The functional integral in the Hubbard model
TTHE FUNCTIONAL INTEGRAL IN THEHUBBARD MODEL
V.M. ZharkovInstitute for Natural Sciences,Perm State University, Perm, Russia, e-mail: [email protected] 29, 2018
Abstract
In a new functional integral approach proposed for the model, we findthe regime with a deformed integration measure in which the standard in-tegral is replaced with the Jackson integral. We indicate the relation to ap-adic functional integral. For the magnetic and electronic subsystems inthe effective functional that results from the operator formulation of theHubbard model, we find the two-parametric quantum derivative result-ing in the appearance of the quantum SU rq (2) group. We establish therelation to the one-parametric quantum derivative and to the standardderivative. The Hubbard model [1] remains the main object on which to probe new ap-proaches to strongly correlated systems. It is traditionally assumed that ef-fects of strong Coulomb repulsion play an important role in understandingmechanisms of high-temperature superconductivity, the physics of a metal-dielectric transition of the Mott-Hubbard type, and the related magnetic states[2]. The experimental discovery of spin liquid in an organic crystal ( BEDT − T T F ) Cu ( CN ) [3] made the situation even more complicated, establishingthe necessity of taking spin liquid into account in the phase diagram of themodel under study. It recently became clear that we need new approaches forhelp, in particular, in understanding properties manifested by materials andseparate molecules used in nanotechnology devices. Materials with a one- ortwo-dimensional conductivity band, for instance, polymers, are widely used inthese devices. For the TA-PPE polymer (the abbreviation of poly(p-phenyleneethynylene), jumps of two types, step and sawtooth, were observed [7] in thedependence of conductivity on the applied voltage.Jumps of the conductivity and magnetization were observed in other systemsand materials [4, 5, 6]. We proposed [8, 9] describing such a behavior of macro-scopic quantities using functions of the argument taking values in the p-adicnumber field. The argument of this function determines the value of the volt-age, while the numerical simulation has demonstrated that the function itself,1 a r X i v : . [ c ond - m a t . s t r- e l ] O c t Figure 1: Fig. 3. The dependence of the conductivity on the voltage for thepolymer TA - PPE: in the insets, we present magnified jumps in the ranges (a)from -4.0 V to -5.0 V and (b) from 4.5 V to 5.0 V.which depends on a one parameter, results in the two experimentally observeddependence types: step and sawtooth. Increasing the accuracy when calculatingthis function of the p-adic argument (we took the values p = 2 , for the p-adicnumbers) revealed the appearance of a nested systems of jumps in the conduc-tivity behavior. The physical pattern of such a behavior in strongly correlatedsystems was described in [9] on the qualitative level. It was mentioned that inthe regime of strong electron repulsion in one-dimensional systems, the electronsystem splits into fluctuating clusters containing several electrons each. As thetemperature and voltage increase, the viscous motion of these clusters taken asa whole results in processes of splitting them into smaller clusters. We thus ob-tain a hierarchical embedding of jumps of smaller amplitude into the structureof jumps with larger amplitude.This paper is devoted to justifying that a regime in which the studied macro-scopic quantities are described by functions of a p-adic argument can arise instrongly correlated multielectron systems with strong Coulomb repulsion. Wehere justify the following scenario: p-adic numbers appear through a deforma-tion of the integration measure depending on the deformation parameter q. TheJackson measure used when calculating macroscopic quantities appears in thefunctional integral in the strong correlation regime. In the case q = 1 /p , wherep is a prime, the Jackson integral becomes the p-adic integral [18]. We cantherefore use functions of a p-adic argument to describe jumps in the Hubbardmodel. In particular, we here continue the investigation of the new formulationof the functional integral that we proposed in [16, 17]. We note that this ap-proach differs substantially from functional formulations of the Heisenberg andHubbard models common in the foreign literature. Therefore, in what follows,we present a comparison with different versions of functional representations forcreation-annihilation operators proposed in various papers. We show that the2btained expressions for supercoherent states result in more complex compositeexpressions for the operator symbols than those provided by the slave-boson andslave-roton approaches [14, 15]. The expressions for the creation-annihilationoperators obtained here with various group reduction schemes in the Hubbardmodel first proposed in [16], result in all currently known formulations of thefunctional integral for systems with strong Coulomb interaction. Moreover, asshown in [17], the proposed formulation allows studying various cohomologiesof groups and supergroups and thus provides a controlled concretization andexpansion of dynamical groups and symmetries spontaneously appearing whenstrengthening the interaction in multielectron systems. Our proposed represen-tation for the effective functional and creation-annihilation operators is essen-tially nonlinear. This nonlinearity allows segregating those terms that providequantum derivatives determining generators of the quantum algebra in expres-sions belonging to the universal enveloping algebra. In [10], we introduced theapproximation in which radius vectors of vector fields were independent of thedynamical field coordinates. This assumption is equivalent to the approximationin which the values of the fields that correspond to the Casimir operators andto the invariants of the classical groups SU (2) and SU (1 , in our formulationare coordinate independent. Below, we demonstrate that these combinationsdetermine two parameters of deformation of the quantum Lorentz