Fidelity susceptibility in one-dimensional disordered lattice models

We investigate quantum phase transitions in one-dimensional quantum disordered lattice models, the Anderson model and the Aubry-André model, from the fidelity susceptibility approach. First, we find that the fidelity susceptibility and the generalized adiabatic susceptibility are maximum at the quantum critical points of the disordered models, through which one can locate the quantum critical point in disordered lattice models. Second, finite-size scaling analysis of the fidelity susceptibility and of the generalized adiabatic susceptibility show that the correlation length critical exponent and the dynamical critical exponent at the quantum critical point of the one-dimensional Anderson model are respectively 2/3 and 2 and of the Aubry-André model are respectively 1 and 2.375. Thus the quantum phase transitions in the Anderson model and in the Aubry-André model are of different universality classes. Because the fidelity susceptibility and the generalized adiabatic susceptibility are directly connected to the dynamical structure factor which are experimentally accessible in the linear response regime, the fidelity susceptibility in quantum disordered systems may be observed experimentally in the near future.

Figure 1

Universal finite-size scaling of the fidelity susceptibility in the 1D Anderson model. (a) The averaged fidelity susceptibility ${\chi }_F(N,\mathrm{\Delta })$ as a function of disorder strength $\mathrm{\Delta }$ (in units of hopping amplitude $J$) for different lattice sizes, $N=50$ (red circle), $N=100$ (blue square), $N=150$ (green upper triangle), and $N=200$ (magenta lower triangle). (b) The logarithm of the maximum of the fidelity susceptibility (in units of $1/J^2$) as a function of the logarithm of the system sizes. Linear fit shows that $\nu =0.670\pm 0.002$. (c) Scaled fidelity susceptibility ${\chi }_F(N,\mathrm{\Delta })/{\chi }_{F,\max }$ as a function of scaled variable $(\mathrm{\Delta }-{\mathrm{\Delta }}_m)N^{1/\nu }$. All curves for different system sizes collapse into a single curve when we choose the correlation length critical exponents $\nu =0.667\pm 0.002$ and ${\mathrm{\Delta }}_m=0$. (d) Distributions of the fidelity susceptibility generalized by 7000 realizations of uniform disorder with strength $\mathrm{\Delta }=0.03$ for different lattice sizes, $N=50$ (red solid line), $N=100$ (magenta short-dashed line), $N=150$ (green dotted line), and $N=200$ (blue long-dashed line). Here $\mathrm{\Delta }$ is in units of the hopping amplitude $J$ and the fidelity susceptibility is in units of $1/J^2$.

Figure 2

Universal finite-size scaling of the generalized adiabatic susceptibility ${\chi }_4$ in the 1D Anderson model. (a) The natural logarithm of the generalized adiabatic susceptibility ${\chi }_4(N,\mathrm{\Delta })$ as a function of disorder strength $\mathrm{\Delta }$ for different lattice sizes, $N=50$ (red circle), $N=100$ (blue square), $N=150$ (green upper triangle), and $N=200$ (magenta lower triangle). (b) The logarithm of the maximum of generalized adiabatic susceptibility as a function of the logarithm of the system sizes. Linear fit shows that $z=1.982\pm 0.002$. (c) Scaled fidelity susceptibility ${\chi }_4(N,\mathrm{\Delta })/{\chi }_{4,\max }$ as a function of scaled variable $(\mathrm{\Delta }-{\mathrm{\Delta }}_m)N^{1/\nu }$. All curves for different system sizes collapse into a single curve when we choose the correlation length critical exponents $\nu =0.667\pm 0.002$ and ${\mathrm{\Delta }}_m=0$. Here $\mathrm{\Delta }$ is in units of the hopping amplitude $J$ and the generalized adiabatic susceptibility ${\chi }_4$ is in units of $1/J^4$.

Figure 3

Universal finite-size scaling of the fidelity susceptibility in the Aubry-André model with odd number of lattice sizes. (a) The logarithm of the fidelity susceptibility ${\chi }_F(N,\mathrm{\Delta })$ as a function of disorder strength $\mathrm{\Delta }$ for different odd number of lattice sizes, $N=89$ (red circle), $N=233$ (blue square), $N=377$ (green upper triangle), and $N=987$ (magenta lower triangle). (b) The logarithm of the maximum of fidelity susceptibility as a function of the logarithm of the system sizes. Linear fit shows that $\nu =0.98\pm 0.01$. (c) Scaled fidelity susceptibility ${\chi }_F(N,\mathrm{\Delta })/{\chi }_{F,\max }$ as a function of scaled variable $(\mathrm{\Delta }-{\mathrm{\Delta }}_m)N^{1/\nu }$ with ${\mathrm{\Delta }}_m$ being the position of the maximum of the fidelity susceptibility. All curves for odd number of lattice sizes collapse into a single curve when we choose the correlation length critical exponents $\nu =1.00\pm 0.01$. Here $\mathrm{\Delta }$ is in units of the hopping amplitude $J$ and the fidelity susceptibility is in units of $1/J^2$.

