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We present a method for the numerical calculation of derivatives of functions of general complex matrices. The method can be used in combination with any algorithm that evaluates or approximates the desired matrix function, in particular with implicit Krylov–Ritz-type approximations. An important use case for the method is the evaluation of the overlap Dirac operator in lattice Quantum Chromodynamics (QCD) at finite chemical potential, which requires the application of the sign function of a non-Hermitian matrix to some source vector. While the sign function of non-Hermitian matrices in practice cannot be efficiently approximated with source-independent polynomials or rational functions, sufficiently good approximating polynomials can still be constructed for each particular source vector. Our method allows for an efficient calculation of the derivatives of such implicit approximations with respect to the gauge field or other external parameters, which is necessary for the calculation of conserved lattice currents or the fermionic force in Hybrid Monte-Carlo or Langevin simulations. We also give an explicit deflation prescription for the case when one knows several eigenvalues and eigenvectors of the matrix being the argument of the differentiated function. We test the method for the two-sided Lanczos approximation of the finite-density overlap Dirac operator on realistic S U ( 3 ) gauge field configurations on lattices with sizes as large as 14 × 14 3 and 6 × 18 3 . [ABSTRACT FROM AUTHOR]

Abstract We present a non-iterative solver based on the Schur complement method for sparse linear systems of special form which appear in Quantum Monte-Carlo (QMC) simulations of strongly interacting fermions on the lattice. While the number of floating-point operations for this solver scales as the cube of the number of lattice sites, for practically relevant lattice sizes it is still significantly faster than iterative solvers such as the Conjugate Gradient method in the regime of strong inter-fermion interactions, for example, in the vicinity of quantum phase transitions. The speed-up is even more dramatic for the solution of multiple linear systems with different right-hand sides. We present benchmark results for QMC simulations of the tight-binding models on the hexagonal graphene lattice with on-site (Hubbard) and non-local (Coulomb) interactions, and demonstrate the potential for further speed-up using GPU. [ABSTRACT FROM AUTHOR]

We consider the second Rényi entropy S(2) in pure lattice gauge theory with SU(2), SU(3), and SU(4) gauge groups, which serves as a first approximation for the entanglement entropy and the entropic C -function. We compare the results for different gauge groups using scale setting via the string tension. We confirm that at small distances l our approximation for the entropic C -functioNc (l), calculated for the slab-shaped entangled region of width l, scales as Nc² - 1 in accordance with its interpretation in terms of free gluons. At larger distances l C(l) is found to approach zero for Nc = 3, 4, somewhat more rapidly for Nc = 4 than for Nc = 3. This finding supports the conjectured discontinuity of the entropic C -function in the large- N limit, which was found in the context of AdS/CFT correspondence and which can be interpreted as transition between colorful quarks and gluons at small distances and colorless confined states at long distances. On the other hand, for SU(2) gauge group the long-distance behavior of the entropic C-function is inconclusive so far. There exists a small region of lattice spacings yielding results consistent with Nc = 3, 4, while results from other lattice spacings deviate without clear systematics. We discuss several possible causes for discrepancies between our results and the behavior of entanglement entropy in holographic models. [ABSTRACT FROM AUTHOR]