Quantum Criticality

Quantum criticality in a material is the finite-temperature and finite-frequency organization produced by a continuous phase transition whose endpoint lies at zero temperature. The endpoint itself is an ideal limit. Experiments infer it from a controlled tuning trajectory, the fate of the adjacent phases, vanishing crossover scales, singular response, and scaling shared by several observables.

That standard is deliberately stronger than observing a resistivity exponent different from two. Linear resistivity, a divergent effective mass, an $E/T$ collapse, or a broad thermodynamic maximum can each occur for more than one reason. A quantum-critical interpretation connects these observations to the same nearby phases and vanishing energy scales.

Required background. Quantum Phase Transitions supplies the zero-temperature endpoint and thermodynamic-limit criteria. Critical Exponents and Scaling supplies homogeneity laws and corrections to scaling; Susceptibilities supplies the source, channel, and limit defining a response. Fermi Liquid Theory Preview supplies the metallic baseline, and Kondo Lattices supplies the screening-versus-ordering competition used in heavy-fermion examples.

From a Control Knob to a Zero-Temperature Endpoint

Let $\lambda$ denote the experimentally controlled variable and define a dimensionless detuning

$$r = \frac{\lambda-\lambda_c}{\lambda_0}.$$

The scale $\lambda_0$ and the sign of $r$ must be stated. A magnetic field in tesla, pressure in gigapascals, and substitution fraction are not interchangeable scaling fields. Even when each moves the same ordering temperature, it may also change carrier density, crystal-field splittings, dimensionality, disorder, or symmetry in a different proportion.

Control	What changes, and what needs calibration
Hydrostatic pressure	Tunes bandwidth and hybridization; measure the pressure and its gradients, including nonhydrostatic stress.
Magnetic field	Produces Zeeman and orbital effects; track orientation, demagnetization and sample heating.
Composition	Changes electronic structure together with disorder and local strain; measure the actual composition and homogeneity.
Uniaxial stress or strain	Selects symmetry channels; determine strain transfer and the full relevant tensor.
Gate voltage	Tunes thin-device density through electrostatics; calibrate capacitance, contacts and filling inhomogeneity.

Locate both phases before locating the point

A critical endpoint separates phases, not merely two fitting regimes. The ordered side requires a thermodynamic or symmetry-resolved diagnosis: diffraction, local magnetic probes, calorimetry, elastic anomalies, or a conjugate response. The other side likewise needs a low-temperature identity, such as a Fermi liquid, polarized phase, superconductor, paramagnet, or distinct topological state.

For an ordering line $T_{\mathrm{ord}}(\lambda)$, choose the convention that the ordered side has $\lambda<\lambda_c$. The first necessary observation is

$$\lim_{\lambda\to\lambda_c^-} T_{\mathrm{ord}}(\lambda) = 0.$$

This limit is not sufficient. The transition can become first order, split, terminate at nonzero temperature, disappear under a superconducting dome, or give way to another ordered phase. Latent heat, hysteresis, phase coexistence, discontinuous volume or magnetization, and protocol dependence must therefore be tracked as the line is suppressed.

Clean tuning is not one-dimensional tuning

The scaling field controlling criticality is generally a linear combination of laboratory variables. In this page’s magnetic tuning notation, $H$ denotes the applied field coordinate $\mu_0H_{\mathrm{SI}}$ measured in tesla, not $H_{\mathrm{SI}}$ in amperes per metre. Near the endpoint,

$$r = a_p(p-p_c) + a_H(H-H_c) + a_\varepsilon\varepsilon + a_n(n-n_c) + \cdots .$$

This matters whenever pressure and field produce different phase diagrams or when uniaxial stress couples strongly to an apparently hydrostatic transition. A single scalar axis is a local coordinate choice, not proof that all other relevant perturbations vanish.

The Quantum-Critical Fan Is a Crossover Map

For a continuous endpoint with correlation-length exponent $\nu$ and dynamical exponent $z$, detuning generates a characteristic energy

$$E_r = E_0 |r|^{\nu z}.$$

The adjacent ground-state regimes dominate when $k_{\mathrm B}T\ll E_r$. The quantum-critical regime is entered when thermal energy is comparable to or exceeds this detuning scale. Within this single-scale description, an operational crossover width obeys

$$|\lambda-\lambda_c| \sim \lambda_0 \left( \frac{k_{\mathrm B}T}{E_0} \right)^{1/(\nu z)}.$$

The familiar fan is therefore a statement about scaling trajectories. Its sides are usually crossovers, not phase boundaries. Different observables can define different order-one crossover contours because each weights the scaling function and regular background differently.

