A Some aspects concerning a more general single-band Hamiltonian than the single-band Hubbard Hamiltonian

# Some rigorous results concerning the uniform metallic ground states of single-band Hamiltonians in arbitrary dimensions

## Abstract

We reproduce and review some of the main results of three of our earlier papers, utilizing in doing so a considerably more transparent formalism than originally utilized. The most fundamental result to which we pay especial attention in this paper, is that the exact Fermi surface of the -particle uniform metallic ground state of any single-band Hamiltonian, describing fermions interacting through an isotropic two-body potential whose Fourier transform exists, is a subset of the Fermi surface within the framework of the exact Hartree-Fock theory, in general to be distinguished from the one corresponding to a single-Slater-determinant approximation of the ground-state wave function. We also review some of the physical implications of the latter result. Our considerations reveal that the interacting Fermi surface of a uniform metallic ground state (whether isotropic or anisotropic) cannot be calculated exactly to order , with , in the coupling constant of the interaction potential in terms of the self-energy calculated to order in a non-self-consistent fashion. We show this to be interlinked with the failure of the Luttinger-Ward identity, and thus of the Luttinger theorem, for a self-energy that is not appropriately (e.g., self-consistently) related to the single-particle Green function from which the Fermi surface is deduced. We further show that the same mechanism that embodies the Luttinger theorem within the framework of the exact theory, accounts for a non-trivial dependence of the exact self-energy on that cannot be captured within a non-self-consistent framework. We thus establish that the extant calculations that purportedly prove deformation of the interacting Fermi surface of the -particle metallic ground state of the single-band Hubbard Hamiltonian with respect to its Hartree-Fock counterpart at the second order in the on-site interaction energy , are fundamentally deficient. In an appendix we show that the number-density distribution function, to be distinguished from the site-occupation distribution function, corresponding to the -particle ground state of the Hubbard Hamiltonian is not non-interacting -representable, a fact established earlier numerically. This property is of particular relevance in respect of the zero-temperature formalism of the many-body perturbation theory.

###### pacs:
71.10.-w, 71.10.Fd, 71.10.Hf, 71.27.+a
1

## I Introduction

The main purpose of this paper is to present an overview of some of the salient findings of Refs. (1); (2); (3) that have thus far not received the attention that we believe they deserve. Here we provide, with the advantage of hindsight, simplified demonstrations of these findings. For this purpose, in this paper we explicitly deal with the single-band Hubbard Hamiltonian (4); (5); (6), for which one has

 ˆH=∑k,σεk^a†k;σ^ak;σ+U2N\textscs∑σ,σ′∑k,p,q^a†k+q;σ^a†p−q;σ′^ap;σ′^ak;σ. (1)

Where appropriate, we shall indicate the way in which the results corresponding to this Hamiltonian are extended and made to correspond to a more general single-band Hamiltonian that accounts for an arbitrary isotropic interaction potential, Eq. (202) (2). In Eq. (1), is the non-interacting energy dispersion, which may or may not be of the strictly tight-binding form, the on-site interaction energy, and the number of the lattice sites on which is defined. Since we are interested in the -particle metallic ground state (GS) of , in the following will be macroscopically large.

For definiteness, we assume that is a Bravais lattice embedded in so that where we do not state otherwise, the summations over wave vectors, such as those in Eq. (1), are over the -dimensional first Brillouin zone (1BZ) corresponding to . The operators and are canonical annihilation and creation operators in the Schrödinger picture, corresponding to fermions with spin index . They are periodic over the complete wave-vector space, with the 1BZ the fundamental region of periodicity. This property is enforced by identifying, for instance, with , where is a reciprocal-lattice vector for which .

A byproduct of the considerations in this paper is some new (from the perspective of either Refs. (1); (2); (3) or other earlier relevant publications by others known to us) insights regarding a number of properties of the exact self-energy and the failure of non-self-consistent many-body perturbation expansions, to arbitrary order in the coupling constant of interaction, to reproduce these correctly. Of particular interest is our explicit demonstration of the vital role that satisfaction of the Luttinger theorem (7); (8); (9); (10); (11) plays in correctly, albeit qualitatively, reproducing the dependence of the exact on the coupling constant of interaction in approximate calculations.

### i.1 Generalities

The -particle uniform GS of for spin- fermions is characterized by two site-occupation numbers , where

 nσ=NσN\textscs,σ∈{↑,↓}, (2)

in which is the total number of particles with spin index in the GS. One has

 N=∑σNσ,n=∑σnσ≡NN\textscs. (3)

With assumed to be macroscopically large, a non-vanishing corresponds to a macroscopically large .

