Pis’ma v ZhETF, vol. 110, iss. 4, pp. 266 - 267

Fermion condensation, T -linear resistivity and Planckian limit

V. R. Shaginyan^+∗1), M. Ya. Amusia^×◦, A. Z. Msezane^∗, V. A. Stephanovich^∇, G. S. Japaridze^∗, S. A. Artamonov⁺

⁺Petersburg Nuclear Physics Institute of NRC “Kurchatov Institute”, 188300 Gatchina, Russia

^∗Clark Atlanta University, Atlanta, GA 30314, USA

^×Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel

^◦Ioffe Physical Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia

^∇Institute of Physics, Opole University, Oleska 48, 45-052 Opole, Poland

Submitted 26 June 2019

Resubmitted 14 July 2019

Accepted 14

July 2019

DOI: 10.1134/S0370274X19160100

Exotic experimentally observable properties of dif-

metals at their quantum critical region. This is because

ferent classes of strongly correlated Fermi systems are

flat bands, responsible for quantum criticality, generate

still remain largely unexplained due to the lack of uni-

transverse zero-sound mode, reminiscent of the phonon

versal underlying physical mechanism. It is customary

mode in solids, with the Debye temperature T_D [3]. At

to attribute these properties to so-called non-Fermi-

T ≥ T_D the mechanism of the linear temperature de-

liquid (NFL) behavior. Latter behavior is widely ob-

pendence of the resistivity is the same both in conven-

served in heavy-fermion (HF) metals, graphene, and

tional metals and strongly correlated ones, and is rep-

high-T_c superconductors (HTSC). Experimental data

resented by electron-phonon scattering. Therefore, it is

collected on many of these systems show that at T = 0

electron-phonon scattering at T ≥ T_D, which yields the

a portion of their excitation spectrum becomes disper-

near material-independence of the lifetime τ. It is ex-

sionless, giving rise to so-called flat bands, see, e.g., [1-

pressed as 1/(τT ) ∼ k_B /ℏ. The observed scattering rate

4]. The presence of flat band indicates that the system

is well explained by the emergence of flat bands formed

is close to the topological fermion-condensation quan-

by the topological FQCPT, rather than by the so called

tum phase transition (FCQPT) [1, 2, 4], leading to flat

Planckian limit at which the assumed Planckian scatter-

bands (ε(k) = µ, where ε(k) is quasiparticle energy and

ing rate takes place. At low temperatures, the observed

µ is a chemical potential) formation. Recent challenging

resistivity in HTSC and HF metals obeys linear law (so-

experimental findings of linear temperature T depen-

called linear T-resistivity)

dence of the resistivity ρ(T ) ∝ T collected on HTSC,

ρ(T ) = ρ₀ + AT.

(1)

graphene, HF and conventional metals, have revealed

that the scattering rate 1/τ of charge carriers reaches

Here ρ₀

is the residual resistivity and A is a T -

the so-called universal Planckian limit 1/(T τ) = k_B /ℏ

independent coefficient. On the other hand, at room

(k_B and ℏ = h/2π are the Boltzmann and Plank con-

temperature the T -linear resistivity is exhibited by con-

stants, respectively) [5-8]. Note that above Planckian

ventional metals such as Al, Ag or Cu. In case of a simple

limit, used to explain the universal scattering rate in

metal with a single Fermi surface pocket the resistivity

the so-called Planckian metals [5-8], can occur acci-

reads e²nρ = p_F /(τv_F ), where e is the electronic charge,

dentally since its experimental manifestations in other

τ is the lifetime, n is the carrier concentration, and p_F

(than Planckian) metals may be equally well explained

and v_F are the Fermi momentum and velocity respec-

by more conventional physical mechanisms like those

tively. Representing the lifetime τ (inverse scattering

related to phonon contribution [3].

rate) of quasiparticles in the form

Within the framework of the fermion condensation

(FC) theory, we show that the quasi-classical physics

ℏ

k_BT

≃a₁ +

(2)

is still applicable to describe the universal scattering

a₂

rate 1/τ, experimentally observed in strongly correlated

we obtain [3]

e²nℏ

∂ρ

a₂

(3)

