arXiv:2107.07802

Planckian Dissipation in Metals

Sean A. Hartnoll, Andrew Peter Mackenzie

Published Jul 16, 2021• 17 citations

Abstract

We review the appearance of the Planckian time $τ_\text{Pl} = \hbar/(k_B T)$ in both conventional and unconventional metals. We give a pedagogical discussion of the various different timescales (quasiparticle, transport, many-body) that characterize metals, emphasizing conditions under which these times are the same or different. Throughout, we have attempted to clear up aspects of the problem that had been confusing us, in the hope that this helps the reader as well. We discuss the possibility of a Planckian bound on dissipation from both a quasiparticle and a many-body perspective. Planckian quasiparticles can arise naturally from a combination of inelastic scattering and mass renormalization. Many-body dynamics, on the other hand, is constrained by the basic time- and length- scales of local thermalization.

DOI: 10.48550/arxiv.2107.07802

Related papers

Colloquium : Planckian dissipation in metals

Sean A. Hartnoll, Andrew Peter Mackenzie

Reviews of Modern PhysicsNov 30, 2022158 citations

Theory of a Planckian Metal

Aavishkar A. Patel, Subir Sachdev A

arXiv:1906.03265Physical Review LettersAug 7, 2019137 citations

A stability bound on the $T$-linear resistivity of conventional metals

Chaitanya Murthy, Akshat Pandey, Ilya Esterlis +1 more

arXiv:2112.06966Dec 13, 20210 citations

A stability bound on the T-linear resistivity of conventional metals

Chaitanya Murthy, Akshat Pandey, Ilya Esterlis +1 more

Proceedings of the National Academy of SciencesJan 12, 20230 citations

Planckian Bounds via Spectral Moments of Optical Conductivity

Debanjan Chowdhury

arXiv:2512.09979Dec 10, 20250 citations

Planckian bound on the local equilibration time

Marvin Qi, Alexey Milekhin, Luca V. Delacrétaz

arXiv:2602.17638Feb 19, 20260 citations

arXiv:2107.07802

Planckian Dissipation in Metals

Sean A. Hartnoll, Andrew Peter Mackenzie

Published Jul 16, 2021• 17 citations

arXiv HTML PDF

Abstract

We review the appearance of the Planckian time

τ_\text{Pl} = \hbar/(k_B T)

in both conventional and unconventional metals. We give a pedagogical discussion of the various different timescales (quasiparticle, transport, many-body) that characterize metals, emphasizing conditions under which these times are the same or different. Throughout, we have attempted to clear up aspects of the problem that had been confusing us, in the hope that this helps the reader as well. We discuss the possibility of a Planckian bound on dissipation from both a quasiparticle and a many-body perspective. Planckian quasiparticles can arise naturally from a combination of inelastic scattering and mass renormalization. Many-body dynamics, on the other hand, is constrained by the basic time- and length- scales of local thermalization.

DOI: 10.48550/arxiv.2107.07802

Related papers

Colloquium : Planckian dissipation in metals

Sean A. Hartnoll, Andrew Peter Mackenzie

Relaxational processes for conduction electrons in metals come with characteristic temperature dependences; classic examples at low temperatures are the $1/{T}^{2}$ law for the relaxation time in a Fermi liquid, and the Bloch $1/{T}^{5}$ law for electron-phonon scattering. The strange metal phase of cuprate superconductors is governed instead by the Planckian time, which is given in terms of fundamental constants only. This Colloquium discusses the concept of such a fundamental timescale and the degree to which it provides a lower bound for relaxation times, and illustrates these ideas with examples from conventional and unconventional metals.Relaxational processes for conduction electrons in metals come with characteristic temperature dependences; classic examples at low temperatures are the $1/{T}^{2}$ law for the relaxation time in a Fermi liquid, and the Bloch $1/{T}^{5}$ law for electron-phonon scattering. The strange metal phase of cuprate superconductors is governed instead by the Planckian time, which is given in terms of fundamental constants only. This Colloquium discusses the concept of such a fundamental timescale and the degree to which it provides a lower bound for relaxation times, and illustrates these ideas with examples from conventional and unconventional metals.

