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arXiv:2012.09339 · Physical review. B./Physical review. B

Low-temperature T -linear resistivity due to umklapp scattering from a critical mode

Patrick A. Lee

Published Jul 19, 2021• 32 citations
arXivHTMLPDF

Abstract

We consider the transport properties of a model of fermions scattered by a critical bosonic mode. The mode is overdamped and scattering is mainly in the forward direction. Such a mode appears at the quantum critical point for an electronic nematic phase transition and in gauge theories for a U(1) spin liquid. It leads to a short fermion lifetime, violating Landau's criterion for a Fermi liquid. In spite of this, transport can be described by a Boltzmann equation. We include momentum relaxation by umklapp scattering, supplemented by weak impurity scattering. We find that above a very low temperature which scales with ${\mathrm{\ensuremath{\Delta}}}_{q}^{3}$, where ${\mathrm{\ensuremath{\Delta}}}_{q}$ is the minimum umklapp scattering vector, the resistivity is linear in $T$ with a coefficient which is independent of the amount of disorder. This picture holds until the temperature dependent part of the resistivity exceeds that due to impurity scattering. We compare the relaxation time approximation with an exact numerical solution of the Boltzmann equation. Surprisingly we find that, unlike the resistivity, the Hall coefficient strongly deviates from the relaxation time approximation and shows a strong reduction with increasing temperature. We comment on possible comparisons with experiments on high ${T}_{c}$ cuprates.

