to the onset of the bound state thebinding energy starts quadratically ( d = 3) resp. expo-nentially ( d = 2), due to the square root nonanalyticity n = 0 n = 2 E b /J z U/Jz
FIG. 3: Dimensionless binding energy of dimers ( n = 0) anddi-holes ( n = 2) as a function of the dimensionless interactionstrength. The upper lines (black online) denote the meanfield results, the lower/middle curves (red/blue online) are theexact binding energies in dimensions 2 and 3, respectively. resp. logarithmic divergence of the fluctuation integral: d = 3 : ˜ E b ≈ − (cid:16) | ˜ U | − | ˜ U | σ ˜ U (cid:17) , (52) d = 2 : ˜ E b ≈ − Λ exp( − π | ˜ U | ) , ˜ E b, mf = ˜ U mf − | ˜ U | = 2 − | ˜ U | , with numbers σ ≈ . , Λ ≈ .
50 determined numeri-cally. We have added the linear mean field result. Thisdimensionally invariant behavior is approached for largenegative couplings, cf. Fig. 3.The fact that the Schr¨odinger equation is repro-duced by our nonperturbative calculation is an impor-tant benchmark for our theory. Note that any deviatingimplementation of the constraint, such as an expansionof the square root Eq. (13) to leading order, would gener-ate incorrect prefactors in the scattering equations, andthe Schr¨odinger equation could not be reproduced.
D. Perturbative Limit: Effective Two-Body HardCore Dimer Gas
While the perturbative limit for the two-body prob-lem is straightforwardly obtained to any order from theexact solution above, for higher order vertices, nonper-turbative calculations are hard – the computation of thefull dimer-dimer scattering vertex would require the com-plete solution of the four-body problem. However, in thelimit
J/U (cid:28) J only diagrams with at most two ver-tices have to be taken into account. We find that a loopexpansion to one-loop order is insufficient even at order J . We may understand that qualitatively from the fact3that on short ranges, we are dealing with a full quantummechanical problem with strong fluctuations – the occu-pation of a site is either 0 or 1. Thus, a loop expansion(in orders of (cid:126) ) cannot be expected to be reliable.In this section, we calculate the effective Hamiltonianfor dimers. There dimers will obviously have a two-bodyhardcore constraint, as they are made up of two atomseach. The relation to a spin 1/2 model is made in [14].There, we also show that a symmetry enhancement fromthe conventional U (1) ∼ SO (2) for bosons to SO (3) istaking place in the strongly correlated limit, and discussits physical implications.We will first calculate the effective Hamiltonian up tosecond order perturbation theory. We then perform a partial calculation of the fourth order. These results arecrucially needed when discussing the many-body phasesin the strongly coupled regime in [14], as well as for thediscussion of the nature of the phase transition.
1. Second Order
We are interested in extracting the effective action fordimers in the perturbative limit. If restricting to secondorder in J , the effective theory can only contain nearestneighbour terms. The most general form compatible withthe constraint and symmetry principles is given by S [ t ] = (cid:90) dτ (cid:16) (cid:88) i Zt † ,i ( τ )( ∂ τ − µ d ) t ,i ( τ ) − t (cid:88) (cid:104) i,j (cid:105) t † ,i ( τ ) X i ( τ ) X j ( τ ) t ,j ( τ ) + v (cid:88) (cid:104) i,j (cid:105) ˆ n ,i ( τ )ˆ n ,j ( τ ) (cid:17) . (53)The X terms are introduced in order to satisfy the con-straint principle, and there cannot be an onsite dimerinteraction for the same reason. Z is a wave functionrenormalization factor. It accounts for the energy de-pendence of the perturbative expansion in x = J/ | U | . µ d = 2 µ − U + ∆ µ d ≤ t, v are the constrained hopping and interac-tion constants to be determined. Kinetic Terms – The desired information on the localand the constrained hopping term can be extracted fromthe solution of the two-body problem, Eq. (49). For thispurpose we take µ → −∞ , G − d ( ω ; µ, k ) = i ω − µ + U − J i ω − µ (cid:88) λ (1 + cos ke λ ) . (54)In this form it is apparent that the full inverse propaga-tor contains the microscopic one exactly, but is now sup-plemented by a qualitatively new hopping term. To beconsistent at second order, in the denominator we haveto insert µ = −| U | /
