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

The density matrix, i.e., the Fourier transform of the momentum distribution, is obtained analytically for all values of the magnetization of the Gutzwiller wave function in one dimension with the exclusion of double occupancy per site. The present result complements the previous analytic derivation of the density matrix for the majority spin. The derivation makes use of a determinantal form of the squared wave function, and multiple integrals over particle coordinates are performed with the help of a diagrammatic representation. In the thermodynamic limit, the density matrix at distance $x$ is completely characterized by quantities $v_{c}x$ and $v_{s}x$, where $v_s$ and $v_c$ are spin and charge velocities in the supersymmetric $t\text{−}J$ model for which the Gutzwiller wave function gives the exact ground state. The present result then gives the exact density matrix of the $t\text{−}J$ model for all densities and all magnetization at zero temperature. Discontinuity, slope, and curvature singularities in the momentum distribution are identified. The momentum distribution obtained by numerical Fourier transform is in excellent agreement with existing results.

Figure 1

(a) Diagram for evaluating the product of Vandermonde and alternant determinants. The white circles denote the exponents in the alternant determinant, integer spaced from $-(Q+2M-1)∕2$ to $(Q+2M-1)∕2$. The horizontal positions of the circles give the exponents, which are equally spaced horizontally. The black circles denote the exponents in the Vandermonde determinant and are slightly separated from the white circles for clarity. 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 coordinate. 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 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

Type-I diagrams. The jokers have shaded black circles. The two diagrams shown in the figure are a complementary pair. Both jokers are partnered with white circles below the magnon-hole boundary. All magnon lines are horizontal or dimerized.

Figure 3

Type-II diagrams. The jokers have shaded black circles. The two diagrams shown in the figure are a complementary pair. One joker is partnered with a white circle above the line $l$ levels above the magnon-hole boundary. All magnon lines that are not shown are above this, and are horizontal or dimerized.

Figure 4

Type-III diagrams. Both jokers are partnered with white circles above the line $l_1$ and $l_2$ levels above the magnon-hole boundary. All magnon lines that are not shown are below or above the $[l_1,l_2]$ block, and are horizontal or dimerized.

Figure 5

Type-IV diagrams. Both jokers are partnered with white circles above the line $l_1$ and $l_2$ levels above the magnon-hole boundary. The white circles that they leave unpaired connect with each other, forming a magnon line below the magnon-hole boundary. This line must be horizontal, or ascending or descending by one level. All three cases are shown in the figure.

Figure 6

Distance dependence of $G_{\downarrow }\left(x\right)=-{\rho }_{\downarrow }\left(x\right)+{\delta }_{x,0}(n-m)∕2$, with $n=0.9,0.7,0.5,0.3$ and $m=0.1$ for all cases. Only integer values of $x$ are physically relevant. For comparison, the data obtained by the inverse Fourier transform of Kollar-Vollhardt results are also shown by points. A wriggle for $1<x<2$ in the case of $n=0.3$ comes from the analytic form of Eq. (53), but only integer values of $x$ are physically relevant. The wriggles between integer valus of $x$ in the ensuing figures are of the same character.

Figure 7

Distance dependence of $G_{\downarrow }\left(x\right)$ in the singlet limit with $n=0.3$ and $m=0$ (solid line and dots). The inset shows the expanded view for small $x$.

Figure 8

Distance dependence of $G_{\downarrow }\left(x\right)$ in the singlet limit with $n=0.1$ and $m=0$ (solid line and dots). The shape is reasonably well fitted by $-\sin (\pi nx∕2)∕\left(\pi x\right)$ (dashed line), which is the case for noninteracting electrons. The inset shows the expanded view for small $x$.

Figure 9

Distance dependence of $G_{\uparrow }\left(x\right)$ with $n=0.9,0.7,0.5,0.3$ and $m=0.1$ for all cases. Only integer values of $x$ are physically relevant, and the lines are shown only for easy tracking of $G_{\uparrow }\left(x\right)$ at integers.

Figure 10

Plot of $n_{\downarrow }\left(k\right)$ versus $k$ with $n=0.9,0.7,0.5,0.3$ and $m=0.1$ for all cases. For comparison, the Kollar-Vollhardt results (Ref. 8) are also shown by solid lines.