¹⁾e-mail: vrshag@thd.pnpi.spb.ru

p_F k_B ∂T

v_F

266

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2019

Fermion condensation, T -linear resistivity and Planckian limit

267

where a₁ and a₂ are T -independent parameters. A chal-

dependence of the resistivity. On the other hand, it has

lenging point for a theory is that experimental data

been shown that the quasi-classical physics describes

confirm Eq.(3) for both strongly correlated metals (HF

the T -linear dependence of the resistivity of strongly

metals and HTSC) and ordinary ones, provided that

correlated metals at T > T_D, since flat bands, forming

these demonstrate the linear T -dependence of their re-

the quantum criticality, generate transverse zero-sound

sistivity [5], see Fig. 1.

mode with the Debye temperature T_D located within

the quantum criticality area [3, 9]. Therefore, the T -

linear dependence is formed by electron-phonon scat-

tering in both ordinary metals and strongly correlated

ones. As a result, it is electron-phonon scattering that

leads to the near material-independence of the lifetime

τ that is expressed as

ℏ

τT ∼

(4)

We stress, that the Planckian limit may occur acciden-

tally: it is highly improbable that it would be realized

in conventional metals, which, obviously, cannot be rec-

ognized as Planckian ones with quantum criticality at

high or low temperatures. The fact, that we observe the

same universal behavior of the scattering rate in micro-

scopically different strongly correlated compounds like

HTSC, HF and conventional metals, suggests that some

general theory is needed to explain the above body of

materials and their behavior in the uniform manner. We

Fig. 1. (Color online) Scattering rates per kelvin of dif-

may conclude that the FC theory is a suitable candidate.

ferent strongly correlated metals like HF ones, HTSC, or-

We thank V. A. Khodel for stimulating discussions.

ganic and conventional metals [5]. All these metals exhibit

This work was partly supported by U.S. DOE

ρ(T) ∝ T and demonstrate two orders of magnitude vari-

(United States Department of Energy), Division of

ations in their Fermi velocities vF . The parameter a2 ≃ 1

Chemical Sciences, Office of Basic Energy Sciences, Of-

gives the best fit shown by the solid line, and corresponds

fice of Energy Research.

to the scattering rate τT = h/(2πkB ), with h = 2πℏ,

Full text of the paper is published in JETP Letters

see Eqs. (3) and (4). The region occupied by the conven-

tional metals is highlighted by two (blue) arrows. The sin-

journal. DOI: 10.1134/S002136401916001X

gle (green) arrow shows the region of strongly correlated

metals, including organic ones. Note, that at low temper-

1. V. A. Khodel and V. R. Shaginyan, JETP Lett. 51, 553

atures T ≪ TD , the scattering rate per kelvin of a conven-

(1990).

tional metal is orders of magnitude lower, and does not

2. G. E. Volovik, JETP Lett. 53, 222 (1991).

correspond to the Planckian limit

3. V. R. Shaginyan, K. G. Popov, and V. A. Khodel, Phys.

Rev. B 88, 115103 (2013).

The coefficient a₂ is always close to unity, 0.7 ≤ a₂ ≤

4. Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi,

≤ 2.7, notwithstanding huge distinction in the absolute

E. Kaxiras, and P. Jarillo-Herrero, Nature 556,

value of ρ, T and Fermi velocities v_F , varying by two or-

(2018).

ders of magnitude [5]. As a result, it follows from Eq. (2)

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1/(τT) ∼ k_B/ℏ.

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As it is seen from Fig. 1, this scaling relation spans

Nat. Phys. 15, 142 (2019).

two orders of magnitude in v_F , attesting to the robust-

7. Y. Cao, D. Chowdhury, D. Rodan-Legrain, O. Rubies-

Bigordá, K. Watanabe, T. Taniguchi, T. Senthil, and

ness of the observed empirical law [5]. This behavior is

P. Jarillo-Herrero, arXiv:1901.03710.

explained within the framework of the FC theory, since

8. Y. Nakajima, T. Metz, C. Eckberg et al. (Collabora-

for both conventional metals and strongly correlated

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ones the scattering rate is defined by phonons [3, 9]. In

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case of conventional metals at T > T_D it is well known

M. V. Zverev, JETP Lett. 92, 532 (2010).

that phonons make the main contribution to the linear

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2019