2. After Fourier transformation ofthe quadratic part of the action with inverse Green func-tion (54), we obtain ∆ µ d = 2 J z/ | U | , t = 2 J z/ | U | . Thewave function renormalization factor Z is extracted fromEq. (54) from expanding in i ω ; there is no second or-der contribution at second order perturbation theory, i.e. Z = 1. Dimer-Dimer Interaction – At short ranges, one ex-pects a dimer density-density repulsion due to the re-duced decay and recombination possibilities of one dimerif there is another one sitting close by. The Dyson-Schwinger Equation governing the dimer-dimer scatter-ing in the perturbative regime is displayed in Fig. 4.The one-particle irreducible graphs give rise to a nearest- neighbour (second order in J ) density-density repulsion,due to the presence of the constraint, manifesting itselfvia the five-point splitting vertex. As anticipated above,it is interesting to note that the equation is two-loop evenin the leading order perturbation theory. For the deriva-tion of the symmetry factors and the explicit calculationwe refer to App. A. Here we indicate the second orderresult of the momentum space calculation and discuss it: v { k i } J | U | (cid:0) (cid:15) k − k + (cid:15) k − k + (cid:15) k + (cid:15) k + (cid:15) k + (cid:15) k (cid:1) . (55)The two qualitatively different momentum dependencescorrespond to different interaction processes in positionspace. After Fourier transform we find the contributionto the effective action2 J | U | (cid:90) dτ (cid:88) (cid:104) i,j (cid:105) (cid:104) ˆ n ,i ( τ )ˆ n ,j ( τ ) + t † ,i ( τ ) t ,j ( τ ) (cid:0) ˆ n ,i + ˆ n ,j (cid:1)(cid:105) . (56)The terms may be interpreted as follows. The first termis a true dimer-dimer interaction describing the exchangeof dimers on adjacent sites and is repulsive. The secondone is the explicit manifestation of the constraint beinginherited by the effective theory of dimers – they preciselycontribute the terms linear in ˆ n which are contained in X i X j in Eq. (53).In summary, the effective couplings to second orderread Z = 1 , ∆ µ d = t = v J | U | . (57)Both effective hopping and interaction are not present inthe mean field approximation. Conceptually and prac-tically, they are however of high importance. The first4 FIG. 4: The dimer-dimer interaction vertex to second orderin the hopping J . one makes the dimers true physical, i.e. spatially prop-agating degrees of freedom, while the second one, withthe positive sign, is very important for the many-bodyand long-wavelength calculations carried out in [14], asit stabilizes the thermodynamic potential for the dimersuperfluid. It also gives rise to the stiffness of the su-perfluid. Since the second order contributions are due tofluctuations on a single link of nearest neighbours, theresults are dimension independent.
2. Fourth Order
At fourth order, we obtain not only nearest neighbour(nn) contributions to the constrained hopping and inter-action, but also next-to-nearest neighbour (nnn) contri-butions. However, due to the reduced number of path-ways connecting these more distant sites, the coefficientsare substantially smaller then for nn terms. In the fol-lowing, we thus concentrate on the latter. In particular,for later purposes we will be interested in the deviationof the ratio of interaction vs. kinetic energy λ = v/ (2 t )from the second order result λ = 1.A brute force diagrammatic fourth order calculation israther complex. Here we present a way to perform thefourth order calculation of the nn coeffcients for the in-teraction from a combination of geometric and diagram-matic arguments: First we argue that based on the ge-ometry of the contributing processes, the repulsive partof the fourth order contribution must equal the fourthorder hopping contribution. The latter, in turn, can becalculated straightforwardly from the exact solution ofthe two-body problem. This argument