Diagnosing phase transitions through time scale entanglement

Abstract

Spatial entanglement of wave functions has matured into an enthralling and very active research area. Here, we unearth a completely different kind of entanglement, the entanglement between different time scales. This is feasible through quantics tensor train diagnostics (QTTD), wherein the bond dimension for an $n$-particle correlation function allows diagnosing the temporal entanglement. As examples, we study time-scale entanglement of the Hubbard dimer, the four-site Hubbard ring with and without next-nearest neighbor hopping and the single-impurity Anderson model. Besides introducing the QTTD method, our major finding is that the time-scale entanglement is generically maximal at phase transitions and crossovers. This is independent of the correlation function studied. Thus, QTTD is a universal tool for detecting quantum phase transitions, ground state crossings in finite systems, and thermal crossovers.

Figures (5)

• Figure 1: Entanglement of exponentially different time scales for the Hubbard dimer at $T=0.02$: (a) Local spin susceptibility $\chi_s$ indicating different ground states filled with one, two and three electrons; (b) QTT maximum bond dimension $D_{\max}:= \max_{\ell} D_{\ell}$ of the imaginary-time four-point Green's function $G_{1111}^{\uparrow \uparrow \uparrow \uparrow}$ as a function of interaction strength $U$ and chemical potential $\mu/U$, where red-dashed lines mark crossings of the ground state; and (c) QTT bond dimension $D_\ell$ between timescale $2^{-\ell}$ and $2^{-\ell -1}$ for different values of $\mu/U$ at $U=8$, marked by the same symbol and color in (a) and (b).
• Figure 2: Four-site Hubbard ring with nearest-neighbor hopping only: (a) $D_\mathrm{max}$ of the QTT for $G_{1111}^{\uparrow \uparrow \uparrow \uparrow}(\tau_1, \tau_2, \tau_3)$ at $1/T=\beta=50$, $\epsilon=10^{-14}$ and $R=6$ in comparison to (b) the quantum Fisher information $F_Q$, where the white dashed line indicates $F_Q=1$ , (c) spin susceptibility $\chi_S$ and (d) fidelity susceptibility. (e) and (f): $U=4$ and $\mu=U/2$ slice of (a), where red dashed lines in (e) indicate QPTs and dashed lines in (f) indicate $F_Q=1$. In (f), two additional $\beta$'s are shown.
• Figure 3: Four-site Hubbard ring with next nearest-neighbor hopping: Red dots (dotted lines) indicate crossings of s(inglet), d(oublet) and q(uartet) ground states. (a) $D_\mathrm{max}$ of QTT of $G_{1111}^{\uparrow \uparrow \uparrow \uparrow}(\tau_1,\tau_2,\tau_3)$ at $\beta=50$ with $\epsilon=10^{-5}$ and $R=6$. (b)-(c) $D_\mathrm{max}$ for various $\mu/U$-slices at $\epsilon=10^{-8}$.
• Figure 4: (a) $D_\mathrm{max}$ of the two-particle Green's function of the SIAM at $\beta=100$ in the $U$-$V$ plane [red dots indicate $T_K(U,V)=1/\beta$]. (b) Spin susceptibility $\chi_{S}$ indicating the different regimes.
• Figure 5: QTT bond dimension $D_\mathrm{max}$ of different imaginary-time correlation functions for the Hubbard dimer as a function of electronic repulsion $U$ and chemical potential $\mu/U$: (a) $D_\mathrm{max}$ for the one-particle Green's function $G_{11}^{\uparrow \uparrow}$ at temperature $T=\beta^{-1}=1/50$, and (b) for the two-particle Green's function $G_{1111}^{\uparrow \uparrow \uparrow \uparrow}$ at $T=1/30$. Red dashed lines mark crossings of the ground state of the model. These crossings can be diagnosed as a sharp maximum in $D_\mathrm{max}$.