Non-Perturbative Renormalization Group for Ising-Nematic Criticality: A Closed-Form Nonlocal Ansatz
Abstract
The two-dimensional metallic quantum critical problem is a long-standing puzzle that is widely believed to hold the key to resolving ubiquitous non-Fermi liquid behavior in strongly correlated electronic systems. In this study, we present a non-perturbative renormalization group (RG) analysis of the metallic Ising-nematic quantum critical point in two dimensions, formulated directly around an intrinsically nonlocal infrared (IR) boson propagator. Rather than treating the anomalous dynamical critical exponent a as a fixed phenomenological parameter, we regard it as an intrinsic component of the fixed-point data to be determined from the internal consistency of the low-energy patch field theory under highly anisotropic scaling dimensions ([k0]=a+1, [kx]=2, [ky]=1). While the leading two-loop diagrammatics vanish identically due to kinematic pole configurations, our three-loop evaluation reveals a profound structural asymmetry between the sectors: the fermion self-energy and Yukawa vertex receive non-vanishing logarithmic corrections, whereas the corresponding bosonic counter-term remains strictly zero. Consequently, we find that no self-consistent, intersecting fixed-point solution for the exponent a exists within the three-loop truncation, failing to reproduce the physical value of a ≈ 1.85 observed in quantum Monte Carlo simulations. We conjecture that the cross-linked topology of the four-loop boson self-energy diagrams is exactly marginal and yields the minimal, mandatory bosonic counter-term required to restore multi-sector self-consistency. Our framework establishes a rigid multi-loop matching scheme necessary to uniquely pin down the critical exponent, and uncovers a stable phase space for field anomalous dimensions.
Versions
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BibTeX
@misc{kim2026nonperturbative,
title = {{Non-Perturbative Renormalization Group for Ising-Nematic Criticality: A Closed-Form Nonlocal Ansatz}},
author = {Hyeon Jung Kim and Kyoung-Min Kim and Ki-Seok Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.30715},
eprint = {2605.30715},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.30715},
}
RIS
TY - UNPB
TI - Non-Perturbative Renormalization Group for Ising-Nematic Criticality: A Closed-Form Nonlocal Ansatz
AU - Hyeon Jung Kim
AU - Kyoung-Min Kim
AU - Ki-Seok Kim
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2605.30715
UR - https://arxiv.org/abs/2605.30715
AB - The two-dimensional metallic quantum critical problem is a long-standing puzzle that is widely believed to hold the key to resolving ubiquitous non-Fermi liquid behavior in strongly correlated electronic systems. In this study, we present a non-perturbative renormalization group (RG) analysis of the metallic Ising-nematic quantum critical point in two dimensions, formulated directly around an intrinsically nonlocal infrared (IR) boson propagator. Rather than treating the anomalous dynamical critical exponent a as a fixed phenomenological parameter, we regard it as an intrinsic component of the fixed-point data to be determined from the internal consistency of the low-energy patch field theory under highly anisotropic scaling dimensions ([k_0]=a+1, [k_x]=2, [k_y]=1). While the leading two-loop diagrammatics vanish identically due to kinematic pole configurations, our three-loop evaluation reveals a profound structural asymmetry between the sectors: the fermion self-energy and Yukawa vertex receive non-vanishing logarithmic corrections, whereas the corresponding bosonic counter-term remains strictly zero. Consequently, we find that no self-consistent, intersecting fixed-point solution for the exponent a exists within the three-loop truncation, failing to reproduce the physical value of a ≈ 1.85 observed in quantum Monte Carlo simulations. We conjecture that the cross-linked topology of the four-loop boson self-energy diagrams is exactly marginal and yields the minimal, mandatory bosonic counter-term required to restore multi-sector self-consistency. Our framework establishes a rigid multi-loop matching scheme necessary to uniquely pin down the critical exponent, and uncovers a stable phase space for field anomalous dimensions.
ER -