can be appliedfor processes of arbitrary intersite distance, and we willpresent it in its general form. Then we refocus on near-est neighbours and identify processes contributing to theinteraction which have no analog in the hopping process.We calculate the corresponding reduced set of diagramsexplicitly. They yield an attractive contribution to theinteraction strength such that λ < J/ | U | and at a fixed distance, we canfind this contribution in principle by drawing all possiblepathways with a fixed number of hops which connect ini-tial and final site. We call these paths connecting initialand final state via hopping the hopping paths. Next weconsider the interaction coefficient. The physical originof this interaction emerges from the constraint: if there isa dimer at site j , then another particle cannot hop on thissite. This can be calculated as the energy difference thatemerges when comparing the number of paths throughwhich a dimer at site i can decay and come back to thissite without a dimer sitting at j , and the correspondingnumber when there is a dimer on j . The energy differ-ence obviously is always positive, because the constraintalways excludes a stet of paths, leading to a repulsive in-teraction. Thus, the energy contribution can be obtainedby just counting the number of paths where at least oneof the travelling particles hits the site j . We call thesepaths interaction paths.Now we observe that any hopping path can be trans-formed into an interaction path by reversing the direc-tion of the arrows of one of the travelling particles onthe shortest path which connects initial and final site(there may be several of these shortest paths) for anyoverall length of the paths, i.e. at any order of perturba-tion theory. Therefore, the number of interaction pathsis larger or equal compared to the number of hoppingpaths. The reverse is also true. Thus the number of in-teraction and hopping paths is equal, and therefore theso-obtained contribution to the interaction must equalthe fourth order hopping contribution.However, in general there are pathways contributing tothe nn interaction which have no analog in the hoppingpathways . These are processes in which none of thedimers is static, and we have to calculate them diagram-matically. For nn, the corresponding diagrams are pro-vided in Fig. 5. We discuss the corresponding processes(the explicit calculations are performed in App. A): Inthe left diagrams, the process starts with the decay ofa dimer into atoms, where the sum over nearest neigh-bours indicates the z = 2 d possibilities resulting fromthe unconstrained splitting vertex. The splitting is thenfollowed by a double swap of atom and dimer. Finally,the atom on k recombines with the one on the targetsite i or j into a dimer. The position indices of theseprocesses are fixed by the nn range of the couplings, andno further summation occurs. Performing the frequencyintegrations for zero external frequencies, the left sideof Fig. 5 evaluates to − zJx , with x = J/ | U | , thusproviding an attraction piece to the dimer-dimer interac-tion. The three-loop diagrams on the left take care of theconstraint: one decay possibility described by the uncon- At second order, no other interaction processes can occur. Thisis the reason for λ = 1 at second order. FIG. 5: Attractive contribution to the nearest neighbourinteraction at fourth order. The three-loop graphs implementthe constraint, as discussed in the text. strained splitting vertex is not allowed, since the otherdimer is located there. Indeed, the diagrams evaluate to+16 Jx , such that the net result of the processes in Fig.5 is v a − z − Jx . (58)This is an attractive contribution to the nn interactionin any dimension.We are now in the position to provide the full fourthorder contribution. The repulsive part to the interactionis given by the fourth order contribution to the nn hop-ping, which we can obtain from the expansion of Eq. (49)up to fourth order with the result v r t = 2 J [ x + (12( z − − x ] . (59)We thus obtain the final result for the