Complex Fermion Determinant Research Articles

Applying complex Langevin simulations to lattice QCD at finite density

We study the use of the complex-Langevin equation (CLE) to simulate lattice QCD at a finite chemical potential ($\mu$) for quark-number, which has a complex fermion determinant that prevents the use of standard simulation methods based on importance sampling. Recent enhancements to the CLE specific to lattice QCD inhibit runaway solutions which had foiled earlier attempts to use it for such simulations. However, it is not guaranteed to produce correct results. Our goal is to determine under what conditions the CLE yields correct values for the observables of interest. Zero temperature simulations indicate that for moderate couplings, good agreement with expected results is obtained for small $\mu$ and for $\mu$ large enough to reach saturation, and that this agreement improves as we go to weaker coupling. For intermediate $\mu$ values these simulations do not produce the correct physics. We compare our results with those of the phase-quenched approximation. Since there are indications that correct results might be obtained if the CLE trajectories remain close to the $SU(3)$ manifold, we study how the distance from this manifold depends on the quark mass and on the coupling. We find that this distance decreases with decreasing quark mass and as the coupling decreases, i.e. as the simulations approach the continuum limit.

The complex Langevin analysis of spontaneous symmetry breaking induced by complex fermion determinant

In many interesting physical systems, the determinant which appears from integrating out fermions becomes complex, and its phase plays a crucial role in the determination of the vacuum. An example of this is QCD at low temperature and high density, where various exotic fermion condensates are conjectured to form. Another example is the Euclidean version of the type IIB matrix model for 10d superstring theory, where spontaneous breaking of the SO(10) rotational symmetry down to SO(4) is expected to occur. When one applies the complex Langevin method to these systems, one encounters the singular-drift problem associated with the appearance of nearly zero eigenvalues of the Dirac operator. Here we propose to avoid this problem by deforming the action with a fermion bilinear term. The results for the original system are obtained by extrapolations with respect to the deformation parameter. We demonstrate the power of this approach by applying it to a simple matrix model, in which spontaneous symmetry breaking from SO(4) to SO(2) is expected to occur due to the phase of the complex fermion determinant. Unlike previous work based on a reweighting-type method, we are able to determine the true vacuum by calculating the order parameters, which agree with the prediction by the Gaussian expansion method.

A subset solution to the sign problem in simulations at non-zero chemical potential

At nonzero quark chemical potential lattice QCD simulations with dynamical quarks are hampered by the sign problem caused by the complex fermion determinant. In this talk we discuss the relation between QCD and random matrix theory and present a novel solution to the sign problem in dynamical random matrix simulations. The sign problem is solved by gathering matrices in subsets, whose sums of complex determinants are real and positive. Markov chains of relevant subsets were generated in Monte Carlo simulations and various observables, like the chiral condensate and quark number, were accurately computed. The subset formula, which is at the basis of the positivity proof for the subset weights, also provides a simple solution to the Silver Blaze puzzle.

Holographic realization of large-N c orbifold equivalence with non-zero chemical potential

Recently, it has been suggested that large-Nc orbifold equivalences may be applicable to certain theories with chemical potentials, including QCD, in certain portions of their phase diagram. When valid, such an equivalence offers the possibility of relating large-Nc QCD at non-zero baryon chemical potential, a theory with a complex fermion determinant, to a related theory whose fermion determinant is real and positive. In this paper, we provide a test of this large Nc equivalence using a holographic realization of a supersymmetric theory with baryon chemical potential and a related theory with isospin chemical potential. We show that the two strongly-coupled, large-Nc theories are equivalent in a large region of the phase diagram.

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.

Stochastic quantization at finite chemical potential

A nonperturbative lattice study of QCD at finite chemical potential is complicated due to the complex fermion determinant and the sign problem. Here we apply the method of stochastic quantization and complex Langevin dynamics to this problem. We present results for U(1) and SU(3) one link models and QCD at finite chemical potential using the hopping expansion. The phase of the determinant is studied in detail. Even in the region where the sign problem is severe, we find excellent agreement between the Langevin results and exact expressions, if available. We give a partial understanding of this in terms of classical flow diagrams and eigenvalues of the Fokker-Planck equation.

