费米液体中的第一松原频率规则第一部分：费米子自能
First-Matsubara-frequency rule in a Fermi liquid. Part I: Fermionic self-energy
摘要：
详细分析了有限温度T和频率ω下费米液体（FL）中的费米自能ε（ω，T）。我们考虑了空间维度D >2的正则FLS --系统，其中费米子自能中的首项是解析的[延迟的Im ^R（ω，T）= C（ω^2 +π^2 T^2）]，以及1<D <2的非正则FLS，其中Im ^R（ω，T）中的首项标度为T^D或ω^D。我们将ω^2 + π^2 T^2形式与自能的一个特殊性质--“第一松原频率规则”联系起来，该规则规定，在正则FL中，π ^R（iπT，T）包含O（T）但不包含T^2项。我们证明了在任意D >1中，在T ^R（iπT，T）中O（T）之后的下一项的阶为T^D（T^3 \ln T in D=3）。这个T^D项只来自前向和后向散射，并且用这些过程的完全重整化振幅表示。在“局域近似”中，T^D项的总前因子为零，此时相互作用可以通过费米面上的初始和最终费米态的值来近似。局部近似是合理的附近的Pomeranchuk不稳定性，即使顶点修正是不可忽略的。我们证明了第一松原频率规则的强度在局部近似中被放大了，它指出不仅T^D项为零，而且^R（iπT，T）不包含任何超过O（T）的项。这个规则对自能的标度形式施加了两个约束：当用iπT替换ω时，Im ^R（ω，T）必须为零，Re ^R（ω，T）必须降为O（T）。在从实验和数值数据中提取τ ^R（ω，T）的标度形式时，应考虑这两个约束。
英文摘要：
We analyze in detail the fermionic self-energy Σ(ω, T) in a Fermi liquid (FL) at finite temperature T and frequency ω. We consider both canonical FLs -- systems in spatial dimension D >2, where the leading term in the fermionic self-energy is analytic [the retarded ImΣ^R(ω,T) = C(ω^2 +π^2 T^2)], and non-canonical FLs in 1<D <2, where the leading term in ImΣ^R(ω,T) scales as T^D or ω^D. We relate the ω^2 + π^2 T^2 form to a special property of the self-energy -"the first-Matsubara-frequency rule", which stipulates that Σ^R(iπT,T) in a canonical FL contains an O(T) but no T^2 term. We show that in any D >1 the next term after O(T) in Σ^R(iπT,T) is of order T^D (T^3\ln T in D=3). This T^D term comes from only forward- and backward scattering, and is expressed in terms of fully renormalized amplitudes for these processes. The overall prefactor of the T^D term vanishes in the "local approximation", when the interaction can be approximated by its value for the initial and final fermionic states right on the Fermi surface. The local approximation is justified near a Pomeranchuk instability, even if the vertex corrections are non-negligible. We show that the strength of the first-Matsubara-frequency rule is amplified in the local approximation, where it states that not only the T^D term vanishes but also that Σ^R(iπT,T) does not contain any terms beyond O(T). This rule imposes two constraints on the scaling form of the self-energy: upon replacing ωby iπT, ImΣ^R(ω,T) must vanish and ReΣ^R (ω, T) must reduce to O(T). These two constraints should be taken into consideration in extracting scaling forms of Σ^R(ω,T) from experimental and numerical data.
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