Let denote the -particle GS of and the corresponding eigenenergy, where is the spin index complementary to ; for , . In this paper, the numbers , and thus , are special in that is minimal with respect to variations of , , in . In contrast, in dealing with such eigenstates of as , we shall be merely considering the lowest-lying -particle eigenstates of corresponding to the specific to . In general, the exact -particle GSs of coincide with , where ((10), §B.1.1). It should be evident, however, that the eigenenergies are variational upper bounds to the energies of the -particle GSs of . Below we shall use and interchangeably, and similarly as regards and .

## Ii On the Fermi surface S\textscf;σ

In this section we introduce two energy dispersions, and , which we demonstrate to satisfy

 εk;σ,∀k∈1BZ, (4)

where is the chemical potential. The locus of the points of the for which and are up to a microscopic deviation of the order of equal, defines a -dimensional subset of the , which we denote by and which may in principle be empty. The inequalities in Eq. (4) imply that at any the energies and must be up to errors of the order of equal to . We demonstrate that not only is equal to the exact Fermi surface of the -particle metallic GS of , but also it is a subset of , the Fermi surface within the framework of the exact Hartree-Fock theory. For the cases where is a proper subset of , the difference set constitutes the pseudogap region ((1), §10) of the Fermi surface of the -particle metallic GS under consideration (1); (2). It is interesting to note that the property is known to be exact for the Hubbard model in (12) (13); (14); (15); (16), on account of the constancy of the self-energy in this limit with respect to variations of (14); (12).

### ii.1 Preliminaries

Defining

 μ−N;σ ≐ EN;0−ENσ−1,N¯σ;0, (5) μ+N;σ ≐ ENσ+1,N¯σ;0−EN;0, (6)

on account of the Jensen inequality (17)

 EN;0<12(EN−1;0+EN+1;0), (7)

one arrives at

 EN;0<12(ENσ−1,N¯σ;0+ENσ+1,N¯σ;0)⇔μ−N;σ<μ+N;σ. (8)

The Jensen inequality, signifying the strict convexity of as a function of (for the consequence of not being strictly convex, but merely convex, see Ref. ((10), §B.1)), applies for thermodynamically stable systems (18). Hence, the last inequality in Eq. (8) amounts to an exact relationship for the system under consideration, which by assumption is thermodynamically stable ((10), §B.1). One can more generally show that the chemical potential at zero temperature, , specific to the grand canonical ensemble of the states spanning the Fock space of , in which the mean value of the number of particles is equal to , satisfies the following inequalities ((10), §B.1):

 μ−N;σ<μ<μ+N;σ,∀σ. (9)

Since in this paper we are dealing with metallic -particle GSs, up to microscopic corrections of the order of ((10), §B.1) one has

 μ=ε\textscf=maxσμ−N;σ=minσμ+N;σ, (10)

where is the Fermi energy corresponding to the -particle metallic GS of . While deviating from the most general formulation ((10), §B.1), in the following we shall for simplicity assume that . In the following we shall therefore denote more concisely by .

With

 nσ(k)≐⟨ΨN;0|^a†k;σ^ak;σ|ΨN;0⟩, (11)

the GS momentum-distribution function corresponding to particles with spin index , for and we introduce the following normalized -particle states (1):

 |ΦNσ−1,N¯σ;k⟩≐1√nσ(k)^ak;σ|ΨN;0⟩, (12)
 |ΦNσ+1,N¯σ;k⟩≐1√1−nσ(k)^a†k;σ|ΨN;0⟩. (13)

With reference to the latter expression, we note that following the anti-commutation relation , one has

 ⟨ΨN;0|^ak;σ^a†k;σ|ΨN;0⟩≡1−nσ(k). (14)

With denoting the total-momentum operator ((19), Eq. (7.50)), making use of the property , one verifies that are eigenstates of corresponding to eigenvalues .

