Perturbation theory of the Fermi surface in a quantum liquid. A general quasiparticle formalism and one-dimensional systems
Perturbation theory of the Fermi surface in a quantum liquid. A general quasiparticle formalism and one-dimensional systems
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量子液体中费米面的微扰理论。
DOI：
10.1007/bf01025844
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发表时间：
1990
影响因子：
1.6
通讯作者：
G. Gallavotti
中科院分区：
文献类型：
--
作者：
G. Benfatto;G. Gallavotti
我们发展了一个微扰理论形式的费米液体中的费米面理论的粒子通过一个有界的短程排斥对势相互作用。形式主义是基于重整化群，并提供了一个正式的扩展的大距离Schwinger函数在一个家庭的运行耦合组成的一个和两个体准粒子的潜力。流动的运行耦合描述的β函数，这是研究的所有订单的扰动理论，并服从，在thenth秩序，n！界限。为简单起见，对于无自旋情况，流动方程用一般量纲表示.出现的图像是，在大尺度上，该系统看起来像是一个费米子系统，通过δ类相互作用势相互作用（即，一个潜在的接近0除了在原点，在那里它发散，虽然保持积分有界的任何地方）;理论不是渐近自由在通常意义上和自由机制，因此比通常更微妙：通过引入准粒子的数学精确概念解决了处理无界有效势的技术问题，即使当物理势作为δ函数发散时，它们也是具有有限相互作用的自然物体。准粒子具有一种引人注目的规范对称性。为了证实与准粒子理论的类比，我们使用我们的准粒子概念讨论平均场理论：由此产生的自洽关系与BCS模型的自洽关系密切相关。这种形式主义似乎适合于费米液体和BCS态的正常态的联合理论：第一个与流的平凡不动点或附近的非平凡不动点有关（或不变集），第二个可能自然对应于真正的非平凡不动点（然而，由于BCS状态是准自由状态，因此非常简单，不同于场论的非平凡不动点）。对于我们的无自旋费米子，Thed=1的情况与thed> 1的情况有很大的不同：我们可以基本上完全将其视为小耦合。该系统不是渐近自由的，并提出了反常的重整化群流与消失的β函数，并在费米表面的占领数的不连续性是平滑的相互作用（保持奇异性与耦合依赖的奇异性的幂型与指数识别的异常尺寸）。最后，我们提出了一个启发式的讨论理论的流动的spinlessd> 1系统的耦合常数：其结构被进一步简化和相关部分的运行的相互作用，正是对准粒子之间的相互作用，我们确定的库珀对超导。形式微扰理论似乎只有在库珀对之间的相互作用是排斥的情况下才有可能起作用：而到了二阶，我们证明了在自旋为0的情况下，如果物理势是排斥的，就会发生这种情况。我们的结果表明，只有当相互作用是排斥的，才有可能存在正常的费米面。
We develop a perturbation theory formalism for the theory of the Fermi surface in a Fermi liquid of particles interacting via a bounded short-range repulsive pair potential. The formalism is based on the renormalization group and provides a formal expansion of the large-distance Schwinger functions in terms of a family of running couplings consisting of one- and two-body quasiparticle potentials. The flow of the running couplings is described in terms of a beta function, which is studied to all orders of perturbation theory and shown to obey, in thenth order,n! bounds. The flow equations are written in general dimensiond⩾1 for the spinless case (for simplicity). The picture that emerges is that on a large scale the system looks like a system of fermions interacting via aδ-like interaction potential (i.e., a potential approaching 0 everywhere except at the origin, where it diverges, although keeping the integral bounded); the theory is not asymptotically free in the usual sense and the freedom mechanism is thus more delicate than usual: the technical problem of dealing with unbounded effective potentials is solved by introducing a mathematically precise notion ofquasiparticles, which turn out to be natural objects with finite interaction even when the physical potential diverges as a deltalike function. A remarkable kind of gauge symmetry is associated with the quasiparticles. To substantiate the analogy with the quasiparticle theory we discuss the mean field theory using our notion of quasiparticles: the resulting self-consistency relations are closely reminiscent of those of the BCS model. The formalism seems suited for a joint theory of normal states of Fermi liquids and of BCS states: the first are associated with the trivial fixed point of our flow or with nearby nontrivial fixed points (or invariant sets) and the second may naturally correspond to really nontrivial fixed points (which may nevertheless turn out to be accessible to analysis because the BCS state is a quasi free state, hence quite simple, unlike the nontrivial fixed points of field theory). Thed=1 case is deeply different from thed> 1 case, for our spinless fermions: we can treat it essentially completely for small coupling. The system is not asymptotically free and presents anomalous renormalization group flow with a vanishing beta function, and the discontinuity of the occupation number at the Fermi surface is smoothed by the interaction (remaining singular with a coupling-dependent singularity of power type with exponent identified with the anomalous dimension). Finally, we present a heuristic discussion of the theory for the flow of the running coupling constants in spinlessd> 1 systems: their structure is simplified further and the relevant part of the running interaction is precisely the interaction between pairs of quasiparticles which we identify with the Cooper pairs of superconductivity. The formal perturbation theory seems to have a chance to work only if the interaction between the Cooper pairs is repulsive: and to second order we show that in the spin-0 case this happens if the physical potential is repulsive. Our results indicate the possibility of the existence of a normal Fermi surface only if the interaction is repulsive.