Top: dashed crossover contours indicate where thermal and detuning scales become comparable. A competing phase can hide the putative zero-temperature critical point; its continuation beneath that phase is hypothetical. Bottom: at fixed temperature, an entropy maximum gives $\alpha_V<0$ on its rising side and $\alpha_V>0$ on its falling side. The maximum is drawn at $p_c$ for simplicity; it can shift at finite temperature. Both panels are schematic, not quantitative phase diagrams.

A fan can be hidden or truncated

Superconductivity, nematic order, charge order, or another low-temperature instability can preempt the putative endpoint. The normal state may then be accessed only by field, pressure, disorder, or extrapolation, each of which changes the Hamiltonian. A fan drawn through a competing phase is a hypothesis about an underlying normal state, not a directly observed region.

A first-order transition also has a broad finite-temperature fluctuation regime when its discontinuity is weak. Likewise, a narrow-band crossover can create a V-shaped map without any singular zero-temperature endpoint. The evidence must reach beyond the geometry of a color plot.

Scaling as a Material Test

Introduce the dimensionless temperature $t=k_{\mathrm B}T/E_0$. After subtracting an explicitly modeled regular contribution, a singular observable may take the form

$$X_s(r,T) = X_0 t^{\kappa_X} \Phi_X\!\left( \frac{r}{t^{1/(\nu z)}} \right).$$

The exponent $\kappa_X$ depends on the observable and on whether dangerously irrelevant variables or conservation laws modify naive hyperscaling. This equation is a testable ansatz, not permission to optimize every parameter until curves overlap.

Crossover sharpening

Suppose a field or pressure sweep shows a crossover of width $w_\lambda(T)$. Single-parameter scaling predicts, up to corrections,

$$\frac{w_\lambda(T)}{\lambda_0} = C_w t^{1/(\nu z)} \left[ 1+c_w t^{\omega/z}+\cdots \right],$$

where $\omega>0$ is a correction exponent. The center, width, step height, and background should be fitted separately. A width extrapolating to zero while the step height also vanishes does not establish a discontinuity in the zero-temperature observable.

Dynamical scaling

At the critical detuning $r=0$, a single-scale theory of a mode near $\mathbf Q$ identifies the thermal correlation length as

$$\xi_T = \xi_0 t^{-1/z}.$$

This identification requires care: an interaction can be irrelevant at the zero-temperature fixed point yet control the finite-temperature static correlation length. In that case, often called dangerous irrelevance, the static length need not scale as $T^{-1/z}$ and simple $\omega/T$ scaling can fail (Millis, 1993). Under the single-scale assumption, a dynamical test is

$$\chi''(\mathbf q,\omega,T) = \chi_0 t^{-a} \Phi_\chi\!\left( (\mathbf q-\mathbf Q)\xi_T, \frac{\hbar\omega}{k_{\mathrm B}T} \right).$$

An $\hbar\omega/(k_{\mathrm B}T)$ collapse is often called $E/T$ scaling. It is strongest when the same exponents describe several momenta and temperatures, instrumental resolution is folded into the fit, detailed balance is respected, and sum rules remain consistent. A collapse over less than a decade or after a temperature-dependent background subtraction is only suggestive.

Testing a scaling collapse

Use an independently determined critical tuning value where possible, and model regular backgrounds explicitly. Shared calibration and subtraction errors correlate data points; those correlations belong in the fit uncertainty. Fit one window and test nearby temperatures or detunings without resetting every parameter. Stability against changing the fit window and comparison with nonsingular crossover models are more informative than a visually impressive overlap alone.

Probe	Critical information and common ambiguity
Calorimetry	Entropy and free-energy derivatives; separate nuclear, phonon and addenda terms.
Thermal expansion	Pressure derivative of entropy; distinguish volume response from anisotropic strain.
Scattering	Ordering wavevector, correlation length and dynamics; include resolution and form factors.
Local magnetic probes	Static fraction and relaxation distribution; hyperfine filtering and disorder affect the response.
Quantum oscillations	Orbit areas and masses; a lost amplitude need not mean a vanished orbit.
Hall and thermoelectric response	Carrier-sensitive crossovers; multiband and anomalous terms prevent direct density counting.
Electrical and thermal transport	Current relaxation; cold regions, phonons and other heat carriers can dominate.

