Determinant Diagrammatic Monte Carlo Algorithm in the Thermodynamic Limit

Abstract

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, such that the thermodynamic limit can be taken analytically. With our approach one can achieve superior performance compared to conventional Diagrammatic Monte Carlo, while rendering the algorithmic part dramatically simpler. By considering the sum of all connected diagrams at once, we allow for massive cancellations between different diagrams, greatly reducing the sign problem. In the end, the computational effort increase only exponentially with the order of the expansion, which should be contrasted with the factorial growth of the standard diagrammatic technique. We illustrate the efficiency of the technique for the two-dimensional Fermi-Hubbard model. Finding an efficient method to solve the quantum many-body problem is of fundamental physical importance. A promising strategy to gain insight is represented by quantum simulators. For example, experimentalists working with cold atoms in optical lattices are able to realise one of the most prominent models of stronglycorrelated electrons, the Hubbard model. A study of its equation of state at low-temperature has been reported in [1]. Moreover, short-range antiferromagnetic correlations have been observed [2][3][4][5][6], and, very recently, even a long-range antiferromagnetic state was realized [7].From the theoretical side, making predictions for strongly-correlated fermionic systems is a very challenging problem. Quantum Monte Carlo methods are affected by the infamous sign problem when dealing with fermionic models, which in general precludes reaching low-temperature and large system sizes (see [8,9] and references therein). The simulation time to obtain a given precision scales exponentially with the system size and the inverse of the temperature. In the strongly correlated regime, it is then very difficult to extrapolate to the Thermodynamic Limit (TL). However, the fermionic sign gives an unexpected advantage when considering fermionic perturbation theory. For bosonic theories, typically, the perturbative expansion has a zero radius of convergence. This is due to the factorial number of Feynman diagrams all contributing with the same sign. For fermions, strong cancellations between different diagrams at fixed order lead in general to a finite convergence radius on a lattice at finite temperature. This happens because for sufficiently small interactions of whatever sign (or phase) the system is stable on a lattice at non-zero temperature. This was seen numerically [10,11], and even proved mathematically for the Hubbard model at high enough temperature [12]. Perturbation theory is therefore a powerful tool to study fermionic theories. By * riccardo.rossi@ens.fr analytic continuation, any point of the phase diagram which is in the same phase as the perturbative starting point can be reached in principle, provided that one has a method to compute the diagrammatic series to high order. At the moment...

Real-frequency diagrammatic Monte Carlo at finite temperature

“…The control parameters of the calculations are most commonly the lattice size and the maximal perturbation order. Some algorithms [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] are very efficient for small systems but have not yet reached very large lattice sizes, while others [17][18][19][20][21][22][23][24] can address the thermodynamic limit directly but are limited in the number of perturbation orders that can be computed.…”

Spin and Charge Correlations across the Metal-to-Insulator Crossover in the Half-Filled 2D Hubbard Model

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Cancellation of vacuum diagrams and the long-time limit in out-of-equilibrium diagrammatic quantum Monte Carlo

“…Recent developments in imaginarytime diagrammatic QMC also achieved, through an iterative procedure, the cancellation of vacuum (and, later on, non oneparticle irreducible) diagrams at every Monte Carlo step at an exponential cost in the perturbation order. [38][39][40]…”

“…At a given perturbation order n, its key idea is to regroup a factorial number of Feynman dia-grams in a sum over Keldysh indices of 2 n determinants. This exponential sum has been shown to cancel vacuum diagrams, a property also used in recent diagrammatic QMC methods in imaginary-time [38][39][40] . As a direct consequence, the Monte Carlo sampling only involves interaction times in a neighborhood around the measurement time t max : we talk about the clusterization of times.…”