Figure 4

Universal finite-size scaling of the fidelity susceptibility in the Aubry-André model with even number of lattice sizes. (a) The logarithm of the fidelity susceptibility ${\chi }_F(N,\mathrm{\Delta })$ as a function of disorder strength $\mathrm{\Delta }$ for different even number of lattice sizes, $N=34$ (red circle), $N=144$ (blue square), and $N=610$ (green upper triangle). (b) The logarithm of the maximum of fidelity susceptibility as a function of the logarithm of the system sizes. Linear fit shows that $\nu =1.01\pm 0.01$. (c) Scaled fidelity susceptibility ${\chi }_F(N,\mathrm{\Delta })/{\chi }_{F,\max }$ as a function of scaled variable $(\mathrm{\Delta }-{\mathrm{\Delta }}_m)N^{1/\nu }$ with ${\mathrm{\Delta }}_m$ being the position of the maximum of the fidelity susceptibility. All curves for even number of lattice sizes collapse into a single curve when we choose the correlation length critical exponents $\nu =1.00\pm 0.01$. Here $\mathrm{\Delta }$ is in units of the hopping amplitude $J$ and the fidelity susceptibility is in units of $1/J^2$.

Figure 5

Universal finite-size scaling of the generalized adiabatic susceptibility ${\chi }_4$ in the Aubry-André model with odd number of lattice sizes. (a) The logarithm of the generalized adiabatic susceptibility ${\chi }_4(N,\mathrm{\Delta })$ as a function of disorder strength $\mathrm{\Delta }$ for different odd number of lattice sizes, $N=89$ (red circle), $N=233$ (blue square), $N=377$ (green upper triangle), and $N=987$ (magenta lower triangle). (b) The logarithm of the maximum of generalized adiabatic susceptibility for odd number of lattice sizes as a function of the logarithm of the system sizes. Linear fit shows that $z\approx 2.38\pm 0.01$. (c) Scaled generalized adiabatic susceptibility ${\chi }_4(N,\mathrm{\Delta })/{\chi }_{4,\max }$ as a function of scaled variable $(\mathrm{\Delta }-{\mathrm{\Delta }}_m)N^{1/\nu }$. All curves for odd number of lattice sizes collapse into a single curve when we choose the correlation length critical exponents $\nu =1.00\pm 0.01$. Here $\mathrm{\Delta }$ is in units of the hopping amplitude $J$ and the generalized adiabatic susceptibility ${\chi }_4$ is in units of $1/J^4$.

Figure 6

Universal finite-size scaling of the generalized adiabatic susceptibility ${\chi }_4$ in the Aubry-André model with even number of lattice sizes. (a) The logarithm of the generalized adiabatic susceptibility ${\chi }_4(N,\mathrm{\Delta })$ as a function of disorder strength $\mathrm{\Delta }$ for different even number of lattice sizes, $N=34$ (red circle), $N=144$ (blue square), and $N=610$ (green upper triangle). (b) The logarithm of the maximum of generalized adiabatic susceptibility for even number of lattice sizes as a function of the logarithm of the system sizes. Linear fit shows that $z\approx 2.37\pm 0.01$. (c) Scaled generalized adiabatic susceptibility ${\chi }_4(N,\mathrm{\Delta })/{\chi }_{4,\max }$ as a function of scaled variable $(\mathrm{\Delta }-{\mathrm{\Delta }}_m)N^{1/\nu }$. All curves for even number of lattice sizes collapse into a single curve when we choose the correlation length critical exponents $\nu =1.00\pm 0.01$. Here $\mathrm{\Delta }$ is in units of the hopping amplitude $J$ and the generalized adiabatic susceptibility ${\chi }_4$ is in units of $1/J^4$.

Figure 7

Fidelity susceptibility ${\chi }_F$ in the Aubry-André model as a function of disorder strength $\mathrm{\Delta }$ for lattice sizes $N=13$. Here $\mathrm{\Delta }$ is in units of the hopping amplitude $J$ and the fidelity susceptibility ${\chi }_F$ is in units of $1/J^2$.