nn hopping andinteraction terms, t = 2 J ( x + 2(6( z − − x ) , (60) v v a + v r J ( x + 2(2( z − − x ) ,λ = v t = 1 + 2(2( z − − x z − − x ≈ − z − x < . Finally, we discuss an additional effect at fourth orderwhich is associated to the wave function renormalization.The leading term, relevant at fourth order perturbationtheory, can be extracted from the frequency expansion ofEq. (54) at k = 0, yielding Z = 1 + 4 zx . (61)The correction to the dimensionless quantity Z is O ( x )and has effects at fourth order only. For example,if we are interested in the dispersion relation, ob-tained from the pole condition on the Green’s function for real frequencies at a finite molecule momentum k , G − d (i ω ( k ); µ b = E b / , k ) = 0, we find ω ( k ) = 4 J Z | U | (cid:88) λ (1 − cos ke λ ) . (62)The wave function contributes a fourth order term to thedispersion relation. In order to see the effect on the ra-tio of interaction vs. kinetic energy, we absorb the wavefunction renormalization factor into a redefinition of thefield t → √ Zt and into the remaining couplings, e.g. t → ˜ t = t/Z, v → ˜ v = v/Z , where the redefinition ofthe couplings is performed in such a way as to keep thefull effective action invariant under the transformation.Since Z > λ = ˜ v/ (2˜ t ) < λ , such thatthe ratio of interaction vs. kinetic energy is effectivelydecreased additionally. We finally note that the effectsof the wave function renormalization correspond to anenergy dependence of an effective Hamiltonian obtainedin a Brillouin-Wigner perturbation theory for a micro-scopic Hamiltonian. It is well known that these effectsoccur at fourth order perturbation theory only. E. Relation of n = 0 and n = 2 : Particle-HoleMapping At density n = 2, the physics is expected to be similarto the case n = 0. Similar to the latter, the effect of spon-taneous symmetry breaking is absent due to the completefilling. Moreover, due to the constraint, there is only asingle microscopic configuration, in full analogy to thecompletely empty lattice for n = 0. Importantly, there-fore the n = 2 state in the constrained model must not beviewed as a Mott insulator phase, such as e.g. n = 1 andstrongly positive U , but rather as a constraint inducedband insulator: For n = 1 the motion of the particlessimply costs energy (providing for the gap in the Mottspectrum), while for n = 2 this is impossible in principle.Therefore, we may expect another “vacuum problem”.The low lying excitations on top of this vacuum will beholes and doublets of holes or di-holes instead of atomsand dimers, and the complete amplitude resides in the t mode. From this reasoning, we now replace t ,i → − t † ,i t ,i − t † ,i t ,i ≡ X (cid:48) i . (63)Alternatively, we could have adopted the formal point ofview, applying the procedure described in Sec. IV C forthe angles χ = θ = π . Here we want to stress the physicalsimilarities between n = 0 ,
2, but of course, the formalprocedure generates exactly the same result. The propagation of the dimers in vacuum is gapless, and we haveadjusted the zero of energy at k = 0 via appropriate choice of µ . H kin = − J (cid:88) (cid:104) i,j (cid:105) (cid:2) t † ,i X (cid:48) i X (cid:48) j t ,j + t † ,i t ,i t † ,j t ,j (64)+ √ X (cid:48) i t ,i t † ,j t ,j + t † ,i t ,i t † ,j X (cid:48) j ) (cid:105) ,H pot = + µ (cid:88) i t † ,i t ,i + t † ,i t ,i − U (cid:88) i t † ,i t ,i + t † ,i t ,i +( − µ + U ) M d , where M is the number of lattice sites in each latticedirection. The potential energy term can be written as H pot = − µ (cid:48) (cid:88) i t † ,i t ,i + t † ,i t ,i + U (cid:88) i t † ,i t ,i , (65) µ (cid:48) = − µ + U. Working with the action in the following we may permutethe operators at will. In this case, the mapping of theaction for zero density to the one at n = 2 is given, upto constants (the mean field energies) by the followingreplacements t → t , µ → µ (cid:48) , U → U, g split ≡ √ J → √ J,J hop ≡ J → J, g exchange ≡ J → J. (66)While the coupling strength of the splitting vertex re-mains invariant under the transformation, the roles of K (10) and K (21) (cf. Eqs. (12,18)) are exchanged, andso is the role of the corresponding coupling constants.Based on Eq. (66) we observe that the scattering equa-tion for the holes can be obtained from simple replace-ments in the scattering equations (41). The equation forthe hole splitting vertex,Γ k ( p ) = Γ (0) k ( p ) + (cid:90) d d q (2 π ) d Γ k ( q )2(Γ (0) k ( p ) + Γ (0) k ( q )) E + 2Γ (0) k ( q ) (67)has the identical form to the one for dimers if we redefinethe energy variable E = √ ω − µ ) → E (cid:48) = √
