Exact full counting statistics for the staggered magnetization and the domain walls in the $XY$ spin chain

We calculate exactly cumulant generating functions (full counting statistics) for the transverse, staggered magnetization, and the domain walls at zero temperature for a finite interval of the $XY$ spin chain. In particular, we also derive a universal interpolation formula in the scaling limit for the full counting statistics of the transverse magnetization and the domain walls which is based on the solution of a Painlevé V equation. By further determining subleading corrections in a large interval asymptotics, we are able to test the applicability of conformal field theory predictions at criticality. As a by-product, we also obtain exact results for the probability of formation of ferromagnetic and antiferromagnetic domains in both the ${\sigma }^z$ and ${\sigma }^x$ basis in the ground state. The analysis hinges upon asymptotic expansions of block Toeplitz determinants, for which we formulate and check numerically a different conjecture.

Figure 1

Numerical check of the interpolation formula of Eq. (32) between the noncritical and the critical regimes of the FCS of the transverse magnetization ${\chi }_{{\mathfrak{M}}_z}\left(\lambda \right)$. We represent ${\mathrm{\Delta }}_{\mathrm{P}}$, defined in Eq. (C2), as a function of $x=2L|ln|h\left|\right|/\gamma$, for different fixed values of $\gamma$ and $L$, and varying $h$. The dots have been calculated numerically using Eq. (17) for ${\chi }_{{\mathfrak{M}}_z}\left(\lambda \right)$. As we explain in Appendix pp3, ${\mathrm{\Delta }}_{\text{P}}\sim ln{\tau }_V\left(x\right)$ for large $L$. The solid curves represent the expansion (34) around $x=0$ up to order $O\left(x^4\right)$ of $ln{\tau }_V\left(x\right)$. The dashed curves correspond to the asymptotic behavior (35) for large $x$ of this $\tau$ function.

Figure 2

Numerical check of the expansion conjectured in Eq. (52) for the formation probability of an antiferromagnetic domain in an $XX$ spin chain with Fermi momentum $k_F\ne \pi /2$. The dots have been obtained by calculating numerically the determinant in Eq. (39) in the limit $\lambda \to -\infty$. The curves correspond to the conjectured expansion (52); the $O\left(1\right)$ term has been determined from $ln{\mathcal{E}}_s(L=500)-{500}^{2}lnsin{k}_F-500ln2-1/4ln\left(500\right)$, taking as ${\mathcal{E}}_s(L=500)$ the value obtained numerically for this quantity at $L=500$ for each Fermi momentum considered.

Figure 3

Numerical check of the $O(L^{-1}lnL)$ term in the expansion of $ln{\chi }_{\mathfrak{K}}\left(\lambda \right)$ along the critical line $h=1$. We plot ${\mathrm{\Delta }}_{\mathfrak{K}}$, defined in Eq. (C7), as a function of $lnL$ for several fixed values of $\gamma$ and $\lambda$, and choosing $L_0={10}^3$. The dots have been obtained by calculating numerically ${\chi }_{\mathfrak{K}}\left(\lambda \right)$ through Eq. (55). The lines represent $d_{\mathfrak{K}}ln(L/L_0)$, taking for $d_{\mathfrak{K}}$ the coefficient of the $O(L^{-1}lnL)$ term predicted in Eq. (60).

Figure 4

Numerical analysis of the crossover between the noncritical and the critical behavior of the FCS of the number of domain walls, ${\chi }_{\mathfrak{K}}\left(\lambda \right)$. We study the quantity ${\mathrm{\Delta }}_{\mathrm{P}}$, defined in Eq. (C2) but replacing ${\chi }_{{\mathfrak{M}}_z}$ and $g_{{\mathfrak{M}}_z}$ by ${\chi }_{\mathfrak{K}},g_{\mathfrak{K}}$, vs the ratio $x=2L|ln|h\left|\right|/\gamma$; we vary $h$, keeping $\gamma , \lambda$, and $L$ fixed. The dots correspond to calculate numerically ${\chi }_{\mathfrak{K}}\left(\lambda \right)$ using Eq. (55). As we point out in Appendix pp3, we expect ${\mathrm{\Delta }}_{\text{P}}\sim ln{\tau }_V\left(x\right)$ when $L$ is large enough. The solid curves represent the expansion (34) of the logarithm of the Painlevé V $\tau$ function around $x=0$ up to order $O\left(x^4\right)$. The dashed curves are the asymptotic behavior (35) of this function for $x\to \infty$.

Figure 5

Probability distribution ${\mathcal{P}}_K\left(W\right)$ of domain walls in an interval of length $L$ along the critical line $h=1$. The dots correspond to the direct numerical integration of Eq. (61), considering for ${\chi }_{\mathfrak{K}}\left(\lambda \right)$ the asymptotic expansion obtained by applying the Fisher-Hartwig conjecture. The solid and dashed curves represent the analytical approximation found in Eq. (62) for $W=4n$ and $W=4n+2$, respectively, with $n\in \mathbb{N}$. For the case $\gamma =0.2$ and $L=100, B=0.327387$, while for $\gamma =1.3$ and $L=50, B=0.682844$.

Figure 6

Numerical check of the term of order $O(L^{-1}lnL)$ in the expansion of Eq. (43) for the FCS of the staggered magnetization along the critical line $h=1$. We plot ${\mathrm{\Delta }}_s$ [defined in Eq. (C5)] with $L_0=2000$, subtracting $c_s^{\mathrm{fit}}(L-L_0)$, and taking for $c_s^{\text{fit}}$ the values collected in the corresponding column of Table 1. The dots correspond to ${\mathrm{\Delta }}_s$ obtained by computing numerically ${\chi }_{{\mathfrak{M}}_s}\left(\lambda \right)$ from Eq. (39). The lines represent $d_{s}ln(L/L_0)$, where $d_s$ is the coefficient of the $O(L^{-1}lnL)$ term predicted in Eq. (45).

Figure 7

Contours in the complex $z$ plane discussed in the main text.