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HD-THEP-01-29

arXiv:cond-mat/0107390v1 [cond-mat.str-el] 18 Jul 2001

Correlation functions for composite operators in the Hubbard model

Tobias Baier1 , Eike Bick2 , Christof Wetterich3 Institut f¨ ur Theoretische Physik Universit¨ at Heidelberg Philosophenweg 16, D-69120 Heidelberg

Abstract We compute the correlation functions for antiferromagnetic and d-wave superconducting fermion bilinears in a generalized mean ?eld type approximation for the Hubbard model. For high temperature our explicit expressions show that homogeneous ?eld con?gurations are preferred for these composite bosons. Below a critical temperature we ?nd spontaneous symmetry breaking with homogeneous expectation values of the composite ?elds. Our results can be used to device a nonperturbative ?ow equation for the exploration of the low temperature regime.

1 2

e-mail: baier@thphys.uni-heidelberg.de e-mail: bick@thphys.uni-heidelberg.de 3 e-mail: C.Wetterich@thphys.uni-heidelberg.de

1

Introduction

In the attempt to come towards a better understanding of the properties and basic mechanisms of high temperature superconductors, the Hubbard model [1] has become one of the most studied models for systems with strongly correlated electrons. However, it has turned out that it is very di?cult to extract any thermodynamic properties of this model in the interesting range of temperatures, couplings and carrier densities even on a numerical level. What renders this problem so di?cult is the fact that the most prominent physical degrees of freedom (such as antiferromagnetic or superconducting behavior) emerge as a complicated momentum dependence of the e?ective multi-fermion couplings induced by ?uctuations and that di?erent length scales are involved. We suggest that an easier understanding can be gained if the interesting degrees of freedom are included in a more explicit way. In an earlier paper we have presented a reformulation of the 2d Hubbard model, the so called colored Hubbard model [2]. In this reformulation, the interesting physical properties are described by bosonic degrees of freedom. These bosonic ?elds correspond to fermionic composite operators in the appropriate antiferromagnetic or superconducting channels. In particular, this formulation allows us to extract the antiferromagnetic or superconducting properties from a calculation of the two point correlation functions of the bosonic ?elds instead of the more traditional investigation of quartic fermionic vertices. It thus becomes possible to apply standard calculation procedures to the problem. In this paper, we compute the one loop corrections to the bosonic propagators to get a ?rst impression of how antiferromagnetic and superconducting behavior come into play when ?uctuations are included. The intention to do this is twofold: First, we show that in the colored Hubbard model we are able to calculate a mean ?eld type approximation for the correlation functions for composite operators analytically. Our results are expected to be quantitatively reliable at high temperature if the interaction strength is not too large. Furthermore, they are expected to serve as a good qualitative guide even at relatively low temperature, where the complicated physics near the Fermi surface becomes dominant. We emphasize that the one loop calculation of the bosonic propagator accounts for contributions to the e?ective four fermion interaction which involve arbitrarily high powers of the coupling constant. Second, the calculations in this paper may serve as a starting point for a renormalization group analysis in the frame of the colored Hubbard model. Within an exact renormalization group approach [3] they allow us to motivate truncation schemes for the bosonic propagators by identifying the kinetic terms that emerge in our one loop calculation. Our work is organized as follows. In the next section, we present our results for the one loop correlation functions of the two most prominent composite degrees of freedom of the Hubbard model in the antiferromagnetic and d-wave-superconducting 2

channels. We interprete the relevance of these results for the onset of spontaneous symmetry breaking. In the following two more technical sections, we give a brief review of the colored Hubbard model and present a mean ?eld type approximation for the propagators of a whole set of bosonic ?elds. Additionally, in sect. 5 we give the high temperature limit, which has a particularly simple form and is well suited for the truncation ansatz for the renormalization group analysis mentioned above. Sect. 6 sketches brie?y how nonperturbative ?ow equations can be derived from our results and sect. 7 contains our conclusions.

2

Correlation functions for antiferromagnetic and superconducting behavior

The Hubbard–model is de?ned for electrons on a lattice by the Hamiltonian H=

ij,σ

tij a? i,σ aj,σ + U

i

ni,↑ ni,↓

(1)

where a? i,σ and ai,σ are creation-/annihilation-operators for an electron at site i with spin σ and obey the usual anticommutation relations {a? i,σ , aj,τ } = δij δστ . The probability amplitude of an electron for tunneling from site i to site j is denoted by tij . We take a square lattice in 2 dimensions with tij = ?t for neighboring lattice sites and 0 otherwise. The interaction term ? U mimics a screened Coulomb–like interaction, with ni,σ = a? i,σ ai,σ the number operator. Before turning to the technical details, we would like to illustrate our point by discussing our results for antiferromagnetic and superconducting properties of the 2d Hubbard model on a square lattice. Obviously, antiferromagnetic or superconducting behavior is non-local in nature (e.g. antiferromagnetism emerges if spins of electrons situated on neighboring lattice sites are opposite in sign). We include this nonlocality of the most interesting physical degrees of freedom by dividing up the lattice into square plaquettes of 4 lattice sites each, which we enumerate clockwise. In particular, we de?ne electron-hole and electron-electron bilinears ?? (x)τ ψ ?1 (x) ? ψ ?? (x)τ ψ ?2 (x) + ψ ?? (x)τ ψ ?3 (x) ? ψ ?? (x)τ ψ ?4 (x) a ?(x) = ψ 1 2 3 4 ?(x) = i[ψ ?1 (x)τ2 ψ ?2 (x) ? ψ ?2 (x)τ2 ψ ?3 (x) + ψ ?3 (x)τ2 ψ ?4 (x) ? ψ ?4 (x)τ2 ψ ?1 (x)], d (2)

where x = (2an, 2am), n, m ∈ , is the position vector of the corresponding plaque? tte (cf. ?g. (1)), with a the lattice distance. Nonzero expectation values of a ? or d correspond to antiferromagnetic and (d-wave)-superconducting states, respectively. In [2] we have shown that it is possible to rewrite the original partition function of the Hubbard model in a partially bosonized form by introducing a set of fermion bilinears 3

?Γ(2) a

0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8 -2 3π 0 π 2π

?Γ(2) d

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 3π 0 π 2π

k1

2π

3π

π 0

k2

k1

2π

3π

π 0

k2

Figure 2: The one loop correction ?Γ(2) to the bosonic kinetic term in the e?ective action for the boson a and the boson d at high temperature, for T /U = 10, t/U = 1, h2 /U = 10. similar to the two we wrote down explicitely in eq. (2). The full set of fermion bilinears is given in appendix B. Additionally, to preserve translational invariance ?c (x), c = on the original lattice, we introduce a color index for the bilinears a ?c (x), d 1 . . . 4. Di?erent values of c indicate a shift of the corresponding bilinear in position space by one lattice unit a. The exact de?nition is provided in appendix B. In the ?c , d ?c , . . . partially bosonized reformulation of the Hubbard model the bosonic ?elds a ? couple to their fermion–bilinear counterparts a ?c , dc , . . . and therefore describe the intended degrees of freedom explicitly. Before ?uctuations are included the e?ective bosonic interactions are purely quadratic. The inverse bosonic propagators contain only the bare mass term which is the same for all bosons. It is of great interest to investigate the behavior of the quadratic (kinetic) bosonic terms once ?uctuations are included. They describe the propagation of the corresponding composite degrees of freedom. Possible instabilities and the onset of spontaneous symmetry breaking are indicated by zeros of the quadratic terms. In this paper we have calculated the one loop corrections to each of the bosonic propagators in our partially bosonized theory. They correspond to a “mean ?eld type” analysis where the e?ects of fermion ?uctuations are included whereas boson ?uctuations are omitted. All our results are for half ?lling (? = 0). In this section we concentrate on appropriate averages over the position index c (see eq. (25)) which we denote by a and d. The results can be expressed in terms of the quadratic piece in the e?ective action Γ for bosons which reads in momentum space (we take the lattice spacing a = 1/2)

π

Γ2,B =

m

T

?π

d2 k (2) d? (m, k ) 4π 2 + ?Γd (m, k ) d(m, k ) (2π )2 4

?Γ(2) a

0 -10 -20 -30 -40 -50 -60 3π 0 π 2π

?Γ(2) d

0 -5 -10 -15 -20 -25 -30 -35 -40 -45 -50 3π 0 π 2π

k1

2π

3π

π 0

k2

k1

2π

3π

π 0

k2

Figure 3: The one loop correction ?Γ(2) to the bosonic kinetic term in the e?ective action for the boson a and the boson d at low temperature, for T /U = 0.1, t/U = 1, h2 /U = 10. We observe large negative values for k = (0, 0) indicating instabilities. 1 + a(?m, ?k ) 4π 2 + ?Γ(2) a (m, k ) a(m, k ) + · · · . 2 (3)

For convenience, we suppress the dependence on the Matsubara frequency m in the following.

