Fidelity as a probe for a deconfined quantum critical point

The deconfined quantum critical point was proposed as a second-order quantum phase transition between two broken-symmetry phases beyond the Landau-Ginzburg-Wilson paradigm. However, numerical studies cannot completely rule out a weakly-first-order transition because of strong violations of finite-size scaling. We demonstrate that the fidelity is a simple probe to study the deconfined quantum critical point. We study the ground-state fidelity susceptibility close to the deconfined quantum critical point in a spin chain using the large-scale finite-size density-matrix renormalization-group method. We find that the finite-size scaling of the fidelity susceptibility obeys the conventional scaling behavior for continuous phase transitions, supporting the idea that the deconfined quantum phase transition is continuous. We numerically determine the deconfined quantum critical point and the associated correlation length critical exponent from the finite-size-scaling theory of the fidelity susceptibility. Our results are consistent with recent results obtained directly from the matrix product states for infinite-size lattices using others observables. Our work provides a useful probe to study critical behaviors at the deconfined quantum critical point from the concept of quantum information.

Figure 1

Fidelity susceptibility per site ${\chi }_N$ near the DQCP of the spin chain as a function of $J_z$ for $N=288$ (blue dots), $N=336$ (red diamonds), $N=384$ (green triangles), $N=432$ (magenta squares), and $N=512$ (yellow stars) lattice sites. We set $J_x=1, K_x=1/2$, and $K_z=1/2$. Symbols denote DMRG numerical results and the solid lines are to guide the eye.

Figure 2

Fidelity susceptibility peak and critical exponent $\nu$. (a) Scaling of the amplitude of fidelity susceptibility per site ${\chi }_N^{*}$ at the peak position $J_z^{*}$ of the spin chain as a function of the lattice size with $J_x=1, K_x=1/2$, and $K_z=1/2$. Blue circle symbols denote DMRG numerical results and the solid line shows the linear fit from the data points. The exponent of the correlation function $\nu =1.446$ is obtained from the linear fit. (b) Fitted critical exponent $\nu$ as a function of the maximum system size used in fitting. One can see a drift of critical exponents for bigger system sizes.

Figure 3

Data collapse of the fidelity susceptibility ${\chi }_F=N{\chi }_L$ shown in Fig. 1. (a) Data collapse for $N=288,336,384,432$ sites, where the exponent of the correlation function $\nu =1.431$ and the critical value $J_z^c=1.475$ are chosen to achieve perfect data collapse. (b) Data collapse for $N=336,384,432,512$ sites, where the critical exponent $\nu =1.443$ and the critical value $J_z^c=1.473$ are chosen to achieve perfect data collapse. Symbols denote the rescaled DMRG results.