Monte Carlo Simulations of Many-Body Quantum Systems with Global Constraints

Global constraints arise naturally in many-body quantum systems, e.g., systems related by duality, boundary Hamiltonians of topologically ordered states, etc. Such constraints pose a key difficulty in that the basis states for the description of the system do not admit a local product decomposition and carry non-trivial entanglement, posing challenges such as the inapplicability of Monte Carlo methods based on local variable updates. Focussing on systems constructed out of qubit (spin-½) degrees of freedom, with a global “singlet” constraint that allows only those states that are invariant under a global spin flip, we formulate a Euclidean quantum Monte-Carlo method for their study. We show that such systems can be mapped to a classical statistical mechanics model in one higher dimension, which possesses a new subsystem symmetry. We show that the non-breaking of the subsystem symmetry is equivalent to the global singlet constraint of the quantum problem and devise a generalized Metropolis Monte-Carlo method (with unconstrained local updates) to obtain equilibrium properties. We apply this to the transverse field Ising model with the singlet constraint and show excellent agreement for the values of critical exponents compared to analytical results. We discuss how this method can be applied to the study of gauge theories and fracton systems.