Unbiased Monte Carlo Cluster Updates with Autoregressive Neural Networks

Efficient sampling of complex high-dimensional probability densities is a central task in computational science. Machine Learning techniques based on autoregressive neural networks have been recently shown to provide good approximations of probability distributions of interest in physics. In this work, we propose a systematic way to remove the intrinsic bias associated with these variational approximations, combining it with Markov-chain Monte Carlo in an automatic scheme to efficiently generate cluster updates, which is particularly useful for models for which no efficient cluster update scheme is known. Our approach is based on symmetry-enforced cluster updates building on the neural-network representation of conditional probabilities. We demonstrate that such finite-cluster updates are crucial to circumvent ergodicity problems associated with global neural updates. We test our method for first- and second-order phase transitions in classical spin systems, proving in particular its viability for critical systems, or in the presence of metastable states.

Mott insulating states with competing orders in the triangular lattice Hubbard model

The physics of the triangular lattice Hubbard model exhibits a rich phenomenology, ranging from a metal-insulator transition, intriguing thermodynamic behavior, and a putative spin liquid phase at intermediate coupling, ultimately becoming a magnetic insulator at strong coupling. In this multi-method study, we combine a finite-temperature tensor network method, minimally entangled thermal typical states (METTS), with two Green function-based methods, connected-determinant diagrammatic Monte Carlo (DiagMC) and cellular dynamical mean-field theory (CDMFT), to establish several aspects of this model. We elucidate the evolution from the metallic to the insulating regime from the complementary perspectives brought by these different methods. We compute the full thermodynamics of the model on a width-4 cylinder using METTS in the intermediate to strong coupling regime. We find that the insulating state hosts a large entropy at intermediate temperatures, which increases with the strength of the coupling. Correspondingly, and consistently with a thermodynamic Maxwell relation, the double occupancy has a minimum as a function of temperature which is the manifestation of the Pomeranchuk effect of increased localisation upon heating. The intermediate coupling regime is found to exhibit both pronounced chiral as well as stripy antiferromagnetic spin correlations. We propose a scenario in which time-reversal symmetry broken states compete with nematic, lattice rotational symmetry breaking orders at lowest temperatures.

Many-Configuration Markov-Chain Monte Carlo

We propose a minimal generalization of the celebrated Markov-Chain Monte Carlo algorithm which allows for an arbitrary number of configurations to be visited at every Monte Carlo step. This is advantageous when a parallel computing machine is available, or when many biased configurations can be evaluated at little additional computational cost. As an example of the former case, we report a significant reduction of the thermalization time for the paradigmatic Sherrington-Kirkpatrick spin-glass model. For the latter case, we show that, by leveraging on the exponential number of biased configurations automatically computed by Diagrammatic Monte Carlo, we can speed up computations in the Fermi-Hubbard model by two orders of magnitude.

Efficient One-Loop-Renormalized Vertex Expansions with Connected Determinant Diagrammatic Monte Carlo

We present a technique that enables the evaluation of perturbative expansions based on one-loop-renormalized vertices up to large expansion orders. Specifically, we show how to compute large-order corrections to the random phase approximation in either the particle-hole or particle-particle channels. The algorithms efficiency is achieved by the summation over contributions of all symmetrized Feynman diagram topologies using determinants, and by integrating out analytically the two-body long-range interactions in order to yield an effective zero-range interaction. Notably, the exponential scaling of the algorithm as a function of perturbation order leads to a polynomial scaling of the approximation error with computational time for a convergent series. To assess the performance of our approach, we apply it to the non-perturbative regime of the square-lattice fermionic Hubbard model away from half-filling and report, as compared to the bare interaction expansion algorithm, significant improvements of the Monte Carlo variance as well as the convergence properties of the resulting perturbative series.

Renormalized perturbation theory at large expansion orders

We present a general formalism that allows for the computation of large-order renormalized expansions in the spacetime representation, effectively doubling the numerically attainable perturbation order of renormalized Feynman diagrams. We show that this formulation compares advantageously to the currently standard techniques due to its high efficiency, simplicity, and broad range of applicability. Our formalism permits to easily complement perturbation theory with non-perturbative information, which we illustrate by implementing expansions renormalized by the addition of a gap or the inclusion of Dynamical Mean-Field Theory. As a result, we present numerically-exact results for the square-lattice Fermi-Hubbard model in the low temperature non-Fermi-liquid regime and show the momentum-dependent suppression of fermionic excitations in the antinodal region.

Direct sampling of the self-energy with Connected Determinant Monte Carlo

In this note, we present an efficient algorithm to sample directly the self-energy in the framework of the Connected Determinant technique. The introduction of the formalism of many-variable formal power series is essential to the proof, and more generally it is a natural mathematical tool for diagrammatic expansions.

Determinant Diagrammatic Monte Carlo in the Thermodynamic Limit

We present a simple trick that allows to consider the sum of all connected Feynman diagrams at fixed position of interaction vertices for general fermionic models. With our approach one achieves superior performance compared to Diagrammatic Monte Carlo, while rendering the algorithmic part dramatically simpler. As we consider the sum of all connected diagrams at once, we allow for cancellations between diagrams with different signs, alleviating the sign problem. Moreover, the complexity of the calculation grows exponentially with the order of the expansion, which should be constrasted with the factorial growth of the standard diagrammatic technique. We illustrate the efficiency of the technique for the two-dimensional Fermi-Hubbard model.