11.3 Heavy fermions

Characteristics of heavy fermions

Heavy fermion systems are metallic compounds where conduction electrons behave as though they carry an effective mass 100 to 1000 times the free electron mass. This dramatic mass enhancement arises from strong correlations between conduction electrons and localized f-electrons, placing these materials at the boundary between ordinary metals and strongly correlated systems.

Understanding heavy fermions matters because they host a remarkable range of quantum phenomena: unconventional superconductivity, quantum criticality, and non-Fermi liquid behavior all show up in these compounds. The physics here connects single-impurity quantum mechanics (the Kondo effect) to collective lattice behavior in ways that remain an active area of research.

Effective mass enhancement

The defining feature of heavy fermion compounds is that their quasiparticles act as if they're enormously heavy. This isn't a literal mass increase; it's a renormalization caused by strong interactions between conduction electrons and localized f-electrons (typically from Ce, U, or Yb ions).

• The effective mass $m^*$ can reach $100$ to $1000$ times the bare electron mass $m_e$
• This enhancement shows up directly in the electronic specific heat coefficient $\gamma$, which scales linearly with $m^*$. Typical values reach $\gamma \sim 1000 \, \text{mJ/mol·K}^2$, compared to $\sim 1\text{–}10 \, \text{mJ/mol·K}^2$ in normal metals
• The Fermi velocity is correspondingly reduced, since $v_F \propto 1/m^*$
• Landau Fermi liquid theory still applies at low temperatures: the quasiparticles are well-defined, just with heavily modified parameters

Low temperature behavior

At sufficiently low temperatures, most heavy fermion systems settle into a Fermi liquid ground state with characteristic signatures:

• Specific heat: $C = \gamma T + \beta T^3$, where the electronic term $\gamma T$ dominates at low $T$ because $\gamma$ is so large
• Resistivity: follows $\rho = \rho_0 + AT^2$, the hallmark $T^2$ dependence of electron-electron scattering in a Fermi liquid. The coefficient $A$ is also enhanced, and the Kadowaki-Woods ratio $A/\gamma^2$ is roughly constant across many heavy fermion compounds
• Magnetic susceptibility: shows enhanced, temperature-independent Pauli paramagnetism at low $T$, with a Wilson ratio that reflects the degree of correlation

These properties all deviate from what you'd expect in a simple metal, but they remain internally consistent within the Fermi liquid framework.

Kondo effect in heavy fermions

The Kondo effect is the microscopic engine behind heavy fermion behavior. In a single-impurity Kondo system, a localized magnetic moment (from an f-electron) is progressively screened by surrounding conduction electrons as the temperature drops below the Kondo temperature $T_K$.

In heavy fermion compounds, you have a lattice of such magnetic ions, so the physics becomes collective:

1. Above $T_K$, the f-electron moments act like independent local moments, and the resistivity rises logarithmically with decreasing temperature (single-impurity Kondo scattering).
2. Near $T_K$, coherent scattering develops across the lattice. The resistivity peaks and then drops.
3. Below $T_K$, a coherent heavy fermion state forms. The f-electrons become part of the Fermi sea as composite quasiparticles with large effective mass.

The ground state depends on a competition between two energy scales:

• Kondo screening (scale $\sim T_K$), which favors a nonmagnetic singlet ground state
• RKKY interaction (scale $\sim J^2 N(E_F)$), an indirect exchange between local moments mediated by conduction electrons, which favors magnetic order

The Doniach phase diagram captures this competition: when $T_K$ dominates, you get a heavy Fermi liquid; when RKKY dominates, you get magnetic order. The boundary between these regimes is where quantum critical behavior emerges.

Electronic structure

Hybridization of f-electrons

The f-electrons in heavy fermion compounds have a dual character. They're localized enough to carry magnetic moments but hybridize with the conduction band strongly enough to participate in the Fermi surface.

• Hybridization between f-states and conduction electrons creates composite quasiparticles. The hybridization strength $V$ controls how localized or itinerant the f-electrons are.
• A hybridization gap opens near the Fermi level $E_F$. In Kondo insulators like $\text{SmB}_6$, this gap is complete; in heavy fermion metals, it's a partial or indirect gap that reshapes the low-energy density of states.
• The crossover from localized to itinerant f-electron behavior is gradual and depends on temperature, pressure, and composition.

Fermi surface properties

One of the most striking results in heavy fermion physics is that the Fermi surface is large, meaning it includes the f-electrons in the count.

