Universal subleading terms in ground-state fidelity from boundary conformal field theory

The study of the (logarithm of the) fidelity, i.e., of the overlap amplitude, between ground states of Hamiltonians corresponding to different coupling constants provides a valuable insight on critical phenomena. When the parameters are infinitesimally close, it is known that the leading term behaves as $O\left(L^{\alpha }\right)$ ($L$ system size), where $\alpha$ is equal to the spatial dimension $d$ for gapped systems, and otherwise depends on the critical exponents. Here we show that when parameters are changed along a critical manifold, a subleading $O\left(1\right)$ term can appear. This term, somewhat similar to the topological entanglement entropy, depends only on the system’s universality class and encodes nontrivial information about the topology of the system. We relate it to universal $g$ factors and partition functions of (boundary) conformal field theory in $d=1$ and $d=2$ dimensions. Numerical checks are presented on the simple example of the $XXZ$ chain.

Figure 1

(Color online) Universal $g$ factor in the fidelity for the $XXZ$ model together with BCFT predictions. The fidelity is computed between ground states with different anisotropies ${\Delta }_{1,2}$ in the critical region $\left(\left|{\Delta }_{1,2}\right|<1\right)$, and we fixed ${\Delta }_1=0.20$. Plus signs are extrapolations of data obtained with Lanczos diagonalization on very small lattices (length $L\le 22$) and periodic boundary conditions. These data agree perfectly with BCFT predictions [Eq. (3)] together with Eq. (4). Instead, toroidal BCs given by ${\sigma }_{L+1}^{\pm }=e^{\pm i\theta }{\sigma }_1^{\mp }$ and ${\sigma }_{L+1}^z=-{\sigma }_1^z$ induce antiperiodic BCs on the field $\varphi$ and consequently there is no term of order one in this case (i.e., $g=1$). Note that such BC [they belong to conjugacy class (IV) of Ref. 16] breaks the conservation of total magnetization. In the inset, the fidelity is computed when one ground state is critical and the other is deep; the massive (Néel) phase ${\Delta }_1⪢1$. Solid line gives the BCFT prediction. The small discrepancy around ${\Delta }_2\approx 1$ is due to finite-size effects which are more pronounced near the Kosterlitz-Thouless point $\Delta =1$.

Figure 2

(Color online) Universal $g$ factor in the fidelity of the quantum eight-vertex model with periodic BC when all the theories are in the disordered region, i.e., $c^4<4$, ${\left(c^{\prime }\right)}^4<4$, and ${\left(c{c}^{\prime }\right)}^2<4$. We fixed the ratio $L_1/L_2=1$. The $g$ factor is smooth at the border of the region $c$, $c^{\prime }\to 0$, $c$, and $c^{\prime }\to \sqrt{2}$.