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arXiv:2308.11057 · Physical review. B./Physical review. B

Nonanalytic corrections to the Landau diamagnetic susceptibility in a two-dimensional Fermi liquid

R. David Mayrhofer, Andrey V. Chubukov

Published Dec 4, 2023• 2 citations
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Abstract

We analyze potential nonanalytic terms in the Landau diamagnetic susceptibility, ${\ensuremath{\chi}}_{\text{dia}}$, at a finite temperature $T$ and/or in-plane magnetic field $H$ in a two-dimensional (2D) Fermi liquid. To do this, we express the diamagnetic susceptibility as ${\ensuremath{\chi}}_{\text{dia}}={(e/c)}^{2}{lim}_{Q\ensuremath{\rightarrow}0}{\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}(Q)/{Q}^{2}$, where ${\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}$ is the transverse component of the static current-current correlator, and evaluate ${\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}(Q)$ for a system of fermions with Hubbard interaction to second order in Hubbard $U$ by combining self-energy, Maki-Thompson, and Aslamazov-Larkin diagrams. We find that at $T=H=0$, the expansion of ${\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}(Q)/{Q}^{2}$ in $U$ is regular, but at a finite $T$ and/or $H$, it contains ${U}^{2}T$ and/or ${U}^{2}|H|$ terms. Similar terms have been previously found for the paramagnetic Pauli susceptibility. We obtain the full expression for the nonanalytic $\ensuremath{\delta}{\ensuremath{\chi}}_{\text{dia}}(H,T)$ when both $T$ and $H$ are finite, and we show that the $H/T$ dependence is similar to that for the Pauli susceptibility.