22 (i ω − µ (cid:48) ) (68)and therefore has the solution (47) with the replace-ment E → E (cid:48) . The bare inverse di-hole propagator is G (0) − h ( E (cid:48) , k ) = i ω + ( − µ (cid:48) + U ) = i ω − ( − µ + U ), andthe di-hole self energy Σ h = G − h − G (0) − h is found to beΣ h ( E (cid:48) , k ) = 12 1 √ I − ( E (cid:48) ) − E (cid:48) ) (69)= 14 (cid:16) − (i ω − µ (cid:48) )+ (cid:104) (cid:90) d d q (2 π ) d − (cid:15) q + (cid:15) q − k ) + i ω − µ (cid:48) (cid:105) − (cid:17) . Bound state formation – The generalized “Schr¨odingerequation”, governing the formation of a bound state ofholes, is given by the pole condition G − h ( ω = 0; µ (cid:48) , k = 0) = 0 . (70)With definitions analogous to the zero density limit ( µ (cid:48) = µ (cid:48) b − Jz, E (cid:48) b = 2 µ (cid:48) b ; this choice of µ (cid:48) b ensures the zeroof kinetic energy to appear at zero momentum), and indimensionless variables, we find the explicit form14 | ˜ U | −
12 + 3 ˜ E (cid:48) b = (cid:90) d d q (2 π ) d − ˜ E (cid:48) b + 4 /d (cid:80) λ (1 − cos qe λ ) . (71)The bound state forms at the critical dimensionless inter-action strength defined by ˜ E (cid:48) b = 0. Due to the differentstructure of the scattering equations at n = 2, we expectdifferent results from the n = 0 case. Indeed, we find thedimensionless critical interaction strengths in three andtwo dimensions˜ U (cid:48) ≈ − ≈ − . , ˜ U (cid:48) = − . (72)An interesting feature occurs in d = 2: Despite the log-arithmic infrared divergence of the fluctuation integral,the di-hole bound state formation occurs at a finite at-tractive interaction strength. For the dependence of thebinding energy on the interaction strength close to thethreshold we find d = 3 : ˜ E (cid:48) b = − (cid:16) | ˜ U | − | ˜ U (cid:48) | ) σ ( | ˜ U (cid:48) | − (cid:17) , (73) d = 2 : ˜ E (cid:48) b ≈ − Λ exp( − π | ˜ U |− ) , with σ, Λ given below Eq. (52). In contrast, the meanfield binding energy reads ˜ E b = −| ˜ U | + 4. The exactresults Eq. (72) for the onset of the di-hole bound statecan be compared to the mean field answer ˜ U mf = − n = 0limit. Furthermore, from Fig. 3 we observe that the non-perturbative fluctuation dominated quadratic and expo-nential regimes are smaller than in the low density limitdescribed by (52). We conclude that fluctuations effectsare weaker in the maximum density region n ≈ n = 0.The finite value in Eq. (72) in two dimensions is sur-prising, and some comments are in order. An under-standing is obtained from the fact that the coupling con-stants are different in the band insulator n = 2 thanin the vacuum n = 0, and so the coefficients in theeffective propagators are different. This, in particu-lar, implies a shift in the effective interaction strength | U | → | U | −
12 + 3 ˜ E (cid:48) b ; note that the integrals are identi-cal in Eq. (50) and Eq. (71) up to a coefficient 4 insteadof 2 in front of the kinetic term in the integral – thisreflects an enhanced mobility of the holes compared tothe atoms. Therefore, we also have the log divergence in7 d = 2 and n = 2 for ˜ E (cid:48) b →
