Critical Exponent of Dynamical Quantum Phase Transition in One-Dimensional Bose-Hubbard Model in the Strong Interacting Limit

TL;DR

The paper addresses dynamical quantum phase transitions in the strongly interacting Bose-Hubbard model by analyzing the Loschmidt echo following a quench. It develops a P-matrix analytical framework, mapping to spinless fermions via the Jordan-Wigner transformation, to obtain exact Loschmidt echoes for charge-density wave initial states and harmonic confinement driving. The key finding is a universal zero exponent with a logarithmic divergence of the rate function near the critical time, establishing the first analytical DQPT exponent for this bosonic lattice model and showing tunable transition timings through the external potential. This work provides a robust theoretical foundation for observing DQPTs in bosonic systems and guides experimental exploration of dynamical universality and control.

Abstract

We analytically investigated the dynamical quantum phase transitions in the Bose-Hubbard model using the Loschmidt echo as an observable, revealing that after a quench, the global Loschmidt echo exhibits cusp singularities with a logarithmically divergent rate function near criticality and a critical exponent of zero. Through extensive calculations across various system sizes and initial states, we have demonstrated that in the strongly interacting regime, the critical singularity of dynamical quantum phase transitions exhibits consistency across different model details and initial product states (charge-density wave states). Moreover, we find that modifying the harmonic potential well not only preserves the phase transition but also enables precise control over the transition timing.

Figures (4)

• Figure 1: Short-time and long-time dynamical properties of the full system ($L = 32,100,200$) are shown for two distinct initial states. All panels sharing a common legend. (a) and (c) $\left| {1010 \cdots } \right\rangle$ with $V_0=0$. (b) and (d) $\left| {1100 \cdots } \right\rangle$ with $V_0=10^{-2}$.
• Figure 2: The singular behavior of the rate function near the phase transition point is systematically investigated under different conditions: (i) two initial states (panels (a) and (b): $\left| {1010 \cdots } \right\rangle$; other panels: $\left| {1100 \cdots } \right\rangle$); (ii) two potential strengths (panels (e) and (f): ${V_0} = {10^{ - 2}}$; others: 0); with fixed system size $L = 400$. (a), (c) and (e) with timescale $2 \times {10^{ - 4}}$, ${f_L}\left( t \right)$ exhibits a sharply divergent peak at ${t_0}$ (characterized by derivative jumps from $+\infty$ to $-\infty$, distinct from finite jumps in spin systems), whose magnitude increases with temporal resolution-a hallmark of divergence. (b), (d) and (f) with timescale $2 \times {10^{ - 7}}$. All data points represent actual computational results, with the black line showing the linear regression of ${f_L}\left( \tau \right)$ vs $\ln \tau$, and the blue line indicating the linear regression of $\left| \alpha \right|$ vs $1/\ln \tau$-both conclusively demonstrating a zero critical exponent. Identical critical exponents are obtained on both sides of the transition.
• Figure 3: (a) The timing of the first phase transition versus potential well strength $V_0$ (occurring for all ${V_0} \ge 0$). (b) and (c) Rate functions for the $\left| {1100 \cdots } \right\rangle$ state at ${V_0} = 1.05$ and 1.06, respectively. The lattice length $L$ was fixed at 32 with a potential resolution of $V_0$ set to 1/300.
• Figure 4: Evolution of the $\left| {1100 \cdots } \right\rangle$ state's rate function under different statistical parameters: (a) $U/J=1000$, (b) $U/J=0.1$. Key two-site TDVP algorithm parameters: lattice length $L=20$, maximum occupancy per site $n_{max}=3$, maximum number of singular values $\chi=512$, time step $dt=0.005$.