DOI
10.1103/physrevb.108.235108

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We analyze potential nonanalytic terms in the Landau diamagnetic susceptibility, \ensuremathχdia{\ensuremath{\chi}}_{\text{dia}}\ensuremathχdia​, at a finite temperature TTT and/or in-plane magnetic field HHH in a two-dimensional (2D) Fermi liquid. To do this, we express the diamagnetic susceptibility as \ensuremathχdia=(e/c)2limQ\ensuremath→0\ensuremathΠ\ensuremath⊥JJ(Q)/Q2{\ensuremath{\chi}}_{\text{dia}}={(e/c)}^{2}{lim}_{Q\ensuremath{\rightarrow}0}{\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}(Q)/{Q}^{2}\ensuremathχdia​=(e/c)2limQ\ensuremath→0​\ensuremathΠ\ensuremath⊥JJ​(Q)/Q2, where \ensuremathΠ\ensuremath⊥JJ{\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}\ensuremathΠ\ensuremath⊥JJ​ is the transverse component of the static current-current correlator, and evaluate \ensuremathΠ\ensuremath⊥JJ(Q){\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}(Q)\ensuremathΠ\ensuremath⊥JJ​(Q) for a system of fermions with Hubbard interaction to second order in Hubbard UUU by combining self-energy, Maki-Thompson, and Aslamazov-Larkin diagrams. We find that at T=H=0T=H=0T=H=0, the expansion of \ensuremathΠ\ensuremath⊥JJ(Q)/Q2{\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}(Q)/{Q}^{2}\ensuremathΠ\ensuremath⊥JJ​(Q)/Q2 in UUU is regular, but at a finite TTT and/or HHH, it contains U2T{U}^{2}TU2T and/or U2∣H∣{U}^{2}|H|U2∣H∣ terms. Similar terms have been previously found for the paramagnetic Pauli susceptibility. We obtain the full expression for the nonanalytic \ensuremathδ\ensuremathχdia(H,T)\ensuremath{\delta}{\ensuremath{\chi}}_{\text{dia}}(H,T)\ensuremathδ\ensuremathχdia​(H,T) when both TTT and HHH are finite, and we show that the H/TH/TH/T dependence is similar to that for the Pauli susceptibility.
We calculate the basic parameters of the Fermi liquid: the scattering vertex, the Landau interaction function, the effective mass, and physical susceptibilities for a model of two-dimensional (2D) fermions with a short-ranged interaction at nonzero temperature. The leading temperature dependences of the spin components of the scattering vertex, the Landau function, and the spin susceptibility are found to be linear. T-linear terms in the effective mass and in the "charge-sector" quantities are found to cancel to second order in the interaction, but the cancellation is argued not to be generic. The connection with previous studies of the 2D Fermi-liquid parameters is discussed.
We calculate, using perturbative calculations and Ward identities, the basic parameters of the Fermi liquid: the scattering vertex, the Landau interaction function, the effective mass, specific heat, and physical susceptibilities for a model of two-dimensional (2D) fermions with a short-ranged interaction at nonzero temperature. The leading temperature dependence of the spin components of the scattering vertex, the Landau function, and the spin susceptibility is found to be linear. We find that the standard T=0T=0T=0 relationships for a Galilean-invariant Fermi liquid are violated by finite-temperature terms. The coefficients in the temperature corrections to these relationships involve a subtle interplay between contributions from small and large $(\ensuremath{\sim}{2k}_{F})$ momentum processes. A connection with previous studies of the 2D Fermi-liquid parameters is discussed. We conclude that the linear leading temperature dependence of the parameters is a generic feature of the 2D Fermi liquid.
We show that the singularities in the dynamical bosonic response functions of a two-dimensional Fermi liquid give rise to universal nonanalytic corrections to the results of the Fermi-liquid theory. In particular, we find a T2{T}^{2}T2 term in the specific heat, linear-in-TTT terms in the effective mass and in the uniform spin susceptibility ${\ensuremath{\chi}}_{s}(Q=0,T),$ and ∣Q∣|Q|∣Q∣ term in ${\ensuremath{\chi}}_{s}(Q,T=0).$ The existence of these terms has been the subject of recent controversy, which is resolved in this paper. We present exact expressions for all nonanalytic terms to second order in a generic interaction U(Q)U(Q)U(Q) and show that the nonanalytic terms originate exclusively from forward- and backward-scattering of particles with zero total momentum.
We extend previous calculations of the nonanalytic terms in the spin susceptibility ${\ensuremath{\chi}}_{s}(T)$ and the specific heat C(T)C(T)C(T) to systems in a magnetic field. Without a field, ${\ensuremath{\chi}}_{s}(T)$ and C(T)∕TC(T)∕TC(T)∕T are linear in TTT in two dimensions (2D), while in 3D, ${\ensuremath{\chi}}_{s}(T)\ensuremath{\propto}{T}^{2}$ and $C(T)∕T\ensuremath{\propto}{T}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}T$. We show that in a magnetic field, the linear in TTT terms in 2D become scaling functions of ${\ensuremath{\mu}}_{B}H∕T$. We present explicit expressions for these functions and show that at high fields ${\ensuremath{\mu}}_{B}H⪢T$, ${\ensuremath{\chi}}_{s}(T,H)$ scales as $\ensuremath{\mid}H\ensuremath{\mid}$. We also show that in 3D, ${\ensuremath{\chi}}_{s}(T,H)$ becomes nonanalytic in a field and at high fields scales as ${H}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\ensuremath{\mid}H\ensuremath{\mid}$.
We present the expression for the quasiparticle vertex function ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ (proportional to the Landau interaction function) in a 2D Fermi liquid (FL) near an instability towards antiferromagnetism. This function is relevant in many ways in the context of metallic quantum criticality. Previous studies have found that near a quantum critical point, the system enters into a regime in which the fermionic self-energy is large near hot spots on the Fermi surface [points on the Fermi surface connected by the antiferromagnetic ordering vector ${q}_{\ensuremath{\pi}}=(\ensuremath{\pi},\ensuremath{\pi})$] and has much stronger dependence on frequency than on momentum. We show that in this regime, which we termed a critical FL, the conventional random-phase-approximation- (RPA) type approach breaks down, and to properly calculate the vertex function one has to sum up an infinite series of terms which were explicitly excluded in the conventional treatment. Besides, we show that, to properly describe the spin component of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ even in an ordinary FL, one has to add Aslamazov-Larkin (AL) terms to the RPA vertex. We show that the total ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ is larger in a critical FL than in an ordinary FL, roughly by an extra power of magnetic correlation length $\ensuremath{\xi}$, which diverges at the quantum critical point. However, the enhancement of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ is highly nonuniform: It holds only when, for one of the two momentum variables, the distance from a hot spot along the Fermi surface is much larger than for the other one. This fact renders our case different from quantum criticality at small momentum, where the enhancement of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ was found to be homogeneous. We show that the charge and spin components of the total vertex function satisfy the universal relations following from the Ward identities related to the conservation of the particle number and the total spin. We show that in a critical FL, the Ward identity involves ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ taken between particles on the FS. We find that the charge and spin components of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ are identical to leading order in the magnetic correlation length. We use our results for ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ and for the quasiparticle residue to derive the Landau parameters Fcl=0=Fsl=0{F}_{c}^{l=0}={F}_{s}^{l=0}Fcl=0​=Fsl=0​, the density of states, and the uniform (q=0q=0q=0) charge and spin susceptibilities ${\ensuremath{\chi}}_{c}^{l=0}={\ensuremath{\chi}}_{s}^{l=0}$. We show that the density of states NF{N}_{F}NF​ diverges as $log\ensuremath{\xi}$; however, Fc,sl=0{F}_{c,s}^{l=0}Fc,sl=0​ also diverge as $log\ensuremath{\xi}$, such that the total ${\ensuremath{\chi}}_{c,s}^{(l=0)}\ensuremath{\propto}{N}_{F}/(1+{F}_{c}^{l=0})$ remain finite at $\ensuremath{\xi}=\ensuremath{\infty}$. We show that at weak coupling these susceptibilities are parametrically smaller than for free fermions.
We show that the singularities in the dynamical bosonic response functions of a two-dimensional Fermi liquid give rise to universal nonanalytic corrections to the results of the Fermi-liquid theory. In particular, we find a T2{T}^{2}T2 term in the specific heat, linear-in-TTT terms in the effective mass and in the uniform spin susceptibility ${\ensuremath{\chi}}_{s}(Q=0,T),$ and ∣Q∣|Q|∣Q∣ term in ${\ensuremath{\chi}}_{s}(Q,T=0).$ The existence of these terms has been the subject of recent controversy, which is resolved in this paper. We present exact expressions for all nonanalytic terms to second order in a generic interaction U(Q)U(Q)U(Q) and show that the nonanalytic terms originate exclusively from forward- and backward-scattering of particles with zero total momentum.