• Luttinger's theorem requires that the Fermi surface volume accounts for all electrons, including the f-electrons, even though they started as localized. This has been confirmed by de Haas-van Alphen experiments.
• Multiple Fermi surface sheets are common, reflecting the complex band structure created by hybridization.
• The quasiparticle bands near $E_F$ are extremely flat, giving rise to very low Fermi velocities and a large density of states $N(E_F)$.

Band structure anomalies

Strong correlations reshape the band structure in ways that single-particle theory can't capture without renormalization:

• Flat bands near $E_F$ are the direct signature of heavy quasiparticles. These are the renormalized f-derived bands.
• Hybridization gaps and pseudogaps appear in the density of states, affecting optical conductivity and tunneling spectra.
• Spin-orbit coupling is large for f-electrons (especially in uranium compounds), and crystal electric field splitting further breaks the degeneracy of f-levels. Both effects must be included for realistic modeling.

Heavy fermion materials

Cerium compounds

Cerium-based compounds ($4f^1$ configuration) are the most widely studied heavy fermion systems.

• $\text{CeAl}_3$ was the first recognized heavy fermion compound, with $\gamma \approx 1600 \, \text{mJ/mol·K}^2$
• $\text{CeCu}_6$ shows non-Fermi liquid behavior when tuned to a quantum critical point by doping (e.g., $\text{CeCu}_{6-x}\text{Au}_x$)
• $\text{CeRhIn}_5$ is an antiferromagnet at ambient pressure, but superconductivity emerges under pressure as the magnetic order is suppressed
• $\text{CeCoIn}_5$ is a heavy fermion superconductor with $T_c \approx 2.3 \, \text{K}$, relatively high for this class, and shows d-wave pairing symmetry

Uranium compounds

Uranium compounds ($5f$ electrons) tend to have stronger hybridization than cerium systems because the 5f orbitals are more extended.

• $\text{UPt}_3$ exhibits multiple superconducting phases (at least two distinct transitions), pointing to a multi-component order parameter
• $\text{URu}_2\text{Si}_2$ has a phase transition at $17.5 \, \text{K}$ whose order parameter remains unidentified after decades of study, hence the name "hidden order"
• $\text{UBe}_{13}$ shows superconductivity coexisting with strong spin fluctuations
• The stronger 5f hybridization generally makes uranium systems more itinerant than their cerium counterparts

Ytterbium compounds

Ytterbium ($4f^{13}$) is the hole analogue of cerium ($4f^1$), so many of the same concepts apply with the roles of electrons and holes swapped.

• $\text{YbRh}_2\text{Si}_2$ is one of the cleanest examples of quantum criticality, with a field-tuned quantum critical point and clear non-Fermi liquid signatures
• $\text{YbAlB}_4$ was the first Yb-based heavy fermion superconductor discovered
• $\text{YbNi}_2\text{Ge}_2$ shows valence fluctuation behavior alongside Kondo lattice physics
• Yb compounds are particularly useful for studying quantum criticality because magnetic fields (a clean tuning parameter) can often access the critical point

Quantum critical phenomena

Non-Fermi liquid behavior

Near a quantum critical point (QCP), the Fermi liquid description breaks down. The quasiparticle concept itself becomes ill-defined as scattering rates diverge.

• Thermodynamic and transport properties show anomalous power laws instead of the standard Fermi liquid forms. For example, resistivity may go as $\rho \sim T$ (linear) instead of $T^2$, and specific heat may show $C/T \sim -\ln T$.
• The effective mass appears to diverge as the QCP is approached, signaling the destruction of coherent quasiparticles.
• Two theoretical scenarios compete: spin-density-wave (SDW) criticality, where the Fermi surface remains intact and only the magnetic order parameter fluctuates, and local quantum criticality, where the Kondo effect itself is destroyed at the QCP and the Fermi surface reconstructs. Experiments on $\text{YbRh}_2\text{Si}_2$ and $\text{CeCu}_{6-x}\text{Au}_x$ provide evidence for the local scenario.

Quantum phase transitions

A quantum phase transition occurs at $T = 0$ and is driven by a non-thermal control parameter such as pressure, magnetic field, or chemical doping.