(0,2)

11 00 00 11

N

(1,2)

(2,2)

11 00 00 11

N

The inverse bosonic propagators Γd,a (k ) = 4π 2 + ?Γd,a (k ) are directly related to momentum dependent four fermion interactions of the form ?? (k ) Γ(2) (k ) d d

?1 (2)

(2)

(3,2)

11 00

(0,1) (0,0)

y

(1,1)

(2,1)

1 0 1 0 0 1

x

N

(3,1)

?(k ), d a ? (k ), . . . (4)

a

which arise from the exchange of the com?(k ) posite boson (see eq. (19)). Here a ? (k ), d (0,-1) (1,-1) (2,-1) (3,-1) are the Fourier transforms of color aver4 3 ages of the bosons in eq. (2) (see eq. (25)). (2) In particular, the vanishing of ΓB for a Figure 1: The introduction of plaquegiven momentum k corresponds to a diverttes and the enumeration of the colors. gence of the four fermion interaction in the corresponding channel. In the bosonic language this instability is easily interpreted: In addition to the quadratic terms (3) the ?uctuations also induce higher order bosonic interactions. They stabilize the e?ective scalar potential for large values of

1

11 00 00 11

2

(1,0)

(2,0)

11 00 00 11

(3,0)

a ?(?k ) Γ(2) a (k )

a

?1

11 00 00 11

N

a

5

3 2 1

10 8 6

(2) M2 a = Γa (0)

-1 -2 -3 -4 -5 -6 -7 0.14 0.16 0.18 0.2 0.22 0.24 0.26

(2) M2 a = Γa (0)

0

4 2 0 -2 -4 -6 -8 -10 1.4 1.6 1.8 2 2.2 2.4 2.6

T/U

(2)

T/U

Figure 4: The inverse propagator at zero momentum Γa (0) as a function of temperature. The uncertainty of our mean ?eld approximation is demonstrated by two di?erent e?ective couplings ha . The left plot is for h2 a /U = 10, the right plot for 2 ha /U = 40. We have indicated the lower bound for the critical temperature by a circle. a and d (see [2]). A negative quadratic term therefore implies that the minimum of Γ occurs for nonvanishing values of the corresponding bosonic ?eld and re?ects spontaneous symmetry breaking. In thermal equilibrium the inverse propagators ΓB depend on the temperature T . They typically decrease for decreasing T . The critical temperature Tc for a phase transition to an antiferromagnet or a superconductor is equal to (for second order (2) transitions) or above (for ?rst order transitions) the temperature for which ΓB (T ) (2) becomes negative ?rst. The ?eld for which ΓB becomes negative ?rst is likely to indicate the order parameter for the low temperature phase. Similarly, the value of (2) the momentum k for which ΓB reaches zero ?rst tells us about the preferred spatial dependence of the order parameter. If in the vicinity of the phase transition the (2) minimum of ΓB (k ) occurs for k = 0 we expect a homogeneous bosonic expectation value. Such a scenario would lead to a considerable simpli?cation since the dominant nonlocality of our problem would be absorbed in the de?nition of appropriate bosons as nonlocal fermion bilinears (2). Our results for the inverse bosonic propagators ΓB , (B = a, d), are shown in ?gs (2) 2–4. In ?gs. 2 and 3 we have plotted ΓB as a function of the bosonic momentum k for di?erent temperatures. We choose the Matsubara frequency ωm = 0 of the bosons which is the dominant term in the propagator. The parameter h2 /U re?ects a freedom in the choice of bosonization [2] and will be explained in section 3. What can we learn from these plots? First of all, observe that for high temperature (?g. 2) we get a simple momentum dependence. In this limit the one loop corrections 6

(2) (2)

are of the form ? cos2

k1 4

cos2

k2 4

for a and ? cos2

k1 4

+ cos2

k2 4

for d, namely

?Γ(2) a (k ) = ?

2h2 a [cos(k1 /4) cos(k2 /4)]2 , T h2 (2) ?Γd (k ) = ? d [cos2 (k1 /4) + cos2 (k2 /4)]. 2T

(5)

This has the nice consequence that the loop corrections in the high temperature limit do not change sign when varying parameters or momenta4 . The periodicity 4π in ki re?ects the original lattice on which the Hubbard model is formulated. The momentum range of the bosons, however, is given by ?π ≤ ki ≤ π so that the propagator for each boson has only one single minimum at k = 0. When T is lowered, the physics near the Fermi surface comes into play (in our formulation, the Fermi surfaces are given by k1 = ±k2 mod 2π ) and the momentum dependence becomes more complicated (?g.3). We ?nd that the minimum of the one loop corrections to the bosonic term in the e?ective action is situated at k = 0 for all values of the temperature. Remarkably, the (2) bosonic propagator (4π 2 + ?ΓB )?1 diverges at low temperature for small values of |k |. This has important consequences once the bosonic ?uctuations (omitted so far) (2) will be included. The propagation of momentum modes near zeros of ΓB is strongly facilitated. One therefore expects that the physical behavior of the system should be dominated by ?uctuations around such zeroes. Our result for the momentum dependence of the bosonic propagator is quite pleasing, because momentum modes with k ≈ 0 correspond to (almost) homogeneous bosonic ?eld con?gurations in position space. For the minimum of ΓB (k ) at k = 0 the relevant parameters for the onset of instabil(2) 2 2 ity are the mass terms MB = ΓB (k = 0). Negative values of MB indicate instability of the “symmetric” state without antiferromagnetism and superconductivity. The one loop result for the mass term can also be inferred from [2]: (ci = 2t cos(qi /2))

π 2 Ma (2)

= 4π ?

2

2h2 a

?π

d2 q (2π )2

tanh

? =±1

+ ? c2 ) . c1 + ? c 2

(2)

1 (c1 2T

(6)

2 In ?g.4 we show the temperature dependence of the mass term Ma = Γa (0). For our 2 parameters the change of sign of Ma indicates a second order phase transition. We have indicated the critical temperature with a circle in ?g.4. This result coincides with the diagrams in [2] for ρ = 0 (or ? = 0) which were derived by using a mean ?eld approximation with constant homogeneous background ?elds.

4 This feature will facilitate the regularization of the bosonic propagators in a renormalization group approach.

7

In the same way as for a and d we have calculated the one loop corrections to each of the bosonic propagators in our partially bosonized theory [2]. As we have included 16 di?erent boson species and each boson carries an additional color index, we get a symmetric matrix of 64(64 + 1)/2 di?erent kinetic terms in the e?ective action. It turns out that the inverse propagators become minimal for homogeneous ?eld con?gurations. Thus it is tempting to analyze ?rst the matrix entries in the limit of homogeneous ?elds. We ?nd that under the assumption of homogeneous ?eld con?gurations most of the entries vanish and nonvanishing mixings between di?erent bosons occur mainly between bosons of the same species but di?erent color (there are some nonvanishing contributions mixing complex bosons of di?erent species which we will discuss below). Diagonalising the 4 × 4 propagator matrices in color space for a given boson species allows us to rewrite the 64 × 64-propagator matrix of the bosons in a block diagonalised form. For general non homogeneous ?eld con?gurations we ?nd that the propagator matrix remains diagonal in color space. However, additional non vanishing terms occur in the propagator matrix mixing bosons of di?erent species and same color. For this set of bosons with propagator matrices which are diagonal in color space we have calculated the one loop correction also for the propagators at nonvanishing momenta which corresponds to the e?ective action for non homogeneous bosonic ?eld con?gurations. We ?nd that the momentum dependence of ?ΓB in the high temperature limit is given by some linear combination of cos2 k41 , cos2 k42 and cos2 k41 cos2 k42 also for the other bosons. The discussion above thus generalizes to the other bosons as well. In conclusion, the explicit notion of nonlocal degrees of freedom in the colored Hubbard model yields a very powerful tool to discuss spontaneous symmetry breaking in a very simple and illuminating way. Although our one loop results for strong interactions will not produce a quantitatively correct picture at low temperature, they nevertheless give a ?rst impression of how the composite ?elds associated with antiferromagnetism and superconductivity propagate.

