The kinetic theory of ultra-subsonic fermion systems and applications to flat band magic angle twisted bilayer graphene

The kinetic theory of ultra-subsonic fermion systems and applications to flat band magic angle twisted bilayer graphene explores how phonon scattering events in subsonic fermions lead to unique low-temperature transport physics, offering a new explanation for strange metal behavior observed in flat band systems like magic angle twisted bilayer graphene.

The Gist

The theory predicts a linear-in-T resistivity down to a temperature much lower than the Bloch-Grüneisen paradigm of supersonic fermion bands would suggest the kinetic theory of ultra-subsonic fermion systems and applications to flat band magic angle twisted bilayer graphene explores how phonon scattering events in subsonic fermions lead to unique low-temperature transport physics, offering a new explanation for strange metal behavior observed in flat band systems like magic angle twisted bilayer graphene.

Kinematic Constraints and Scattering Mechanisms

The only kinematically-allowed phonon-scattering events for bands of subsonic fermions (vF < vp) are interband transitions, leading to different low-T transport physics than in the typical supersonic case. This kinematic requirement means that all single-phonon scattering processes must be interband transitions. In a multiband subsonic fermion system, this implies that phonon scattering is necessarily interband. The geometry of isolated NFBs can manifest a mid-T ρ ∝ T2 power law.

Transport Regimes Predicted by Kinetic Theory

The kinetic theory predicts several distinct transport regimes based on temperature and doping level. These predicted regimes include:

1. A linear-in-T resistivity over a wide range of T, down to a temperature much lower than the BG paradigm of supersonic fermion bands would suggest.

2. Regimes of perfect conductivity that mimic the behavior of superconductivity.

3. A mid-T ρ ∝ T2 power law, which is generated entirely by phonon scattering in NFB systems.

The Ultrasubsonic (USS) Limit and Relaxation Times

The theory develops a transport theory for an “ultrasubsonic” (USS) two-band fermion system, which is accurate in the asymptotic double-limit of small vF /vp and small band separation. The interband nature of subsonic fermion scattering manifests an exponentially divergent relaxation time at low temperatures, in contrast with the familiar τ ∝ T−4 Bloch-Grüneisen (BG) power law applicable to supersonic fermions. This divergence in relaxation time is capable of perfectly balancing the thermodynamic suppression of states away from the Fermi level, leading to a non-intuitive physical picture in which states far from the Fermi level contribute meaningfully to transport.

Extreme Low-T Behavior and Crossover Temperature

In the extreme low-T limit, where all the sinh and cosh functions blow up exponentially, we find a crossover to perfect conductivity with exponentially suppressed scattering when kBT2µ < 1. The crossover temperature is parametrically suppressed by the small parameter, vF /vp the subsonic case is thus expected to host linear-in-T resistivity scaling down to a significantly lower temperature than one would estimate based on intuition from supersonic fermion scattering.

Application to Magic Angle Twisted Bilayer Graphene (MATBLG)

The theory provides an alternative theoretical explanation for low-T “strange metal” resistivity scaling in NFB systems. The results suggest that low-T ρ ∝ T, ρ ∝ T2, and perfect conductivity regimes may all arise in NFB systems from the same universal phonon scattering physics. The theory is particularly germane to the ongoing debate on the presence of a strange metal state in MATBLG.

Quantitative Results for the USS Dirac Cone

Applying the theory to the ultrasubsonic Dirac cone yields a concrete formula for resistivity, showing robust linear-in-T resistivity partially generated by states far from the Fermi level. Specializing to the µ = 0 case, we find purely linear-in-T phononinduced resistivity down to T = 0. Away from µ = 0 but in the extreme low-T limit, the low-T resistivity is proportional to ρ ∝ exp[-2µ/(kBT)], and is exponentially suppressed when kBT2µ, giving a crossover to perfect conductivity. The prefactor contains nontrivial power-law dependencies on the velocities vF and vp that could prove useful for experimental probes of USS transport physics in systems for which the Dirac cone is a useful model.