Reviews of Modern PhysicsNov 30, 2022158 citations

Theory of a Planckian Metal

Aavishkar A. Patel, Subir Sachdev A

We present a lattice model of fermions with N flavors and random interactions that describes a Planckian metal at low temperatures T→0 in the solvable limit of large N. We begin with quasiparticles around a Fermi surface with effective mass m^{*} and then include random interactions that lead to fermion spectral functions with frequency scaling with k_{B}T/ℏ. The resistivity ρ obeys the Drude formula ρ=m^{*}/(ne^{2}τ_{tr}), where n is the density of fermions, and the transport scattering rate is 1/τ_{tr}=fk_{B}T/ℏ; we find f of order unity and essentially independent of the strength and form of the interactions. The random interactions are a generalization of the Sachdev-Ye-Kitaev models; it is assumed that processes nonresonant in the bare quasiparticle energies only renormalize m^{*}, while resonant processes are shown to produce the Planckian behavior.We present a lattice model of fermions with N flavors and random interactions that describes a Planckian metal at low temperatures T→0 in the solvable limit of large N. We begin with quasiparticles around a Fermi surface with effective mass m^{*} and then include random interactions that lead to fermion spectral functions with frequency scaling with k_{B}T/ℏ. The resistivity ρ obeys the Drude formula ρ=m^{*}/(ne^{2}τ_{tr}), where n is the density of fermions, and the transport scattering rate is 1/τ_{tr}=fk_{B}T/ℏ; we find f of order unity and essentially independent of the strength and form of the interactions. The random interactions are a generalization of the Sachdev-Ye-Kitaev models; it is assumed that processes nonresonant in the bare quasiparticle energies only renormalize m^{*}, while resonant processes are shown to produce the Planckian behavior.

arXiv:1906.03265Physical Review LettersAug 7, 2019137 citations

A stability bound on the $T$ -linear resistivity of conventional metalsA stability bound on the $T$-linear resistivity of conventional metals

Chaitanya Murthy, Akshat Pandey, Ilya Esterlis +1 more

Perturbative considerations account for the properties of conventional metals, including the range of temperatures where the transport scattering rate is $1/τ_\text{tr} = 2πλT$ , where $λ$ is a dimensionless strength of the electron-phonon coupling. The fact that measured values satisfy $λ\lesssim 1$ has been noted in the context of a possible "Planckian" bound on transport. However, since the electron-phonon scattering is quasi-elastic in this regime, no such Planckian considerations can be relevant. We present and analyze Monte Carlo results on the Holstein model which show that a different sort of bound is at play: a "stability" bound on $λ$ consistent with metallic transport. We conjecture that a qualitatively similar bound on the strength of residual interactions, which is often stronger than Planckian, may apply to metals more generally.Perturbative considerations account for the properties of conventional metals, including the range of temperatures where the transport scattering rate is $1/τ_\text{tr} = 2πλT$, where $λ$ is a dimensionless strength of the electron-phonon coupling. The fact that measured values satisfy $λ\lesssim 1$ has been noted in the context of a possible "Planckian" bound on transport. However, since the electron-phonon scattering is quasi-elastic in this regime, no such Planckian considerations can be relevant. We present and analyze Monte Carlo results on the Holstein model which show that a different sort of bound is at play: a "stability" bound on $λ$ consistent with metallic transport. We conjecture that a qualitatively similar bound on the strength of residual interactions, which is often stronger than Planckian, may apply to metals more generally.

