Exact density matrix of the Gutzwiller wave function as the ground state of the inverse-square supersymmetric $t\text{−}J$ model

The density matrix—i.e., the Fourier transform of the momentum distribution—is obtained analytically in closed form for the Gutzwiller wave function with exclusion of double occupancy per site. The density matrix for the majority spin is obtained for all magnetizations including the singlet case. Since the wave function gives the ground state of the supersymmetric $t\text{−}J$ model with the $1∕r^2$ exchange and transfer, the result gives the exact density matrix of the model at zero temperature. From the oscillating behavior of the density matrix, the discontinuity of the momentum distribution at the Fermi momentum $k_F$ is identified. The form of the weaker singularity at $3k_F$ is also obtained; there is a discontinuity in the second derivative of the momentum distribution, whose magnitude is calculated analytically. The momentum distribution over the whole Brillouin zone is obtained numerically from the analytic solution of the density matrix. The result is in excellent agreement with previous results derived by different methods.

Figure 1

(a) Diagram showing how the product of the Vandermonde determinant and confluent alternant is evaluated. The white circles denote the exponents in the confluent alternant, integer spaced from $-(Q+2M-1)∕2$ to $(Q+2M-1)∕2$. The horizontal positions of the circles give the exponents; therefore, the circles are equally spaced horizontally. The black circles denote the exponents in the Vandermonde determinant and are slightly separated from the white circles for clarity. The diagram shown corresponds to $Q=6$, $M=3$. A line connecting a black circle to a white one represents a hole coordinate, and one connecting a white circle to a white one represents a magnon. Each magnon line introduces a factor in the diagram equal to its horizontal extent. All lines are constrained to connect points equidistant from the symmetry axis, because of the integrals over hole and magnon coordinates, and are therefore horizontal. For clarity, some of the hole lines have not been shown: every black circle should be connected to the white circle on the opposite side at the same level. (b) Schematic of the same diagram, where a vertical projection has been taken and the points are equally spaced on the vertical axis. The actual horizontal separation between two points, which is of significance for magnon lines, can be calculated by counting how many levels down one side and up the other separate them. The dashed line is the magnon-hole boundary.

Figure 2

(a) Diagram showing a pair of adjacent ascending and descending hole lines. (b) Diagram showing a pair of adjacent ascending and descending magnon lines.

Figure 3

The three possible ways in which a pair of ascending and descending lines can straddle the magnon-hole boundary. For clarity, the horizontal lines are shifted slightly downwards in the first two diagrams. Notice that in all three diagrams, there is one magnon line partially or entirely below what we are calling the magnon-hole boundary.

Figure 4

(a) Diagram showing an ascending line above the magnon-hole boundary. The circle marked 1 must connect to $1^{″}$, since its only other option is to connect to $1^{\prime }$, in which case the circle 3 has no partner. (b) Diagram showing an ascending line below the magnon-hole boundary, connecting a white circle to a black circle. Again, if 1 is not connected to $1^{″}$, the circle 3 has no partner.

Figure 5

Diagram showing an ascending line below the magnon-hole boundary, connecting two white circles. Proceeding in order, we can see that circle 1 has to be connected to $1^{\prime }$, 2 to $2^{\prime }$, and so on until 5 is connected to $5^{\prime }$ and the cluster terminates. For clarity, circle 5 has not been labeled. The right-hand side shows the same cluster, with the connecting lines visible.

Figure 6

Diagram showing an ascending line below the magnon-hole boundary, connecting a black circle to a white one. If circle 1 is not connected to $1^{″}$, it has to connect to the white circle beside 3. Proceeding in order, circles 2, 3, and 4 have only one possible partner. Circle 5 has two possible circles to connect to, both shown in the figure. In either case, the cluster does not go any further down.

Figure 7

Plot of $n\left(k\right)$ vs $k$ for filling fractions $n=0.1$, 0.5, and 0.9. All the plots have zero magnetization—i.e., $Q+2M=L$. The line is obtained from Eq. (79), evaluated with a large upper and lower cutoff, which corresponds to the $L\to \infty$ limit of $g\left(k\right)$. The points are obtained from evaluating the existing dynamical result with $L=100$ (Ref. 10).