The Hilbert space of electrons in a local orbital is spanned by four states: two with single occupancy (representing a local spin 1/2) and the empty and doubly occupied states. Obviously, the singly occupied states have fermionic character, while the remaining two states have bosonic character. One may now envisage to create these states out of a vacuum state |vac), which is defined by the absence of any of the four occupation number states. These four states may then be created by fermionic or bosonic auxiliary operators. This may be done in a multitude of ways. We will concentrate here on the representations introduced by Barnes [9] and by Kotliar and Ruckenstein [12].

3.2.2.1 Barnes's Representation

The basic idea consists in locally decomposing the electronic excitations into spin and charge components. This can be achieved in many different ways. A suitable

Hubbard-Stratonovich decoupling of the interaction term could reach this goal, but would likely be limited to weak interaction. Instead, in the pioneering Barnes approach [9] the spin and charge degrees of freedom (DoF) are represented by fermionic and bosonic operators, respectively. Being more numerous than the original (physical) operators, the auxiliary operators span a Fock space that is larger than the physical one. Consequently they need to fulfill an appropriate set of constraints for such a representation to be faithful. Specifically Barnes considered the single impurity Anderson model (SIAM):

H = Yu skcLcka+efJ2a°a°+vJ2(ckva°C +Uai ataW • (3.5)

ka a ka

Clearly, this problem may not be treated by means of perturbation theory in U, especially in the U limit. Instead, Barnes introduced the auxiliary fermionic (fa) and bosonic (e, d) operators in terms of which the physical electron operators aa read aa = efa C °flad. (3.6)

The aa-operators obey the ordinary Fermion anticommutation relations. This property is not automatically preserved when using the representation (3.6), even when the fermionic and bosonic auxiliary operators obey canonical commutation relations. In addition the constraint

a must be satisfied. Equation (3.6) together with (3.7) constitutes a faithful representation of the physical electron operator in the sense that both have the same matrix elements in the physical Hilbert subspace with Q = 1. The above representation has been widely used, in particular, in the U limit where the operator d (linked to double occupancy) drops out. The constraint can be implemented by means of a functional integral representation. For example, for U the partition function, projected onto the Q = 1 subspace, reads

with the fermionic and bosonic Lagrangians

L/(r) = E cL(t)(9^ + C £ ft.rXB, C e< - M C iA)/ (r)

ko o

Here the A intégration enforces the constraint, and the Lagrangian is bilinear in the fermionic fields. Remarkably, this has been achieved without decoupling the interaction term. Besides, the correctness of the representation can be verified by carrying out all integrals in, e.g., the V ! 0 limit. By virtue of the substitution z = e-iPA, PdA = idz/z, the A integral in (3.8) is transformed into a contour integral along the complex unit circle. Observing that this substitution implies exactly a second-order pole at z = 0 (i.e., at iA ! Ci, real), it is seen that the projection of Z amounts to calculating the grand canonical Q expectation value in the limit of infinite, real chemical potential, Z = limiA!1(g)iA [23], equivalent to the Abrikosov projection. Equation (3.8) may also be viewed as the projection of the non-interacting partition function onto the "U = i"-subspace. Indeed, (3.8) may be rewritten as

o with det [So[e(r),A]] the fermionic determinant for one spin species involving an effective time-dependent hybridization (Vet(r)), and the projection operator

Yet, there is an asymmetry in the representation of spin and charge DoF. While the latter can be expressed in terms of bosons, this is not the case of the former, and may cause unnecessary errors in any approximate treatment (for details see [14]).

With this motivation Kotliar and Ruckenstein introduced a representation where spin and charge DoF may be expressed by bosons.

In the Kotliar and Ruckenstein (KR) representation two additional Bose operators linked to the spin DoF are introduced, p# and P" [12]. In this approach the physical electron operators are represented as:

where the first term corresponds to the transition from the singly occupied state to the empty one, and the second term to the transition from the doubly occupied state to the singly occupied one. Again the representation is faithful provided the auxiliary operators obey canonical commutation relations and satisfy constraints. They read efe + J2 PlPo + dfd = 1 (3.13)

They may be enforced in a functional integral representation with Lagrange multipliers in a fashion analogous to the one we encountered with the Barnes representation. Besides, the density operator (J] a pi pa + 2dtd) and the z-component of the spin operator (1 =+ apjpa) may be expressed in terms of bosons. Spin and charge DoF may, therefore, be treated on equal footing. This procedure can be extended to multiband models [15].

