We study numerically the saddle point structure of two-dimensional (2D) lattice gauge theory, represented by the Gross-Witten-Wadia unitary matrix model. The saddle points are in general complex-valued, even though the original integration variables and action are real. We confirm the trans-series/instanton gas structure in the weak-coupling phase, and identify a new complex-saddle interpretation of non-perturbative effects in the strong-coupling phase. In both phases, eigenvalue tunneling refers to eigenvalues moving off the real interval, into the complex plane, and the weakto-strong coupling phase transition is driven by saddle condensation. Introduction: Path integral saddle points are physically important in quantum mechanics, matrix models, quantum field theory (QFT) and string theory, and are deeply related to the typical asymptotic nature of weak coupling perturbative expansions. Such relations are central to the concept of resurgence, whereby different saddles are intertwined by monodromy properties that connect them and account for Stokes phases. The theory of resurgence has recently provided new insights into matrix models and string theories [1][2][3][4][5][6], and has been applied to asymptotically free QFTs and sigma models [7][8][9][10], and localizable SUSY QFTs [11]. One motivation for such QFT studies is to find a practical numerical implementation of a semiclassical expansion that could provide a Picard-Lefschetz thimble decomposition of gauge theory, either in the continuum or on the lattice, especially for theories with a sign problem [12,13]. A unifying theme in these studies, and in related work [14][15][16][17], is the appreciation that complex saddles are important, in particular in the context of phase transitions, even though the original 'path integral' may be a sum over only real configurations.

We report on the mean-field study of the Chiral Magnetic Effect (CME) in static magnetic fields within a simple model of a parity-breaking Weyl semimetal given by the lattice Wilson-Dirac Hamiltonian with constant chiral chemical potential. We consider both the mean-field renormalization of the model parameters and nontrivial corrections to the CME originating from re-summed ladder diagrams with arbitrary number of loops. We find that on-site repulsive interactions affect the chiral magnetic conductivity almost exclusively through the enhancement of the renormalized chiral chemical potential. Our results suggest that nontrivial corrections to the chiral magnetic conductivity due to inter-fermion interactions are not relevant in practice, since they only become important when the CME response is strongly suppressed by the large gap in the energy spectrum.

We study the static electric current due to the Chiral Magnetic Effect in samples of Weyl semimetals with slab geometry, where the magnetic field is parallel to the boundaries of the slab. We use the Wilson-Dirac Hamiltonian as a simplest model of parity-breaking Weyl semimetal with two-band structure. We find that the CME current is strongly localized at the open boundaries of the slab, where the current density in the direction of the magnetic field approaches the conventional value j = µ 5 B 2π 2 at sufficiently small values of the chiral chemical potential µ 5 and magnetic field B. On the other hand, very large values of magnetic field tend to suppress the CME response. We observe that the localization width of the current is independent of the slab width and is given by the magnetic length l B = 1/ √ B when µ 5 √ B. In the opposite regime when µ 5 √ B the localization width is determined solely by µ 5 .

The concept of Lefschetz thimble decomposition is one of the most promising possible modifications of Quantum Monte Carlo (QMC) algorithms aimed at alleviating the sign problem which appears in many interesting physical situations, e.g. in the Hubbard model away from half filling. In this approach one utilizes the fact that the integral over real variables with an integrand containing a complex fluctuating phase is equivalent to the sum of integrals over special manifolds in complex space ("Lefschetz thimbles"), each of them having a fixed complex phase factor. Thus, the sign problem can be reduced if the resulting sum contains terms with only a few different phases. We explore the complexity of the sign problem for the few-site Hubbard model on square lattice combining a semi-analytical study of saddle points and thimbles in small lattices with several steps in Euclidean time with results of test QMC calculations. We check different variants of conventional Hubbard-Stratonovich transformation based on Gaussian integrals. On the basis of our analysis we reveal a regime with minimal number of relevant thimbles in the vicinity of half filling. In this regime we found only two relevant thimbles for the few-site lattices studied in the paper. There is also indirect evidence of the existence of this regime in more realistic systems with large number of Euclidean time slices. In addition, we derive a new non-Gaussian representation of the interaction term, where the number of relevant Lefschetz is also reduced in comparison with conventional Gaussian Hubbard-Stratonovich transformation.