Matrix Product State Simulation of Quantum Phase Transition

Abstract

The fundamental computational challenge in Quantum many body physics is about the exponential growth of the Hilbert space with the system size, therefore making it impossible to do exact simulation for more than a few tens of qubits. Matrix Product States (MPS) is a efficient way used to represent quantum states that obey the entanglement area law, using which one can accurately simulate one-dimensional quantum systems in polynomial computational cost. In this work, we develop a GPU-based MPS simulator in PyTorch and use it to study phase transitions in the Transverse Field Ising Model (TFIM) and its non-integrable version, the Mixed-Field Ising Model (MFIM). We also implement imaginary time evolution using the Time-Evolving Block Decimation (TEBD) algorithm with a second-order Suzuki-Trotter decomposition and SVD-based bond dimension truncation. We check the accuracy against exact diagonalisation for system sizes N =6 to N = 12, giving errors of order 10→6 and 10→4 respectively at the critical point. The ground state phase diagram of the TFIM is described by computing six observables, energy density, longitudinal magnetisation <Z>, transverse magnetisation <X>, entanglement entropy, maximum bond dimension and a susceptibility proxy. The finite-size scaling of the half-chain entanglement entropy over system sizes N =6 to N = 20 gives a central charge c = 0.530, which is nearly in accordance with the exact value c =0.5 for the 2D Ising universality class. Moreover the power-law decay of ZZ spin correlations at criticality is also validated along with computing the entanglement spectrum and showing Rényi entropies for multiple indices. Finally, we study the MFIM in the ordered phase (h/J = 0.5), where the longitudinal field forces the existing ferromagnetic order, causing the ground state to move toward a classical product state and thereby reducing entanglement entropy and bond dimension requirements as g increases. This demonstrates that the effect of non-integrability on entanglement is strongly phase-dependent, near the critical point, a longitudinal field would instead compete with quantum fluctuations and increase entanglement, a direction left for future work.