EMMI Workshop: SIGN 2014

I will review recent results on fluctuations of conserved charges at finite temperature and discuss how they can be used to estimate the temperature and chemical potential at the time of chemical freeze-out.

We will review recent progress on the calculation of conserved charge fluctuations with highly improved staggered quarks (HISQ). In particular we will focus on higher order cumulants of net baryon number, net electric charge and net strangeness fluctuations. We will discuss how these quantities approach the hadron resonance gas model at low temperatures and analyze to what extent they show sensitivity to universal scaling behavior, i.e. we estimate the relative strength of contributions from the regular and singular part of the free energy. Based on this analysis we discuss consequences for the convergence radius of the Taylor expansion of the pressure and possible experimental observables of the QCD critical point.

We investigate the phase diagram in the temperature, imaginary chemical potential plane for Nf=3 QCD using Wilson type fermions. While more expensive than the staggered fermions used in past studies in this area, Wilson fermions can be used safely to simulate systems with three quark flavors. In this talk, we focus on the (pseudo)critical line that extends from mu=0 in the imaginary chemical potential plane, trace it to the Roberge-Weiss line, and determine its location relative to the Roberge-Weiss transition point. Our results are compatible with the standard expectations.

I give an overview of recent results for using dual variables in lattice field theories at finite density. I review systems where the complex action problem was solved with dual variables and discuss the role of these reference models for assessing other approaches to finite density.

Rewriting the partition sum in its dual representation allows us to solve the sign problem exactly in several systems. Therefore we can use the results obtained with the dual representation as a reference test for other approaches. For example, I will show a comparison between the dual representation and different expansion methods, which are also used in full QCD at finite density.

We discuss the obvious generalization of the successful 'worm' simulation methods from scalar field theories to abelian gauge models. Due to the geometrically different situation, gauge simulations along these lines (so-far) still suffer from critical slowing down. Other aspects like the low noise estimation of observables -- here the Polyakov lines and partition function ratios -- do generalize. We exploit these to compare the 3D Z(2) gauge theory with effective string models.

In three different fermionic cases---repulsive Hubbard model, resonant fermions, and fermionized spins-1/2 (on triangular lattice)---we observe the phenomenon of sign blessing: Feynman diagrammatic series features finite convergence radius despite factorial growth of the number of diagrams with diagram order. Bold diagrammatic Monte Carlo technique allows us to sample millions of skeleton Feynman diagrams. With the universal fermionization trick we can fermionize essentially any (bosonic, spin, mixed, etc.) lattice system. The combination of fermionization and Bold diagrammatic Monte Carlo yields a universal first-principle approach to strongly correlated lattice systems, provided the sign blessing is a generic fermionic phenomenon.

Properties of geometrically frustrated spin systems in various dimensions, geometries, and temperature regimes constitute one of the main research directions of modern condensed matter physics. These studies however are severely hampered by the infamous sign problem synonymous with frustrated geometry. Here I summarize key physical features of frustrated quantum antiferromagnet, using triangular as an example, and then proceed to describe first successful application of the bold diagrammatic Monte Carlo technique, based on exact Popov-Fedotov trick of representing spin operators with fermions, to this important problem. I present results for the static spin correlation function at various temperatures and describe unexpected approximate quantum-to-classical correspondence between short-distance behaviors of the quantum and classical spin models.

We solve the sign problem in a class of particle-hole symmetric spin-less fermion models using the idea of fermion bags. The solutions emerge in the Hamilton formulation of lattice field theory where time can be continuous. The solution allows us for the first time to explore quantum phase transitions in models with a single four-component Dirac fermion in three dimensions.

Over the last two years, a 3d effective lattice theory has been developed by means of strong coupling and hopping expansions, which is a valid description of QCD with large quark masses and has only a mild sign problem. This contribution summarises the extension of the effective theory to order kappa^4 and from one to two flavours. The theory is applied to a description of the nuclear liquid gas transition which is compared to the behaviour at finite isospin chemical potential.

We briefly review the derivation of an effective Polyakov Loop theory for finite density Lattice QCD and discuss recent developments. As a further test of the effective theory, we compute several observables numerically and compare with results from full (3+1)d simulations.

The "relative weights" method is used to derive the effective Polyakov line action corresponding to an underlying lattice gauge theory at any chemical potential. Results for SU(2) and SU(3) gauge theories are presented. The motivation is that it may be easier to address the sign problem in the effective model, e.g. by reweighting or other techniques, than in the original lattice gauge theory.

I summarize the results obtained on the (mu,T) phase diagram of lattice QCD with staggered fermions in the strong coupling limit, beta=0, and show how to determine at least the leading, O(beta) corrections.