Lattice QCD at finite temperature and density in the phase-quenched approximation

QCD at a finite quark-number chemical potential $\ensuremath{\mu}$ has a complex fermion determinant, which precludes its study by standard lattice QCD simulations. We therefore simulate lattice QCD at finite $\ensuremath{\mu}$ in the phase-quenched approximation, replacing the fermion determinant with its magnitude. (The phase-quenched approximation can be considered as simulating at finite isospin chemical potential $2\ensuremath{\mu}$ for ${N}_{f}/2$ $u$-type and ${N}_{f}/2$ $d$-type quark flavors.) These simulations are used to study the finite-temperature transition for small $\ensuremath{\mu}$, where there is some evidence that the position (and possibly the nature) of this transition is unchanged by this approximation. We look for the expected critical endpoint for 3-flavor QCD. Here, it has been argued that the critical point at zero $\ensuremath{\mu}$ would become the critical endpoint at small $\ensuremath{\mu}$, for quark masses just above the critical mass. Our simulations indicate that this does not happen, and there is no such critical endpoint for small $\ensuremath{\mu}$. We discuss how we might adapt techniques used for imaginary $\ensuremath{\mu}$ to improve the signal/noise ratio and strengthen our conclusions, using results from relatively low statistics studies.

Finite Density Simulations: Comparison of Various Approaches

This is a short overview of the lattice QCD simulations of finite density systems. We first describe a brief introduction of the lattice QCD at finite density, including the minimum necessary formulation, where we show why an annoying complex fermion determinant appears, and why in some cases it does not appear. Then we review several approaches of present and past days. We conclude possible directions of lattice QCD simulations at finite density in near future.

Phase structure of lattice QCD at finite density with dynamical fermions

We compare the chemical potential associated with the onset of non-zero baryon number density on 6 4 and 8 4 lattices at β = 5.1 and ma = 0.01. We provide evidence for Z(3) tunnelling. We determine a critical chemical potential of μ a ≅ 0.1 which is unexpectedly low. We also determine the dependence of the onset of the observed phase transition on the quark mass. The physically misleading result of the quenched theory is shown to persist despite the inclusion of the complex fermion determinant.

Self-consistent solution to a complex fermion determinant with space dependent fields

A numerical routine is provided which allows to compute and minimize a mesonic energy functional involving a complex fermion determinant. The energy functional is evaluated by summing over the real and imaginary parts of the eigenvalues of a non-Hermitian one-particle Dirac Hamiltonian. In order to minimize the energy functional the corresponding eigenfunctions are used to set up the equations of motion for the meson fields. Iteration of these equations yields the self-consistent configuration of the meson fields.

Bounds on the couplingsin an SU (2) L ⊗ SU (2) R Yukawa model and simulations with decoupled mirror fermions

The upper and lower bounds on the mass of the Higgs boson are numerically investigated in an SU(2) L ⊗ SU(2) R symmetric Yukawa model, which describes two heavy degenerate fermion doublets in the limit of vanishing gauge couplings. The values for the bounds are in good agreement with those obtained from an integration of the one-loop β-functions, although the Yukawa couplings are strong. Simulations in a U(1) L ⊗ U(1) R symmetric Yukawa model with decoupled mirror fermions and a single fermion doublet have been performed. This theory, whose interacting sector is truly chiral, has a complex fermion determinant.

Yukawa models with mirror fermions

Scalar-fermion models with mirror pairs of fermion fields on the lattice are discussed. Numerical simulations have been performed both for models with U (1) L ⊗ U (1) R and SU (2) L ⊗ SU (2) R symmetry. For the model with U (1) L ⊗ U (1) R symmetry the phase diagram has been investigated and various phases have been identified. In both the U (1) and the SU (2) case the models appear to allow for strong Yukawa couplings. Results on lower and upper bounds for the Higgs-boson mass and the effect of heavy fermions on these bounds are obtained. In the U (1) L ⊗ U (1) R model the complex fermion determinant has been studied in view of simulations with an odd number of fermion generations.