金属量子临界点附近的自由能与比热
Free energy and specific heat near a quantum critical point of a metal
AI总结：
本文用Luttinger-Ward-Eliashberg自由能框架研究金属QCP处费米子与无隙玻色子耦合的比热，证明γ模型对截断的虚假依赖可由玻色子贡献抵消，且正常态在QCP仍稳定。
摘要：
我们分析了金属中一个量子临界点（QCP）处与无隙玻色子相互作用的费米子的自由能和比热。我们采用正常态自由能的Luttinger-Ward-Eliashberg公式，该公式包含来自玻色子、费米子及其相互作用的贡献，所有这些均通过完全修饰的费米子和玻色子传播子表示。后两项贡献之和是有效低能模型的自由能F_γ，在该模型中，费米子具有由玻色子介导的动力学四费米子相互作用V(Ω_m)∝1/|Ω_m|^γ（即γ模型）。这一纯电子模型已被用于分析QCP附近非费米液体（non-FL）行为与配对之间的相互作用，二者均与高能截断Λ无关。然而，由F_γ得到的比热C_γ(T)却依赖于Λ。我们认为，一旦把玻色子的贡献包括进来，这种依赖性就是虚假的并会被抵消。我们将自己的C(T)与γ模型中利用最近提出的F_γ正则化方法所得结果进行比较。我们指出，对于γ<1，完整C(T)与正则化后的C_γ(T)相差一个依赖于γ的前因子；而对于γ>1，完整C(T)是C_γ(T)与具有完全修饰质量的自由玻色子比热之和。对于这些γ，C_γ(T)为负。文献[1]的作者认为，负的C_γ(T)意味着正常态在距QCP一定距离处变得不稳定。在我们的计算中，C(T)中的两项来自同一来源，并且只要顶点修正可以被安全忽略，C_γ(T)就更小。我们进而认为，即使在QCP处，正常态也保持稳定。
英文摘要：
We analyze free energy and specific heat for fermions interacting with gapless bosons at a quantum-critical point (QCP) in a metal. We use the Luttinger-Ward-Eliashberg formula for the free energy in the normal state, which includes contributions from bosons, fermions, and their interaction, all expressed via fully dressed fermionic and bosonic propagators. The sum of the last two contributions is the free energy $F_γ$ of an effective low-energy model of fermions with boson-mediated dynamical 4-fermion interaction $V (Ω_m) \propto 1/|Ω_m|^γ$ (the $γ-$model). This purely electronic model has been used to analyze the interplay between non-Fermi liquid (non-FL) behavior and pairing near a QCP, which are both independent of the upper energy cutoff $Λ$. However, the specific heat $C_γ(T)$, obtained from $F_γ$, does depend on $Λ$. We argue that this dependence is spurious and cancels out, once we include the contribution from bosons. We compare our $C(T)$ with the one obtained within the $γ-$model using recently proposed regularization of $F_γ$. We argue that for $γ<1$, the full $C(T)$ and the regularized $C_γ(T)$ differ by a $γ-$dependent prefactor, while for $γ>1$, the full $C(T)$ is the sum of $ C_γ(T)$ and the specific heat of free bosons with fully dressed mass. For these $γ$, $C_γ(T)$ is negative. The authors of Ref. [1] argued that a negative $ C_γ(T)$ implies that the normal state becomes unstable at some distance to a QCP. In our calculation, both terms in $C(T)$ come from the same source, and $ C_γ(T)$ is smaller as long as vertex corrections can be safely neglected. We then argue that the normal state remains stable even at a QCP.
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