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We propose a new strategy to evaluate the partition function of lattice QCD with Wilson gauge action coupled to staggered fermions, based on a strong coupling expansion in the inverse bare gauge coupling

Lattice QCD at finite baryon density suffers from the notorious sign problem

The idea of dual representations is old, and in the last decade, many different sign problems have been solved in this way. Some of the hallmarks in the context of spin models are the O(N) and CP(N-1) models

The focus of this paper is whether dual representations can be successfully applied to lattice QCD at finite baryon density, which has a severe sign problem in the usual representation, where fermions are integrated out, resulting in the fermion determinant. The standard approach is then hybrid Monte Carlo. At finite baryon chemical potential

A dual representation of lattice QCD has only been derived in the strong coupling regime: the classical formulation in terms of a monomer-dimer-polymer system has been both addressed via mean field

The paper is organized as follows: in Sec.

We consider the finite density partition function of lattice gauge theory with

The first step in the dualization process is to perform a combined Taylor expansion of Eq.

Our dualization corresponds to exactly integrate out the gauge links

The traces appearing in Eq.

After gauge integration, some of the open color indices need to be contracted between links that share a common site such that the plaquette terms are recovered. The remaining indices are contracted with the Grassmann-integrated quark fields. We postpone the description of this step to Sec.

The

By inspecting Eq.

Every finite group admits a unitary irrep. In the case of the symmetric group, the matrix elements can be also chosen to be real. This basis is known as the Young’s orthogonal form.

The quantities in the brackets of Eq.Given the result in Eq.

The key insight is that fixing the values of the DOI

The dependency of

Illustration of the contraction step in two dimensions: on each of the four links attached to the central lattice site the DOIs

The tensor network resulting from the dual description: depending on the dual d.o.f. at any lattice site, the tensor

In some cases, the contraction of different operators produces the same tensor elements. For instance, two operators with DOIs

As we already mentioned, not all sets of dual d.o.f. are allowed. On each lattice link, they have to combine in a way that the corresponding

Using the quantities defined in the previous sections, the partition function Eq.

An allowed configuration in

Left: a typical strong coupling configuration where dimers are attached to a given site. The tensor

A great simplification occurring is that the strong coupling contributions always decouple from those corresponding to nonzero

Therefore, to evaluate the total weight of a configuration, it is sufficient to use the more involved structure based on the tensor network contraction on the sublattice where the plaquette occupation numbers are nonzero, exploiting the factorization of the tensor network for disconnected plaquette contributions. The strong coupling part can be evaluated using the standard combinatorial formulae [e.g., Eq.

We now want to comment on the complexity of the dual partition function Eq.

Two

The second possibility is to consider the DOIs as an additional d.o.f. along with

Having discussed the partition function, we now turn to the computation of

At finite

For SU(2), only single quark fluxes can produce a negative

As we already mentioned, another potential source of negative signs, which does not depend on the fermion fields, is caused by the lack of positivity of the tensor elements

As both the fermion field and the gauge links have been integrated out, the observables in the dual representation take a different form. The ones defined as derivatives of

To perform finite temperature calculations at nonzero

In a finite volume, the partition function Eq.

Number of distinct configurations on a

Comparison between exact enumeration and HMC simulations for U(2) (upper plots) and U(3) (lower plots). For both gauge groups, the average plaquette

Similar comparison between exact enumeration and HMC on a

Another relevant information we can extract from the exact enumeration concerns the magnitude of the sign problem. A measure of its severity is given by the average sign

Sign problem on a

Although our numerical results are preliminary and only based on an exact enumeration of the partition function on a

In this work, we proposed a new strategy for the evaluation of higher order contributions in the strong coupling expansion of lattice QCD with staggered fermion discretization. The dual representation in terms of local tensorial weights improves on the sign problem as compared to evaluations in a Weingarten function basis. The color constraints from gauge and Grassmann integration combine to yield admissible configurations that after contracting the tensors are intersecting plaquette surfaces that are either closed or bounded by fermion fluxes. The configuration space is thus a worldline and worldsheet representation with the additional multi-indices

The prospects of Monte Carlo simulations of lattice QCD at finite density in the strong coupling regime are encouraging: the weights in the partition functions are local, and various strategies to sample the partition function Eq.

Number of distinct nonzero tensor elements for various gauge groups and truncations of

A finite chemical potential does not introduce an additional sign problem as the zero-density Boltzmann weights get multiplied only by positive factors. Moreover, at fixed values of

We thank Jangho Kim for helpful discussions on fast exact enumeration. We acknowledge support by the Deutsche Forschungsgemeinschaft (German Research Foundation) through the Emmy Noether Program under Grant No. UN 370/1 and through the CRC-TR 211 strong-interaction matter under extreme conditions—Project No. 315477589—TRR 211.

Equation

Thanks to the Schur-Weyl duality

Not to be confused with the characters

Given the expression

The partition function

An incoming (strong coupling) baryon splits, at a corner of the plaquette, into a single quark flux and a

An incoming

As in (2) with the external dimer or monomer replaced by a dimer on one of the two excited links [Fig.

To each

To each

For

To each site corresponding to a

To each site corresponding to a

The four different types of tensor at a corner of the excited plaquette. (a) The two excited links are occupied by dimers and a single quark flux. It represents the most general