As mentioned in the Introduction, the first auxiliary particle representations of quantum operators proposed and successfully applied are the bosonic representations of spin operators by Holstein-Primakoff and by Schwinger. Since both are well documented in the literature, we do not consider them here. Generally speaking, bosonic representations are useful to describe ordered states and fluctuations of collective variables. They are less useful if the fermionic character of spin 1/2 particles is of importance. In the following we will concentrate on fermionic representations.

3.2.1.1 Fermionic Representations of S=1/2 Spins

The spin 1/2 operator S has a faithful representation in terms of fermion operators fa (a ="> #)

where r is the vector of Pauli matrices. The Hilbert space obtained by the creation operators fOi acting on the vacuum state |vac) is spanned by four states. The two unphysical states, the empty and the doubly occupied one, are eliminated by requiring that all states considered are eigenstates of the occupation number operator

Abrikosov Projection

A first projection scheme involves adding a term XQ to the Hamiltonian, and taking the limit X ! i [7]. In that case double occupancy is forbidden, while empty states are not involved in expectation values of physical spin operators. The projection of the pseudofermion Green's function G<(!), for example, is effected by taking the following limit G<(!) = limX—1[G<(!)ZX/ hQ)X]. Here G<(rn) is the pseudofermion Green's function at a finite chemical potential X, ZX is the partition function and (Q)X = J2 (fa fa) is the total number of pseudofermions at given X. While taking the limit X ! i removes the contribution of doubly occupied states, the unphysical contribution of the empty level in the partition function ZX is removed in the limit X ! i by replacing ZX by (Q)X. In equilibrium, we may now use that —iG<(!) = f(! + X)A(!)X—1 e~X/Te~m/T A(rn), where f(a>) is the Fermi function, to express the projected Green's function as G<(!) = ie~®/TA(!)/[2/d!e"®/TA(o>)]. Out of equilibrium the occupation function fX(!) has to be calculated from the quantum Boltzmann equation.

The above projection scheme has the disadvantage that particle-hole symmetry is maximally broken. The spectral functions are, therefore, very unsymmetric under a sign change of !, causing difficulty in numerical evaluations.

If one takes A = 0 the pseudofermion spectral function is particle-hole symmetric, A(—!) = A(!), which facilitates calculations considerably. In calculating any physical quantity, which only involves spin operators, unphysical states only come in through the partition function, which enters quantities described by diagrams with closed pseudofermion loops. In the case of quantum impurity models, diagrams with more than one pseudofermion loop do not contribute, as each loop introduces a factor l/NL with NL the number of lattice sites in the system. The pseudofermion self-energy is not affected, because its relevant diagrams do not contain loops. By contrast, the diagrams for response functions like the conductance and the spin susceptibility necessarily contain one loop. The latter quantities have to be corrected by a normalization factor, as discussed by Larsen [17]. The correction amounts to multiplying the pseudofermion occupation factor /A(!) by a factor

where Z(A) = Tr[e"(H +AQ)/T], z = Z(A = 0), and Z0 = Tr[e"Ho/r] are the partition functions of the (unprojected) interacting system and noninteracting system (taking J ! 0 ), respectively, and Zp is the physical (projected) partition function. We now calculate (Z/Z0) by using the relations Z = Tr[e_H/T] = -Z0 + Zp and Z (g) = -Z0 + Zp. From these two relations we conclude that (Q) = 1 and that Y is indeed given by the above relation. One may calculate (Z/Z0) by integrating the total pseudofermion occupation (g)A = Z;j"1Tr[ge~(H +AQ)/T] = —Td; lnTr[e_(H+AQ)/T] with respect to A from 0 to 1. As the spin levels are not occupied in the limit A the system is, therefore, noninteracting in this limit. Hence limA!1 ZA = Z0/4 (keeping in mind that the trace over the pseudofermion states at A = 0 gives a factor of 4), leading to the result

In this setup, the total occupation number may be calculated approximately from

/dA (g)A = -jdA2/d!/(!CA)A(!) ^ 2/d!ln(l + e"!/T)A(!), neglecting

the A-dependence of the spectral function A(!). In the case of the spin l/2 Kondo model the factor Y increases from Y = l at low temperatures (T ^ TK, the Kondo temperature) to Y = 2 at T ^ TK.

A different approach allowing for an exact treatment of the constraint even for lattice systems has been proposed by Popov and Fedotov [18]. It amounts to applying a homogeneous, imaginary-valued chemical potential ^ppv = —i4rT, where T is the temperature. Thus, within this scheme, the Hamiltonian H is replaced by

Note that H denotes a given spin 1/2 Hamiltonian using the fermionic representation of spin operators. Given a physical operator O (i.e., an arbitrary sum or product of spin operators) it can be shown[19] that the expectation value (O)ppv, calculated with Hppv and the entire Hilbert space, is identical to the physical expectation value (O), where the average is performed with the original Hamiltonian H. The projection works by virtue of a mutual cancellation of the unphysical contributions of the sectors Q, = 0 and Q, = 2, at each site. It should be emphasized that, although the Hamiltonian Hppv is no longer hermitian, the quantity (O)ppv comes out real-valued. If, on the other hand, O is unphysical in the sense that it is nonzero in the unphysical sector, e.g., the operator O = Q,-, the expectation value (Q,)ppv is meaningless and one has (Q,) ^ (Q,)ppv.

This approach is applicable to spin models [19-22], but unfortunately it cannot be extended to cases away from half-filling. Although ^ppv vanishes in the limit T ! 0, in principle the exact projection with ^ = ^ppv and the average projection with ^ = 0 are not equivalent at T = 0. This is due to the fact that the computation of an average (... )ppv does not necessarily commute with the limit T ! 0. Nevertheless it can be expected that in usual quantum spin models both projection schemes are equivalent at T = 0. This can be understood with the following argument: Starting from the physical ("true") ground state, a fluctuation of one fermion charge results in two sites with unphysical occupation numbers, one with no and one with two fermions. Since these sites carry spin zero the sector of the Hamiltonian with that occupation is identical to the physical Hamiltonian where the two sites are effectively missing. Thus, a fluctuation from the ground state into this sector costs the binding energy of the two sites which is of the order of the exchange coupling, even in the case of strong frustration. Consequently, at T = 0 charge fluctuations are not allowed and it is sufficient to use the simpler average projection with ^ = 0.

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