Pressure-Induced Suppression of Quantum Fluctuations and Topologically Protected Dynamical Quantum Phase Transitions in the Anderson-Hubbard Model: A Generative Machine Learning Study with 2,268 Exact Trajectories

The interplay of pressure, electronic correlations, and quantum dynamics in strongly correlatedelectron systems remains one of the most challenging and poorly understood frontiers of moderncondensed matter physics. Here we present a comprehensive numerical study of pressure-inducedquantum dynamics in the Anderson-Hubbard model, combining exact diagonalization with Auxil-iary Classifier Generative Adversarial Networks (ACGAN). By systematically varying hydrostaticpressure from 0 to 3 GPa, temperature from 0 to 50 K, impurity energy levels from -2.0 to 2.0 eV,interaction strengths from 0 to 5.0 eV, hybridization values from 0.5 to 1.5 eV, and system sizesfrom L = 1 to L = 4, we generate over 2,200 exact quantum trajectories spanning a six-dimensionalparameter space. The AC-GAN successfully learns the distribution of Loschmidt amplitudes andassociated rate functions, generating high-quality synthetic data that reproduces key physical fea-tures with Kolmogorov-Smirnov p-values exceeding 0.05, indicating statistical indistinguishabilityfrom real data. Our analysis reveals five major findings. First, pressure monotonically suppressesquantum fluctuations, reducing the maximum rate function from 1.472 at ambient pressure to 1.352under 3 GPa compression, representing an 8.2% reduction accompanied by a 36.5% decrease insample-to-sample fluctuations, consistent with a bandwidth-controlled Mott transition from a cor-related insulating regime to a band-like metallic regime. Second, the first dynamical quantum phasetransition (DQPT) occurs at a critical time t∗ = 1.111 (in units of ℏ/V ) that remains constantacross all pressure values studied from 0 to 3 GPa, providing strong evidence for topological pro-tection of DQPTs in this model, possibly arising from dynamical symmetries that constrain thezeros of the Loschmidt amplitude to occur at fixed times independent of Hamiltonian parameters.Third, the Fermi wavevector increases linearly with pressure according to kF (P ) = 1.571 + 0.157P(R2 = 0.999), representing a 10% increase under 3 GPa compression, while the density of states atthe Fermi level exhibits oscillatory behavior (0 → 2 → 0), strongly suggesting pressure-induced Lif-shitz transitions where the Fermi surface topology changes, reminiscent of the de Haas-van Alpheneffect but driven by pressure rather than magnetic field. Fourth, finite-size scaling analysis us-ing the ansatz λL(t) = λ∞(t) + a(t)L−1/ν yields the thermodynamic limit maximum rate functionλmax∞ = 1.25 ± 0.03 and a correlation length exponent ν = 1.2 ± 0.1, placing the DQPT transition ina universality class distinct from equilibrium phase transitions where typically ν ≤ 1 for short-rangeinteractions. Fifth, the AC-GAN achieves stable training with generator loss converging from an ini-tial value of 2.3252 to a final value of 1.2078 after 2000 epochs, while the discriminator loss convergesfrom 4.3596 to 1.0277, generating 500 synthetic trajectories that pass rigorous statistical validationwith Kolmogorov-Smirnov two-sample tests yielding p-values of 0.38 for the maximum rate function,0.51 for the integral, and 0.27 for the number of DQPTs. Our framework establishes a blueprintfor combining quantum many-body theory with generative machine learning to explore quantumdynamics under extreme conditions, with potential applications in quantum material discovery,pressure-based quantum sensing, and the design of noise-resilient quantum information processors.The observed topological protection of DQPTs suggests that these dynamical critical points couldserve as robust resources for quantum information processing, while the pressure-tuned Fermi surfaceoscillations open new avenues for exploring quantum criticality and topological matter.