[Paper Review] Ising nematic quantum critical point in a metal: a Monte Carlo study

Yoni Schattner, Samuel Lederer+2 more|DSpace@MIT (Massachusetts Institute of Technology)|Nov 10, 2015

Physics of Superconductivity and Magnetism6 citations

TL;DR

This study presents a sign-problem-free determinantal quantum Monte Carlo simulation of an Ising nematic quantum critical point in a two-dimensional metal, revealing scaling behavior in correlation functions near the critical point. The results show critical exponents λ = 1.0 ± 0.1 and γ = 1.0 ± 0.1, consistent with theoretical predictions and exhibiting striking similarity to nematic fluctuations observed in iron-based superconductors.

ABSTRACT

The Ising nematic quantum critical point (QCP) associated with the zero temperature transition from a symmetric to a nematic {\it metal} is an exemplar of metallic quantum criticality. We have carried out a minus sign-free quantum Monte Carlo study of this QCP for a two dimensional lattice model with sizes up to $24 imes 24$ sites. The system remains non-superconducting down to the lowest accessible temperatures. The results exhibit critical scaling behavior over the accessible ranges of temperature, (imaginary) time, and distance. This scaling behavior has remarkable similarities with recently measured properties of the Fe-based superconductors proximate to their putative nematic QCP.

Motivation & Objective

• To investigate the nature of the Ising nematic quantum critical point in a metal using a sign-problem-free quantum Monte Carlo method.
• To determine whether scaling behavior emerges near the quantum critical point in a metallic system with finite fermion density.
• To compare the observed critical behavior with experimental data from iron-based superconductors near a putative nematic quantum critical point.
• To assess the stability of the quantum critical point against superconducting or other instabilities in the metallic phase.
• To extract critical exponents and test their consistency with field-theoretic predictions for metallic quantum criticality.

Proposed method

• Employed determinantal quantum Monte Carlo (DQMC) on a two-dimensional lattice model with up to 24×24 sites to simulate the Ising nematic transition.
• Used a sign-problem-free Hamiltonian with a transverse field h controlling the nematic order parameter and finite fermion density via chemical potential μ.
• Computed thermodynamic nematic correlation functions D(h, T, q, iωₙ=0) and analyzed their scaling behavior using a functional approximant A(h, T, q) = [A / (T^λ + b(h−h_c) + κ|q|²)]^γ.
• Extracted critical exponents λ and γ from fits to the correlation functions across varying temperatures, momenta, and system sizes.
• Analyzed momentum and frequency dependence of the nematic and quadrupolar correlators D⁻¹ and Q⁻¹ to probe critical scaling and anomalous behavior.
• Performed finite-size scaling and checked consistency across system sizes (L=16, 20, 24) to confirm convergence to the thermodynamic limit.

Experimental results

Research questions

• RQ1 Does the Ising nematic quantum critical point in a metal exhibit scaling behavior in correlation functions over accessible energy, temperature, and length scales?
• RQ2 What are the critical exponents λ and γ for the thermodynamic nematic susceptibility near the quantum critical point, and do they match field-theoretic predictions?
• RQ3 How does the nematic correlation function scale with temperature, momentum, and distance in the critical regime?
• RQ4 Is the quantum critical point stable against superconducting or other instabilities in the metallic phase at the lowest accessible temperatures?
• RQ5 To what extent do the observed critical correlations resemble those measured in iron-based superconductors near a putative nematic QCP?

Key findings

• The thermodynamic nematic correlation function D(h, T, q, ωₙ=0) exhibits scaling behavior consistent with the functional form [A / (T^λ + b(h−h_c) + κ|q|²)]^γ over a wide range of T, h, and q.
• The critical exponent λ = 1.0 ± 0.1 and γ = 1.0 ± 0.1 were extracted, matching predictions from Hertz-Millis theory and other field-theoretic approaches.
• The uniform nematic susceptibility χ(h, T) follows a Curie-Weiss form with an effective Weiss temperature linearly dependent on (h_c − h), confirming critical scaling.
• The inverse nematic correlator D⁻¹ shows quadratic momentum dependence, while Q⁻¹ exhibits apparent linear momentum dependence at low temperatures, suggesting distinct critical behavior.
• The system remains non-superconducting down to the lowest accessible temperatures (T = 0.025t), indicating the stability of the quantum critical point.
• Data from multiple system sizes (L=16, 20, 24) collapse onto the same curve, confirming convergence to the thermodynamic limit for the studied observables.