3

The colored Hubbard model

The colored Hubbard model is a reformulation of the Hubbard model for electrons on a square lattice with next neighbor hopping5 . Let us brie?y review the basic relations in the colored Hubbard model as described in [2]. The partition function6 reads Z = exp ?

5 6

2 π 2 V2 ? 2 h2 ρT

?? D ψD ? u Dψ ?? D u ?D w ? exp

? (Skin + SY + Sj )

(7)

It can also be used for a generalization of the Hubbard model [2] See appendix A for details of the Fourier transforms. We set the lattice distance of the coarse lattice to unity: 2a = 1.

8

with Skin =

Q

?? (Q)P ψ (Q)ψ ?b (Q) ψ a ab +π 2

β

u ?? uβc (Q) + βc (Q)?

′ ′′

π2 2

γ

w ?γc (?Q)w ?γc (Q)

SY

= ?

QQ′ Q′′

δ (Q ? Q ? Q )

?

uβ ′ ′′ ? ′′ ′ ?? (Q′′ ) ?? (Q′ )V uβ (Q′ , Q′′ )ψ ? ?βc (Q)ψ u ?? b a βc (Q)ψa (Q )Vab,c (Q , Q )ψb (Q ) + u ab,c β

+

γ

?b (?Q′′ ) ?? (Q′ )V wγ (Q′ , ?Q′′ )ψ w ?γc (Q)ψ a ab,c ?

? Jβc (Q)? uβc (Q) + Jβc (Q)? u? βc (Q) ?

Sj =

Q

β

γ

Lγc (?Q)w ?γc (Q) (8)

? ?a (Q) + ηa (Q)ψ ?? (Q) ? ηa (Q)ψ a

and inverse classical fermion propagators P ψ P ψ (Q) = iωn ? 2t cos(q1 /2) τ1 0 0 τ1 + cos(q2 /2) 0 τ1 τ1 0 . (9)

The momentum vector Q = (ωn , q ) involves the Matsubara frequencies ωn = 2πnT with n integer for bosons and half integer for fermions. The action S = Skin + SY + Sj ? ψ ?? describes a coupling of electrons and holes denoted by Grassmann variables ψ, ? 7 to bosonic ?elds u ?, u ? and w ? . The temperature T is a free parameter . The original four fermion interaction of the Hubbard model is encoded in the vertex factors V which are proportional to the Yukawa couplings h with h2 ? U . It can be recovered by solving the ?eld equations for the bosons as functionals of the fermions. The sums over β run over the set of complex bosons, the sum over γ over the set of real bosons. These sets are listed in appendix B and de?ned more precisely in [2]. We use a and b for fermion color and V are also matrices in spinor space. Respectively, c denotes bosonic color. The partition function eq. (7) reduces to the one of the Hubbard model if the Yukawa couplings are chosen appropriately (cf. appendix B). The explicit vertices corresponding to the Hubbard model eq. (1) are shown in appendix C. ? ?] in the partially bosonized formulation is introduced by The e?ective action Γ[ψ, the usual Legendre transform ? ?] = ? ln Z [? Γ[ψ, η, K ] +

7

?+ η ?ψ

K?

(10)

A nonvanishing chemical potential ? could be included in the source of ρ ? as Lρc = L′ ρc + ?

9

? and ? by where η ? and K are expressed in terms of ψ ? = δ ln Z , ψ δη ? ?= δ ln Z . δK (11)

? and sources η ? includes ψ , Here we use a collective notation for fermionic ?elds ψ ? (ψ ? ? ψ ,η ? includes η , η ) and for bosonic ?elds ? and sources K (? includes the real and complex bosons). The mapping of our results for the partially bosonized e?ective ?] of the one particle irreducible correlation action to the generating functional Γ(ψ) [ψ functions within the original fermionic theory is straightforward [5]. In presence of arbitrary bosonic sources K we may de?ne ? ?, K ] = Γ[ψ, ? ?] ? Γ(K ) [ψ, The ?eld equations for ? δ Γ(K ) δ? =0

? ψ,K

K?.

(12)

(13)

? K ]. The e?ective action in the purely fermionic language is obis solved by ?0 [ψ, tained by insertion of the solution ? K ] = Γ(K ) [ψ, ? ?0 [ψ, ? K ], K ] Γ(ψ) [ψ, This is easily veri?ed by the fermionic ?eld equation δ Γ(ψ) ? δψ =

K

(14)

δ Γ(K ) ? δψ

+

?,K

δ?0 ? δψ

K

δ Γ(K ) δ?

=η ?

? ψ,K

(15)

where the rhs has to be evaluated for ? = ?0 so that the second term vanishes. In the approximation of this paper ? ?] = Γ[ψ, 1? ?B + 1 ?σ (Γ(2) )στ ?τ ? VAB,σ ψ ?A ψ ?B ?σ ψA (P ψ )AB ψ B 2 2 (16)

the classical solution of (13) for K = 0 reads

1 ? ? ?0σ = (ΓB )? στ VAB,τ ψA ψB (2)

(17)

and we obtain the 1PI-generating functional in the fermionic language ?] = 1 ψ ?A (Pψ )AB ψ ?B ? 1 VAB,σ (Γ(2) )?1 VCD,τ ψ ?A ψ ?B ψ ?C ψ ?D . Γ(ψ) [ψ B στ 2 2

(2)

(18)

Our computation of ΓB therefore amounts to an approximation to the 1PI-four fermion vertex which can be compared with the results from other methods [4]. 10

Finally we would like to stress the simple connection between ΓB and the fermionic (2) ? (x)B ? (y ) ?1 , where B ? ∈ 4-point Green functions. First note that ΓB = B conn ? d, ? . . . }. The right hand side is the inverse of the the connected bosonic 2-point {a, function. From the explicit expressions for the original fermionic partition function and our partially bosonized partition function [2], one ?nds ? ? (X )B ? B (Y ) B A

conn.

(2)

=

1 hA hB δAB δ (X ? Y ) + 2 π (4π 2 )2

? ? (X )B ? B (Y ) ? B ? ? (X ) B ? B (Y ) B A A (19)

? and B ? . As a second order phase where hA , hB are the Yukawa couplings between B transition into the antiferromagnetic or d-wave superconducting phase occurs for (2) diverging correlation length, we expect ΓB to vanish when such a phase transition takes place.

4

One loop corrections

In the partially bosonized formulation eq. (7) the action S = Skin + SY + Sj is purely quadratic in the fermionic variables (in absence of the fermionic sources, η = η = 0). The fermionic part of the functional integration is therefore Gaussian and can be performed in a standard way. For this purpose it is convenient to de?ne ?(Q) = ψ ?(Q) ψ ?? (Q) ψ . (20)

so that the fermionic part of the action takes the form Sψ = ?0 = P 1 ?(Q′ )P ?(Q′′ ), ? (Q′ , Q′′ )ψ ψ 2 Q′Q′′ 0 ?(P ψ )T Pψ 0 , ?= ?P ?=P ?0 ? ?P ? P C ?AT A B , (21)

A = V w (Q′ , Q′′ ) w ? (Q′ ? Q′′ ), B = 2V u (Q′ , Q′′ ) u ?(Q′ + Q′′ ), C = 2V u (Q′ , Q′′ ) u ?? (Q′ + Q′′ ). The interaction terms A, B , C are to be interpreted as matrices with momentum labels Q′ , Q′′ , fermion colors a, b and spinor indices inherited from the vertices V . Summation over the boson species and their indices is implied in the products V w ? etc. 11

?

