[Paper Review] Domain wall fermions in vector theories

Pavlos Vranas|ArXiv.org|Jan 11, 2000

Physics of Superconductivity and Magnetism9 citations

TL;DR

This paper reviews domain wall fermions (DWF) in vector-like lattice gauge theories, emphasizing their ability to achieve near-exact chiral symmetry at finite lattice spacing by increasing the extra dimension size $L_s$, thus separating chiral and Lorentz symmetry limits. Key results show that DWF correctly reproduce anomalous Ward identities and topological zero modes, with improved chiral properties enabling reliable simulations of QCD thermodynamics and scalar interactions at smaller $L_s$ values through optimized coupling schemes.

ABSTRACT

Applications of Domain Wall fermions to various vector-like lattice theories are reviewed with an emphasis on QCD thermodynamics. Methods for improving their chiral properties at strong coupling are discussed and results from implementing them are presented.

Motivation & Objective

• To explore the application of domain wall fermions (DWF) in vector-like lattice gauge theories, particularly for QCD thermodynamics.
• To address the challenge of maintaining good chiral symmetry at finite lattice spacing, where traditional fermion formulations break chiral symmetry explicitly.
• To improve DWF's chiral properties at strong coupling by reducing the required $L_s$, minimizing computational cost.
• To investigate the behavior of DWF in topologically non-trivial gauge backgrounds and their zero-mode structure.
• To develop consistent coupling schemes for DWF with scalar fields, avoiding incorrect flavor-space interpretations.

Proposed method

• DWF are formulated in a five-dimensional space-time with an extra dimension $s$, where a domain wall mass $m_0$ localizes chiral modes at opposite boundaries.
• The fermion action is constructed using the Furman-Shamir formalism with a mass term $m_f$ that mixes chiral components, with residual mass $m_{\rm res} \propto (1-m_0)^{L_s}$, exponentially small for large $L_s$.
• Numerical simulations are performed at finite $m_f$, with extrapolation to $L_s = \infty$ using exponential fits of the form $A + B e^{-cL_s}$ to recover the chiral limit.
• For scalar coupling, a boundary-localized interaction is used: $\overline{\Psi}_R(x,0)\sigma(x)\Psi_L(x,L_s-1) + \text{h.c.}$, avoiding incorrect flavor-space coupling.
• Large-$N$ techniques and numerical studies are used to analyze phase structure in four-Fermi models, identifying parity-flavor broken phases at small $L_s$.
• Classical instanton backgrounds are used to test zero-mode properties, verifying the presence of exact zero modes at $m_f=0$ and $L_s=\infty$, and assessing stability at finite $m_f$ and $L_s$.

Experimental results

Research questions

• RQ1 Can domain wall fermions reliably reproduce anomalous Ward identities in QCD thermodynamics at finite lattice spacing?
• RQ2 How do the chiral properties of DWF depend on $L_s$ and $m_f$, and can they be improved to reduce $L_s$ requirements at strong coupling?
• RQ3 What is the correct way to couple scalar fields to DWF, and how does this affect the effective number of flavors?
• RQ4 Do DWF correctly reproduce exact zero modes in topologically non-trivial gauge backgrounds at finite $L_s$ and small $m_f$?
• RQ5 What is the phase structure of four-Fermi models with DWF at small $L_s$, and how does it compare to Aoki-like phases in Wilson fermions?

Key findings

• The effective mass in free theory is $m_{\rm eff} = m_0(2-m_0)[m_f + (1-m_0)^{L_s}]$, with residual mass $m_{\rm res} = m_0(2-m_0)(1-m_0)^{L_s}$, confirming exponential chiral symmetry breaking suppression.
• Numerical extrapolation of $w$ to $L_s = \infty$ using $A + B e^{-cL_s}$ fits agrees with the overlap formalism, validating DWF for anomalous processes.
• The minimum decay rate for scalar-coupled DWF models is $-\ln(2 - \sqrt{2}) \approx 0.535$, confirmed both analytically and numerically.
• A parity-flavor broken phase is observed in the $SU(2)\times SU(2)$ four-Fermi model at small $L_s$ and negative $m_f$, resembling the Aoki phase of Wilson fermions.
• In classical instanton backgrounds, zero modes are stable at $m_f=0$ and $L_s=\infty$, but finite-$L_s$ and finite-$m_f$ configurations show slow decay rates near topology-changing fluctuations.
• The scalar coupling via boundary terms correctly preserves chiral structure and avoids spurious flavor mixing, unlike naive flavor-space coupling.