A partition function for strong coupling lattice QCD valid at O(beta) has been derived recently. We present numerical results based on the reweighting of fermionic observables to finite beta. In this way we obtain the beta-dependence of the phase boundary in the mu-T plane, based on the chiral susceptibility and the baryon number density. We also present gauge observables such as the Polyakov loop correlator and Wilson loops and sketch how to obtain higher order corrections in general.

The pion condensation phase of QCD at large isospin chemical potential is studied in the presence of an external magnetic field. Since the up and down quark electric charges differ, the external field induces a sign problem, which is circumvented through a Taylor-expansion method.

We consider the sign problem for classical spin models at complex beta =1/g^2 on square lattices. We show that the tensor renormalization group method allows reliable calculations for larger Im beta than the reweighting Monte Carlo method. For the Ising model with complex beta we compare our results with the exact Onsager-Kaufman solution at finite volume. We show that the Fisher zeros can be determined precisely with the TRG method. We check the convergence of the TRG method for the O(2) model when the number of states increases. We show that the finite size scaling of the calculated Fisher zeros agrees very well with the Kosterlitz-Thouless transition assumption and predicts the locations for larger volume. We present new applications of the TRG method for the O(2) model with a chemical potential. It provides robust estimations of the eigenvalues of the transfer matrix. The results are in good agreement with results obtained with the worm algorithm developed by Banerjee and Chandrasekharan. We discuss the phase diagram in the beta-mu plane. We discuss the possibility of using this framework to describe real time evolution.

The dependence of 4D SU(N) gauge theories on the topological theta term is investigated at finite temperature, and in particular in the large-N limit. General arguments and numerical analyses exploiting the lattice formulation show that it drastically changes across the deconfinement transition. The low-T phase is characterized by a large-N scaling with theta/N as relevant variable, while in the high-T phase the scaling variable is just theta and the free energy is essentially determined by the instanton-gas approximation.

I discuss recent results regarding the phase structure non-Abelian gauge theories in the presence of a topological theta term, which are based on numerical simulations performed at imaginary values of theta.

The complex Langevin method is extended to full QCD at non-zero chemical potential. The use of gauge cooling stabilizes the simulations at small enough lattice spacings. The method allows simulations also at high densities, all the way up to saturation. Results are compared to the reweighting approach.

Abstract: Complex Langevin dynamics is the most appealing and challenging approach to solve the sign problem in QCD at nonzero chemical potential. We apply complex Langevin dynamics to chiral random matrix theory. In the microscopic limit this random matrix theory is equivalent to QCD and the sign problem due to the complex valued fermion determinant is equally severe. We discuss how the Langevin force is affected by the presence of the logarithm of the fermion determinant in the action and test the complex Langevin simulations against the exact analytic results for chiral random matrix theory.

The formal justification of the Langevin method for the solution of sign problems does not allow singularities in the drift. This is a problem for QCD at finite density where zeroes of the fermion determinant produce poles in the drift. We review briefly the mathematical origin of the problem and describe various strategies to deal with it, as well as their limitations.

We study the various aspects of the method on simple, soluble cases used as effective models, then extend the analysis to more complex lattice models with non-trivial phase structure. We finally apply the method to study HQCD, a full QCD-approximation for heavy quarks which allows putting questions of direct physical interest.

In this talk, I summarize the Lefschetz thimble approach to the sign problem of Quantum Field Theories. In particular, I review its motivations, and summarize the results of its first applications

We consider a hybrid Monte Carlo algorithm which is applicable to lattice theories defined on Lefschetz thimbles. In the algorithm, any point (field configuration) on a thimble is parametrized uniquely by the flow-direction and the flow-time defined at a certain asymptotic region close to the critical point, and it is generated by solving the gradient flow equation downward. The associated complete set of tangent vectors is also generated in the same manner. Molecular dynamics is then formulated as a constrained dynamical system, where the equations of motion with Lagrange multipliers, are solved by the second-order constraint-preserving symmetric integrator. The algorithm is tested in the lambda phi^4 model at finite density, by choosing the thimbles associated with the classical vacua for subcritical and supercritical values of chemical potential. For the lattice size L=4, we find that the residual sign factors are safely included by reweighting and that the results of the number density are consistent with those obtained by the complex Langevin simulations.

I present a subset solution of the sign problem in one-dimensional QCD with various numbers of flavors.

We present properties of Z3 subsets for QCD in two or more dimensions.

