Quantum phase transitions of metals in two spatial dimensions: I. Ising-nematic order (1001.1153v5)

Abstract: We present a renormalization group theory for the onset of Ising-nematic order in a Fermi liquid in two spatial dimensions. This is a quantum phase transition, driven by electron interactions, which spontaneously reduces the point-group symmetry from square to rectangular. The critical point is described by an infinite set of 2+1 dimensional local field theories, labeled by points on the Fermi surface. Each field theory contains a real scalar field representing the Ising order parameter, and fermionic fields representing a time-reversed pair of patches on the Fermi surface. We demonstrate that the field theories obey compatibility constraints required by our redundant representation of the underlying degrees of freedom. Scaling forms for the response functions are proposed, and supported by computations by up to three loops. Extensions of our results to other transitions of two-dimensional Fermi liquids with broken point-group and/or time-reversal symmetry are noted. Our results extend also to the problem of a Fermi surface coupled to a U(1) gauge field.

• The paper develops a local field theory framework that models the critical dynamics and symmetry reduction in 2D metals undergoing Ising-nematic transitions.
• It employs advanced renormalization group techniques, deriving scaling forms up to three-loop order and confirming a robust dynamic critical exponent (z=3).
• The findings extend to diverse symmetry-breaking transitions, predicting Fermi surface anisotropies with significant implications for high-temperature superconductivity.

Analysis of Quantum Phase Transitions in Two-Dimensional Metals: Ising-Nematic Order

The paper in question addresses the intricate phenomena of quantum phase transitions within two-dimensional metallic systems, focusing specifically on the emergence of Ising-nematic order. This order is of significant interest due to its implications in systems such as the enigmatic cuprate superconductors, wherein it plays a crucial role in delineating the 'pseudogap' region from the 'strange metal' phase. The work by Metlitski and Sachdev offers a rigorous renormalization group (RG) analysis of these transitions, providing insights into the critical behavior and universality classes associated with such phases.

Key Contributions

The primary contribution of this paper is the development of a local field theory framework that captures the critical dynamics of a metal transitioning into a phase characterized by Ising-nematic order. This involves a symmetry reduction from square to rectangular lattices, leading to anisotropies observable experimentally, such as in the Nernst effect in cuprates.

1. Field Theory Construction: The authors construct a field theory using a real scalar field for the Ising order parameter and fermionic fields to represent time-reversed patches on the Fermi surface. This complex interplay results in a system of 2+1-dimensional field theories with each being characterized by points on the Fermi surface.
2. Scaling Behavior: One of the significant technical achievements is the derivation of scaling forms for the response functions, which are supported by computations up to three-loop order. The scaling analysis highlights the dynamic critical exponent $z$, which remains at 3, implying robustness against higher-order quantum corrections in typical metallic systems.
3. Universality and Extensions: The findings are not strictly limited to Ising-nematic transitions. The framework is adaptable to other symmetry-breaking transitions, including superconducting states and scenarios involving U(1) gauge coupling with a Fermi surface, indicating a broad applicability of their model.
4. Non-Fermi Liquid Behavior: The theory successfully integrates the peculiarities of non-Fermi liquid behavior endemic to such transitions, predicting a significant impact on Fermi surface properties such as anisotropies and shifts, indicative of the intricate dynamics at play.

Numerical Results and Predictions

The RG calculations in the paper, extending to three-loop order, reveal critical insights into the behavior of fermionic and bosonic correlations. Notably, while some corrections are finite at three loops, implying stability at large $N$, the authors also speculate on potential subtleties that could arise in real systems, emphasizing the non-trivial nature of these transitions.

For researchers, the precise predictions about scaling and universality serve as a roadmap for experimental verifications and potential simulations, posing questions about measurements in varying dimensions of the parameter space. The non-renormalization of the mass term stands out as a bold theoretical claim that warrants further investigation, both theoretically and experimentally.

Implications and Future Outlook

From a theoretical standpoint, the paper provides a solid foundation for understanding complex quantum phase transitions in condensed matter systems, extending its implications to other strongly correlated phenomena. Practically, it feeds directly into the ongoing puzzle of high-temperature superconductors, offering a microscopic lens to view phenomena like nematicity.

Future research directions could involve exploring more general order parameter symmetries and coupling to other exotic types of fermionic excitations beyond the Ising or U(1) limits. Additionally, integrating these findings with novel computational techniques and experimental platforms, including quantum simulations, could unlock new realms of understanding in quantum criticality.

In conclusion, Metlitski and Sachdev's work stands as a pillar in the paper of quantum phase transitions, blending rich theoretical frameworks with computational rigor to elucidate critical processes in quantum materials. It sets the stage for further explorations into the quantum states of matter that lie at the frontier of condensed matter physics.