and the normalization of the spectral function is not affected. If Gab is the cluster Green function, this corresponds to the C-scheme; if it is the CPT Green function (8.4), this corresponds to the G-scheme; if it is the cluster self-energy , then r a,r the resulting periodized self-energy also has the correct analytic properties and so does the E-scheme Green function. Last, the M-scheme too has the correct analytic properties, since removing and reinserting later the off-diagonal part t off(k) do not affect the analytic properties of the Green function.

In many applications that use the CPT Green function, whether in VCA or C-DMFT, it is necessary to compute various physical quantities, which often can be expressed as the expectation value of a one-body operator of the general form

In the simple case of the number of electrons, the matrix s is diagonal:

but one could be interested in the expectation value of the kinetic energy, or some order parameter.

We will show in this section how to practically calculate the density O = •V )/N with the help of a Green function defined along the imaginary axis, as obtained for instance from the ED solver and the CPT (we assume zero temperature).

From the Lehmann representation of the Green function, we see that ) is given by the integral of the Green function along a contour C< surrounding the negative real frequency axis counterclockwise:

Therefore, the expectation value we are looking for is

The trace includes a sum over lattice sites, spin and band. In the (a, k) basis, this becomes

where we assumed that thé matrix s is diagonal in k (translation invariance over thé superlatticé).

Next, let us consider the asymptotic behavior of the Green function as z !i: G (z) ! 1/z. This allows us to modify (8.87) as follows:

where p > 0 (in practice, we use p ~ 1). The term we added does not contribute, since its unique pole lies outside the contour. However, it modifies the asymptotic behavior of the integrand, which now decays as 1/z2. This allows us to replace thé contour C< by an integral along the imaginary axis, plus an infinite semi-circle that does not contribute, since the integrand falls faster than 1/z.

Consider next the part of the contour C< that lies above the real axis, and let us follow this contour clockwise and call it C. Let C be the mirror image of C below the real axis, followed counterclockwise. To each z and dz of C correspond the mirror images z* and dz* on C ', so that

If, in addition, the integrand is such that f(z*) = f *(z), then

The integral of f(z) along the counterclockwise contour C< would then be

I[C<] = I[C'] - I[C] = I*[C] - I[C] = -21 ImI[C], (8.91)

One of the properties of the Green function is its hermiticity: Gap(z* ) = G**a(z). In the mixed Fourier representation, this is rather expressed as G (k, z*) = G^(-k, z). We also assume that s is Hermitian: s(k) = s^(-k) so that the expectation value is real. This means that the integrand of the expectation value respects the condition f(z*) = f *(z).

Finally, the expectation value becomes

C,= I X p d! Re I tr hs(k)G(k,iœ)] - i^î , (8.92)

This expression can be used in practice (i.e. numerically) to compute the desired quantity.

Let us make an important remark on the computation of expectation values with the periodized Green functions described in the previous section. The result (8.86) is quite general and could formally be expressed as

where the trace ( Tr ) stands for a functional trace, i.e. includes an integral over frequency as well as a trace over site and band indices. The above expression is basis-independent; in the full wavevector basis of one-particle states (see, e.g., (8.70)), the frequency summand would take the following form:

N J2 Re {s(k + K,k' C K0)G(k' + K0 ,k + K,/!)}. (8.94)

If the operator O is translation invariant, as it usually is, then s(k C K , k0 C K0) = Ikk' 1kk's(k) = Ikk's(k) (8.95)

and the above reduces to

k where Gper(k , i!) is the Green function (8.71), periodized in the G-scheme. This means that expectation values of translation invariant, one-body operators, computed in the G-scheme of periodization, coincide with those computed without periodization, i.e. with (8.92). This does not hold for the other periodization schemes, as it crucially depends on our discarding the off-diagonal elements of G in the full wavevector basis, which is possible because we take the trace of G against a matrix s that is itself diagonal in that basis. The G-scheme is therefore the best periodization scheme, in the sense of expectation values of one-body operators.

The CPT was devised at first to be applied on the Hubbard model, in particular to calculate an approximate spectral function that could be compared with ARPES data. The main advantage of CPT in this context is that it provides momentum-resolved spectral information. This is particularly useful in investigating the pseudogap observed in high-temperature superconductors in their normal phase, as was done in [20] with a one-band Hubbard model appropriate to cuprates (in particular YBCO). Figure 8.3, borrowed from [20], shows the spectral function a a

Fig. 8.3 Spectral weight for wavevectors along thé high-symmetry directions shown in thé inset. Thé band parameters are t = 1, t' = —0.3 and t'' = 0.2. (a): CPT calculations on a 3 x 4 cluster with 10 électrons on thé cluster (17% hole doped). (b): thé same, with 14 électrons (17% électron doped). From [20]

Fig. 8.3 Spectral weight for wavevectors along thé high-symmetry directions shown in thé inset. Thé band parameters are t = 1, t' = —0.3 and t'' = 0.2. (a): CPT calculations on a 3 x 4 cluster with 10 électrons on thé cluster (17% hole doped). (b): thé same, with 14 électrons (17% électron doped). From [20]

obtained from CPT for the hole- and electron-doped one-band Hubbard model, at various values of U, on a 3 x 4 cluster. The separation of the two Hubbard bands as U increases is clearly visible, as well as the suppression of the quasi-particle weight along the X-H direction in the hole-doped case, and along the diagonal in the electron-doped case. This weight suppression constitutes the pseudogap, i.e., a gap that opens only along a certain portion of the Fermi surface. The pseudogap is also clearly seen from plots of the spectral function as a function of wavevector (Fig. 8.4). The intersection of the anti-ferromagnetic zone boundary (white diagonal) with the non-interacting Fermi surface (black curve) defines "hot spots" where the scattering of quasi-particles by short-range anti-ferromagnetic fluctuations depletes the spectral weight. This compares well with ARPES results for electron-doped (Fig. 3 of [21]) and hole-doped (Fig. 8 of [22]) cuprates.