Entropy, Thermal Expansion, and Grüneisen Tests

A continuous endpoint concentrates low-energy states. At fixed nonzero temperature, the entropy often forms a ridge near the critical trajectory. This provides a thermodynamic diagnostic independent of a transport relaxation model.

The volume thermal-expansion coefficient obeys the Maxwell relation

$$\alpha_V = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p = -\frac{1}{V} \left( \frac{\partial S}{\partial p} \right)_T.$$

If pressure moves the system through an entropy maximum, $\alpha_V$ changes sign across the ridge (Garst and Rosch, 2005). At nonzero temperature, the ridge need not sit exactly at $p_c$. The critical pressure Grüneisen parameter may be written

$$\Gamma_p^{\mathrm{cr}} = \frac{V_m}{\kappa_T} \frac{\alpha_V^{\mathrm{cr}}}{C_p^{\mathrm{cr}}},$$

where $V_m$ is molar volume, $C_p$ is molar heat capacity, $\alpha_V$ is the fractional volume-expansion coefficient, and $\kappa_T$ is isothermal compressibility. This convention makes $\Gamma_p^{\mathrm{cr}}$ dimensionless. For a generic pressure-tuned point governed by one leading relevant field, with a nonzero coupling to pressure and a nonsingular conversion factor $V_m/\kappa_T$, scaling predicts (Zhu and collaborators, 2003)

$$\Gamma_p^{\mathrm{cr}}(p_c,T) \propto T^{-1/(\nu z)}.$$

Away from $p_c$, its low-temperature singular part commonly approaches a signed inverse detuning. These predictions require the critical pieces of both numerator and denominator, a known pressure coupling, and no more singular competing contribution. Special symmetry can make the leading thermal-expansion amplitude vanish at the critical point; a vanishing numerator is not ruled out by the exponent alone. Anisotropic crystals can have different linear-expansion signs even when the volume response follows the expected entropy flow.

For a field-tuned endpoint, use molar Gibbs energy $g$ with $dg=-s\,dT-M\,dH$ at fixed pressure: $M$ is the conjugate molar magnetic moment, $s$ molar entropy, and $C_H=T(\partial s/\partial T)_H$ molar heat capacity. With the tesla field coordinate defined above, the magnetic Grüneisen parameter is

$$\Gamma_H = -\frac{(\partial M/\partial T)_H}{C_H} = \frac{1}{T} \left( \frac{\partial T}{\partial H} \right)_S.$$

It has units of inverse tesla and measures the magnetocaloric slope of an isentrope. If one instead differentiates with respect to $H_{\mathrm{SI}}$ in amperes per metre, the conjugate moment term and response require the corresponding factor of $\mu_0$. A divergence and sign change near $H_c$ can locate entropy accumulation, but eddy-current heating, nuclear contributions, hysteresis, and finite sweep rate must be controlled.

Transport Anomalies: Powerful but Nonunique

A Landau Fermi liquid has an electronic resistivity that often approaches

$$\rho(T,\lambda) = \rho_0(\lambda) + A(\lambda)T^2$$

when electron–electron scattering dominates and momentum relaxation is available. Near many candidate endpoints, the empirical form

$$\rho(T,\lambda) = \rho_0(\lambda) + A_n(\lambda)T^{n(\lambda)}$$

has $n<2$, sometimes close to one. The local logarithmic slope

$$n_{\mathrm{eff}}(T) = \frac{\partial\ln[\rho(T)-\rho_0]}{ \partial\ln T }$$

is more informative than fitting one exponent across a broad crossover. Its uncertainty is dominated by the residual-resistivity estimate at the lowest temperatures.

Why the exponent is not universal

Transport weights momentum relaxation, velocities, Fermi-surface geometry, impurity mixing, and current vertices. In an itinerant antiferromagnetic scenario, critical scattering can be concentrated near hot spots joined by $\mathbf Q$, while cold regions conduct in parallel. Disorder can mix the two. Phonons, valence fluctuations, a nearby Lifshitz transition, two-level systems, and hydrodynamic flow can also produce nonquadratic laws.

Linear-$T$ resistivity is therefore a phenomenon to explain, not a standalone measurement of $z$ or proof of a quantum critical point. The canonical treatments of the Boltzmann and Kubo frameworks are Boltzmann Transport and Kubo Formula.

Charge versus heat

The Lorenz ratio is

$$L(T) = \frac{\kappa}{\sigma T}, \qquad L_0 = \frac{\pi^2k_{\mathrm B}^2}{3e^2}.$$

Testing the Wiedemann–Franz limit requires $T\to0$, the electronic thermal conductivity, matched electrical contacts, and subtraction or bounding of phonon, magnon, neutral-fermion, and nuclear heat channels. A finite-temperature deviation can reflect inelastic scattering even when charged quasiparticles survive asymptotically.

