Fermionic Hubbard Model the Hamiltonian of the Fermionic Hubbard Model Reads: ˆ X † X H = −T Aîσaˆjσ + U Nî↑Nî↓

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Fermionic Hubbard model The Hamiltonian of the fermionic Hubbard model reads: ˆ X † X H = −t aîσaˆjσ + U nî↑nî↓. (1) hi,ji,σ i

† Herea îσ/âiσ creates/annihilates a fermion with spin projection σ =↑, ↓ on site i of a certain lattice (most typically, the square lattice in 2D and cu- † bic lattice in 3D);n îσ =a îσaîσ; the symbol h...i restricts summation to neighboring lattice sites; t and U are, respectively, the amplitudes of the (nearest-neighbor) hopping amplitude and (on-site) repulsion. The fermionic Hubbard model is one of “standard models” of condensed matter physics. The metal-insulator transition at half filling hnî↑ +n î↓i = 1, along with the antiferromagnetism promoted by it, was the main context of the original formulation of the model and subsequent two decades of its intensive theoretical studies. The advent of high-temperature supercon- ductivity dramatically enhanced (and changed the focus of) the interest to Eq. (1). The model became paradigmatic for high-temperature supercon- ductors, at least as a minimalistic Hamiltonian featuring (not far from half filling) the relevant dx2−y2 Cooper instability, solely due to repulsive interaction between fermions. The third wave of interest to the model has been generated recently by its direct realization with ultracold atoms in optical lattices.

Classiﬁcation of Cooper channels. D4h nomenclature

Cooper instability leads to the anomalous averages haˆi1σ1 aˆi2σ2 i 6= 0. (In what follows, we use the integer-vector notation, i1 ≡ n1, i2 ≡ n2, conve- nient for taking into account translation symmetry.) In view of the SU(2)- invariance1 of the model Eq. (1), along with the (discrete) translation invariance, as well as the absence of spin-dependent hopping and spin-dependent interaction, we expect the following simple structure of the anomalous cor- relator:

haˆn1σ1 aˆn2σ2 i = Sσ1σ2 f(n1 − n2). (2) The dependence on the spin and spatial variables factors into a spin tensor

Sσ1σ2 and a scalar translation-invariant function of the two coordinates, f(n1−n2). The anti-commutativity of the fermionic operators in (2) requires

Sσ2σ1 f(−n) = −Sσ1σ2 f(n), (3) 1The invariance with respect to spin rotations.

1 implying two rather diﬀerent options—the so-called singlet and triplet pairings:

Sσ2σ1 = −Sσ1σ2 , f(−n) = f(n), (singlet) (4)

Sσ2σ1 = Sσ1σ2 , f(−n) = −f(n). (triplet) (5) With respect to macroscopic properties, the singlet case corresponds to the Bose-Einstein condensate (BEC) of spin-less bosons (referred to as singlet Cooper pairs2), while the triplet case is macroscopically equivalent to the BEC of spin-1 bosons (referred as as Cooper pairs in the triplet states).

The form of the tensor Sσ1σ2 in the singlet case is very simple. Up to a global factor that can alway be absorbed into the function f, there is no parametric freedom:

S↑↑ = S↓↓ = 0, S↑↓ = −S↓↑ = 1. (singlet) (6)

The triplet case is more subtle. Here the state can have a magnetic order, which should be reﬂected in the structure of Sσ1σ2 . The relations (2)–(10) can be also written in the quasi-momentum rep- resentation. Applying the lattice Fourier transform to the annihilations operators in (2), we see that

haˆk1σ1 aˆk2σ2 i = haˆk1σ1 aˆ−k1σ2 i δk1,−k2 , (7) and that

haˆkσ1 aˆ−kσ2 i = Sσ1σ2 f(k), (8) where f(k) is the lattice Fourier transform of f(n), implying, in particular, that for the singlet and triplet pairings we have

f(−k) = f(k), (singlet) (9)

f(−k) = −f(k). (triplet) (10) The symmetry of the function f(k) is supposed to reflect the symmetry of the Hamiltonian (1). All qualitatively different cases thus can be classified based on the point group and other symmetry properties of the Fermi Hubbard model on a given lattice. The point-group symmetries im- pose distinct constraints on the dependence of f(k) on the direction kˆ of the

2The notion of the Cooper pair here is essentially jargonic. By no means it should be interpreted as a compact two-fermion molecule. And it is really the “condensate of Cooper pairs” rather than the Cooper pair per se that has reasonably well deﬁned physical meaning.

2 vector k, while the dependence on the absolute value k normally has little to do with the symmetries, and will be omitted in our discussion of the D4h nomenclature in the next section.

D4h nomenclature

Consider 2D fermionic Hubbard model on the square lattice. In 2D, f(kˆ) ≡ f(θ), with θ the polar angle in the reciprocal space, and the desired symmetry classiﬁcation of the functions f(θ) is naturally done within the frame- work of Fourier series. The point group we are dealing with is called D4h. It involves the following symmetry operations.