Fig. 8.4 Density plot of the spectral function at the Fermi level, in the first quadrant of the Brillouin zone, for an electron doped system at weak coupling (left), and a hole-doped system at stronger coupling (right). The non-interacting Fermi surface is shown in black, and the anti-ferromagnetic zone boundary in white. A 4 x 4 cluster was used

Comparisons with the known results from the one-dimensional Hubbard model are particularly instructive. We have shown the spectral function in various periodization schemes for the half-filled, one-dimensional Hubbard model on Fig. 8.2. On that figure (top panel), the spin charge separation is clearly visible, especially far from kF, where the energy difference between the holon and spinon is largest. Access to a continuum of wavevectors by CPT allows for a natural smoothing of the computed density of states (DoS), compared to the same quantity obtained from the cluster Green function alone. Figure 8.5 shows results for clusters of sizes 4 and 16. The CPT density of states, when extrapolated to zero Lorenzian broadening display a set of (sometimes overlapping) bands that correspond to the dispersing poles of the CPT Green function. Overall, the CPT DoS, when calculated with a natural value of is less spiky than the corresponding cluster DoS. Non-zero spectral weights at the tails of the distribution and within the Mott gap are due to the finite value of

Let us now consider the spectral gap at half-filling. Estimates of this gap from a 12-site cluster, and extrapolations to infinite size are shown in Fig. 8.6 (see caption for details). We see that the CPT provides a better estimate of the gap than the cluster alone, even more so when an infinite-size extrapolation can be computed. The behavior as U ! 0 merits attention: cluster data alone, at a fixed cluster size, do not show the vanishing of the gap, but the CPT result does. However, both the raw cluster and CPT gaps tend to 0 as U ! 0 when an extrapolation is performed, although the gap does not go to zero nearly as fast as it should, from the exact solution. Long-range fluctuations (much longer than the cluster size) are likely important to set the correct gap value in this regime.

Fig. 8.4 Density plot of the spectral function at the Fermi level, in the first quadrant of the Brillouin zone, for an electron doped system at weak coupling (left), and a hole-doped system at stronger coupling (right). The non-interacting Fermi surface is shown in black, and the anti-ferromagnetic zone boundary in white. A 4 x 4 cluster was used

L |
H |
Li |
! |
0 |
1 |
b |
J |
L | ||

/ a |
1 1 |
Fig. 8.5 Density of States (DoS) for thé half-filled, one-dimensional Hubbard model (U = 4 ,t = 1). Top panel: results for the 4-site cluster; the DoS on the cluster is shown with a Lorenzian broadening q = 0.1, as well as the DoS from the CPT Green function with the same q. In addition, an extrapolation to q ! 0 is shown. Bottom panel: same, for the 16-site cluster (no extrapolation shown) 8.8 Applications to Other Models 8.8.1 Multi-Band Hubbard Models Hubbard models with more than one band can be treated with CPT just like the one-band model. Of course, the computational burden depends on the total number of orbitals on the cluster, and therefore a three-band model on a 4-site cluster will require the same resources as a one-band model on a 12-site cluster, everything else being equal. The CPT was first applied to the three-band Hubbard model for high-temperature superconductors in [24] and [2]. The three-band Hubbard model was studied more recently using VCA [25]. The periodic Anderson model (or Kondo lattice) studied in [26] with the VCA can be viewed as a two-band Hubbard model in which only one band is correlated. Likewise, the CPT was applied in [27] to a two-band Hubbard model used to model an organic polymer coupled to a quantum wire: one (uncorrelated) band describing the wire and the other (correlated) describing Fig. 8.6 Mott gap of the half-filled, one-dimensional Hubbard model, as a function of U. The open blue circles are obtained by looking at the lowest energy level of a 12-site cluster, in the sectors with 12 and 13 electrons respectively. The filled blue circles are obtained by extrapolating this estimate to L from L = 4, 6, 8, 10 and 12. The open red circles are obtained from the Lehmann representation (8.41) of the CPT Green function based on a 12-site cluster, at k = w/2. The filled red circles are obtained by extrapolating this estimate to L from L = 4, 6, 8, 10 and 12. The dashed line is the known exact result from the Lieb and Wu solution [23] Fig. 8.6 Mott gap of the half-filled, one-dimensional Hubbard model, as a function of U. The open blue circles are obtained by looking at the lowest energy level of a 12-site cluster, in the sectors with 12 and 13 electrons respectively. The filled blue circles are obtained by extrapolating this estimate to L from L = 4, 6, 8, 10 and 12. The open red circles are obtained from the Lehmann representation (8.41) of the CPT Green function based on a 12-site cluster, at k = w/2. The filled red circles are obtained by extrapolating this estimate to L from L = 4, 6, 8, 10 and 12. The dashed line is the known exact result from the Lieb and Wu solution [23] the polymer, the goal of this work being to explore the possibility of a spintronics device in which spin channels could be controlled by a gate instead of an external magnetic field. |

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