0. However, this divergencenow causes | ˜ U | → | U | → U is a microscopic quantity,and the shift to it in Eq. (71) is due to the effects ofthe n = 2 band insulator. The finite threshold, thus, isan observable prediction of our theory. In addition, wewould like to stress that there is no strong reason to ex-pect identical critical interaction strengths at n = 0 and n = 2, as there is, in this regime of moderate interactionstrengths, no particle-hole symmetry suggesting such be-havior. Effective di-hole theory – We have already noted thatthe mean field result for the binding energy is approachedin the limit U → −∞ . The leading correction is given byΣ h ( E (cid:48) → ∞ , k ) → − J | U | (cid:88) λ (1 + cos ke λ ) . (74)This precisely coincides with the leading perturbativeself-energy correction for the dimers (cf. Eq. (54)at ω = 0 , − µ = | U | ). Furthermore, the di-hole-di-hole interaction is identical to the zero density case at O ( J / | U | ), since the splitting and the five-point verticesentering the dimer-dimer scattering vertex (cf. Fig. 4)have the same value, and only the atomic onsite propaga-tor enters at this order. Thus, to leading order we recoverprecisely the effective hardcore theory given by Eq. (53)with t (dimers) replaced by t (di-holes) but identicalinteraction constants, though in general the original mi-croscopic theories at n = 0 , n = 0 and n = 2 must be expected: When the atom-atom attraction U is the largest scale in the problem,the integration of the high energy atom degrees of free-dom in perturbation theory can be performed prior tothe inclusion of the effects of, e.g. a finite density in thesystem. Due to the decoupling from the atomic degreesof freedom, the theories in the perturbative regime butat different densities must then be directly related to areference density, e.g. n = 0, by a rotation defined byEq. (29). The mapping of the theories for n = 0 and n = 2 may be understood as performing the rotationwith θ = χ = π . This is different from the case wherethe atoms do not decouple completely, such that the den-sity might have implicit fluctuation effects adding to theexplicit effect of the rotation (29) on the dimer degreesof freedom. This hints at enhanced symmetry proper-ties in the perturbative limit, which are discussed in thecompanion paper [14].Finally, we note that the low lying di-hole excita-tions on top of the n = 2 band insulator state dispersesquadratically, as expected for nonrelativistic excitations.Once the density is lowered away from n = 2 and sponta-neous symmetry breaking sets in, there will be a linearlydispersing Goldstone mode [14]. VI. CONCLUSION
In this paper, we have developed a method which al-lows to exactly map a bosonic lattice model with three-body onsite constraint to a theory for two unconstrainedbosonic degrees of freedom with conventional polyno-mial interactions. The Gutzwiller mean field theory isrecovered as zero order contribution to the thermody-namic potential. The quadratic fluctuations around themean field solution reproduce precisely spin wave the-ory. However, due to the exact nature of the mappingwe can also systematically access the non-linearities. Aconvenient framework for our analysis is found to be thequantum effective action, the generating functional of theone-particle irreducible correlation functions. We estab-lish that the usual symmetry principles are supplementedwith a new “constraint principle”, which depends onscale, being important at short distances but irrelevant atlong wavelength. This setting allows to address fluctua-tions on various length scales in a unified framework. Theapplication of the formalism to the full many-body prob-lem is performed in the related work [14], where we iden-tify various quantitative and qualitative aspects whichare uniquely tied to the presence of interactions. Here,in order to demonstrate the validity of the formalism,and to prepare for the many-body analysis in [14], we in-vestigate the scattering properties of few particles in thelimit of vanishing density. Furthermore, we address thecomplementary problem in the limit of maximum filling,where the low lying excitations are holes and di-holes,and calculate the effective theory for hardcore dimers inthe strong coupling limit.We believe that the formalism developed here hasstrong potential applications beyond the particularmodel analyzed here and in the related Ref. [14]. Weconclude by giving an outlook on problems that may beaddressed within this framework.