group, whichis the group of four-dimensional rotations. We treat the transition to the one-parameter quantum derivative in detail and show that the magnetic and electricsubsystems of the Hubbard model are described by deformed versions of nonlin-ear sigma models. We further calculate the part of the functional integral for theHubbard model that results from expressions containing this quantum deriva-tive. We realize our consideration on model contributions naturally present inthe Hubbard model. We consider the limiting cases with respect to the defor-mation parameter and calculate the functional integral in these limit regimes.We calculate the effective functional following from the kinetic energy of theHubbard model using supercoherent states containing no more than the firstpowers of fermionic fields. We fix the dynamical fields such that the residualexpressions indicate the difference between our approach and other approachesin the literature. The main distinct feature of our representation is its non-linearity, which allows introducing a two-parameter quantum derivative intothe problem. When passing to a one-parameter quantum derivative, we obtaina deformed nonlinear sigma model in the auxiliary dimension determined bythe scale factor. Further, by analogy with the diagram technique for atomicX-operators [2, 16], we consider the integral of the scale factor linear in thequantum number. This integral determines the integration measure. Directlycalculating the corresponding series, we show that in the limit case as the de-formation parameter tends to zero, this series converges to the Jackson integral.Obtaining deformations of the integration measure when varying the deforma-tion parameter is our main result. In the limit case of small q , the standardintegral becomes the Jackson integral. We briefly describe the general scheme for constructing the effective functionalfor the Hubbard model proposed in [10] and recently further developed, for3xample, in [17]. We begin with the Hubbard model written in terms of thestandard creation-annihilation operators, H = − W (cid:88) ijσ α + σ,i α σ,j + U (cid:88) i,σ n σ,i n − σ,i − µ (cid:88) σ,i n σ,i , (1)where α + σ,i , α σ,j are the creation and annihilation operators, n σ,i is the elec-tron density operator, W, U, µ are the respective width of the conductivity band,the one-site repulsion of two electrons, and the chemical potential, and σ deter-mines the spin value. We sum over the indices i and j labeling the lattice sites.For instance, for a one-dimensional lattice, we have ¯ N/ < i < N/ , where N is the total number of atoms. For electrons, we sum over the indices σ = ± / .In the weak correlation regime with a small on-site repulsion, we take the zerothapproximation for the kinetic energy and treat the repulsion perturbatively. Inthe "atomic" approach with strong repulsion, we take a one-site contribution ofthe form U n σ,i n − σ,i − µn σ,i . as the zeroth approximation. In what follows, we omit the index i in order toconcentrate on the internal structure of the appearing fiber bundle. This termis diagonal in the "atomic" basis and has the eigenfunctions and eigenvalues (cid:15) : | (cid:31) ; | + (cid:31) = α + ↑ | (cid:31) ; |− (cid:31) = α + ↓ | (cid:31) ; | (cid:31) = α + ↑ α + ↓ | (cid:31) ,(cid:15) = 0 , (cid:15) + = − µ, (cid:15) − = − µ, (cid:15) = U − µ. Calculating the matrix elements for α + σ , α σ in this basis, we obtain the fol-lowing matrix representation. For instance, α + ↑ is: α + ↑ = − = X +0 − X − . (2)The operators α + σ therefore contain two nonzero matrix elements. We introducethe operators X rs , r, s = 0 , + , − , containing only one nonzero matrix element.We then obtain the expansion of the creation and annihilation operators interms of the Hubbard operators [1]: α + ↑ = X +0 − X − , α + ↓ = X − + X , α ↑ = X − X − , α ↓ = X − + X +2 . Because we have a four-dimensional basis and the creation and annihilationoperators are expressed as 4x4 matrices, we have a basis comprising 16 opera-tors in the general case. From this set we remove the unity operators and theoperators of the form γ = − − .
4e use this notation for this operator because it coincides with the γ op-erator written in the chiral basis in quantum field theory (QFT). It also equals γ = ( X − X ) − ( X ++ − X −− ) . We can express the one-site Hubbard repulsion in terms this operator. Be-cause we can express it in terms of other operators, we do not take it intoaccount in what follows. The remaining operators on a site in the given basiscan then be separated into the fermionic operators of the form ( X , X − , X +0 , X − , X +2 , X − , X , X − ) , and the bosonic operators of the form ( X + − , X − + , X ++ − X −− , X , X , X − X ) . We omit the lattice index of these operators to indicate the possibility of ob-taining generators of the global dynamical algebra. We can endow the Hubbardoperators with the lattice index by taking N their copies, where N is the totalnumber of lattice sites, and constituting the direct product from these copies.A contemporary review of the atomic approach was presented in [11]. In termsof these operators, the Hubbard model becomes H = U (cid:88) i,r X rri − W (cid:88) ijαβ X − αi X βj (3)In what follows, we use the functional formulation for multielectron systemsproposed in [10]. The evolution operator between the initial and final statesin this formulation is given by the following functional integral with the actionexpressed in terms of the effective functional, which is to be calculated usingoperator expression (3) using the supercoherent states (we note that the indexlabeling the lattice in this formulation is lacking for the set of operators X rs ,which therefore constitute the set of generators of the dynamical superalgebra): < G f | e − iH ( t f − t i ) | G i > = (cid:90) | G ( t f ) > = | G f > | G ( t i ) > = | G i > D ( G, G ∗ ) e − iS [ G,G ∗ ] ; (4)The action here has the form S [ G, G ∗ ] = (cid:90) t f t i dt (cid:90) V d r < G ( r, t ) | i ∂∂t − H | G ( r, t >< G ( r, t | G ( r, t > (5)and the integration measure is D ( G, G ∗ ) = (cid:89) t i