• Unlike classical phase transitions (driven by thermal fluctuations), quantum phase transitions are governed by quantum fluctuations consistent with the uncertainty principle.
• The influence of the QCP extends to finite temperatures, creating a quantum critical fan in the temperature vs. control-parameter phase diagram. Inside this fan, non-Fermi liquid behavior dominates.
• Critical slowing down occurs: the characteristic energy scale $T^*$ vanishes as the QCP is approached, meaning fluctuations become arbitrarily slow.

Magnetic instabilities

Most heavy fermion QCPs involve the suppression of antiferromagnetic order.

• The Doniach phase diagram predicts that as the ratio of Kondo coupling to RKKY interaction increases, the system crosses from a magnetically ordered ground state to a paramagnetic heavy Fermi liquid.
• Antiferromagnetic order is the most common magnetic ground state, but more exotic phases like multipolar order (ordering of orbital degrees of freedom rather than dipole moments) also occur.
• Spin density wave (SDW) instabilities can arise from Fermi surface nesting, providing a weak-coupling route to magnetism that competes with the strong-coupling local-moment picture.

Experimental techniques

Specific heat measurements

Specific heat is the most direct probe of the enhanced density of states in heavy fermion systems.

• The electronic specific heat coefficient $\gamma = C/T$ at low temperature directly measures $N(E_F)$ and hence $m^*$
• Near a QCP, $C/T$ often diverges logarithmically or as a power law instead of saturating to a constant
• Integrating $C/T$ gives the entropy, which reveals how many degrees of freedom are being quenched as the heavy fermion state forms (typically approaching $R \ln 2$ per mole for a doublet ground state)
• Jumps in specific heat at superconducting or magnetic transitions provide thermodynamic evidence for phase changes

de Haas-van Alphen effect

Quantum oscillations in magnetization as a function of applied field directly map the Fermi surface.

• The oscillation frequency is proportional to the extremal cross-sectional area of the Fermi surface perpendicular to the field: $F = (\hbar/2\pi e) A_k$
• Measuring how the oscillation amplitude decays with temperature gives the cyclotron effective mass $m^*_c$ through the Lifshitz-Kosevich formula. In heavy fermion systems, $m^*_c$ can exceed $100 \, m_e$.
• Rotating the sample maps out the full three-dimensional Fermi surface topology
• These measurements require very clean single crystals and low temperatures (often below $100 \, \text{mK}$), which is one reason crystal growth is so important in this field

Neutron scattering

Neutron scattering probes both the static magnetic structure and the dynamic spin fluctuation spectrum.

• Elastic neutron scattering detects magnetic Bragg peaks, confirming long-range magnetic order and determining the magnetic structure
• Inelastic neutron scattering measures the spin fluctuation spectrum $S(\mathbf{q}, \omega)$, revealing the energy and momentum dependence of magnetic excitations. The Kondo resonance appears as a broad feature at energy $\sim k_B T_K$.
• In the superconducting state, a spin resonance peak can appear below $T_c$ at a specific wavevector, providing evidence for sign-changing (unconventional) pairing symmetry

Theoretical models

Anderson lattice model

The Anderson lattice model is the most complete starting point for heavy fermion theory. It describes a periodic array of f-electron sites, each hybridizing with a conduction band.

The Hamiltonian includes three key ingredients:

• A conduction electron band with dispersion $\epsilon_k$
• Localized f-levels at energy $\epsilon_f$ with strong on-site Coulomb repulsion $U$ (this $U$ is what makes the problem strongly correlated)
• A hybridization matrix element $V$ coupling f-states to conduction electrons

Solving this model is extremely difficult because of the large $U$. Common approaches include large-N expansion (generalizing from SU(2) spin symmetry to SU(N) and expanding in $1/N$) and slave boson techniques (decomposing the f-electron operator into auxiliary bosonic and fermionic fields to enforce the constraint of no double occupancy).

Kondo lattice model

The Kondo lattice model is a low-energy effective model obtained from the Anderson lattice in the limit of strong $U$ and singly occupied f-levels. It keeps only the spin degree of freedom of the f-electrons.

• Each site has a localized spin $\mathbf{S}_i$ coupled to the conduction electron spin density via an antiferromagnetic exchange $J_K$
• The RKKY interaction between local moments emerges at second order in $J_K$: it's an indirect exchange mediated by conduction electrons, oscillating in sign with distance ($\sim \cos(2k_F r)/r^3$)
• The Doniach phase diagram plots the ground state as a function of $J_K N(E_F)$: small coupling gives RKKY-dominated magnetic order, large coupling gives Kondo-screened paramagnetic heavy Fermi liquid

Periodic Anderson model

The periodic Anderson model is essentially the Anderson lattice model by another name, though the term is sometimes used to emphasize specific solution methods.