Applicability to NFB Systems

The theory is general and could apply widely to NFB systems in other 2D heterostructures or heavy fermion materials. The analysis shows that much of the band is made up of subsonic states, with energy spacing below 0.0005 eV and vF /vp < 0.01 for some states. While a direct quantitative application to the BM model might be limited due to possible band renormalizations, it is possible that large portions of the NFB are well-captured by the ultrasubsonic limit and exhibit the related low-T transport phenomena. The coexistence of subsonic and supersonic states in the band would need to be taken into account to correctly apply USS scattering theory to the BM band structure in a quantitative way.

Microscopic Model Details

The microscopic model describes the system using the Hamiltonian H = He + Hph + He−ph. The model is described by the Hamiltonian Hph = X l,q ωqa† l,q al,q, and He−ph = s D2 2ρMA X l,q nˆq,l √ωq (−iq · eˆq)(aq,l + a† −q,l). The standard values of the above parameters for monolayer graphene are D = 25 eV, ρM = 7.6·10−8 g/cm2, and vp = 2.0·106 cm/s.

Boltzmann Kinetic Theory Framework

The resistivity of the system follows from the relaxation times via δ ij ρ = 1 / 4 e2 kBT / A X S τSv i S v j S cosh ε−µ2kBT2. The relaxation times are determined from a self-consistent integral equation derived from the Boltzmann equation. The function F µ,T (S, S0) encodes all the thermodynamics, including the occupation data for the fermions and phonons and all dependence on µ and T.

Ultrasubsonic Transport Theory Derivation

The derivation of the master formula for resistivity involves solving a 2x2 matrix equation determining both τ (S) and τ (S¯) for the USS relaxation times. Combining this with the resistivity formula gives the master formula for the resistivity of the ultra-subsonic fermion system. The PH-symmetric resistivity master formula is given by Eq. (7) in the main text.

Hierarchy of Approximations

The hierarchy of approximations collectively referred to as “ultra subsonic” (USS) includes:

1. “USS 1”: τ (S0) ≈ τ (S¯) along the scattering manifold.

2. “USS 2”: In addition to USS 1, we further assume that F µ,T S,S0 ≈ F µ,T S,S¯ along the scattering manifold.

3. “USS 3”: In addition to USS 1 and USS 2, we also approximate the scattering manifold as a small circle around the point S¯.

Results in the PH-Symmetric Case

In the case of a particle-hole (PH) symmetric system, the formulae for the USS relaxation time simplifies to Eq.

1. The improved AI system can accurately predict low-temperature resistivity scaling in magic angle twisted bilayer graphene (MATBLG) by applying ultrasubsonic (USS) transport theory to its noninteracting band structure, as described in the text: This result provides a concrete mechanism for the linear-in-T resistivity over a wide range of temperatures based only on familiar concepts of solid state physics and the distinct kinematics of NFB systems.

2. The system can distinguish between different low-temperature transport regimes by analyzing the temperature dependence derived from Eq. (10), which shows at the charge neutrality point (µ = 0), we have purely linear-in-T phononinduced resistivity down to T = 0 and a crossover to perfect conductivity when kBT 2µ, compared to supersonic fermion predictions.

3. The AI can quantify the suppression of the linear-in-T scaling by the subsonic parameter, predicting that The crossover temperature is parametrically suppressed by the small parameter, vF /vp, which offers a new explanation for low-T “strange metal” behavior in flat band systems.

The only kinematically-allowed phonon-scattering events for bands of subsonic fermions (v F < v p) are interband transitions, leading to different low- T transport physics than in the typical supersonic case. We apply a kinetic theory of phonon-limited transport to a generic two-band system of subsonic fermions, deriving formulae for relaxation times and resistivity that are accurate in the limit of close bands and small v F/v p. We predict regimes of ρ proportional to T, ρ proportional to T squared, and perfect conductivity. Our theory predicts linear-in- T resistivity down to a crossover temperature that is suppressed from its supersonic analogue by a factor of v F/v p, offering a new explanation for low- T "strange metal" behavior observed in flat band systems.