[Paper Review] Inhomogeneous chiral condensates and non-analyticity under an external magnetic field

Kouji Kashiwa, Tong-Gyu Lee+2 more|arXiv (Cornell University)|Jul 30, 2015

Quantum chaos and dynamical systems4 citations

TL;DR

This paper demonstrates that inhomogeneous chiral condensates—such as the dual chiral density wave (DCDW) phase—induce non-analyticity at zero chemical potential under an external magnetic field, disrupting analytic continuation from imaginary to real chemical potential. As a result, lattice QCD simulations using Taylor expansion or analytic continuation methods may miss critical information about inhomogeneous phases unless non-analytic behavior at μ²=0 is properly accounted for.

ABSTRACT

We investigate inhomogeneous chiral condensates, such as the so-called dual chiral density wave of dense quark matter, under an external magnetic field at finite real and imaginary chemical potentials. In a model-independent manner, we find that analytic continuation from imaginary to real chemical potential is not possible due to the singularity induced by inhomogeneous chiral condensates at zero chemical potential. From the discussion on the non-analyticity and methods used in lattice QCD simulations, e.g., Taylor expansion, and the analytic continuation with an imaginary chemical potential, it turns out that information on an inhomogeneous chiral condensed phase is missed in the lattice simulations at finite baryon chemical potentials unless the non-analyticity at zero chemical potential is correctly considered. We also discuss an exceptional case without such non-analyticity at zero chemical potential.

Motivation & Objective

• To investigate the impact of inhomogeneous chiral condensates on analytic continuation from imaginary to real chemical potential in the presence of an external magnetic field.
• To identify why standard lattice QCD methods—Taylor expansion and analytic continuation—fail to capture inhomogeneous chiral condensed phases.
• To clarify the conditions under which the information missing problem arises and to propose a viable path forward for effective model construction.
• To examine the role of non-analytic singularities at μ²=0 in obstructing reliable extrapolation from μ_I to μ_R.

Proposed method

• Analyzing the grand canonical partition function's reality and symmetry properties under inhomogeneous chiral condensates and external magnetic fields.
• Examining two scenarios for DCDW condensates at finite imaginary chemical potential (μ_I), identifying a singularity at μ²=0 due to spatially modulated chiral order.
• Applying holomorphic function theory to show that analytic continuation fails when μ²=0 forms a singular line.
• Using model-independent field-theoretic arguments to establish that non-analyticity at μ²=0 prevents reliable Taylor series extrapolation.
• Proposing a hybrid method combining imaginary chemical potential matching with input of inhomogeneous condensate solutions to restore missing information.
• Considering the exceptional case where q² becomes negative, which avoids the singularity and allows analytic continuation.

Experimental results

Research questions

• RQ1 Why do standard lattice QCD methods fail to detect inhomogeneous chiral condensates like the DCDW phase?
• RQ2 How does the presence of an external magnetic field affect the analytic structure of the partition function at zero chemical potential?
• RQ3 What is the role of non-analyticity at μ²=0 in obstructing analytic continuation from imaginary to real chemical potential?
• RQ4 Can the information missing problem be resolved by modifying the simulation approach or incorporating inhomogeneous condensate solutions?
• RQ5 Under what conditions does the singularity at μ²=0 not arise, allowing reliable analytic continuation?

Key findings

• The DCDW phase induces a non-analytic singularity at μ²=0 in the presence of an external magnetic field, disrupting analytic continuation from imaginary to real chemical potential.
• Lattice QCD simulations using Taylor expansion or analytic continuation methods may miss inhomogeneous chiral condensates unless the non-analytic behavior at μ²=0 is explicitly considered.
• The singularity at μ²=0 arises due to the different form of chiral condensates and the imaginary part of the partition function, making analytic continuation impossible across this line.
• For finite current quark masses, the singular line may shift to smaller μ_R/T, worsening the information missing problem.
• An exceptional case exists where q² becomes negative, avoiding the singularity and allowing well-defined analytic continuation.
• A promising approach to overcome the problem involves unifying the imaginary chemical potential matching method with input of inhomogeneous condensate solutions, enabling reliable effective model construction.