Omitting the bosonic ?uctuations, the e?ective action8 becomes Γ = S + ?Γ, where the arguments of S are now the “background ?elds” ψ , u, w . The bosonic ?elds u, ? w appear in the contribution from the fermionic ?uctuations ?Γ = ?1/2 Tr ln P since A, B and C are linear in these ?elds. Therefore ?Γ is a functional of the bosonic ?elds. We emphasize that ?Γ accounts for the full one loop correction9 to the bosonic part of Γ since S does not contain purely bosonic vertices. We next expand the loop correction ?Γ in the number of boson ?elds 1 ? = ? 1 Tr ln[P ?0 (? ? P ? ?1?P ? )] ?Γ = ? Tr ln P 0 2 2 1 ?0 ? Tr(P ? ?1?P ? ) ? 1 Tr(P ? ?1?P ? )2 + · · · = ? Tr ln P 0 0 2 2

,

(22)

The ?rst ?eld independent term is discarded and the second linear “tadpole” term does not contribute to the propagators of interest here. It vanishes for half ?lling (? = 0). The bosonic propagator corrections are described by the third term. ?Γ2 =

QQ′

1 wγ wγc (?Q) tr (P ψ )?1 (Q′ )V,c (Q′ , Q + Q′ )(P ψ )?1 (Q + Q′ ) 2 V,c′γ (Q + Q′ , Q′ ) wγ ′ c′ (Q)

′

w

β ψ ?1 ′ ′ ′ ψ ?1 T ′ ?2u? βc (Q) tr (P ) (Q )V,c (Q , Q ? Q )((P ) ) (Q ? Q )

u

V,c′β (Q ? Q′ , Q′ ) uβ ′ c′ (Q) .

u? ′

(23)

Here tr refers to a trace over color and spinor indices and V,c denotes the matrix [V,c ]ab = Vab,c . We note that Q in ?Γ2 involves a summation over momenta and Matsubara frequencies m. We extract the inverse propagator matrices (K = (ωm , k )) ?Γ(2) wγc wγ ′ c′ (K ) =

Q wγ (Q, K + Q)(P ψ )?1 (K + Q)V,c′ γ (K + Q, Q) tr (P ψ )?1 (Q)V,c

′

w

?Γ(2) uβc uβ ′ c′ (K ) = ?2

8 9

(24)

u u? β′ ,c′

Q

tr (P ψ )?1 (Q)V,c β (Q, K ? Q)((P ψ )?1 )T (K ? Q)V

(K ? Q, Q)

Field independent additive constants are neglected in Γ. One loop corrections to Γ which involve fermion ?elds are not described by ?Γ.

12

These one loop expressions

w

involve a momentum integration over q and a sum over fermionic Matsubara frequencies n, with K = (ωm , k ), Q = (ωn , q). The computation of the inverse propagators Γ(2) (K ) is the aim of this note. We note that the one loop expression ?Γ(2) is ? h2 and therefore ? U . Retranslating our result to the fermionic language (eq. (4)) it corresponds to a resummation involving arbitrarily high powers of U .

4.1

Representation with respect to translations

When calculating the loop corrections for a given boson-species one observes that di?erent colors are mixed. It is favorable to use color combinations that render the propagator matrix diagonal. This can be achieved by using combinations which are simple representations with respect to translations in x- and y - direction, i.e. the color-combinations

1 1 (B1 + B2 ? B3 ? B4 ), B 4 = 4 (B1 ? B2 ? B3 + B4 ). B3 = 4 1 1 B1 = 4 (B1 + B2 + B3 + B4 ), B 2 = 4 (B1 ? B2 + B3 ? B4 )

?

K +Q K V V K Q 13

w

(25)

To motivate this choice, note that for homogeneous bosonic ?elds (Bc (x) = Bc (x + ex/y ) the group of translations by a = 1/2 in the x- and y -direction is isomorphic to G ≡ Z2 × Z2 . G is Abelian, has order 4 and therefore 4 irreducible (necessarily one dimensional) representations given by the parity of the two Z2 factors. Four linear independent basis functions for the irreducible representations of G which yield eigenvalues ±1 are easily written down and are exactly the B i . In particular, a homogeneous ?eld B 1 is invariant under translations by a. It turns out that the combinations eq. (25) diagonalize the propagator matrix in color space even in the case of nonvanishing momenta. This requires that the Fourier transforms are chosen as in appendix A. Writing eq. (25) as B a = Mab Bb /4, we de?ne the vertices in the new basis as B B = Mab V,b . With M = M T , M 2 = 4, the Yukawa interaction SY then looks the V,a

same as in eq. (7), with B → B and V B → V B . Hence the above calculation and (2) (23) stays the same with these replacements. The “classical” mass terms ΓB (k = 0) of the bosons B are given by 4π 2 .

4.2

Loop corrections to the bosonic propagators

We can now insert the explicit formulae for the vertices from appendix C and evaluate the inverse propagators for the various boson species. The boson ρ corresponds ? to the charge density ρ ? ? ψa ψa (see appendix B for further details). In the basis (ρ1 , ρ2 , ρ3 , ρ4 ) one ?nds the correction

(2) ?Γρ (K ) π

=

4h2 ρ

T

?π

d2 q (2π )2 (26)

r r diag [cos(k1 /4) cos(k2 /4)]2 g? (+, +) + g? (?, ?) , r r [cos(k1 /4) sin(k2 /4)]2 g? (+, ?) + g? (?, +) ,

r r ? [sin(k1 /4) sin(k2 /4)]2 g+ (+, +) + g+ (?, ?) , r r ? [sin(k1 /4) cos(k2 /4)]2 g+ (+, ?) + g+ (?, +)

.

Here we have de?ned the Matsubara sums (m, n ∈ Z, ?i ∈ {+, ?}):

r,c g? (?1 , ?2 ) = 3 n

(c1 + ?1 c2 )(c′1 + ?2 c′2 ) + ?3 ωω ′ [(c1 + ?1 c2 )2 + ω 2 ][(c′1 + ?2 c′2 )2 + ω ′ 2 ]

S1 (m, a?1 , b?2 ) S2 (m, a?1 , b?2 ) = (c1 + ?1 c2 )(c′1 + ?2 c′2 ) ± ?3 , 4 (πT ) (πT )2

(27)

which depend on m, k and q . The upper sign applies to real bosons (r) and the lower sign for complex bosons (c). The sums S1,2 can be found in eq. (50) in the appendix A. The frequencies ω , ω ′ appearing in the de?nition of g read ω = 2π (n + 1/2)T, ω′ = )T for real bosons 2 π (m + n + 1 2 1 2π (m ? n ? 2 )T for complex bosons (28)

For the arguments of S1,2 we use the abbreviations (?i ∈ {1, ?1}) a?i = (c1 + ?i c2 )/(πT ) b?i = (c′1 + ?i c′2 )/(πT ), where ci and c′i are given by ci = 2t cos(qi /2), c′i = 2t cos((ki + qi )/2) for real bosons . 2t cos((ki ? qi )/2) for complex bosons 14 (30) (29)

From eq. (26) it is obvious that only ρ1 receives a propagator correction for k = 0. This re?ects the fact that only ρ1 has a nonvanishing vertex in this limit. A similar formula is found for the boson p, if one replaces cos(ki /4) ? sin(ki/4), cos(k1 /4) ? sin(k1 /4), and for q y by cos(k2 /4) ? sin(k2 /4), hρ → hqy . hρ → hp . hρ → hqx Similarly, the expressions for q x are obtained by

In the case of the bosons with spin index, m, a, g x,y , one obtains the same result as for the above scalar bosons ρ, p, q x , q y since tr(τi τj ) = 2δij = δij tr?2 spin . The bosons s, c, t again receive a loop correction with the same structure as the corresponding scalar ones because the vertices are the same – one only has to multiply by ?2 and to replace c′1 , c′2 and ω ′ by the corresponding expressions for the complex bosons (30),(27). For the boson d one ?nds: ?Γd (K ) = ?2h2 d T

(2) π ?π

d2 q (2π )2 ?=±1

c diag [cos(k1 /4) cos((k2 ? 2q2 )/4) ? ? cos(k2 /4) cos((k1 ? 2q1 )/4)]2 g? (?, ?),

c ? [sin(k1 /4) sin((k2 ? 2q2 )/4) ? ? sin(k2 /4) sin((k1 ? 2q1 )/4)]2 g+ (?, ?), 2 c [cos(k1 /4) sin((k2 ? 2q2 )/4) ? ? sin(k2 /4) cos((k1 ? 2q1 )/4)] g? (?, ??),

c ? [sin(k1 /4) cos((k2 ? 2q2 )/4) ? ? cos(k2 /4) sin((k1 ? 2q1 )/4)]2 g+ (?, ??) .