Rotation by the angle π/2: f(θ) → f(θ + π/2), (11)

Reﬂections with respect to four special axes: the axis kx, the axis ky, the axes ky = ±kx. In terms of the θ-dependence, those operations read (respectively):

f(θ) → f(2θ0 − θ) , θ0 = 0, π/2, ±π/4, (12) and for briefness we will be referring to them as θ0-reﬂections, corresponding axes being labeled by the four values of θ0. Consider the eigenfunctions of the rotation by the angle π/2:

i πm fm(θ + π/2) = e 2 fm(θ) , m = 0, ±1, 2 . (13)

i πm The four diﬀerent eigenvalues e 2 follow from the condition of 2π-periodicity of the function f(θ), implying ei2πm = 1. To satisfy the requirement (13), the Fourier expansion of the functions fm should have the form (n is integer)

∞ X i(4n+m)θ fm(θ) = Cn e . (14) n=−∞

Note that the period of f0(θ) is π/2, the period of f2(θ) is π, while the period of f±1(θ) is generic. Applying reflections (12) to functions fm, we see—for example, from i πm (14)—that (i) for m = 0, 2, reflections preserve the (real) eigenvalue e 2 ; i πm (ii) for m = ±1, reflections change the (imaginary) eigenvalue e 2 to its negative (complex conjugate). In accordance with the theory of symmetry, this means that in each of the sectors m = 0 and m = 2 can be consistently classified not only with m, but also with a reflection eigenvalue. Below we

3 employ the subscript showing the eigenvalue for the reﬂection about the x- axis, corresponding reﬂectional eigenvalue (+1 or −1) being denoted with corresponding sign subscript: “+” or “−.” Then, combining the identity

fm,±(θ) ≡ [fm,±(θ) ± fm,±(−θ)]/2 (m = 0, 2) (15) with (14), we get

∞ X fm,+(θ) = An cos[(4n + m)θ](m = 0, 2) , (16) n=0 ∞ X fm,−(θ) = Bn sin[(4n + m)θ](m = 0, 2) . (17) n=0 From Eqs. (16)–(17) we see that: (i) The state f0,+ is symmetric with respect to all the reflections (12). (ii) The state f0,− is antisymmetric with respect to all the reflections (12). (iii) the state f2,+ is symmetric with respect to reflections θ0 = 0, π/2 and antisymmetric with respect to reflections θ0 = ±π/4. (iv) the state f2,− is antisymmetric with respect to reflections θ0 = 0, π/2 and symmetric with respect to reflections θ0 = ±π/4. In the literature, the state f0,+ is referred to as s-wave, the state f2,+ is known as dx2−y2 -wave, and the state f2,− is called dxy-wave. These three states are very important in the context of the pairing channels of the fermionic Hubbard model; let us write them separately:

∞ X fs(θ) ≡ f0,+(θ) = An cos(4nθ), (18) n=0 ∞ f (θ) ≡ f (θ) = X B cos[(4n + 2)θ], (19) dx2−y2 2,+ n n=0 ∞ X fdxy (θ) ≡ f2,−(θ) = Cn sin[(4n + 2)θ], (20) n=0 In contrast, the state

∞ X f0,−(θ) = Dn sin(4nθ) (21) n=1 proves rather exotic. There is a simple mnemonics behind the notations x2 −y2 and xy: equa- 2 2 tions kx − ky = 0 and kxky = 0 deﬁne the positions of the reﬂection axes

4 of the eigenvalues (−1), and thus the positions of zeros of corresponding function. The states f±1 are referred to as p-states. It actually makes sense to consider the subspace of all linear superpositions of the states f±1, the so- called p-sector (or p-channel, in the context of Cooper pairing). Indeed, the fact that the D4h reﬂections change m to −m implies that the Cooper pairing in the p-channel breaks not only the global U(1), but also this or that symmetry of the D4h group. Consider two most typical examples of such states. The ﬁrst example, typically referred to as fpx±ipy pairing, is nothing but the states f±1:

fpx±ipy ≡ f±1 . (22)

As we discussed above, these states feature the (π/2)-rotational symmetry, but break reﬂectional symmetries of the D4h group. Another typical example is given by the two similar states, fpx and fpy , such that

fpx/py ∝ f+1(θ) ± f+1(−θ) ∝ f−1(θ) ± f−1(−θ) , (23) that is,

∞ ∞ X X fpx = Dn cos[(2n + 1)θ] , fpy = Dn sin[(2n + 1)θ] . (24) n=0 n=0 Note that here we optimized parametrization in such a way that the summation is over non-negative n’s only, which is precisely the reason why we have the term 2n (instead of 4n, in the case when negative n’s are allowed). With Eq. (24) we see that

fpx (θ) = fpy (θ + π/2) . (25)

Finally, observing that

f0(θ + π) = f0(θ) , f2(θ + π) = f2(θ) , f±1(θ + π) = −f±(θ) , (26) we conclude that the m = 0, 2 functions correspond to the singlet case, while the p-channel corresponds to the triplet case.