Defining (recall that for all (20))

 E′Nσ±1,N¯σ;k≐⟨ΦNσ±1,N¯σ;k|ˆH|ΦNσ±1,N¯σ;k⟩, (15)

by the variational principle one has

 ENσ±1,N¯σ;0≤E′Nσ±1,N¯σ;k,∀k∈1BZ. (16)

Hence, by writing

 E′Nσ±1,N¯σ;k≡ENσ±1,N¯σ;0+δE±(k), (17)

in the light of the inequality in Eq. (16) one has , . Introducing the single-particle energy dispersions (cf. Eqs. (5) and (6))

 εk;σ ≐ E′Nσ+1,N¯σ;k−EN;0, (19)

on account of , , the following inequalities apply (see Eqs. (4) and (9)):

 εk;σ,∀k∈1BZ. (20)

Making use of the expression for in Eq. (1), and of the canonical anti-commutation relations for and , one readily obtains that (1); (2)

 εk;σ=εk+Uβ>k;σ1−nσ(k), (21)

where

 βk;σ ≐ 1N\textscs∑σ′∑p,q⟨ΨN;0|^a†p−q;σ′^ap;σ′^ak−q;σ^a†k;σ|ΨN;0⟩ (23) ≡ n¯σ−β

### ii.2 The surface S′\textscf;σ and its relation to the exact Hartree-Fock Fermi surface S\textschf\textscf;σ

Let

 S′\textscf;σ≐{k∥εk;σ}, (24)

where, in the light of the inequalities in Eq. (20), is meant to signify an equality up to a microscopic correction of the order of . It cannot a priori be ruled out that the set may be empty.

From the expressions in Eq. (21) and the identity in Eq. (23), one deduces that

 εk;σ⟺β

Combining the equality on the right-hand side (RHS) of the in Eq. (25) with the first expression in Eq. (21), in view of the results in Eqs. (10) and (20) one immediately obtains

 εk;σ⟹εk+Un¯σ=ε\textscf. (26)

The in Eq. (26) and the in Eq. (25) should be noted. Thus, whereas is a necessary and sufficient condition for , is only a necessary condition for . With reference to the expression in Eq. (27) below, this is underlined by the right-most exact inequalities in Eq. (56) below.

We point out that for the exact Hartree-Fock self-energy corresponding to the -particle uniform GS of , one has (see Eq. (207))

 Σ\textschfσ(k)=1ℏUn¯σ, (27)

which is independent of . Thus, the on the RHS of Eq. (26) may be replaced by . Interestingly, one can demonstrate that in dealing with the -particle uniform GS of the Hamiltonian in Eq. (202) one similarly has (2)

 εk;σ⟹εk+ℏΣ\textschfσ(k)=ε\textscf, (28)

where non-trivially depends on for non-contact-type two-particle interaction functions, Eqs. (204) – (206). For the explicit expressions of the and specific to the -particle uniform GS of the in Eq. (202), the reader is referred to Eq. (6) in Ref. (2). Although in the present paper we are explicitly dealing with the -particle uniform GS of the Hamiltonian in Eq. (1), below we shall often denote by as a reminder that many of the results to be presented in this paper are applicable to the -particle uniform metallic GS of the more general Hamiltonian in Eq. (202) (2).

Defining

 S\textschf\textscf;σ≐{k∥εk+ℏΣ\textschfσ(k)=ε\textscf}, (29)

from the expressions in Eqs. (24) and (28) one immediately observes that

 S′\textscf;σ⊆S\textschf\textscf;σ. (30)

That is not necessarily identical to , is a direct consequence of the in Eq. (28) (or Eq. (26)).

We should emphasize that at this stage of the considerations, differs from the Hartree-Fock Fermi surface corresponding to particles with spin index , by the fact that the in Eq. (29) is the exact Fermi energy, Eq. (10), to be in principle distinguished from the Fermi energy within the framework of the exact Hartree-Fock theory. The latter energy is obtained by solving the following equation:

 1N\textscs∑σ∑k∈1BZΘ(ε\textschf\textscf−[εk+ℏΣ\textschfσ(k)])=n. (31)

Later, Sec. II.6, we shall demonstrate that

 ε\textschf\textscf=ε\textscf. (32)

For now we only mention that the exact is an explicit functional of the exact , Eqs. (204) – (206), to be distinguished from , the GS momentum-distribution functions corresponding to the single-Slater-determinant approximation of the -particle GS of (for which one has , ), so that violation of the equality in Eq. (32) would amount to an internal inconsistency in the exact Hartree-Fock theory (a possibility that cannot a priori be ruled out): in the event of the equality in Eq. (32) failing, on replacing the on the left-hand side (LHS) of Eq. (31) by , the equality in Eq. (31) would fail to hold. This failure should be viewed in the light of the fact that, following the defining expression in Eq. (11), one has

 ∑k∈1BZnσ(k)=Nσ,∀σ, (33)

where are the exact partial particle numbers corresponding to , Eq. (3). We remark that in general is not equal to its counterpart within the framework in which the -particle uniform GS of is approximated by a single Slater determinant, except when and its approximation are paramagnetic, for which one has ; following the equality in Eq. (27), in this case the approximate and exact Hartree-Fock self-energies coincide.