(31)

From this we obtain the inverse propagator for the boson e by the replacements g?3 (?1 , ?2 ) → g?3 (??1 , ??2 ),

π

hd → he

Similarly, the propagator correction for the boson v x is

(2) ?Γvx (K )

=

?2h2 vx

T

?π

d2 q (2π )2 (32)

c c diag [sin(k1 /4) cos((k2 ? 2q2 )/4)]2 g? (+, +) + g? (?, ?) , c c [sin(k1 /4) sin((k2 ? 2q2 )/4)]2 g? (+, ?) + g? (?, +) ,

c c ? [cos(k1 /4) sin((k2 ? 2q2 )/4)]2 g+ (+, +) + g+ (?, ?) , c c ? [cos(k1 /4) cos((k2 ? 2q2 )/4)]2 g+ (+, ?) + g+ (?, +)

15

and for the boson v y one obtains

(2) ?Γvy (K ) π

=

?2h2 vy

T

?π

d2 q (2π )2 (33)

c c diag [sin(k2 /4) cos((k1 ? 2q1 )/4)]2 g? (+, +) + g? (?, ?) ,

c c [cos(k2 /4) cos((k1 ? 2q1 )/4)]2 g? (+, ?) + g? (?, +) , c c ? [sin(k2 /4) sin((k1 ? 2q1 )/4)]2 g+ (+, ?) + g+ (?, +)

c c ? [cos(k2 /4) sin((k1 ? 2q1 )/4)]2 g+ (+, +) + g+ (?, ?) ,

.

Though it is not always immediately apparent, the above expressions for (2) ?Γb (K, K ′ ) are symmetric under re?ection of the external momenta. Note furthermore that the sums Si are symmetric in the Matsubara frequency, Si (m, a, b) = Si (?m, a, b). Up to now, we did not consider the o? diagonal propagator terms involving bosons of di?erent species. Many of these terms are zero due to symmetry arguments: The real and the complex bosons do not mix because of the U (1)-symmetry (which can be interpreted as charge conservation) and the charge waves do not couple to the spin waves because of the SU (2)-symmetry acting on spins. Unfortunately, there is no symmetry prohibiting the mixing of the site and link pairs. The best thing one can do is considering the transformation ? ? ? ? ? ? ? ? ? ? ?ψ1 ψ1 ψ1 ψ1 ? ? ? ψ2 ? ? ? ? ? ? ψ2 ? ? ? ? ? (x, τ ) → ? ?ψ2 ? (x, ?τ ), ? ψ2 ? ( x, τ ) → ? ? (x, ?τ ) (34) ? ? ? ?ψ3 ? ψ3 ? ψ3 ? ? ψ3 ? ? ? ψ4 ψ4 ?ψ4 ψ4 which is no symmetry of the original fermionic partition function for ? = 0 (since this transformation yields a theory with a chemical potential with reversed sign). Nevertheless it is a symmetry of the fermionic e?ective action, since the latter does not depend on the chemical potential (this transformation reverses the sign of ρ ? and ρ ′ and correspondingly the sign of the associated source Lρc +?). To keep this symmetry in the partially bosonized e?ective action, we ?nd the corresponding transformation properties of the bosons Φ(x, τ ) → Φ(x, ?τ ), χ(x, τ ) → ?χ(x, ?τ ).

Formulating this in Fourier space, we ?nd that this symmetry tells us that the propagator matrix elements for mixing between site and link pairs are uneven functions of the Matsubara frequency m. In particular, for m = 0 these propagator matrix elements vanish. 16

Furthermore, we ?nd by explicit calculation that di?erent colors do not mix even for di?erent boson species. The only remaining o? diagonal terms in the inverse propagator occur between charge waves with equal colors and correspondingly spin waves with equal color as well as site pairs and link pairs with equal color. It is tempting to diagonalize further in these remaining non diagonal 4 × 4 propagator blocks. For nonvanishing momentum this is a highly nontrivial task und subject to current work. Su?ce to say that nearly all the non vanishing o?diagonal terms which couple di?erent species vanish in the limit of homogeneous ?elds. To be precise, only the link pairs of di?erent species couple even in the limit of homogeneous ?elds.

4.3

Mean ?eld results

In [2] we have evaluated the fermionic loop correction to the e?ective action for ?elds constant in time and space, thus obtaining the e?ective potential. In particular we have analyzed the bosons ρ1 , a2 and d1 . If one is mainly interested in antiferromagnetic and superconducting behavior we have now shown a posteriori that it was legitimate to choose exactly these color–combinations. They are stable under ?uctuations in the sense that the minimum of Γ(2) is at K = 0 and they have a nonvanishing coupling to the fermion bilinears. To make the connection between the present results for the bosonic propagator and the mean ?eld calculation of [2] more explicit it is interesting to investigate the above expressions for constant ?elds, i.e. in the limit of vanishing outer momenta. (2) 2 0 In this limit, we should have Γa (K = 0) = Ma = 2 ?U , where U0 is ?a2 ρ=d=a=?=0

2 the mean ?eld potential de?ned in [2] and correspondingly Γd (K = 0) = Md = ?U0 . ?d2 ρ=d=a=?=0 (2)

Indeed, as already anticipated in ?gure 4, the results are equal and we ?nd

π 2 Ma

= 4π ?

2

2

2h2 a

?π

d2 q (2π )2

π

tanh

?∈{+1,?1}

+ ? c2 ) c1 + ? c 2 + ? c2 ) (c1 ? ? c 2 )2 . c1 + ? c 2

1 (c1 2T

1 (c1 2T

(35)

2 Md

1 = 4π ? 2 h2 4t d

?π

d2 q (2π )2

tanh

?∈{+1,?1}

In a similar way, mass terms for the whole set (cf. appendix B) of bosons have been calculated in this limit. In ?g. (5) we show the mass of di?erent bosons as a function 2 2 of temperature. We choose h2 = h2 ρ = ha = hd (the couplings for the other bosons are then uniquely determined by the conditions in appendix B). Only those bosons are shown which reach zero mass at T /U > 0.01. It is reassuring to see that the most prominent degrees of freedom as ferromagnetic, antiferromagnetic, s- and dwave superconducting states, which are known to play a role in the Hubbard model 17

near half ?lling, emerge naturally in our framework as the candidates competing in determining the way of symmetry breaking. On the other hand, we see that because of the arbitrariness of the Yukawa couplings we cannot decisively identify the boson that wins the mass run to zero. Getting rid of the arbitrariness of the Yukawa couplings will be a signi?cant premise for correct predictions of phase transitions. This is also necessary for a quantitative comparison with other methods. Qualitatively, our results are compatible with renormalization group investigations of the four fermion interactions in the Hubbard model [4]. We stress that our formalism for partial bosonization is exact and the ?nal result has to be independent of the choice of the Yukawa couplings. This requires the inclusion of the bosonic ?uctuations which we have omitted in the present work. Also note that in our ?gures the bosons are treated as completely independent. The mass of every boson is calculated for a vanishing expectation value of all bosonic ?elds. Going beyond this assumption, one has to take into account the fact that in case of a transition to a phase with a nonvanishing order parameter, the corresponding boson no longer will have a zero expectation value. Often this nonzero expectation value will prevent the masses of the other bosons going to zero — a behavior encountered in [2] for the competition between antiferromagnetic and superconducting phases.

