Order Parameter Statistics in the Critical Quantum Ising Chain

The probability distribution of the order parameter is expected to take a universal scaling form at a phase transition. In a spin system at a quantum critical point, this corresponds to universal statistics in the distribution of the total magnetization in the low-lying states. We obtain this scaling function exactly for the ground state and first excited state of the critical quantum Ising spin chain. This is achieved through a remarkable relation to the partition function of the anisotropic Kondo problem, which can be computed by exploiting the integrability of the system.

Figure 1

${\mathcal{Z}}_K(i\lambda )$ at $g=1/8$ and (dashed line) its approximation (10).

Figure 2

Dots show numerical data for $sP(sX)$ for the ground state of the critical Ising chain for $L=16–23$. The continuous line is the scaling function $f_0(X)$ for the ground state, the Fourier transform of ${\mathcal{Z}}_K(iz)$.

Figure 3

Comparison of numerical and analytical results for the first excited state scaling function $f_1(X)$.