We note in passing that the equality in Eq. (31) amounts to the statement of the Luttinger theorem (7); (8); (9); (10); (11) within the framework where is the total self-energy. With reference to the expression in Eq. (118) specialized to the case of , one observes that indeed in this framework the Luttinger-Ward identity (7) is satisfied.

### ii.3 The exact Fermi surface S\textscf;σ and its relation to S′\textscf;σ

Since and are variational single-particle excitation energies at point , in view of the inequalities in Eq. (20) it trivially follows that

 S′\textscf;σ⊆S\textscf;σ, (34)

where denotes the exact Fermi surface specific to particles with spin index of the metallic -particle GS under consideration. For clarity, is by definition the locus of the points of the at which the single-particle excitation energies, as measured from , are microscopically small, of the order of . With denoting the energy-momentum representation of the exact proper self-energy operator pertaining to the GS under consideration, the exact Fermi surface is mathematically defined according to (cf. Eq. (74))

 S\textscf;σ≐{k∥εk+ℏΣσ(k;ε\textscf)=ε\textscf}. (35)

We note that for all ((10), §2.1.2).

By the Luttinger theorem (7); (8); (9); (10); (11), for the -particle uniform GS under investigation one has (cf. Eqs. (31), (78) and (79))

 1N\textscs∑σ∑k∈1BZΘ(ε\textscf−[εk+ℏΣ\textschfσ(k)+ℏSσ(k)])=n, (36)

where we have used the decomposition (2)

 Σσ(k;ε\textscf)≡Σ\textschfσ(k)+Σ′σ(k;ε\textscf)≡Σ\textschfσ(k)+Sσ(k), (37)

in which, by the Kramers-Krönig relationship, (2)

 Sσ(k)=1π∫∞0dεεIm[Σσ(k;ε\textscf+ε)+Σσ(k;ε\textscf−ε)]. (38)

Since for and for , ((10), §2.1.2), one observes that is comprised of two competing contributions, so that the possibility of for some cannot a priori be ruled out. In fact, in the light of the decomposition in Eq. (37) and the expressions in Eqs. (29) and (35), the fundamental relationship in Eq. (44) below implies that

 Sσ(k)=0,∀k∈S\textscf;σ. (39)

Thus, following the expressions leading to the result in Eq. (39), one can write

 S\textscf;σ=S\textschf\textscf;σ∩{k∥Sσ(k)=0}, (40)

and consequently

 S\textschf\textscf;σ∖S\textscf;σ=S\textschf\textscf;σ∩{k∥Sσ(k)≠0}. (41)

With reference to the remarks in the opening paragraph of Sec. II, the set in Eq. (41), if non-empty, amounts to the pseudogap region ((1), §10) of the Fermi surface of the -particle uniform GS of (1); (2). We note in passing that following the result in Eq. (39), the vector , when it exists, stands normal to for all (2).

The result in Eq. (39) gains additional significance by comparing the expressions in Eqs. (31) and (36), taking into account the equality in Eq. (32). For instance, one observes that the combination of for inside and for outside the Hartree-Fock Fermi sea, amounts to a sufficient condition for the validity of the Luttinger theorem (7); (8); (9); (10); (11) for the -particle uniform metallic GS of .

For illustration, by considering the data for in Figs. 1(b) and 1(c) of Ref. (21), taking into account the expressions in Eqs. (2.5), (2.14), (B.55) and (B.59) of Ref. (10), and the fact that in these figures is the origin of the energy axis, one can readily convince oneself that for at least the calculations reported in Ref. (21) the function behaves as described above. Explicitly, noting that the results displayed in Figs. 1(b) and 1(c) of Ref. (21) correspond to the non-interacting energy dispersion with , and the band-filling , it is evident that for instance is located outside the underlying Fermi sea. Because of the long tail of the corresponding to this for negative values of , it is evident that the function dominates the value of the in Eq. (38), resulting in . In contrast, with being located inside the underlying Fermi sea, by the same reasoning as above, from the long tail of the corresponding to this for positive values of , it follows that . From the data corresponding to one arrives at a similar conclusion, that , and further that is larger at than at . For the points close to the underlying Fermi surface, such as , one clearly observes that is nearly symmetrical with respect to the origin (insofar as is concerned, Eq. (209), nearly anti-symmetrical with respect to ), in conformity with the result in Eq. (39).