5

The high temperature limit

One of the goals of the one loop calculation was to ?nd a useful ansatz for the bosonic propagator that can be used in a renormalization group study of the Hubbard model. For T → ∞, the fermion loop corrections vanish and we end up with the classical bosonic mass terms. Letting T → 0, we have to face the complications of divergencies at the Fermi surface. Consequently, we have to lower T from ∞ to some temperature, that is low enough to reveal nontrivial physical behavior and high enough to keep this behavior so simple that we are able to write down well justi?ed, easy analytical approximations to the one loop expressions at this temperature. This aim in mind, we calculate a high temperature expansion of our one loop expressions, keeping as many terms in an expansion in T ?1 as is necessary for the result to be nontrivial. Because in our expressions the sum S1 always occurs with a factor T ?4 and S2 with a factor T ?2 , it will su?ce to expand S2 up to order T 0 if we want the result to be correct up to order T ?2 , which will turn out to be the lowest interesting order. We ?nd to this order only a contribution for m = 0 (S2 (m = 0, a, b) = O(T ?2 )) S2 (0, a, b) = resulting in

r g? (?1 , ?2 ) = 3

π2 + O (T ? 2 ) 4

(36)

?3 , (2T )2

c g? (?1 , ?2 ) = ? 3

?3 . (2T )2

(37)

18

40

20

M2 B

0

-20

-40 0 0.2 p m 40 20 0.4 0.6 0.8 1 1.2 1.4 1.6 c d1 1.8 2

T/U

a s

M2 B

0 -20 -40 0 2 4 p q1/2 m a 6 8 10 12 14 16 s c d1 18 20

T/U

Figure 5: Mass terms of di?erent bosons as a function of temperature. The upper plot is for h2 /U = 10, the lower plot for h2 /U = 40.

19

We will mainly be interested in bosons that have a nonvanishing loop–correction in the limit k → 0 and thus only list those below. In the high temperature limit one thus obtains: ?Γρ1 (K ) = ?

(2)

2h2 ρ [cos(k1 /4) cos(k2 /4)]2 δm0 . T

(38)

The same result applies — with appropiate replacements of the Yukawa couplings — to p2 , q x,4 and q y,3 and hence also to m1 , a2 , g x4 , g y3 . Noting the extra minus sign in g c relative to g r in the high temperature limit this also, apart from a factor of +2, applies to the result for the complex bosons s1 , c2 , tx,4 , ty,3 . For d and e one ?nds ?Γd,e (K ) = ?

(2)

h2 d,e δm0 2T diag [cos2 (k1 /4) + cos2 (k2 /4)], [sin2 (k1 /4) + sin2 (k2 /4)], [cos2 (k1 /4) + sin2 (k2 /4)], [sin2 (k1 /4) + cos2 (k2 /4)] .

(39)

Finally, for v x,2 v x,4 v y,2 and v y,3 we ?nd solutions with “stripes” in the x- or y direction

(2) ?Γvx,24 (K )

h2 = ? vx cos2 (k1 /4)δm0 2T 2 h v (2) ?Γvy,23 (K ) = ? y cos2 (k2 /4)δm0 . 2T

(40)

We want to stress that nearly all expressions which give a contribution in the limit k → 0 (except e2,3,4 , d2,3,4 ) possess minima at k = 0. This means that all physically important bosonic degrees of freedom have the tendency to prefer homogeneous ?eld con?gurations. As only the term in the Matsubara sum with m = 0 receives a correction in the high temperature limit, we de?ne ?Γ(2) (k ) ≡ ?Γ(2) (ωm = 0, k). (41)

The high temperature limit for the a and d bosons coincides with good accuracy with the graphs shown in ?g. (2). 20

6

Exact renormalization group

The inclusion of the bosonic ?uctuations and the exploration of small T can presumably best be achieved by using non-perturbative ?ow or renormalization group equations for the scale dependence of the e?ective action. This short section argues that the results of the present work may constitute a good starting point both for the formulation of the ?ow equations and the setting of “initial conditions” at microscopic scales. For the formulation of an exact renormalization group equation in the framework of the e?ective average action [3] we add to the action (8) an infrared cuto? ?k S =

Q

?? (Q)RkF (Q)ψ ?(Q) ψ +

β

(42)

wγ ? γ (?Q)RkB ? γ (Q) , (Q)w w

1 uβ ?? ? u β (Q)RkB (Q)uβ (Q) + 2

γ

where we adopt a vector notation for the ?elds, making the color indices implicit. Accordingly, RkF (Q) is a matrix in spin and color space, and RkB (Q) is a matrix in color space. The e?ective action Γ transmutes to the e?ective average action Γk with Γk=0 = Γ if Rk=0 = 0. For the bosons we suppress the low momentum modes by10 RkB (Q) = ZB,k Xk (Q)θ(Xk (Q)) (43)

Here the form of Xk (Q) can be adopted to the kinetic term in the inverse bosonic 2 (2) (2) propagator ΓB such that ΓB,k + RB,k = M k + k 2 becomes independent of Q for small Q2 . The precise form of Xk (Q) and the wave function renormalization ZB,k generically di?er for the various bosons. For the fermions we want to “regularize” the Fermi surface and therefore use an infrared cuto? that guarantees that the square of the inverse average propagator is always positive and di?erent from zero. The cuto? then acts like a gap in the fermion spectrum. Since the temperature has the desired properties, we propose RkF (Q) = 2πinkZF,k . (44)

This cuto? simply replaces the temperature T in the inverse fermionic average propagator T → T + k . The exact evolution equation for the scale dependence of Γk reads [3] 1 ?k Γk = STr ?k Rk (Γ(2) + Rk )?1 2

10

(45)

This cuto? resembles the opimized cuto? [6].

21

where Γ(2) is the matrix of second functional derivatives with respect to both bosonic and fermionic ?elds. The supertrace STr involves a momentum integration as well as sums over Matsubara frequencies and internal indices. It contains a minus sign for the fermions. The cuto? Rk is a block diagonal matrix, where each block is equal to one of the infrared cuto? functions introduced above. The inverse average (2) propagator (Γk + Rk ) becomes particularly simple for our proposal (43), (44). Due to the simple form of the fermionic cuto? we may use the results of this paper for a one loop calculation of Γk for high k . This simply replaces the prefactor T ?1 in ?Γ(2) by T /(T + k )2 in the results of sect. 5. For large enough k the approximation should be reliable and we can use this result as an “initial value” of the (functional-) di?erential equation (45) for large k . The aim will then be an approximate solution for k → 0 whereby the e?ective action is recovered.

7

Discussion and conclusions

The colored Hubbard model [2] is an equivalent reformulation of the usual Hubbard model in two dimensions as a Yukawa theory. The bosonic degrees of freedom couple to fermion bilinears which are chosen such that they re?ect the most interesting physical properties of the system, i.e. antiferromagnetic or superconducting behavior. This has the advantage of representing these properties explicitly as bosonic expectation values instead of dealing with complicated properties of fermionic interactions. A previous mean ?eld analysis [2] suggests the occurrence of spontaneous symmetry breaking in the antiferromagnetic or d-wave superconducting channels for low T . The present computation supports a crucial ingredient for such a mean ?eld calculation, namely that the composite bosonic condensates are spatially homogeneous. In this article, we have calculated the one loop corrections to the bosonic propagators analytically. The remaining momentum integrals cause trouble for low temperature, where the singularities near the Fermi surface come into play. On the other hand, we are able to perform the momentum integrals in a high temperature limit. We ?nd that the kinetic properties of the bosons emerge in order T ?2 . In this order, the momentum dependence of the propagators is extremely simple and shows a number of pleasant properties. The corrections to the inverse propagator Γ(2) are negative de?nite, which means that they do not change sign when the parameters (2) are varied. Furthermore ΓB becomes minimal for homogeneous ?eld con?gurations for all interesting boson species, which means that these ?eld con?gurations should be preferred by the system. This result yields a strong argument in favor of the basic assumption of the mean ?eld approximation in [2]. Our task to extract the 22

