Quantum phase transitions detected by a local probe using time correlations and violations of Leggett-Garg inequalities

In the present paper we introduce a way of identifying quantum phase transitions of many-body systems by means of local time correlations and Leggett-Garg inequalities. This procedure allows us to experimentally determine the quantum critical points not only of finite-order transitions but also those of infinite order, as the Kosterlitz-Thouless transition that is not always easy to detect with current methods. By means of simple analytical arguments for a general spin-$1/2$ Hamiltonian, and matrix product simulations of one-dimensional $XXZ$ and anisotropic $XY$ models, we argue that finite-order quantum phase transitions can be determined by singularities of the time correlations or their derivatives at criticality. The same features are exhibited by corresponding Leggett-Garg functions, which noticeably indicate violation of the Leggett-Garg inequalities for early times and all the Hamiltonian parameters considered. In addition, we find that the infinite-order transition of the $XXZ$ model at the isotropic point can be revealed by the maximal violation of the Leggett-Garg inequalities. We thus show that quantum phase transitions can be identified by purely local measurements and that many-body systems constitute important candidates to observe experimentally the violation of Leggett-Garg inequalities.

Figure 1

STC along the $z$ direction of the $XXZ$ model, as a function $Jt$ and anisotropy parameter $\mathrm{\Delta }$. The dashed green lines indicate the critical point $\mathrm{\Delta }=-1$.

Figure 2

STC along the $z$ direction of the $XXZ$ model, as a function of $\mathrm{\Delta }$, for $Jt=1$ and $Jt=2$.

Figure 3

STC along the $x$ direction of the $XXZ$ model, as a function of $Jt$ and anisotropy parameter $\mathrm{\Delta }$.

Figure 4

STC along the $x$ direction of the $XXZ$ model, as a function of $\mathrm{\Delta }$, for $Jt=1$ and $Jt=2$.

Figure 5

STC along the $z$ direction of the anisotropic $XY$ model for $\gamma =1$ (Ising model), as a function of $Jt$ and $\nu$.

Figure 6

STC (upper panel) and first derivative (lower panel) along the $z$ direction of the anisotropic $XY$ model $\left(\gamma =0.5,1\right)$, as a function of $\nu$, for $Jt=1$ and $Jt=2$.

Figure 7

Leggett-Garg functions for the $XXZ$ model. Upper panel: $K_{-}^z\left(t\right)$. The regions with diagonal lines correspond to the regime of anisotropies $\mathrm{\Delta }$ and $Jt$ in which the $x$ Leggett-Garg inequalities are violated. The white region below $\mathrm{\Delta }=-1$ indicates that there the time correlations remain constant. Lower panel: $K_{-}^x\left(t\right)$. The region with diagonal lines corresponds to the regime of anisotropies $\mathrm{\Delta }$ and $Jt$ in which the $x$ Leggett-Garg inequalities are violated.

Figure 8

Maximum violation of Leggett-Garg inequalities for the $XXZ$ model, along both $z$ and $x$ directions.

Figure 9

Leggett-Garg functions $K_{-}^z\left(t\right)$ for the anisotropic $XY$ model. Upper panel: $\gamma =1$. Lower panel: $\gamma =0.5$. The regions with diagonal lines correspond to the regime of parameter $\nu$ and $Jt$ in which the $z$ Leggett-Garg inequalities are violated.

Figure 10

Upper panel: Maximum violation of Leggett-Garg inequalities for the anisotropic $XY$ model $\left(\gamma =1,0.5\right)$ along the $z$ direction, as a function of $\nu$. Lower panel: First derivative with respect to $\nu$. Inset: scaling of maximum of the derivative with respect to $\nu$.

Figure 11

Total maximum violation of Leggett-Garg inequalities for the $XXZ$ model as a function of $\mathrm{\Delta }$. The light-red zones indicate the regimes in which the maximal violations come from inequalities along the $x$ direction, while the light-blue zone in between shows the regime in which the maximal violations occur along the $z$ direction; see Fig. 8.