We now proceed by demonstrating that

 S\textscf;σ⊆S′\textscf;σ, (42)

which in conjunction with the relationship in Eq. (34) results in

 S′\textscf;σ=S\textscf;σ. (43)

On the basis of this and the exact relationship in Eq. (30), one arrives at the fundamental relationship (1); (2) (cf. Eq. (40))

 S\textscf;σ⊆S\textschf\textscf;σ. (44)

For demonstrating the validity of the relationship in Eq. (42), we consider the annihilation operator in the Heisenberg picture, which we denote by . From the Heisenberg equation of motion ((19), Eq. (6.29)),

 iℏ∂∂t^a\textschk;σ(t)=[^a\textschk;σ(t),ˆH]−, (45)

one readily obtains that the function , defined in Eq. (22), can be expressed as follows (2):

 β

where is the single-particle Green function corresponding to particles with spin index , defined according to ((19), Eq. (7.46))

 Gσ(k;t−t′)≐−i⟨ΨN;0|T{^a\textschk;σ(t)^a\textsch†k;σ(t′)}|ΨN;0⟩, (47)

where is the fermion time-ordering operator. One readily verifies that (2)

 ℏ∂∂tGσ(k;t−t′)∣∣∣t′=t+0+ = 1ℏ∫∞−∞dε2πieiε0+/ℏεGσ(k;ε) (48) ≡ 1ℏ∫μ−∞dεεAσ(k;ε),

where , to be encountered in Eq. (74) below, is the time-Fourier transform of , and the single-particle spectral function, defined according to

 Aσ(k;ε)≐∓1πIm[~Gσ(k;ε±i0+)]. (49)

Here , , is the single-particle Green function in terms of which the ‘physical’ Green function , , is defined according to the prescription in Eq. (209).

One has the following exact sum rules (1); (2); (10):

 1ℏ∫∞−∞dεAσ(k;ε)=1, (50)
 1ℏ∫∞−∞dεεAσ(k;ε)=εk+ℏΣ\textschfσ(k). (51)

For the latter sum rule, see Eq. (50) in Ref. (1); noting that the expression on the LHS of Eq. (51) is the function ((10), Eq. (B.68)), with reference to Eq. (B.72) in Ref. (10), see Eq. (73) in Ref. (22); also compare the expressions in Eqs. (162) and (173) of the latter reference.

Further, the GS momentum-distribution function, defined in Eq. (11), can be expressed as follows (cf. Eq. (208)):

 nσ(k)=1ℏ∫μ−∞dεAσ(k;ε). (52)

The exact result in Eq. (33), which follows from the defining expression in Eq. (11), is seen to hold on employing the expression on the RHS of Eq. (52), provided that the in this expression satisfies the inequalities in Eq. (9). Combining the result in Eq. (52) with that in Eq. (50), one deduces that (cf. Eq. (14))

 1ℏ∫∞μdεAσ(k;ε)=1−nσ(k). (53)

On the basis of the above observations, the expressions in Eq. (21) can be written as follows:

 εk;σ=1ℏ∫∞μdεεAσ(k;ε)1ℏ∫∞μdεAσ(k;ε). (54)

From these expressions and those in Eqs. (51), (52) and (53), one deduces the following identity (1); (2):

 nσ(k)εk;σ=εk+ℏΣ\textschfσ(k),∀k∈1BZ. (55)

On account of the exact property (as regards the equality, up to a correction of the order of ), Eq. (20), from the above identity one infers that

 0≤nσ(k)≤1⟺εk;σ. (56)

The inequalities on the LHS of the being true for all , it follows that the inequalities on the RHS of the must also be true for all . In this light, it is to be noted that the inequalities on the RHS of the are in full conformity with the result in Eq. (30) (see also Eq. (28)). Evidently, the right-most inequalities in Eq. (56) do not rule out the possibility that may indeed be a proper subset of .

Below we demonstrate the validity of the expression in Eq. (42). In doing so, we consider Fermi-liquid and non-Fermi-liquid metallic GSs separately. In the following we consider an arbitrary , which we denote by . As elsewhere in this paper, below by and we signify radial vectors whose end points are displaced infinitesimally from the end point of , with the endpoint of located inside and that of outside the underlying Fermi sea.

#### Fermi liquids

Here we assume that the -particle uniform metallic GS under consideration is a Fermi liquid (not necessarily a conventional one ((1), §11.2.2) (23)), for which one has (3)

 Missing or unrecognized delimiter for \big (57)

where is the Landau quasi-particle weight at . The equality in Eq. (57) follows on account of being infinitesimally small, whereby the incoherent parts of