most interesting degrees of freedom of the Hubbard model from the complicated momentum structure of its vertices has been accomplished to some extent. A more accurate treatment of the low temperature properties of the colored Hubbard model may use exact renormalization group equations to analyze the ?ow of the Yukawa couplings and the e?ective potential. This should reveal the occurrence of spontaneous symmetry breaking in certain ranges of the density and the temperature [4]. A straightforward approach to this task is complicated by the problem to write down a suitable truncation for the e?ective average action, which on one hand should be simple enough to be tractable, but on the other hand should contain the interesting physical behavior. A common way to compromise these two demands is a truncation which copies the classical action, merely introducing wave function renormalization constants, ?ow dependent couplings and some terms for the e?ective potential in agreement with the symmetries of the system. Unfortunately, for the colored Hubbard model this approach is too simple, because the bosonic propagators of the classical action consist of mass terms only. Some kinetic behavior of the bosons obviously has to be included in the truncation. A priori it is far from clear in which way this should be done. We pursue here the way to calculate the one loop corrections of the bosonic kinetic terms to get an impression of how the bosons behave once ?uctuations are included. We ?nally want to comment brie?y on the formal status of our computation as compared to the ?rst order in a perturbative expansion of the Hubbard model in a purely fermionic language. On the one hand our computation goes beyond the perturbative result for the e?ective four fermion coupling. This is apparent from 2 (2) (2) eq. (18) if we realize V ? U 1/2 and ΓB = M + cU . An expansion of ΓB yields a contribution to the ψ 4 -interaction ? U 2 similar to ?rst order perturbation theory but also higher order terms which amount to a resummation. In particular we ?nd a (2) divergence of the four fermion interaction for a critical temperature (when ΓB vanishes) and the onset of instability towards spontaneous symmetry breaking. These features cannot be seen in perturbation theory. On the other hand, our calculation does not reproduce the complete perturbative result ? U 2 . This would require in addition the computation of V in order U 3/2 as well as of four fermion interactions ? U 2 which remain 1PI-irreducible even in the partially bosonized language. Our computation of the e?ective four fermion vertex can therefore be trusted only if the selected channels really dominate as suggested by the low temperature behavior. The present insu?ciency of our setting is also apparent from another perspective. Our results in order U 2 depend strongly on the choice of the Yukawa couplings h whereas no such parameter appears in the perturbative calculation in the fermionic language. This severely limits the predictive power of our appoach so far (cf. ?g. 4, 5). The present one loop calculation may be used for a determination of h by an optimization procedure. In fact, one may compute the four fermion vertex in selected channels (e.g. antiferromagnetic, d-wave-superconducting and charge density) both 23

by our “one loop result” and by perturbation theory. Since the “one loop result” depends on three Yukawa couplings those may be ?xed by matching the term ? U 2 to the perturbative result in these channels. This guarantees that the neglected corrections to h and 1PI-vertices are indeed small for the selected channels. In consequence a mean ?eld or renormalization group calculation with such an optimized choice of h would start close to perturbation theory for high T (at least for the selected channels) and nevertheless go far beyond at low T . One may then hope that further corrections in the other channels may only result in minor modi?cations. We hope that a combination of such an optimization of the choice of Yukawa couplings together with the study of non perturbative renormalization group equations may permit a reliable quantitative computation of the properties of the (colored) Hubbard model.

Appendix

A

A.1

Useful Formulae

Abbreviations

Q ≡ (ω n , q ), QX ≡ ωn τ + xq, ≡

0

X ≡ (τ, x), n∈ ≡T

ωn ≡ 2πnT,

β

bosons + 1/2 fermions

π

dτ

x

,

Q

X

n

d2 q 2 ?π(2π )

δ (Q ? Q′ ) ≡

1 δn,n′ · (2π )2 δ (q ? q ′ ) T δ (X ? X ′ ) ≡ δ (τ ? τ ′ ) · δ (x ? x′ )

(46)

Note that δ (q ? q ′ ) is periodic in 2π i.e. δ (q + 2π e ?i ) = δ (q ). The same applies to δ (τ ) = ±δ (τ + β ) for bosons/fermions. 24

A.2

Fourier transforms

? ψ ?? and bosonic ?elds B ?, B ? ? (generically For the transforms of the fermionic ψ, denoted by χ ?, χ ?? ) we use √ χ ? a (X ) = 2 a exp (i{QX + za q }) χ ?a (Q) χ ?? a (X ) = √

Q

2a

Q

exp (?i{QX + za q }) χ ?? a (Q)

(47)

As a consequence of the choice of za we ?nd simple transformation properties of the fermionic Fourier modes with respect to the translations by a in the x- and y -direction, Tx and Ty . With sx = (a, 0), sy = (0, a), ψ (Q) = (ψ1 (Q), . . . , ψ4 (Q)) one obtains τ1 0 ψ (Q) Tx ψ (Q) = eiqsx 0 τ1 Ty ψ (Q) = eiqsy 0 τ1 τ1 0 ψ (Q) (49)

a a a a z1 = ? , , z2 = , , 2 2 2 2 (48) a a a a z3 = , ? , z4 = ? , ? 2 2 2 2 We set the lattice distance of the coarse lattice (c.f. ?g. 1) to unity in our calculations: 2a ≡ 1.

and similarly for ψ ? (Q) with eiqs replaced by e?iqs .

A.3

Some sums

The following sums are useful when evaluating the Matsubara sum appearing in the one loop corrections to the bosonic propagator: (ωm = 2πmT, m ∈ ) 1 S1 (m, a, b) := 2 2 [(2n + 1) + a ][(2(n + m) + 1)2 + b2 ] n∈ = ) + a(4m2 + a2 ? b2 ) tanh( πb ) π b(4m2 ? a2 + b2 ) tanh( πa 2 2 , 2 ab[4m2 + (a + b)2 ][4m2 + (a ? b)2 ] S2 (m, a, b) :=

2 2 n∈ 2

(50)

(2n + 1)(2(n + m) + 1) [(2n + 1)2 + a2 ][(2(n + m) + 1)2 + b2 ] (51)

) + b(4m2 ? a2 + b2 ) tanh( πb ) π a(4m + a ? b ) tanh( πa 2 2 = 2 [4m2 + (a + b)2 ][4m2 + (a ? b)2 ] 25

B

B.1

Fermion bilinears and bosons

Naming scheme

Neutral real bosons charge waves R ρ, p, qx , qy spin waves S m, a, gx , gy Charged complex bosons site pairs Φ s, c, tx , ty link pairs χ e, d, vx , vy All these boson appear in four distinct colors. The corresponding fermion bilinears are designed by a tilde, e.g. ρ ?, p ?, etc.

B.2

Fermion bilinears

For each of the bosons b(X ) introduced above, we have a corresponding bilinear ? b(X ) which couples to the boson b(X ). Suppressing the dependence on X , we de?ne ?? ψ ? σ ?ab = ψ b a ? ?? ? ? ab = ψb τ ψa ?T (iτ2 )ψ ?a χ ?ab = ψ b ?? (iτ2 )ψ ?? T χ ?? = ?ψ

ab b a

(52)

and the composite bilinears ρ ?= σ ?11 + σ ?22 + σ ?33 + σ ?44 p ?= σ ?11 ? σ ?22 + σ ?33 ? σ ?44 q ?y = σ ?11 + σ ?22 ? σ ?33 ? σ ?44 q ?x = σ ?11 ? σ ?22 ? σ ?33 + σ ?44 s ?= χ ?11 + χ ?22 + χ ?33 + χ ?44 c ?= χ ?11 ? χ ?22 + χ ?33 ? χ ?44 ?y = χ t ?11 + χ ?22 ? χ ?33 ? χ ?44 ? ? ? ? ? =φ m 11 + φ22 + φ33 + φ44 ? ? ? ? ?=φ a 11 ? φ22 + φ33 ? φ44 ? ? ? ? ? =φ g y 11 + φ22 ? φ33 ? φ44 ? ? ? ? ? =φ g x 11 ? φ22 ? φ33 + φ44 e ?= χ ?12 + χ ?23 + χ ?34 + χ ?41 ?= χ d ?12 ? χ ?23 + χ ?34 ? χ ?41

?x = χ t ?11 ? χ ?22 ? χ ?33 + χ ?44

v ?y = χ ?12 ? χ ?34 v ?x = χ ?23 ? χ ?41

(53)

Note that qx/y , gx/y , tx/y are linear combinations of the de?nitions given in [2]. With these new de?nitions, all bosons have a simple transformation behavior under translations by a. 26

B.3

Yukawa couplings

Integrating out the bosons in (7) gives a purely fermionic theory. However, this theory coincides with the Hubbard model only under certain conditions for the π2 Yukawa couplings. These conditions are (with h2 b = 3 Hb U ) Hρ = 3(λ2 ? λ3 ) H m = 2λ1 + λ2 + 3λ3 + 1 Hp = 3(λ2 + λ3 ) H a = 2λ1 + λ2 ? 3 λ3 + 1 H g x = H g y = 2λ1 + λ2 + 1 = H q y = 3λ2

(54)

Hq x

3 Hs = Hc = Htx = Hty = λ1 , 2 where the parameter λi obey

2He = 2Hd = Hvx = Hvy = 6λ3 .

