The imaginary-time Mpemba effect (ITME) describes a counterintuitive relaxation scenario in which a higher-energy initial state approaches the ground state faster than a lower-energy one. While most studies focus on energy relaxation, whether this anomalous hierarchy persists across other observables remains largely unexplored. Here, using exact diagonalization and Gutzwiller mean-field theory, we systematically investigate the ITME in the one- and two-dimensional extended Bose-Hubbard model—and crucially, contrast energy relaxation with that of other physical quantities, including the superfluid order parameter, local density, and trace distance. Exact diagonalization reveals that the imaginary-time Mpemba effect is controlled by the initial state's spectral projection onto low-lying eigenstates. Mean-field calculations further show that the imaginary-time Mpemba effect occurs across distinct quantum phases, including superfluid and supersolid, indicating a degree of universality. Strikingly, the relaxation of energy is not always synchronized with that of other physical quantities: Energy curves may cross even when the superfluid order parameter does not and vice versa. To further clarify the self-consistent mean-field dynamics, we analyze both the nonlinear relaxation stage and the asymptotic regime near the target fixed point. In a representative ITME case, the initially higher-energy state exhibits a stronger second-instantaneous-excited-state contribution and a larger energy-decay rate, while the late-time relaxation near the target fixed point is governed by the amplitudes of the slow relaxation modes. The results connect spectral projection, self-consistent nonlinear evolution, and observable-dependent relaxation, offering a new perspective on anomalous relaxation in strongly correlated many-body systems.

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• Dipolar interaction
• Quantum phase transitions
• Nonequilibrium systems
• Bose-Hubbard model