Though faithful, the Kotliar and Ruckenstein representation is lacking spin rotational invariance as transverse components of the spin operator may not be simply represented in terms of auxiliary operators. Indeed, Sxy is neither related to 11 Haa' /a^Ov / nor to 1 J2aa' pi^OaW. Hence fluctuations associated to the transverse modes are not treated on the same footing as the ones associated to the longitudinal mode. With this motivation a manifestly spin-rotation invariant (SRI) formulation has been introduced [13,14]. In this setup the doublet pa [12] is replaced by a scalar (S = 0) field p0 and a vector (S = 1) field p = (px , py ,pz), in terms of which the state |a) = aj |0) may be represented as

|a) = X paa'/■' |vac) with paa' = 2 X PKa' • (3.15)

The bosons p^ obey canonical commutation relations, and all the auxiliary operators annihilate the vacuum (/a |vac) = e|vac) = ... |vac) = 0). With this at hand the electron operators may be written as aa =X fa'Za'a with Za'a = eVa'a + a'CTp-a,-a'¿. (3.16)

Again, the auxiliary operators need to satisfy constraints. They read

a,a0

While the density operator (n = J]mpmpm C 2d1d) and the density of doubly occupied sites operator (D = ) may be expressed in terms of bosons or fermions, the spin operator reads

aa0ai

This expression is especially useful in the context of the t-J model, in particular, because the spin DoF need not be expressed in terms of the original fermions. Using the above, one can tackle models of correlated electrons such as the SIAM, the Anderson lattice model, the t-J or the Hubbard model. However, while the spin and charge DoF have been mapped onto bosons, anomalous propagators necessarily vanish on a saddle-point level as the Lagrangian is bilinear in the fermionic fields, independent of the model. Here they are not treated on equal footing with the spin and charge DoF. This motivated two of us to introduce a manifestly spin- chargerotation-invariant (SCRI) formulation [14].

3.2.2.4 Spin- and Charge-Rotation-Invariant Formulation

The SCRI representation is motivated both by the need to be able to account for anomalous expectation values (such as the ones arising when investigating excitonic states) and to satisfy the particle-hole rotational symmetry entailed in many models. The generators of these rotations are given by the components of the operators:

which form a spin algebra with the usual commutation relations. One may then replace the doublet e, d by a scalar (vector) b0 (b) field (with respect to rotations in the particle-hole space), all of them satisfying canonical commutation relations. In terms of them the two local occupation number states |2) = |+) and |0) = |— ) may be represented as

with = /jf and = 1- When considering the generalized z-operator

Zpa,p'a' = pp'b_p,_p,p^a + aa'p_a,bp,p (with b\p, = 2 ,yz ) and the matrix operators:

one may write the physical electron operator as

The constraints now read

In particular, when performing the trace of (3.25), one obtains

Therefore, both spin and charge DoF no longer possess a representation in terms of the auxiliary fermions. Instead, correcting (48) in [14], the density operator reads

with the density of doubly occupied sites

The spin operator is still given by (3.20). The SCRI representation of the Hubbard model is thus obtained using (3.24) and (3.28), together with the constraints (3.25).

3.2.2.5 Gauge Symmetry and Radial Slave Boson Fields

When representing the electron operators a a as Za /a, it is immediately clear that the latter expression is invariant under the group of transformations:

This local U(1) gauge symmetry was first realized by Read and Newns [24-26] In the context of the U Barnes representation for the SIAM (with zr = e^). In that case this can be made use of to gauge away the phase of the slave boson, which remains as a purely radial field, while the constraint Lagrange parameter is promoted to a time-dependent field. Yet, standard textbooks do not mention representations of such radial fields that are set up on a discretized time mesh from the beginning.

A scheme specific to radial slave boson fields has been proposed by one of us [27]. In this scheme the partition function takes a form analogous to (3.8). However, the projection operator does not mix the N time steps, and may be written as

Pn = f1 I^ f1 dxne-#(<'A„(x„-1)+Wx„(x„-1)) _ (3.30)

Here the constraint parameter A„ is defined for each time step n, i.e., it is a time-dependent field, and x represents the radial slave boson field. In the discrete time step form, the fermionic part of the action reads

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