(55)

λi > 0 , λ 2 > λ3 , 2 λ1 + λ2 + 1 > 3 λ3 .

i = 1, 2 , 3 , (56)

B.4

Bosonic colors

The color index for the fermion bilinears and the bosons is de?ned by

?1 w ?1γ (x) = Ty Tx w ? γ (x ) , w ?2γ (x) = Ty w ? γ (x ) , ?1 w ?3γ (x) = w ?γ (x) , w ?4γ (x) = Tx w ?γ (x)

(57)

and similar for u ?, w, ? u ?, where the fermion bilinears without color index are given by the de?nitions in sec. (B.2).

C

Vertex factors for the Hubbard model

The vertices V w (Q′ , Q′′ ) for the bosons ρ ?, p ?, q ?x,y depend only on the momentum Q = Q′ ? Q′′ . With ex = (1, 0), ey = (0, 1) and za , a = 1 . . . 4, given in the appendix A, eq. (48), they can be written in the form

w w Vab,c (Q′ , Q′′ ) = Vab,c (Q) =

hw ?iza q izc q w e e Mab,c (Q) ? ?spin 2 , 4 27

(58)

w The color matrices Mc read ρ ρ (Q) = diag{1, 1, e?iey q , e?iey q }, (Q) = diag{1, eiex q , ei(ex ?ey )q , e?iey q }, M2 M1 ρ (Q) = diag{1, 1, 1, 1}, M3 q p ρ Mc (Q) = (?1)c?1 diag(1, ?1, 1, ?1) Mc (Q); q ρ (Q) = diag{1, eiex q , eiex q , 1}; M4

ρ qx ρ qx (Q) · diag(1, ?1, ?1, 1), (Q) = M2 (Q) · diag(?1, 1, 1, ?1), M2 (Q) = M1 M1 qx ρ qx ρ M3 (Q) = M3 (Q) · diag(1, ?1, ?1, 1), M4 (Q) = M4 (Q) · diag(?1, 1, 1, ?1). (59)

ρ ρ (Q) · diag(?1, ?1, 1, 1), (Q) · diag(?1, ?1, 1, 1), M2 y (Q) = M2 M1 y (Q) = M1 qy qy ρ ρ M3 (Q) = M3 (Q) · diag(1, 1, ?1, ?1), M4 (Q) = M4 (Q) · diag(1, 1, ?1, ?1);

The same can be obtained for the bosons with spin index, m, a, gx,y , by substituting → τ spin . ?spin 2 For the bosons s, c, tx,y one ?nds similarly (cspin = iτ2 ):

u Vab,c (Q′ , Q′′ ) =

?

hu iza (q′ +q′′ ) ?izc (q′ +q′′ ) u? e e Mab,c (Q′ , Q′′ ) ? cspin , 4 s? ρ c? p Mc (Q′ , Q′′ ) = Mc (?Q′ ? Q′′ ), Mc (Q′ , Q′′ ) = Mc (?Q′ ? Q′′ ),

t1 q1 Mc (Q′ , Q′′ ) = Mc (?Q′ ? Q′′ ),

? ? ′

t2 q2 Mc (Q′ , Q′′ ) = Mc (?Q′ ? Q′′ ),

′′

(60)

while d, e, vx,y are a bit more complicated. Let us de?ne eij = ei(zi q +zj q ) and a ?-product C = A ? B by Cij := Aij Bij (no sum over indices here!). One then obtains he ?izc (q′ +q′′ ) e? ′ ′′ e Mc (Q , Q ) ? cspin 8 ? ′′ ′′ 0 e12 e?iq ex 0 e14 eiq ey ′ ′′ ′ ′′ ? e21 e?iq ex 0 e23 ei[q ey ?(q +q )ex ] 0 e? M1 (Q′ , Q′′ ) = ? ′ ′ ′′ 32 i[q ey ?(q +q )ex ] 34 i[?ex q ′ +ey (q ′ +q ′′ )] ? 0 e e 0 e e ′ ′′ ′ ′′ e41 eiq ey 0 e43 ei[?ex q +ey (q +q )] 0 ? ? ′′ 0 e12 0 e14 eiq ey ′′ ? ? e21 0 e23 eiq ey 0 e? ?, M2 (Q′ , Q′′ ) = ? ′e ′ +q ′′ )e 32 iq 34 i ( q y ? ? 0 e e y 0 e e ′ ′ ′′ e41 eiq ey 0 e43 ei(q +q )ey 0 ? ? 0 e12 0 e14 ? e21 0 e23 0 ? e? ? M3 (Q′ , Q′′ ) = ? ? 0 e32 0 e34 ? , e41 0 e43 0 ? ? ′′ 0 e12 e?iq ex 0 e14 ′ ′′ ? e21 e?iq′ ex ? 0 e23 e?i(q +q )ex 0 e? ′ ′′ ? ?; M4 (Q , Q ) = ? (61) ′ ′′ ′ 0 e32 e?i(q +q )ex 0 e34 e?iq ex ? ′′ e41 0 e43 e?iq ex 0 Vce (Q′ , Q′′ ) =

?

?

? ?, ?

28

With the aid of the ?-product the other vertices can now be obtained from these ? ? 0 1 0 ?1 ? 1 0 ?1 0 ? d? ? ? M e? (Q′ , Q′′ ); Mc (Q′ , Q′′ ) = ? c ? 0 ?1 0 1 ? ?1 0 1 0 ? ? 0 0 0 ?1 ? ? 0 0 1 0 ? ′ ′′ e? vx ? (Q′ , Q′′ ) = ? Mc ? 0 1 0 0 ? ? Mc (Q , Q ) · λc , λ = (?1, 1, 1, ?1); ?1 0 0 0 ? ? 0 1 0 0 ? 1 0 0 0 ? v? e? ′ ′′ ? Mc y (Q′ , Q′′ ) = ? ? 0 0 0 ?1 ? ? Mc (Q , Q ) · λc , λ = (?1, ?1, 1, 1). (62) 0 0 ?1 0 ?ψ ?-vertices V u to the u ?? ψ ?? -vertices V u can be carried The transition from the u ?? ψ ?ψ out by

?

V u (Q′ , Q′′ ) → V u (Q′ , Q′′ ) = ?(V u (Q′ , Q′′ ))? = ?V u (?Q′ , ?Q′′ ).

?

?

?

References

[1] J. Hubbard, Proc. Roy. Soc. (London) A 276, 238, (1963); J. Kanamori, Prog. Theor. Phys. 30, 275, (1963); M. C. Gutzwiller, Phys. Rev. Lett. 10, 159, (1963) [2] T. Baier, E. Bick and C. Wetterich, Phys.Rev. B62, 23, 15471 (2000), condmat/0005218. [3] J. Berges, N. Tetradis, C. Wetterich, hep-ph/0005122; C. Wetterich, Phys. Lett. 301B, 90, (1993); Z. Phys. C48, 693, (1990); C 57, 451, (1993);C 60, 461, (1993) [4] C. Honerkamp, M. Salmhofer, cond-mat/0105218; C.J. Halboth, W. Metzner, Phys. Rev. B61, 7364 (2000), D. Zanchi and H. J. Schulz, Z. Phys. B103, 339, (1997); Europhys. Lett. 44, 235, (1998) [5] U. Ellwanger, C. Wetterich, Nucl. Phys. B423, 137, (1994); C. Wetterich, Z. Phys. C72, 139, (1996) [6] D. Litim, hep-th/0103195

29

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