Quantum phase transitions of metals in two spatial dimensions: II. Spin density wave order (1005.1288v2)

Abstract: We present a field-theoretic renormalization group analysis of Abanov and Chubukov's model of the spin density wave transition in two dimensional metals. We identify the independent field scale and coupling constant renormalizations in a local field theory, and argue that the damping constant of spin density wave fluctuations tracks the renormalization of the local couplings. The divergences at two-loop order overdetermine the renormalization constants, and are shown to be consistent with our renormalization scheme. We describe the physical consequences of our renormalization group equations, including the breakdown of Fermi liquid behavior near the "hot spots" on the Fermi surface. In particular, we find that the dynamical critical exponent z receives corrections to its mean-field value z = 2. At higher orders in the loop expansion, we find infrared singularities similar to those found by S.-S. Lee for the problem of a Fermi surface coupled to a gauge field. A treatment of these singularities implies that an expansion in 1/N, (where N is the number of fermion flavors) fails for the present problem. We also discuss the renormalization of the pairing vertex, and find an enhancement which scales as logarithm-squared of the energy scale. A similar enhancement is also found for a modulated bond order which is locally an Ising-nematic order.

• The paper presents a detailed two-loop renormalization group analysis of the spin density wave transition in 2D metals.
• It reveals the breakdown of Fermi liquid behavior at hot spots near the quantum critical point with non-mean-field scaling.
• Enhancements in pairing and bond order instabilities highlight the significant impact of quantum critical fluctuations on superconductivity.

Overview of Quantum Phase Transitions of Metals: Spin Density Wave Order

The paper by Metlitski and Sachdev provides a comprehensive analysis of the spin density wave (SDW) transition in metallic systems specifically in two spatial dimensions. Their approach underscores the perturbative renormalization group (RG) treatment of an effective field theory that captures the breakdown of Fermi liquid behavior near certain regions of the Fermi surface termed as “hot spots.” This theory elucidates the complexities appearing at the quantum critical point (QCP) associated with SDW order, offering substantial insights into both theoretical predictions and experimental observations in correlated electron systems like high-temperature superconductors.

Core Contributions

1. Model Framework and RG Analysis: Metlitski and Sachdev employ a renormalization group analysis on the model originally proposed by Abanov and Chubukov. They identify key aspects of spin density wave transitions utilizing a field-theoretic local approach. Specifically, the authors highlight the renormalization of coupling constants and field scales within the two-loop order expansion. Notably, a pivotal advancement of this paper is addressing the infrared singularities at higher loops which are reminiscent of the challenges faced in a Fermi surface interacting with gauge fields—an area thoroughly studied and noted by Lee.
2. Breakdown of Fermi Liquid Theory: The research illustrates a significant breakdown in the conventional Fermi liquid framework upon nearing the SDW critical point. This breakdown is particularly evident in the "hot spot" regions of the Fermi surface, where the dynamical critical exponent $z$ deviates from its mean-field value $z = 2$.
3. Renormalization Constants Consistency: A striking feature of their analysis is the consistency in determining renormalization constants. This establishes robustness in their treatment, especially as they argue the damping of the SDW fluctuations is tied to these renormalizations—diverging from previous results by Chubukov and colleagues.
4. Enhancements in Pairing and Bond Order Instabilities: The paper also presents a detailed inquiry into the renormalization of the pairing vertex, unearthing a logarithm-squared enhancement that implies a notable influence of quantum critical fluctuations on superconductivity. Similarly, enhancements were reported for a modulated bond order resembling an Ising-nematic order, which could lead to novel insights into symmetry-broken phases adjacent to the QCP.

Implications and Future Directions

• Theoretical Insights: The findings open avenues for a deeper theoretical understanding of quantum criticality in two-dimensional metals. The unexpected scaling behaviors and deviations in $z$ point towards the necessity for an adjusted theoretical framework that can accommodate such non-trivial renormalizations.
• Experimental Alignments: The results are particularly intriguing when considering the anomalies observed in experiments with cuprates and pnictides, where interactions between SDW and superconductivity have been a focal point. The results provide a qualitative match to phase transitions observed under conditions such as pressure-induced suppression of SDW order.
• Complex Interactions: Future theoretical work could explore the potentially intricate feedback loops between superconductivity and quantum critical fluctuations hinted at by the renormalization enhancements seen in the paper. Additionally, re-examining the $1/N$ expansion's breakdown in more diverse interacting systems might yield more universally applicable theories.

In conclusion, Metlitski and Sachdev's work marks a significant leap in understanding the SDW transitions in metals and the melting pot of rich quantum phenomena surrounding them. Their rigorous field-theoretic approach coupled with relevant RG techniques paves the way for better alignment between quantum critical theory and emergent experimental phenomena in strongly correlated materials.