We introduce a factorization of the fermion determinant in lattice QCD with Wilson-type fermions that leads to a bosonic action which is local in the block fields. The interaction among gauge fields on distant blocks is mediated by multiboson fields located on the boundaries of the blocks. The resultant multiboson domain-decomposed hybrid Monte Carlo passes extensive numerical tests carried out by measuring standard gluonic observables. The combination of the determinant factorization and of the one of the propagator, that we put forward recently, paves the way for multilevel Monte Carlo integration in the presence of fermions. We test this possibility by computing the disconnected correlator of two flavor-diagonal pseudoscalar densities, and we observe a significant increase of the signal-to-noise ratio due to a two-level integration.
In the last few years it has been proposed a one-dimensional factorization of the fermion determinant in lattice QCD with Wilson-type fermions that leads to a block-local action of the auxiliary bosonic fields. Here we propose a four-dimensional generalization of this factorization. Possible applications are more efficient parallelizations of Monte Carlo algorithms and codes, master field simulations, and multi-level integration.
When lattice QCD is formulated in sectors of fixed quark numbers, the canonical fermion determinants can be expressed explicitly in terms of transfer matrices. This in turn provides a complete factorization of the fermion determinants in temporal direction. Here we present a generic overview of this factorization, apply it to Wilson-type fermions and provide explicit constructions of the transfer matrices. Possible applications of the factorization include multi-level integration schemes and the construction of improved estimators for generic $n$-point correlation functions.
The non-local dependence of the fermion determinant on the gauge field limits our ability of simulating Quantum Chromodynamics on the lattice. Here we present a factorization of the gauge field dependence of the fermion determinant based on an overlapping four-dimensional domain decomposition of the lattice. The resulting action is block-local in the gauge and in the auxiliary bosonic fields. Possible applications are multi-level integration, master field simulations, and more efficient parallelizations of Monte Carlo algorithms and codes.
We discuss the recently proposed multiboson domain-decomposed factorization of the gauge-field dependence of the fermion determinant in lattice QCD. In particular, we focus on the case of a lattice divided in an arbitrary number of thick time slices. As a consequence, multiple space-time regions can be updated independently. This allows to address the exponential degradation of the signal-to-noise ration of correlation functions with multilevel Monte Carlo sampling. We show numerical evidence of the effectiveness of a two-level integration for pseudoscalar propagators with momentum and for vec…
We use canonical formalism to study the fermion determinant in different three dimensional abelian gauge field backgrounds that contain non-zero magnetic and electric flux in order to understand the non-perturbative contributions to the parity-odd and parity-even parts of the phase. We show that a certain phase associated with free fermion propagation along a closed path in a momentum torus is responsible for the parity anomaly in a background with non-zero electric flux. We consider perturbations around backgrounds with non-zero magnetic flux to understand the structure of the parity-breaking…
Temperature in Fermion Systems and the Chiral Fermion Determinant
我们给出了一个解释的问题的手征行列式的热核方法。额外的维度（第5维）被解释为（逆）温度。1+4 dim Dirac方程是由温度的Wick旋转自然导出的。为了定义一个"好“的温度，我们选择那些在额外坐标中以固定方向传播的狄拉克方程的解。这个选择修正了费米子行列式的正则化。1+4维狄拉克质量（$M$）自然地被引入，并且关系式：$|$ 4 dim electron momentum $|$ $\ll$ $|M|$ $\ll$紫外截止，自然出现。对于2维阿贝尔模型，我们导出了手征反常。通常，根据传播算子的选择，会出现两种不同的正则化。一个对应于手征理论，另一个对应于非手征（埃尔米特）理论。
In many interesting systems, the fermion determinant becomes complex and its phase plays a crucial role in the determination of the vacuum. For instance, in finite density QCD at low temperature and high density, exotic fermion condensates are conjectured to form due to such effects. When one applies the complex Langevin method to such a complex action system naively, one cannot obtain the correct results because of the singular-drift problem associated with the appearance of small eigenvalues of the Dirac operator. Here we propose to avoid this problem by adding a fermion bilinear term to the…
Overlap fermion on the lattice has been shown to properly reproduce topological aspects of gauge fields. In this paper, we review the derivation of Overlap fermion formalism in a torus of three space-time dimensions. Using the formalism, we show how to use the Overlap fermion determinants in the massless and infinite mass limits to construct different continuum topological gauge actions, such as the level-$k$ Chern-Simons action, ``half-CS" term and the mixed Chern-Simons (BF) coupling, in a gauge-invariant lattice UV regulated manner. Taking special Abelian and non-Abelian background fields, …
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…
QCD at nonzero baryon chemical potential suffers from the sign problem, due to the complex quark determinant. Complex Langevin dynamics can provide a solution, provided certain conditions are met. One of these conditions, holomorphicity of the Langevin drift, is absent in QCD since zeroes of the determinant result in a meromorphic drift. We first derive how poles in the drift affect the formal justification of the approach and then explore the various possibilities in simple models. The lessons from these are subsequently applied to both heavy dense QCD and full QCD, and we find that the resul…
Lattice QCD at non-vanishing chemical potential is studied using the complex Langevin equation (CLE). One of the conditions for the correctness of the results of the CLE is that the zeroes of the measure coming from the fermionic determinant are outside of the distribution of the configurations, or at least in a region where support for the distribution is very much suppressed. We investigate this issue for Heavy Dense QCD (HDQCD) and full QCD at high temperatures. In HDQCD it is found that the configurations move closest to the zeroes of the measure around the critical chemical potential of t…