DOI
10.1103/physrevb.104.035140

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We consider the transport properties of a model of fermions scattered by a critical bosonic mode. The mode is overdamped and scattering is mainly in the forward direction. Such a mode appears at the quantum critical point for an electronic nematic phase transition and in gauge theories for a U(1) spin liquid. It leads to a short fermion lifetime, violating Landau's criterion for a Fermi liquid. In spite of this, transport can be described by a Boltzmann equation. We include momentum relaxation by umklapp scattering, supplemented by weak impurity scattering. We find that above a very low temperature which scales with \ensuremathΔq3{\mathrm{\ensuremath{\Delta}}}_{q}^{3}\ensuremathΔq3​, where \ensuremathΔq{\mathrm{\ensuremath{\Delta}}}_{q}\ensuremathΔq​ is the minimum umklapp scattering vector, the resistivity is linear in TTT with a coefficient which is independent of the amount of disorder. This picture holds until the temperature dependent part of the resistivity exceeds that due to impurity scattering. We compare the relaxation time approximation with an exact numerical solution of the Boltzmann equation. Surprisingly we find that, unlike the resistivity, the Hall coefficient strongly deviates from the relaxation time approximation and shows a strong reduction with increasing temperature. We comment on possible comparisons with experiments on high Tc{T}_{c}Tc​ cuprates.
Electrical transport properties near an electronic Ising-nematic quantum critical point in two dimensions are of both theoretical and experimental interest. In this work, we derive a kinetic equation valid in a broad regime near the quantum critical point using the memory matrix approach. The formalism is applied to study the effect of the critical fluctuations on the dc resistivity through different scattering mechanisms, including umklapp, impurity scattering, and electron-hole scattering in a compensated metal. We show that electrical transport in the quantum critical regime exhibits a rich behavior that depends sensitively on the scattering mechanism and the band structure. In the case of a single large Fermi surface, the resistivity due to umklapp scattering crosses over from $\ensuremath{\rho}\ensuremath{\sim}{T}^{2}$ at low temperature to sublinear at high temperature. The crossover temperature scales as q03{q}_{0}^{3}q03​, where q0{q}_{0}q0​ is the minimal wave vector for umklapp scattering. Impurity scattering leads to $\ensuremath{\rho}\ensuremath{-}{\ensuremath{\rho}}_{0}\ensuremath{\sim}{T}^{\ensuremath{\alpha}}$ (${\ensuremath{\rho}}_{0}$ being the residual resistivity), where $\ensuremath{\alpha}$ is either larger than 2 if there is only a single Fermi sheet present, or 4/34/34/3 in the case of multiple Fermi sheets. Finally, in a perfectly compensated metal with an equal density of electrons and holes, the low-temperature behavior depends strongly on the structure of ``cold spots'' on the Fermi surface, where the coupling between the quasiparticles and order parameter fluctuations vanishes by symmetry. In particular, for a system where cold spots are present on some (but not all) Fermi sheets, $\ensuremath{\rho}\ensuremath{\sim}{T}^{5/3}$. At higher temperatures, there is a broad crossover regime where $\ensuremath{\rho}$ either saturates or $\ensuremath{\rho}\ensuremath{\sim}T$, depending on microscopic details. We discuss these results in the context of recent quantum Monte Carlo simulations of a metallic Ising nematic critical point, and experiments in certain iron-based superconductors.
We use the Kubo response functions to calculate the electrical and thermal conductivity and Seebeck coefficient at low temperatures and frequencies in the quantum-critical region for fermions on a lattice. The theory uses scattering of the fermions with the previously derived collective fluctuations due to topological defects of the quantum XY model coupled to fermions. The microscopic model is applicable to the fluctuations of the loop-current order in cuprates as well as to a class of quasi-two-dimensional heavy-fermion and other metallic antiferromagnets, and proposed recently also for the possible loop-current order in Moiré twisted bi-layer graphene and bilayer WSe2_22​. All these metals have a linear-in-temperature electrical resistivity in the quantum-critical region of their phase diagrams, often termed ``Planckian" resistivity. The solution of the Kubo equation for transport shows that vertex renormalizations to the external fields, beside those caused by Aslamazov-Larkin (A-L) processes, are absent. A-L appears as an Umklapp scattering matrix, which gives a temperature-independent multiplicative factor for the electrical resistivity but does not affect the thermal conductivity. We also show that the mass renormalization which gives a logarithmic enhancement of the marginal Fermi-liquid specific heat does not appear in the electrical resistivity and, more remarkably, in the thermal conductivity. On the other hand the mass renormalization ∝ln⁡ωc/T\propto \ln ω_c/T∝lnωc​/T appears in the Seebeck coefficient. We also discuss in detail the conservation laws which play a crucial role in all transport properties. We calculate exactly, the numerical coefficients of the transport properties for a circular Fermi surface. The leading temperature dependences is shown to remain the same for a general Fermi surface, but it is too messy to calculate the numerical coefficient.
We study the temperature dependence of the electrical resistivity of interacting two-dimensional Fermi liquids. We perform a numerical simulation of the nonequilibrium state based on semiclassical Boltzmann transport theory. A two-dimensional system of quasiparticles on a square lattice serves as a model system. We use a single-orbital tight-binding model of the dispersion. For conceptual purposes, we choose a simple repulsive onsite interaction that leads to quasiparticle scattering and delta-potential scatterers as a model of nonmagnetic impurity scattering. Through our simulation, we demonstrate that deviations from the predictions of standard Fermi-liquid theory can arise due to the nontrivial scattering geometry of umklapp processes, in special cases even in the ultralow-temperature limit. We show through qualitative arguments how these unconventional scaling properties of the electrical resistivity, which are often interpreted as indication of a non-Fermi-liquid state, can arise due to special geometric conditions of the Fermi surface. The appearance of robust deviations from the predictions of Fermi-liquid theory within our simple model presents a viewpoint in order to interpret unconventional transport properties in electron-electron scattering dominated metallic systems.
We consider two-dimensional metals near a Pomeranchuk instability which breaks ${90}^{\ensuremath{\circ}}$ lattice rotation symmetry. Such metals realize strongly coupled non-Fermi liquids with critical fluctuations of an Ising-nematic order. At low temperatures, impurity scattering provides the dominant source of momentum relaxation and, hence, a nonzero electrical resistivity. We use the memory matrix method to compute the resistivity of this non-Fermi liquid to second order in the impurity potential, without assuming the existence of quasiparticles. Impurity scattering in the ddd-wave channel acts as a random ``field'' on the Ising-nematic order. We find contributions to the resistivity with a nearly linear temperature dependence, along with more singular terms; the most singular is the random-field contribution which diverges in the limit of zero temperature.
We revisit a phenomenological model of fermions coupled to fluctuating bosons that emerges from finite-momentum particle-particle pairs for describing the strange metal phase in the cuprates. The incoherent bosons dominate the transport properties for the resistivity and optical conductivity in the non-Fermi-liquid phase. Within the Kubo formalism, the resistivity is approximately linear in temperature with a Drude form for the optical conductivity, such that the Drude lifetime is inversely proportional to the temperature. Here, we compute the transport properties of such bosons within the memory-matrix approach that successfully captures the hydrodynamic regime. This technique emerges as the appropriate framework for describing the transport coefficients of the strange metal phase. Our analysis confirms the TTT-linear resistivity due to the umklapp scattering that we obtained for this effective model. Finally, we provide new predictions regarding the variation of the thermal conductivity with temperature and examine the validity of the Wiedemann-Franz law.
We compute the particle--hole bubble for an Ising-nematic metal when the critical Fermi surface approaches the Brillouin zone boundary for d=2d=2d=2 dimensions. We find two qualitatively distinct contributions: i)~the standard antipodal piece, which gives ΠATP(q,iΩ)∝Ω/qΠ_{\rm{ATP}}(\mathbf{q}, iΩ)\proptoΩ/qΠATP​(q,iΩ)∝Ω/q and ii)~an additional umklapp piece from electrons near the zone boundary, which gives ΠU(q,iΩ)∝ΩαΠ_{\rm{U}}(\mathbf{q}, iΩ)\propto Ω^αΠU​(q,iΩ)∝Ωα at the minimum umklapp momentum q≈Δqq\approx Δ_qq≈Δq​ with $α= 2/3 $ or 1/21/21/2 depending on the temperature TTT. At high TTT when α=1/2α= 1/2α=1/2, the minimum TTT for the activation of linear/quasi-linear in TTT resistivity, which is expected to be TU∝Δq3T_U \propto Δ_q^3TU​∝Δq3​ from z=3z=3z=3 criticality, could potentially get reduced to TU∝Δq4T_U \propto Δ_q^4TU​∝Δq4​ due to the Ω\sqrtΩΩ​ term and discuss why we find only one hyper-specific scenario where this possibility might be realized. For d=3d=3d=3 the umklapp contribution gives ΠU∼ΩΠ_{\rm{U}}\sim ΩΠU​∼Ω irrespective of TTT therefore TUT_UTU​ is not modified in this case.