• It captures both charge and spin fluctuations, unlike the Kondo lattice model which projects out charge degrees of freedom
• Dynamical mean-field theory (DMFT) has been the most successful approach for solving this model. DMFT maps the lattice problem onto a self-consistent impurity problem, becoming exact in infinite dimensions and providing a good approximation in 3D.
• DMFT and its extensions (cluster DMFT, DMFT + DFT) can describe heavy fermion metals, Kondo insulators (where the hybridization gap is complete), and mixed-valence systems (where the f-occupancy fluctuates significantly from integer values)

Superconductivity in heavy fermions

Unconventional pairing mechanisms

Superconductivity in heavy fermion compounds is almost always unconventional, meaning it's not driven by the electron-phonon interaction and the gap has nodes.

• The pairing glue is widely believed to be spin fluctuations: antiferromagnetic fluctuations near a QCP provide an attractive interaction in certain pairing channels
• The order parameter symmetry is typically non-s-wave. d-wave pairing (as in $\text{CeCoIn}_5$) and p-wave or more exotic symmetries (as proposed for $\text{UPt}_3$) are common.
• Some compounds show evidence for multi-component order parameters, meaning the superconducting state breaks additional symmetries beyond gauge symmetry (e.g., time-reversal symmetry)
• The superconducting gap has nodes (zeros) on the Fermi surface, leading to power-law (rather than exponential) temperature dependences in thermodynamic and transport properties

Coexistence with magnetism

In conventional superconductors, magnetism and superconductivity are antagonistic. In heavy fermion systems, the relationship is more nuanced.

• Microscopic coexistence of superconductivity and antiferromagnetism has been observed in compounds like $\text{UPd}_2\text{Al}_3$ and $\text{CeRhIn}_5$ under pressure
• The magnetic fluctuations that destroy long-range order near a QCP are often the same fluctuations that mediate superconducting pairing
• Whether magnetism helps or hinders superconductivity depends on the symmetry of both order parameters and the nature of the magnetic fluctuations

Critical temperature vs. pressure

Pressure is a particularly clean tuning parameter because it changes the electronic structure without introducing disorder (unlike chemical doping).

• A common pattern: antiferromagnetic order is suppressed with increasing pressure, and a superconducting dome appears centered near the pressure where magnetic order vanishes
• The maximum $T_c$ typically occurs close to the QCP, consistent with the idea that critical magnetic fluctuations enhance pairing
• Some compounds show non-monotonic $T_c(P)$ behavior, sometimes linked to valence transitions where the f-electron occupancy changes abruptly under pressure

Applications and future prospects

Thermoelectric materials

The large effective mass in heavy fermion compounds translates to a large thermopower (Seebeck coefficient), since thermopower scales with $m^*$ through the Mott formula.

• Compounds like $\text{YbAl}_3$ and $\text{CePd}_3$ show large thermoelectric power factors at low temperatures
• The challenge is that high performance currently occurs only at cryogenic temperatures, limiting practical applications
• Achieving competitive thermoelectric efficiency (high $ZT$) at room temperature remains an open problem for this materials class

Quantum computing potential

Some heavy fermion superconductors may host topological superconductivity, which could support Majorana fermions useful for fault-tolerant quantum computation.

• $\text{UPt}_3$ has been proposed as a candidate for chiral topological superconductivity, potentially hosting non-Abelian vortex excitations
• If realized, such states could serve as the basis for topological qubits that are inherently protected against local sources of decoherence
• Major challenges remain: confirming the topological nature of the superconducting state, achieving sufficient control over vortices, and operating at the required cryogenic temperatures

Challenges in heavy fermion research

• Crystal growth: Many heavy fermion compounds require high-purity single crystals for meaningful measurements, and growing these crystals is technically demanding
• Theory: No single theoretical framework handles all aspects of heavy fermion physics. DMFT works well for local correlations but struggles with non-local effects near QCPs. Diagrammatic and renormalization group methods each capture different limits.
• Complexity: Real materials involve the interplay of charge, spin, orbital, and lattice degrees of freedom simultaneously, making it difficult to isolate individual mechanisms
• New frontiers: Non-equilibrium dynamics, ultrafast spectroscopy of heavy fermion states, and the search for new heavy fermion topological phases are active directions