arXiv:2112.06966Dec 13, 20210 citations

A stability bound on the T-linear resistivity of conventional metals

Chaitanya Murthy, Akshat Pandey, Ilya Esterlis +1 more

Perturbative considerations account for the properties of conventional metals, including the range of temperatures where the transport scattering rate is 1/ τ tr = 2 π λ T , where λ is a dimensionless strength of the electron–phonon coupling. The fact that measured values satisfy λ ≲ 1 has been noted in the context of a possible “Planckian” bound on transport. However, since the electron–phonon scattering is quasielastic in this regime, no such Planckian considerations can be relevant. We present and analyze Monte Carlo results on the Holstein model which show that a different sort of bound is at play: a “stability” bound on λ consistent with metallic transport. We conjecture that a qualitatively similar bound on the strength of residual interactions, which is often stronger than Planckian, may apply to metals more generally.Perturbative considerations account for the properties of conventional metals, including the range of temperatures where the transport scattering rate is 1/ τ tr = 2 π λ T , where λ is a dimensionless strength of the electron–phonon coupling. The fact that measured values satisfy λ ≲ 1 has been noted in the context of a possible “Planckian” bound on transport. However, since the electron–phonon scattering is quasielastic in this regime, no such Planckian considerations can be relevant. We present and analyze Monte Carlo results on the Holstein model which show that a different sort of bound is at play: a “stability” bound on λ consistent with metallic transport. We conjecture that a qualitatively similar bound on the strength of residual interactions, which is often stronger than Planckian, may apply to metals more generally.

Proceedings of the National Academy of SciencesJan 12, 20230 citations

Planckian Bounds via Spectral Moments of Optical Conductivity

Debanjan Chowdhury

The observation of Planckian scattering, often inferred from Drude fits in strongly correlated metals, raises the question of how to extract an intrinsic timescale from measurable quantities in a model-independent way. We address this by focusing on a ratio ( ${\cal{B}}$ ) of spectral moments of the dissipative part of the optical conductivity and prove a rigorous upper bound on ${\cal{B}}$ in terms of the Planckian rate. The bound emerges from the analytic structure of thermally weighted response functions of the current operator. Crucially, the bounded quantity is directly accessible via optical spectroscopy and computable from imaginary-time correlators in quantum Monte Carlo simulations, without any need for analytic continuation. We evaluate ${\cal{B}}$ for simplified examples of both Drude and non-Drude forms of the optical conductivity with a single scattering rate in various asymptotic regimes, and find that ${\cal{B}}$ lies far below the saturation value. These findings demonstrate that Planckian bounds can arise from fundamental constraints on equilibrium dynamics, pointing toward a possibly universal structure governing transport in correlated quantum matter.The observation of Planckian scattering, often inferred from Drude fits in strongly correlated metals, raises the question of how to extract an intrinsic timescale from measurable quantities in a model-independent way. We address this by focusing on a ratio (${\cal{B}}$) of spectral moments of the dissipative part of the optical conductivity and prove a rigorous upper bound on ${\cal{B}}$ in terms of the Planckian rate. The bound emerges from the analytic structure of thermally weighted response functions of the current operator. Crucially, the bounded quantity is directly accessible via optical spectroscopy and computable from imaginary-time correlators in quantum Monte Carlo simulations, without any need for analytic continuation. We evaluate ${\cal{B}}$ for simplified examples of both Drude and non-Drude forms of the optical conductivity with a single scattering rate in various asymptotic regimes, and find that ${\cal{B}}$ lies far below the saturation value. These findings demonstrate that Planckian bounds can arise from fundamental constraints on equilibrium dynamics, pointing toward a possibly universal structure governing transport in correlated quantum matter.