We investigate the phase structure of (2+Nf)-flavor QCD, where two light flavors and Nf massive flavors exist, to discuss the feasibility of the electroweak baryogenesis in realistic technicolor scenario and to understand properties of finite density QCD. Because an appearance of a first order phase transition at finite temperature is a necessary condition for the baryogenesis, it is important to study the nature of finite temperature phase transition. Applying the reweighting method, the probability distribution function of the plaquette is calculated in the many-flavor QCD. Through the shape of the distribution function, we determine the critical mass of heavy flavors terminating the first order region, and find it to become larger with Nf. We then study the critical line at finite density and the first order region is found to become wider as increasing the chemical potential. We discuss how the properties of real (2+1)-flavor QCD at finite temperature and density can be understand from simulations of many-flavor QCD.

We apply the histogram method to investigate the critical surface of QCD in the heavy quark region using lattice QCD simulation. In this study, we employ the reweighting technique and cumulant expansion in order to analyze the quark masses dependence of the critical surface and the quark chemical potential dependence. Finally, we show our results that a series of our methods works well to clarify the critical surface of the heavy quark region.

Dense matter quantum field theory is hampered by the notorious sign-problem: the Gibbs factor is not (semi-)positive definite and standard Monte-Carlo simulations are not applicable. We propose to perform Monte-Carlo sampling with respect to the density of states, which is positive even for dense matter systems. I showcase this approach for a SU(2) gauge theory at finite densities of heavy quarks. I then address the Z3 spin model at finite chemical potentials, which features a strong sign problem.

Random matrix theory has had a significant impact on our understanding of the sign problem and nonzero baryon chemical potential. Among others it has explained the nature of the quenched approximation, the relation between Dirac spectra and the sign problem, the Silver Blaze problem and the microscopic Dirac spectrum. Perhaps the most surprising result from random matrix theory is that the discontinuity of the chiral condensate requires an alternative of the Banks-Casher formula. This mechanism occurs both in random matrix theory and one-dimensional QCD and in this lecture we discuss the universality of this mechanism for chiral symmetry breaking. Relations with the sign problem at nonzero $\theta$-angle are discussed.

The existence of the Aoki phase is one of the new structures of the phase space in lattice QCD when introducing Wilson fermions. It is expected that the Aoki phase may have also an impact on Wilson fermions at finite chemical potential. I am going to present new results on these phase space structures for Wilson fermions in a two-flavor theory with a finite real as well as imaginary iso-spin chemical potential. These new structures are manifested in the lowest eigenvalues of the Dirac operator. I am going to point out what the order parameters of these new phases are and to answer the question on the analytic continuation of the chemical potential. Since the Wilson Dirac operator at finite iso-spin chemical potential is unitarily equivalent to the twisted mass Dirac operator for two flavors, the spectrum and, hence, the phase diagram applies to this kind of fermions, too.

Gauge theories play an important role both in particle and condensed matter physics, with applications ranging from QCD to spin liquids, and Kitaev's toric code in quantum information theory. Numerical simulations of such systems on classical computers suffer from very severe sign problems. Quantum simulators are accurately controllable quantum systems that can mimic other quantum systems. They do not suffer from sign problems, because their hardware is intrinsically quantum mechanical. Recently, using ultracold atoms in optical lattices, quantum simulators have been designed for Abelian and non-Abelian gauge theories. Their experimental realization is a challenge for the foreseeable future, which holds the promise to access physical phenomena, as, for example, the evolution of strongly coupled quantum systems in real time, whose understanding has remained beyond reach of the traditional tools of theoretical physics.

After a short introduction about tensor networks and quantum simulators, I will review the recent proposals on how to apply both of them to the characterization of Abelian and non-Abelian lattice gauge theories.

Simulating real-time dynamics of large quantum systems is a notoriously difficult problem. Diagonalizing the Hamiltonian is exponentially expensive, while the configurations contribute a complex weight to the real-time path integral leading to a severe sign-problem. In this particular case, we consider a system of spins whose dynamics is driven by measurements of nearest neighbor spin pair. Remarkably, averaging over the measurement results lead to a sign-problem free real-time path integral which can be simulated with an efficient cluster algorithm.

We explore the possibility of describing lattice QCD in the worldline approach beyond strong coupling. We construct MDP models of lattice gauge theories with staggered fermions for arbitrary values of the lattice coupling, by integrating out all the link variables exactly. We discuss the implications of our constructions to the sign problem at finite coupling and density.

We consider complex saddle points in finite-density QCD, which are constrained by charge and complex conjugations to make the effective action real. The method naturally incorporates color neutrality, and the Polyakov loop and the conjugate loop at the saddle point are real but not identical. Moreover, it gives rise to a complex mass matrix associated with Polyakov loops, reflecting oscillatory behavior in color-charge densities. These results may be experimentally relevant at GSI/FAIR.