Having looked at general consequences of projection, we now turn to the calculation of detailed properties of projected wavefunctions. We should make clear the distinction between Gutzwiller projection and Gutzwiller approximation. Projection P|f0) imposes the "no double occupancy" constraint acting on an arbitrary state |f0), where P was defined in (2.2). The Gutzwiller approximation, to be described below, is a simple approximation scheme to estimate expectation values in projected states.
The key difficulty in analyzing projected wavefunctions is that the projection operator P of (2.2) is a product of single-site operators in real space, while most of the interesting wavefunctions |f0) (e.g., a BCS superconducting state) that one would like to project are naturally written in momentum space. Thus, it is impossible to analytically calculate even the norm of P | f0), let alone expectation values in such a state. One can, however, use the VMC method [7,10,11] to compute properties of projected wavefunctions without any approximation. Here, one works in real space and uses Monte Carlo methods to compute the multiple integrals involved in computing expectation values. We will mention some results of the VMC approach here but mostly for the sake of comparison.
Our main focus will be on the Gutzwiller approximation (GA) [8,14,29], which permits us to evaluate matrix elements of operators in projected states using a simple approximation scheme. The resulting equations have the structure of usual MFT, but with a renormalization of parameters that impacts the results in important ways. Hence, the Gutzwiller approximation scheme has also been called "renormalized MFT" ; for recent reviews, see [12, 13]. As a check on its validity, we note that GA scheme gives results in good qualitative agreement with the exact VMC results. Further, given the simplicity of the GA/RMFT equations one can use them to gain analytical insights into important doping dependent trends, and also generalize them to more complex situations, such as inhomogeneous states [30-36]; see Sect. 2.9. (There have also been attempts to include effects of short range correlations [37,38].)
The essential idea of Gutzwiller approximation is to express the expectation value of an operator A in a projected state as the expectation value in the corresponding unprojected state times a renormalization factor gA called the Gutzwiller factor for that operator:
The approximation provides a prescription for evaluating the Gutzwiller factors for different operators, which depend only on the average electron density in the underlying state. (Here, we focus only on the simplest case of homogeneous systems without spin order.) This seemingly ad-hoc prescription may be rationalized in terms of combinatorial factors for finding different configurations in the projected and unprojected states .
The easiest way to understand the Gutzwiller approximation is to give some explicit examples. Here, we will explain in detail the renormalization of the kinetic energy and superexchange. For a detailed derivation of various other terms, the reader is referred to .
Consider a homogeneous system with density n. In the unprojected states, a single site can be in one of four configurations: (a) \ "), (b) \ #), each of which occurs with probability (n/2)(1 — n/2), (c) doubly occupied with probability n2/4 and (d) empty with probability (1 — n/2)2. In the projected states, each site is in one of three configurations: (a') \ "), (b') \ #), each with probability n/2 and (c') empty with probability (1 — n).
Let us consider the hopping term cj^cj", and look at a single hopping event where an up electron hops from i to j. In the unprojected states, there should be an " spin on site i and no " spin on site j (Pauli exclusion) for this hop to occur. After the event one has an " spin on site j and no " spin on site i. The probability of occurrence of the initial configuration is ni "(1 — nj") and that of the final configuration is nj"(1 — ni"). In projected states, the same hopping event would occur if we have an " spin on site i and a vacancy on site j to begin with and finally lead to an " spin on site j and a vacancy on site i. The probability of these configurations are respectively ni"(1 — nj) and nj"(1 — ni). The initial and final configurations for projected and unprojected states are shown in Fig. 2.4. The Gutzwiller factor for the hopping term is then given by
Fig. 2.4 The initial and final configurations for a spin-" electron to hop from site j to site i in projected and in unprojected states. Initially, site i must be empty for projected states, but can have either a hole or a # spin in unprojected states
Unprojected States ji
n"(1 ~ "j)"; "(1 ~ n) i"(1 - "y ")w j "(1 — ")
We thus have a clear prescription for obtaining the Gutzwiller factors. For a given operator, first neglect the configurations on sites that are not connected by the operator, i.e., ignore the effects of longer-range electronic correlations that would make this probability dependent on the full wavefunction. Then multiply the probabilities of the configurations before and after the event has occurred and take a square root, since we need an amplitude. The ratio in the projected and unprojected states gives the Gutzwiller factor for the corresponding operator. The final answer depends only on the density. The Gutzwiller factor for hopping is given by gt =
where x = 1 — n is the hole doping of the system away from half-filling. Thus, the kinetic energy in projected states is suppressed by a factor of gt, and vanishes ~ x as one approaches the Mott insulator at half filling.
Next, consider the spin-flip terms Si+Sj". In both projected and unprojected states, spin flips require an initial configuration of a single " spin (no double occupancy) on site j and a single # spin on site i. The final configuration has the same requirements with spins interchanged. In projected states, the probability of the initial and final configurations are ni#ny" and ni""j#, respectively. In the unprojected states, one has to explicitly take care of the fact that configurations should avoid double occupancy for the spin flip to occur. The probability of initial and final configurations are then given by ni#nj"(1 — ni")(1 — nj#) and ni"nj#(1 — ni#)(1 — n;") respectively, leading to the Gutzwiller factor for the spinspin interactions
The x-dependence of gs reflects the increasing importance of spin fluctuations as one goes toward the Mott insulator within the Gutzwiller approximation.
Similar considerations can be used to derive the Gutzwiller factors for other operators and we simply state the results without derivation (see ). The renormalization of the remaining terms in the transformed Hamiltonian (2.5) are as follows. The n-ny term is not renormalized, i.e., it has a Gutzwiller factor of 1. For the three-site hopping term with spin flip the Gutzwiller factor is gst = 4x/(1 C x)2, (2.15)
while that for three-site term without spin flip is gt. In certain cases, we will need additional Gutzwiller factors which will be described below as they are needed.
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