Full counting statistics in the Haldane-Shastry chain

We present analytical and numerical results regarding the magnetization full counting statistics (FCS) of a subsystem in the ground state of the Haldane-Shastry chain. Exact Pfaffian expressions are derived for the cumulant generating function, as well as any observable diagonal in the spin basis. In the limit of large systems, the scaling of the FCS is found to be in agreement with the Luttinger liquid theory. The same techniques are also applied to inhomogeneous deformations of the chain. This introduces a certain amount of disorder in the system; however we show numerically that this is not sufficient to flow to the random singlet phase, that corresponds to $XXZ$ chains with uncorrelated bond disorder.

Figure 1

Representation of a particular amplitude $\psi (1,2,5,7,8,10)$ in the ground-state wave function (4). Here the system is of size $L=12$, with $N=L/2=6$ particles (up spins). The particles at sites 1,2,5,7,8,10 are shown as filled blue circles, and the holes at positions 0,3,4,6,9,11 as empty circles.

Figure 2

Convergence of $X_{\ell }\left(\lambda \right)$ to the universal scaling function of Eq. (19) with $K=1/2$ (black thick line). Numerical values for $L=4096$ and $\lambda =1/2,1,3/2$ are shown. The inset shows the difference with the Luttinger liquid prediction. As can be seen, the finite size effects are small but become bigger as $\lambda$ is increased. This is due to the periodicity ${\chi }_{\ell }(\lambda +2\pi )=\chi \left(\lambda \right)$ of the FCS, which is not obeyed by the asymptotic expansion (13). We refer to Ref. [42] for a discussion in the case of free fermions.

Figure 3

(a) Representation of a particular amplitude $\psi (1,2,5,7,8,10)$ in (26), in an inhomogeneous system of size $L=12,N=L/2=6$. (b) The analogous configuration in (27) may be obtained by assigning a $+$ spin to the particles (filled in blue) and a $-$ spin to the holes.

Figure 4

Coefficient of the logarithm in the bipartite fluctuations $C_2\left(L\right)$ for several chains cut into two halves. Each data point is obtained by computing $[C_2\left(L\right)-C_2(L/2)]/ln2$ for a given $L$. The horizontal axis is shown in logarithmic scale. Blue circles represent the clean $XX$ chain, red squares the disordered $XX$ chain (${10}^5$ disorder realizations), brown cross circles the clean Heisenberg chain, orange stars the disordered Heisenberg chain (${10}^3$ realizations), blue lozenges the clean HS chain, and dashed red circles the inhomogeneous HS chain (${10}^5$ realizations). As can be seen the inhomogeneous HS chain still follows the Luttinger liquid prediction, contrary to the disordered XX and Heisenberg chains, that flow to the RSP phase (shown in dashed green). For the nonclean points we choose $\delta =0.5$.

Figure 5

Coefficient of the logarithm for the FCS generating function $ln\chi \left(i\lambda \right)$. Data are for the clean $XX$ chain (green triangles), the disordered $XX$ chain (red circles), and the inhomogeneous Laughlin state (blue stars). Predictions for the Luttinger liquid and RSP universality class are shown for comparison. As can be seen, the bond disordered $XX$ curve flows to the RSP prediction, while the inhomogeneous Laughlin does not.

Figure 6

Two examples of correlations between bond amplitudes $t_{j,k}$ for a chain of size $L=24$. Blue circles represent correlations between two nearest neighbor hoppings $\overline{t_{1,2}{t}_{r,r+1}}-\overline{t_{1,2}}\overline{t_{r,r+1}}$ as a function of their distance $r$. As can be seen these tend to decay rapidly; however neighbor hoppings are strongly anticorrelated. Red squares are the correlations between hoppings emerging from the same site $\overline{t_{1,2}{t}_{1,r+1}}-\overline{t_{1,2}}\overline{t_{1,r+1}}$, one being nearest neighbor, the other of range $r$, as a function of $r$. In both examples the average is performed over ${10}^6$ realizations of the disorder.

Figure 7

Second cumulant for the fluctuations in the ground state of the Hamiltonian (37) with disorder strength $\delta =0.5$ (red squares). The clean case is also shown for comparison (blue circles). We also present data for a truncated version where all couplings further than next-next nearest neighbors are set to zero for clean (brown circles) and disordered (green stars). The last two data sets are shifted by a constant offset $-0.015$ to improve readability. Each data point is averaged over at least ${10}^3$ realizations of the disorder, and the ground state is found with DMRG for each realization.