Exactly constrained theories – Spin models with spin S . The onsite Hilbert spaces can always be formulated interms of 2 S + 1 states. Here, we truncate to three stateson each site ( S = 1), but a generalization is straight-forward. Our construction maps such problems to 2 S coupled bosonic degrees of freedom, and can thus be ef-ficient for small total spin S . More specifically, our for-malism can be particularly useful for XXZ models instrong external fields and a perturbative anisotropy. Themean field plus spin wave approximation is always incor-porated in our setting, but it also offers the opportunityto systematically assess the interaction effects. We alsostress that inhomogeneous ground states, such as e.g. acharge density wave or antiferromagnet, are straightfor-wardly implemented in the formalism by applying sitedependent rotations of the type (29). Furthermore, itwill be interesting to investigate if the limit of infinitespatial dimension d → ∞ leads to a Gaussian fixed pointof the theory, where the quadratic spin wave theory be-comes exact. This would add the possibility of a 1 /d expansion to the conventional 1 /S (number of spin com-8ponents) and 1 /N (number of field components in thecorresponding nonlinear sigma model) expansion used totreat spin models. Models with an approximately realized constraint – Amajor challenge in quantum field theory is the treatmentof strongly coupled systems where the non-linearity pro-vides a larger energy scale than the scales appearing inthe quadratic part. Our theoretical framework is suitedto address a certain class of such strongly coupled sys-tems, namely the one which is spanned by lattice modelswith strong local repulsion or attraction. It is not manda-tory that the onsite repulsion (two-, three- or few- body)is infinitely large. Often, only a few dominant low energydegrees of freedom are necessary to capture the essentialphysics, and these degrees of freedom are explicitly imple-mented in our approach. For example, in the Mott phaseof the Bose-Hubbard model, number fluctuations aroundthe Mott state with given n are small and a truncationto three Fock states n, n ± γ would lead toadditional but massive bosonic degrees of freedom, whichdue to their mass ∼ γ can be taken into account pertur-batively. Fermions – The formalism is ideally suited to studyattractive fermions on the lattice. In particular, recentexperiments with 3-component fermions exhibit strongloss features [13] and are therefore important candidatesfor the observation of loss induced constrained models[7]. A modified scheme for the implementation of theconstraint in one dimension has already been given in[7], followed by the analysis of the model with bosoniza-tion techniques. Our scheme, which explicitly introducesa mean field describing qualitative features of the groundstate, is promising in higher spatial dimensions, and itsimprovements compared to Gutzwiller mean field theorymay be expected to be similar to Dynamic Mean FieldTheory [34]. We note that an application to the fermionicrepulsive Hubbard model is complicated by the choice ofthe qualitative features of the ground state being debat-able. In this case, a treatment of the constraint along thelines of [35] may be preferrable.
Acknowledgements – We thank E. Altman, A. Auer-bach, H. P. B¨uchler, M. Fleischhauer, M. Greiter, A. Mu-ramatsu, N. Lindner, J. M. Pawlowski, L. Radzihovsky,S. Sachdev, J. Taylor and C. Wetterich for interestingdiscussions. This work was supported by the AustrianScience Foundation through SFB F40 FOQUS, by theEuropean union via the integrated project SCALA, bythe Russian Foundation for Basic Research, and by theArmy Research Office with funding from the DARPAOLE program.
Appendix A: Dyson-Schwinger Equations
We calculate the DSE for the two- and three-pointfunctions exactly, and the four-point function perturba-tively. In vacuum, a tremendeous simplification is pro-vided by the fact that density-type contributions vanish.In the loop language, this translates to checking whetherthere is a subdiagram in which the arrows form a closedtour.
