Quantum Phase Transitions and Critical Metals

A zero-temperature critical point can organize broad finite-temperature behavior. The challenge is to identify the critical degrees of freedom and distinguish universal scaling from the extra scales introduced by a Fermi surface or slow relaxation.

Enter this chapter

Begin with Quantum Phase Transitions and Competing Scales to separate tuning, temperature, size, and excitation-gap scales.

Helpful background. Quantum Phase Transitions and Competing Scales and Fermi-Surface Patch Theory and Low-Energy Scaling develop the definitions used by the chapter’s opening pages.

Topics

1. Quantum Phase Transitions and Competing Scales — How do tuning, gaps, correlation scales, temperature, finite size, and irrelevant couplings organize a quantum phase transition?
2. Landau–Ginzburg–Wilson Quantum Criticality — When does an order-parameter Landau–Ginzburg–Wilson action control a quantum transition, and which degrees of freedom invalidate it?
3. Quantum-Critical Fans and Finite-Temperature Scaling — Which finite-temperature scaling functions and crossover boundaries justify a quantum-critical fan rather than a generic incoherent regime?
4. Hertz–Millis Theory and Landau Damping — How does integrating a Fermi surface generate Landau-damped order-parameter dynamics in Hertz–Millis theory, and when is that integration invalid?
5. Spin-Fermion and Hot-Spot Theories — How do hot spots and an order-parameter field form a coupled spin-fermion theory, and which large-N, epsilon, or kinematic limit controls it?
6. Dangerously Irrelevant Couplings and Hyperscaling Violation — How can an RG-irrelevant coupling control an ordered scale, violate naive hyperscaling, or split thermal and quantum crossover exponents?
7. Metallic Non-Fermi Liquids and Quasiparticle Breakdown — Which analytic and scaling criteria establish metallic non-Fermi-liquid behavior near quantum criticality?
8. Critical Fermi Surfaces and Patch Theories — How do gauge or order-parameter bosons coupled to an extended Fermi surface produce critical patch theories and singular fermion dynamics?
9. SYK Models and Local Quantum Criticality — How do SYK-type large-N models realize solvable local criticality, and which features survive in lattice or finite-dimensional quantum matter?
10. Strange-Metal Transport and Planckian Claims — Which transport observables justify “strange metal” or “Planckian” language, and what does a fitted linear rate fail to identify?
11. Evidence and Model Discrimination at Quantum Criticality — How are quantum-critical models discriminated when scaling collapse, thermodynamics, spectroscopy, and transport admit overlapping explanations?

How the ideas fit together

A local order-parameter theory assumes that omitted fields can be integrated out without losing essential infrared information. Gapless metallic fermions challenge that step: they generate damping and can require retaining hot spots or Fermi-surface patches explicitly Hertz 1976, §§ II–IV.

Even when a fixed point controls correlations, a dangerously irrelevant coupling can introduce an additional observable scale. A fitted power law or relaxation rate proportional to temperature therefore does not uniquely identify a critical theory. Compare spectra, thermodynamics, and transport over compatible regimes, including their competing explanations.

The comparison below separates the assumptions and tests needed for these different conclusions.

Mechanism and evidence table

Scroll horizontally to compare all columns. Use the Left and Right arrow keys when the table region is focused.

Low-energy mechanisms, control, and evidence for quantum phase transitions and critical metals.

Mechanism	Retained low-energy fields	Dynamics	Controlled limit	Dangerous variable or hyperscaling issue	Transport assumption	Finite-temperature scope	Supported conclusion
Relativistic LGW	local $N$-component order parameter	$z=1$ in the displayed kernel	$4-(d+z)$ expansion, large $N$, or known fixed point	quartic $u$ dangerous above $d+z=4$	transport requires a separately defined current	fan below UV and above other gaps	identifies an order-parameter universality class
Antiferromagnetic Hertz–Millis	damped order parameter after fermions are removed	$q^2+\gamma\lvert\omega\rvert$, Gaussian $z=2$	regular generated vertices; often above upper critical dimension	$u$ controls thermal mass and $\psi$	momentum relaxation not supplied by the boson action alone	can be preempted by pairing	establishes damping kinematics, not harmless fermion removal
Clean ferromagnetic Hertz theory	small-$q$ order parameter	$q^2+\gamma\lvert\omega\rvert/q$, Gaussian $z=3$	weak-coupling reduction before nonanalytic feedback	$u$ dangerous; soft fermion modes add nonlocality	small-angle scattering weakly relaxes current	first-order or modulated order may intervene	a $z=3$ fit does not prove a continuous local action
Spin-fermion hot spots	order parameter plus discrete hot fermions	coupled, generally beyond fixed $z=2$ one-loop picture	$\epsilon$, emergent velocity ratio, or specified model	pairing and composite channels introduce lower scales	cold regions and momentum sinks must be included	above superconducting and curvature crossovers	identifies hot-region breakdown within the stated control
Gauge or nematic patches	extended Fermi surface plus low-$q$ boson	anisotropic $q_y:q_x:\omega=1:2:3$	controlled deformation; naive large $N$ is nonuniform	patch count produces hyperscaling issues	vertex corrections and antipodal coupling sign matter	often hidden by pairing	supports a critical Fermi surface, not a unique global metal
SYK dot or array	all-to-all large-$N$ fermions; spatial couplings added for arrays	local conformal time, no dot momentum	$N\to\infty$ before deep infrared	residual entropy and finite-$N$ scale depend on limits	dot has no conductivity; arrays require explicit current	$J/N\ll T\ll J$ for conformal dot window	proves solvable local criticality for the specified model
Planckian transport fit	current response and fitted weight	$\alpha=\hbar\Gamma_{\mathrm{tr}}/(k_BT)$	optical/DC consistency and covariance, not an RG limit	changing Drude weight can mimic a rate	explicit umklapp, disorder, phonons, or other momentum sink	declared linear window	establishes an order-one fitted rate, not a universal bound
Evidence comparison	predictions from several mechanisms	model-specific	common covariance and held-out tests	corrections and analytic backgrounds retained	compare DC, optical, Hall, thermal, and one-particle channels	searched windows reported	mechanism only when alternatives make and fail distinct predictions

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Where to go next

Continue to Band Geometry and Symmetry-Protected Matter to construct the occupied Bloch subspace and its real-space interpretation.

References

• Hertz, J. A. “Quantum Critical Phenomena.” Physical Review B 14 (1976): 1165–1184. DOI.