We perform a microscopic analysis of how the constraints imposed by conservation laws affect $q=0$ Pomeranchuk instabilities in a Fermi liquid. The conventional view is that these instabilities are determined by the static interaction between low-energy quasiparticles near the Fermi surface, in the limit of vanishing momentum transfer $q$. The condition for a Pomeranchuk instability is set by $F_l^{c\left(s\right)}=-1$, where $F_l^{c\left(s\right)}$ (a Landau parameter) is a properly normalized partial component of the antisymmetrized static interaction $F(k,k+q;p,p-q)$ in a charge (c) or spin (s) subchannel with angular momentum $l$. However, it is known that conservation laws for total spin and charge prevent Pomeranchuk instabilities for $l=1$ spin- and charge-current order parameters. Our study aims to understand whether this holds only for these special forms of $l=1$ order parameters or is a more generic result. To this end we perform a diagrammatic analysis of spin and charge susceptibilities for charge and spin density order parameters, as well as perturbative calculations to second order in the Hubbard $U$. We argue that for $l=1$ spin-current and charge-current order parameters, certain vertex functions, which are determined by high-energy fermions, vanish at $F_{l=1}^{c\left(s\right)}=-1$, preventing a Pomeranchuk instability from taking place. For an order parameter with a generic $l=1$ form factor, the vertex function is not expressed in terms of $F_{l=1}^{c\left(s\right)}$, and a Pomeranchuk instability may occur when $F_1^{c\left(s\right)}=-1$. We argue that for other values of $l$, a Pomeranchuk instability may occur at $F_l^{c\left(s\right)}=-1$ for an order parameter with any form factor.

中文翻译：

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费米液体中forl = 1Pomeranchuk不稳定性的条件

我们对保护法所施加的约束如何影响进行了微观分析 $q=0$Fermi液体中的Pomeranchuk不稳定性。传统观点认为，这些不稳定性是由费米表面附近的低能准粒子之间的静态相互作用所决定的，在动量传递消失的范围内$q$。Pomeranchuk不稳定性的条件由$F_{升}^{C（s）}=-\mathrm{1个}$， 在哪里 $F_{升}^{C（s）}$ （Landau参数）是反对称静态相互作用的适当归一化的部分分量 $F（ķ，ķ+q;p，p-q）$ 在具有角动量的电荷（c）或自旋（s）子通道中 $升$。但是，众所周知，总自旋和电荷的守恒定律可以防止Pomeranchuk的不稳定性。$升=\mathrm{1个}$自旋电流和充电电流顺序参数。我们的研究旨在了解这是否仅适用于这些特殊形式的$升=\mathrm{1个}$顺序参数还是更通用的结果。为此，我们对电荷和自旋密度阶次参数的自旋和电荷磁化率进行了图解分析，并在Hubbard中对二阶进行微扰计算。$ü$。我们认为$升=\mathrm{1个}$ 由高能费米子决定的自旋电流和电荷电流有序参数（某些顶点函数）在 $F_{升=\mathrm{1个}}^{C（s）}=-\mathrm{1个}$，防止Pomeranchuk发生不稳定。对于具有通用参数的订单参数$升=\mathrm{1个}$ 形状因数，顶点函数不表示为 $F_{升=\mathrm{1个}}^{C（s）}$，并且在以下情况下可能会发生Pomeranchuk不稳定性 $F_{\mathrm{1个}}^{C（s）}=-\mathrm{1个}$。我们认为，对于$升$，Pomeranchuk不稳定性可能会发生在 $F_{升}^{C（s）}=-\mathrm{1个}$ 具有任何形状因数的订单参数。