Formalism – The most important step is to find thecorrect symmetry factors for the loop contributions. Forthat purpose, it is sufficient to consider the field index α to parameterize the field type only; the frequency andmomentum structure of the couplings can then be foundfrom the corresponding conservation laws in a secondstep, reading off the couplings from the microscopic ac-tion (36). Thus we consider χ α = ( t , t † , t , t † ) . (A1)In this formalism, we find the following couplings: S (2) αβ ≡ δ Sδt α δt β = p [ δ α δ β + perm . ] (A2)+ p [ δ α δ β + perm . ] , where ”perm.” means to permute all greek indices whileleaving the explicit numbers in their positions. This cou-pling must be inverted to give the propagators, with theresult G αβ = 1 p [ δ α δ β + perm . ] + 1 p [ δ α δ β + perm . ] . (A3)The higher couplings read S (3) αβγ = [ δ α δ β δ γ + δ α δ β δ γ + perm . ] , (A4) S (5) αβγδ(cid:15) = s (5) [ δ α δ β δ γ δ δ δ (cid:15) + δ α δ β δ γ δ δ δ (cid:15) + perm . ] .p , p , s (3) , s (5) can be read off directly from the classicalaction (36) after proper symmetrization in momentumspace. We can reduce the tensors by contracting withdelta functions. For example, S (3)114 = s (3) δ α δ β δ γ S (3) αβγ = s (3) δ α δ β [ δ α δ β + perm . ]= 2 s (3) δ α δ α = 2 s (3) . (A5)where doubly occuring indices are summed over. Thecouplings are now fully symmetrized in field space, andindices may be permuted at will. Two- and three-point functions – We consider theDSEs for the two- and the three point functions for theinverse propagators and the splitting vertex. In the vac-9uum limit, these equations are one-loop:Γ (2) αβ = S (2) αβ −
12 tr S (3) αµ µ G µ ν G ν µ Γ (3) βν ν , (A6)Γ (2)12 ≡ p = p , , Γ (2)34 ≡ p = p , − s (3) p p γ (3) , Γ (3)114 = S (3)114 −
12 tr S (4)11 µ µ G µ ν G ν µ Γ (3)4 ν ν = 2 γ (3) = 2 s (3) − s (4) p p γ (3) . tr indicates the frequency and momentum integrations,while the discrete index contractions are carried out ex-plicitly. The corresponding integral equations for spe- cific external momentum configurations relevant for two-body scattering are given in Eq. (41) resp. Fig. 2.Γ (3)114 = 2 γ (3) holds under the assumption that the fullFeshbach vertex has the same structure in field space asthe classical one. The atom propagator is not renormal-ized in vacuum. Note that only the atom propagatorenters the diagrams for the dimer self energy and thesplitting vertex. Four-point functions – We further want to computethe fourth order interaction vertices.1) Second order, momentum space calculation – If werestrict to the order J , we can limit ourselves to two-loop order. We findΓ (4) αβγδ = S (4) αβγδ −
12 tr (cid:2) S (5) αβγµ µ G µ ν G ν µ Γ (3) δν ν + S (5) αβδµ µ G µ ν G ν µ Γ (3) γν ν (A7)+ S (5) αγδµ µ G µ ν G ν µ Γ (3) βν ν + Γ (5) βγδµ µ G µ ν G ν µ Γ (3) βν ν (cid:3) −
13! tr (cid:2) S (5) βγµ µ µ G µ ν G ν µ G µ ν Γ (5) αδν ν ν + S (5) βδµ µ µ G µ ν G ν µ G µ ν Γ (5) αγν ν ν + S (5) αβµ µ µ G µ ν G ν µ G µ ν Γ (5) γδν ν ν (cid:3) . Working at second order the full vertices are replacedby the classical ones. The dimer-dimer interaction thenevaluates toΓ (4)4343 = 4 γ (4) = − s (5) p p s (3) + s (3) p p s (5) ] − s (5) p p , p s (5) . (A8)The first two contributions are one-loop, the last one istwo-loop. Now we insert the appropriate momentum structures.At order J the J dependence of the atom propagatoris to be neglected and the dimer propagator equals thebare one, p ( ω ) = i ω − µ = i ω + | U | / , (A9) p ( ω ) = p , ( ω ) = i ω − µ + U = i ω. We then find, for zero external frequency,Γ (4)4343 = +16 J (cid:0) (cid:90) q p ( ω ) 1 p ( − ω ) (cid:2) [ (cid:15) q − q + (cid:15) q + q − q + (cid:15) q − q + (cid:15) q + q − q ][ (cid:15) q − q + (cid:15) q ] + { q ↔ q , q → q } (cid:3) − (cid:90) q,q (cid:48) p ( ω ) 1 p ( ω (cid:48) ) 1 p ( − ω − ω (cid:48) ) [ (cid:15) q − q + (cid:15) q − q (cid:48) + { q → q } ] [ (cid:15) q − q + (cid:15) q − q (cid:48) + { q → q } ] (cid:1) = 8 J | U | [ (cid:15) q + (cid:15) q + (cid:15) q + (cid:15) q + (cid:15) q − q + (cid:15) q − q ] = 4 γ (4) . The contribution to the effective action then reads∆ S = 2 J | U | (cid:90) q ,...,q δ ( q − q + q − q )) t † ( q ) t ( q ) t † ( q ) t ( q )[ (cid:15) q + (cid:15) q + (cid:15) q + (cid:15) q + (cid:15) q − q + (cid:15) q − q ] (A10)= 2 J | U | (cid:88) (cid:104) i,j (cid:105) (cid:2) t † ,i ( τ ) t ,j ( τ ) (cid:2) n ,i ( τ ) + n ,j ( τ )] + n ,i ( τ ) n ,j ( τ ) (cid:3) .
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