2,954 research outputs found
A numerical method to compute derivatives of functions of large complex matrices and its application to the overlap Dirac operator at finite chemical potential
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 gauge field configurations on lattices with
sizes as large as and .Comment: 26 pages elsarticle style, 5 figures minor text changes, journal
versio
New Algebraic Formulation of Density Functional Calculation
This article addresses a fundamental problem faced by the ab initio
community: the lack of an effective formalism for the rapid exploration and
exchange of new methods. To rectify this, we introduce a novel, basis-set
independent, matrix-based formulation of generalized density functional
theories which reduces the development, implementation, and dissemination of
new ab initio techniques to the derivation and transcription of a few lines of
algebra. This new framework enables us to concisely demystify the inner
workings of fully functional, highly efficient modern ab initio codes and to
give complete instructions for the construction of such for calculations
employing arbitrary basis sets. Within this framework, we also discuss in full
detail a variety of leading-edge ab initio techniques, minimization algorithms,
and highly efficient computational kernels for use with scalar as well as
shared and distributed-memory supercomputer architectures
Random matrix analysis of the QCD sign problem for general topology
Motivated by the important role played by the phase of the fermion
determinant in the investigation of the sign problem in lattice QCD at nonzero
baryon density, we derive an analytical formula for the average phase factor of
the fermion determinant for general topology in the microscopic limit of chiral
random matrix theory at nonzero chemical potential, for both the quenched and
the unquenched case. The formula is a nontrivial extension of the expression
for zero topology derived earlier by Splittorff and Verbaarschot. Our
analytical predictions are verified by detailed numerical random matrix
simulations of the quenched theory.Comment: 33 pages, 9 figures; v2: minor corrections, references added, figures
with increased statistics, as published in JHE
The QCD sign problem and dynamical simulations of random matrices
At nonzero quark chemical potential dynamical lattice simulations of QCD are
hindered by the sign problem caused by the complex fermion determinant. The
severity of the sign problem can be assessed by the average phase of the
fermion determinant. In an earlier paper we derived a formula for the
microscopic limit of the average phase for general topology using chiral random
matrix theory. In the current paper we present an alternative derivation of the
same quantity, leading to a simpler expression which is also calculable for
finite-sized matrices, away from the microscopic limit. We explicitly prove the
equivalence of the old and new results in the microscopic limit. The results
for finite-sized matrices illustrate the convergence towards the microscopic
limit. We compare the analytical results with dynamical random matrix
simulations, where various reweighting methods are used to circumvent the sign
problem. We discuss the pros and cons of these reweighting methods.Comment: 34 pages, 3 figures, references added, as published in JHE
Fermionic Operator Mixing in Holographic p-wave Superfluids
We use gauge-gravity duality to compute spectral functions of fermionic
operators in a strongly-coupled defect field theory in p-wave superfluid
states. The field theory is (3+1)-dimensional N=4 supersymmetric SU(Nc)
Yang-Mills theory, in the 't Hooft limit and with large coupling, coupled to
two massless flavors of (2+1)-dimensional N=4 supersymmetric matter. We show
that a sufficiently large chemical potential for a U(1) subgroup of the global
SU(2) isospin symmetry triggers a phase transition to a p-wave superfluid
state, and in that state we compute spectral functions for the fermionic
superpartners of mesons valued in the adjoint of SU(2) isospin. In the spectral
functions we see the breaking of rotational symmetry and the emergence of a
Fermi surface comprised of isolated points as we cool the system through the
superfluid phase transition. The dual gravitational description is two
coincident probe D5-branes in AdS5 x S5 with non-trivial worldvolume SU(2)
gauge fields. We extract spectral functions from solutions of the linearized
equations of motion for the D5-branes' worldvolume fermions, which couple to
one another through the worldvolume gauge field. We develop an efficient method
to compute retarded Green's functions from a system of coupled bulk fermions.
We also perform the holographic renormalization of free bulk fermions in any
asymptotically Euclidean AdS space.Comment: 68 pages, 25 eps files in 9 figures; v2 minor corrections, added two
references, version published in JHE
Machine-learning of atomic-scale properties based on physical principles
We briefly summarize the kernel regression approach, as used recently in
materials modelling, to fitting functions, particularly potential energy
surfaces, and highlight how the linear algebra framework can be used to both
predict and train from linear functionals of the potential energy, such as the
total energy and atomic forces. We then give a detailed account of the Smooth
Overlap of Atomic Positions (SOAP) representation and kernel, showing how it
arises from an abstract representation of smooth atomic densities, and how it
is related to several popular density-based representations of atomic
structure. We also discuss recent generalisations that allow fine control of
correlations between different atomic species, prediction and fitting of
tensorial properties, and also how to construct structural kernels---applicable
to comparing entire molecules or periodic systems---that go beyond an additive
combination of local environments
Transport Properties of Chiral Fermions
Anomalous transport phenomena have their origin in the chiral anomaly, the anomalous non-conservation of the axial charge, and can arise in systems with chiral fermions. The anomalous transport properties of free fermions are well understood, but little is known about possible corrections to the anomalous transport coefficients that can occur if the fermions are strongly interacting.
The main goal of this thesis is to study anomalous transport effects in media with strongly interacting fermions. In particular, we investigate the Chiral Magnetic Effect (CME) in a Weyl Semimetal (WSM) and the Chiral Separation Effect (CSE) in finite-density Quantum Chromodynamics (QCD).
The recently discovered WSMs are solid state crystals with low-energy excitations that behave like Weyl fermions. The inter-electron interaction in WSMs is typically very strong and non-perturbative calculations are needed to connect theory and experiment. To realistically model an interacting, parity-breaking WSM we use a tight-binding lattice Hamiltonian with Wilson-Dirac fermions. This model features a non-trivial phase diagram and has a phase (Aoki phase/axionic insulator phase) with spontaneously broken CP-symmetry, corresponding to the phase with spontaneously broken chiral symmetry for interacting continuum Dirac fermions. We use a mean-field ansatz to study the CME in spatially modulated magnetic fields and find that it vanishes in the Aoki phase. Moreover, our calculations show that outside of the Aoki phase the electron interaction has only a minor influence on the CME. We observe no enhancement of the magnitude of the CME current.
For our non-perturbative study of the CSE in QCD we use the framework of lattice QCD with overlap fermions. We work in the quenched approximation to avoid the sign problem that comes with introducing a finite chemical potential on the lattice. The overlap operator calls for the evaluation of the sign function of a matrix with a dimension proportional to the volume of the lattice. For reasonably large lattices it is not feasible to compute the matrix sign function exactly and one has to resort to approximation methods. To compute conserved currents for the overlap operator it is necessary to take derivatives of the overlap operator with respect to the U(1) lattice gauge field. Depending on which approximation is used to evaluate the overlap operator it is not always clear how to compute this derivative. We develop and implement a new numerical method to take derivatives of matrix functions. This method makes it possible to calculate the conserved currents of the finite-density overlap operator with high precision and opens the way to explore anomalous transport phenomena on the lattice.
We study the CSE in the confining and deconfining phase of QCD. On very small lattices we observe corrections to the CSE in the phase with broken chiral symmetry, which seem to be of topological origin. For larger lattices we find that in both phases the CSE current is the same as for free fermions
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