Optical conductivity adds frequency resolution. A proposed $\omega/T$ scaling or Planckian relaxation rate must state whether the response is Drude-like, which spectral weight and effective mass define the rate, how interband absorption was removed, and whether the sum rule is satisfied. A fitted $\hbar/\tau\sim k_{\mathrm B}T$ cannot be compared meaningfully across systems without controlling these choices, and does not by itself establish universality.

Magnetic Quantum Criticality

Magnetic metals are especially revealing because order-parameter fluctuations coexist with gapless fermions. They are also especially difficult because integrating out those fermions can generate damping and nonlocal interactions.

Itinerant spin-density-wave endpoint

In the simplest antiferromagnetic Hertz–Millis description, the low-energy inverse susceptibility near the ordering vector has the schematic form

$$\chi^{-1}(\mathbf Q+\mathbf q,i\omega_n) = r + cq^2 + \gamma|\omega_n| + \cdots .$$

The damping gives $z=2$ in this common clean antiferromagnetic case. The order parameter reconstructs the Fermi surface by folding the Brillouin zone, but the electronic degrees of freedom counted in the paramagnetic heavy Fermi liquid need not localize. Critical scattering is strongest near hot regions connected by $\mathbf Q$.

This framework is not one formula for every magnet. Ferromagnetic fluctuations have different damping and couple to long-wavelength particle–hole modes. In clean metallic ferromagnets, nonanalytic fermionic corrections often drive the low-temperature transition first order or stabilize modulated order. A material with avoided ferromagnetic criticality should not be forced into the antiferromagnetic $z=2$ template.

Kondo destruction

In a heavy-fermion metal, magnetic order can coincide with the collapse of Kondo entanglement. A second low-energy scale $E^*$ then tends to zero, and the charged Fermi surface can change from a large count including the local moments to a small count excluding them. The critical theory contains more than fluctuations of the magnetic order parameter.

Candidate evidence includes a crossover in Hall or magnetostriction that sharpens on cooling, a Fermi-surface change not accounted for by ordinary zone folding, dynamical scaling with a strong local component, and the merger of the magnetic and Kondo-localization scales. None is decisive alone. Multiband Hall coefficients are not direct carrier counts, quantum-oscillation amplitudes can disappear because masses or scattering diverge, and finite-temperature crossover lines can detach from the magnetic boundary.

Representative Materials

The following examples are landmarks, not a list of universally settled classifications.

Material or family	What it teaches
CeCu$_{6-x}$Au$_x$	Au tuning produces antiferromagnetism near $x\simeq0.1$ and anomalous dynamical scaling; substitution disorder complicates the interpretation.
YbRh$_2$Si$_2$	Low-field magnetic suppression accompanies thermodynamic and Hall crossovers; their zero-temperature connection and multiband interpretation need testing.
CeRhIn$_5$	Pressure produces strong mass and orbit changes near magnetism and superconductivity; high-field access changes the route to the normal state.
Sr$_3$Ru$_2$O$_7$	Field-tuned metamagnetism motivates a critical endpoint, but an intervening nematic phase complicates the clean-sample phase diagram.
MnSi	Pressure suppresses helimagnetism through a first-order transition; its broad unusual transport regime is not a simple continuous-QCP fan.
Spin-dimer magnets	Field can close a singlet–triplet gap, giving relatively direct access to bosonic critical modes; anisotropy and lattice coupling can modify the ideal limit.

These cases illustrate why “proximity to a QCP” must specify the endpoint, tuning path, hidden or intervening phases, and observables. Similar transport exponents across two compounds do not by themselves imply the same universality class.

Exercises

1. Infer a crossover exponent product

A field-sweep crossover has full width $w=0.24\,\mathrm T$ at $8\,\mathrm K$ and $w=0.12\,\mathrm T$ at $2\,\mathrm K$. Assuming $w\propto T^{1/(\nu z)}$ and negligible corrections, infer $\nu z$.

Solution

The ratios obey

$$\frac{0.12}{0.24} = \left( \frac{2}{8} \right)^{1/(\nu z)}.$$

Thus

$$\frac{1}{2} = \left( \frac{1}{4} \right)^{1/(\nu z)},$$

which gives $1/(\nu z)=1/2$ and therefore $\nu z=2$.