arXiv:2512.09979Dec 10, 20250 citations

Planckian bound on the local equilibration time

Marvin Qi, Alexey Milekhin, Luca V. Delacrétaz

The local equilibration time $τ_{\rm eq}$ of quantum many-body systems is conjectured to be bounded below by the Planckian time $\hbar /T$ . We formalize this conjecture by defining $τ_{\rm eq}$ as the time scale at which a hydrodynamic description emerges for conserved densities. Drawing on analytic properties of real time thermal correlators, we establish a rigorous lower bound $τ_{\rm eq} \geq α\hbar /T$ on the onset of hydrodynamic behavior in a `regulated' thermal two-point function. The dimensionless coefficient $α$ depends only on dimensionality and the type of hydrodynamic or diffusive behavior that emerges, and is independent of the thermalization mechanism or other microscopic details. This bound applies universally to local quantum many-body systems, with or without a quasiparticle description, including in the presence of inelastic scattering.The local equilibration time $τ_{\rm eq}$ of quantum many-body systems is conjectured to be bounded below by the Planckian time $\hbar /T$. We formalize this conjecture by defining $τ_{\rm eq}$ as the time scale at which a hydrodynamic description emerges for conserved densities. Drawing on analytic properties of real time thermal correlators, we establish a rigorous lower bound $τ_{\rm eq} \geq α\hbar /T$ on the onset of hydrodynamic behavior in a `regulated' thermal two-point function. The dimensionless coefficient $α$ depends only on dimensionality and the type of hydrodynamic or diffusive behavior that emerges, and is independent of the thermalization mechanism or other microscopic details. This bound applies universally to local quantum many-body systems, with or without a quasiparticle description, including in the presence of inelastic scattering.

arXiv:2602.17638Feb 19, 20260 citations

We review the appearance of the Planckian time

τ_\text{Pl} = \hbar/(k_B T)

Relaxational processes for conduction electrons in metals come with characteristic temperature dependences; classic examples at low temperatures are the

1/{T}^{2}

law for the relaxation time in a Fermi liquid, and the Bloch

1/{T}^{5}

law for electron-phonon scattering. The strange metal phase of cuprate superconductors is governed instead by the Planckian time, which is given in terms of fundamental constants only. This Colloquium discusses the concept of such a fundamental timescale and the degree to which it provides a lower bound for relaxation times, and illustrates these ideas with examples from conventional and unconventional metals.

A stability bound on the

T

-linear resistivity of conventional metals

Perturbative considerations account for the properties of conventional metals, including the range of temperatures where the transport scattering rate is

1/τ_\text{tr} = 2πλT

, where

λ

is a dimensionless strength of the electron-phonon coupling. The fact that measured values satisfy

λ\lesssim 1

has been noted in the context of a possible "Planckian" bound on transport. However, since the electron-phonon scattering is quasi-elastic in this regime, no such Planckian considerations can be relevant. We present and analyze Monte Carlo results on the Holstein model which show that a different sort of bound is at play: a "stability" bound on

λ

consistent with metallic transport. We conjecture that a qualitatively similar bound on the strength of residual interactions, which is often stronger than Planckian, may apply to metals more generally.

{\cal{B}}

) of spectral moments of the dissipative part of the optical conductivity and prove a rigorous upper bound on

{\cal{B}}

in terms of the Planckian rate. The bound emerges from the analytic structure of thermally weighted response functions of the current operator. Crucially, the bounded quantity is directly accessible via optical spectroscopy and computable from imaginary-time correlators in quantum Monte Carlo simulations, without any need for analytic continuation. We evaluate

{\cal{B}}

for simplified examples of both Drude and non-Drude forms of the optical conductivity with a single scattering rate in various asymptotic regimes, and find that

{\cal{B}}

lies far below the saturation value. These findings demonstrate that Planckian bounds can arise from fundamental constraints on equilibrium dynamics, pointing toward a possibly universal structure governing transport in correlated quantum matter.

The local equilibration time

τ_{\rm eq}

of quantum many-body systems is conjectured to be bounded below by the Planckian time

\hbar /T

. We formalize this conjecture by defining

τ_{\rm eq}

as the time scale at which a hydrodynamic description emerges for conserved densities. Drawing on analytic properties of real time thermal correlators, we establish a rigorous lower bound

τ_{\rm eq} \geq α\hbar /T

on the onset of hydrodynamic behavior in a `regulated' thermal two-point function. The dimensionless coefficient

α

depends only on dimensionality and the type of hydrodynamic or diffusive behavior that emerges, and is independent of the thermalization mechanism or other microscopic details. This bound applies universally to local quantum many-body systems, with or without a quasiparticle description, including in the presence of inelastic scattering.