Two points cannot establish a scaling regime. A real analysis needs more temperatures, uncertainty in the width and center, correction terms, and a test that the step shape is governed by one scaling function.

2. Determine the thermal-expansion sign

At fixed low temperature, the entropy $S(p)$ rises as pressure approaches $p_c$ from below and falls after passing through a maximum. What signs of $\alpha_V$ are expected on the two sides?

Solution

Use

$$\alpha_V = -\frac{1}{V} \left( \frac{\partial S}{\partial p} \right)_T.$$

Below $p_c$, $(\partial S/\partial p)_T>0$, so $\alpha_V<0$. Above $p_c$, the entropy decreases with pressure, so $(\partial S/\partial p)_T<0$ and $\alpha_V>0$. The sign change points toward the entropy ridge.

The argument concerns the singular volume response. A measured linear expansion along one axis can have another sign because uniaxial stress couples differently, and a regular lattice background can mask the critical contribution.

3. Extrapolate a Hall crossover carefully

A Hall-related observable is fitted by

$$R_H(H,T) = R_<(T) + \Delta R(T) f\!\left( \frac{H-H^*(T)}{w(T)} \right),$$

where $f$ is a smooth step. Measurements find $w(T)\to0$ and $\Delta R(T)\to\Delta R_0\ne0$ as $T\to0$. What is the strongest immediate inference, and what remains to be checked?

Solution

Within the fitted family and extrapolation window, the result supports a discontinuity of this Hall-related observable at $H^*(0)$. It is stronger than observing a sharpening slope because the limiting step height remains finite.

It is not yet a model-independent measurement of Fermi volume. One must bound anomalous Hall and skew-scattering terms, account for multiband mobilities, verify that the sample temperature follows the bath during field sweeps, and show that $H^*(0)$ coincides with or has a controlled relation to the thermodynamic endpoint. Quantum oscillations, magnetostriction, and spectroscopy can test the Fermi-surface interpretation.

4. Hot and cold regions in parallel

Suppose a simplified metal has independent hot and cold conduction channels with

$$\sigma_h = \frac{C_h}{T}, \qquad \sigma_c = \frac{C_c}{T^2},$$

and $C_h,C_c>0$. Find the low-temperature power of the total resistivity $\rho=1/(\sigma_h+\sigma_c)$.

Solution

The total conductivity is

$$\sigma = \frac{C_hT+C_c}{T^2}.$$

Therefore

$$\rho = \frac{T^2}{C_c+C_hT} \xrightarrow[T\to0]{} \frac{T^2}{C_c}.$$

The cold channel short-circuits the hot channel asymptotically, even though the hot contribution alone has linear resistivity. This toy result explains why a scattering-rate exponent at critical hot spots need not equal the measured bulk resistivity exponent. Impurities and vertex corrections can mix the regions and change the conclusion.

5. A heat-capacity peak without collective criticality

Consider independent two-level units with energies $0$ and $\Delta(H)>0$. Show that their heat capacity per unit is

$$\frac{C}{k_{\mathrm B}}=\frac{x^2}{4\cosh^2(x/2)}, \qquad x=\frac{\Delta}{k_{\mathrm B}T}.$$

How does the peak temperature move as $\Delta$ decreases? Does a peak approaching zero temperature establish a continuous many-body quantum critical point?

Solution

The excited-state probability is $p=1/(1+e^x)$, so the internal energy is $U=\Delta p$. Differentiating at fixed field, or using the energy variance, gives

$$C=\frac{\Delta^2}{k_{\mathrm B}T^2}p(1-p),$$

which is the stated expression. Its nonzero maximum satisfies $x\tanh(x/2)=2$, with $x\simeq2.3994$. Therefore

$$T_{\mathrm{peak}}\simeq0.4168\,\frac{\Delta(H)}{k_{\mathrm B}}.$$

The peak moves toward zero when the local splitting closes, although the units remain uncorrelated. At exact degeneracy this ideal model has independent local degeneracies, not a diverging spatial correlation length. A moving maximum or local gap closure alone therefore cannot distinguish a Schottky crossover from collective criticality. Spatial correlations, the adjacent phases, scaling and tests for a first-order transition supply additional information.

Related Topics

• Quantum Phase Transitions and Critical Exponents and Scaling develop the general transition and scaling theory.
• Kondo Lattices explains heavy-Fermi-liquid and fractionalized alternatives.
• Non-Fermi Liquids and Strange Metals compare failures of quasiparticle descriptions with unusual measured transport.
• Heat Capacity and Thermodynamics explains entropy extraction and backgrounds.

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