2.3 Holography
3.1.1 Fermi liquid theory
Chapter 3
Holography and condensed matter physics
In this chapter we introduce the application of the AdS/CFT correspondence to condensed matter physics, known as AdS/CMT, where CMT stands for condensed matter theory. In the past years, AdS/CMT offered an entirely new perspective on open problems in condensed matter theory [14, 49, 54, 64, 65]. It translates its traditional language to the language of high energy physics. The success of the AdS/CMT duality is an astonishing example of the profound insight that seemingly unrelated fields of physics can be described by the same concepts. In section 3.1 we review basic elements of condensed matter theory formulated in the traditional language and thereby laying the ground for the holographic picture in the remaining sections of this chapter 3.2-3.5.
3.1 Basic elements of condensed matter theory
ment. The source of its quantum nature are its constituents themselves. Fermions are doomed to obey the Pauli exclusion principle and the wave function of the whole system has to be antisymmetric under the exchange of any two fermions.
The principles of quantum physics are thus naturally relevant on a macroscopic scale.
Let us introduce Laundau’s fermi liquid theory as its concepts will reappear when we take the holographic perspective on condensed matter theory. ConsiderN free fermions in a box with volumeV. Their dispersion relation is given by E =k2/2m in terms of the fermion massmand the absolute value of the momentumk =|k|. In order to construct the ground state at zero temperature, we have to fill the energy eigenstates starting at low energies until all of theN fermions are placed. Taking the two options for the spin orientation into account the required momentum space volume is determined by
N = 2 Z
Vk
ddk V
(2π)d. (3.1)
The dispersion relation only depends on the absolute value k of the momentum and thus Vk is a d-dimensional sphere with radius
kF = 2(d−1)/d√
πn1/dΓ
1 + d 2
1/d
, (3.2)
where n = N/V is the particle density. The ground state of the free electron system is therefore a sphere of radius kF in momentum space inside of which all the states are filled while all states outside the sphere are empty. The boundary of the sphere is called Fermi surface. The energy associated to kF according to the dispersion relation is the Fermi energy EF.
If we were to add an extra fermion to the system we would have to invest at least the Fermi energy, since all the states with less energy are already occupied. The energy necessary to add a fermion to a system is called the chemical potential µwhich in this case is equal to the Fermi energy. The chemical potential is a thermodynamic variable of the grand canonical ensemble, a system with a fixed number of particles.
Alternatively ,we could also remove a fermion just inside the Fermi surface. Those lowest energy excitations are called particle and hole, respectively. As they are assumed to be close to the Fermi surface, i.e.k−kF kF their dispersion relation
3.1. Basic elements of condensed matter theory 29
can be obtained by linearising around the Fermi momentum (k) =E(k)−EF =E(k)−µ= kF
m (k−kF) +O (k−kF)2
. (3.3) In general the volumes and particle numbers are assumed to be large, therefore the excitations can cost arbitrarily few energy measured from the chemical potential, the excitations are gapless.
What happens if we allow the fermions to interact? Laundau’s fermi liquid theory postulates that the system behaves qualitatively similar as the non-interacting gas [66]. This implies that the ground state of the system is still given by a Fermi surface and that the dispersion relation for particle excitations (3.3) is still valid for quasiparticle excitations with an effective mass m∗. It can be shown that the generic local interactions in the vicinity of a Fermi surface necessarily result in quasiparticles which makes the second statement self-consistent. Quasiparticles are excitations which live long enough the reveal their particle-like properties, i.e. whose decay rate Γ . They show up as poles in the retarded Green’s function
GR ∼ 1
ω−vF (k−kFF) + Σ(ω, k), (3.4) where vF =kF/m∗ and Σ is the self energy
Σ∼iω2. (3.5)
Note that in this notation ω is measured with from the chemical potential µ.
The success of the Fermi liquid theory reliess on the fact that it can be shown to be a stable fixed point of a generic theory of quasiparticles. It holds even for intrinsically strongly coupled systems as long as the interactions between the quasiparticles are weak. Strong interactions of the fundamental constituents man-ifest themselves in large effective masses m∗. They can be as high as 103 times the electron mass. Such high values for the effective mass indicate that the Fermi surface is on the verge of being destroyed by quantum fluctuations [14].
At finite temperature we expect the sharp spectrum to smoothen out, as thermal fluctuations can excite some fermions to a state above kF such that some of the states below kF are empty. The probability for states of energyE to be occupied
at finite temperature is given by the Fermi-Dirac statistics f(E) =
1 +eE−µT −1
. (3.6)
The logic of the Fermi liquid theory still applies and makes predictions for the low energy behaviour of a system. Two examples that will become important later in this chapter, concern the electrical resistivityρ and the specific heat c[66]:
ρ−ρ0 ∼T2 and c∼T . (3.7)
There are however a number of examples in which the observed low energy proper-ties do not match the predictions from Fermi liquid theory, the so-called non-Fermi liquids. A prominent example are strange metals. At the same time those are also assumed to be the best candidate to be approached with holographic methods.
We will discuss strange metals and their superconducting counterpart, the high temperature superconductors, in more detail at the end of this section 3.1.3.
Photoemission spectroscopy is the canonical experiment to study the Fermi surface of a material. It is based on the photoelectric effect: the material is hit with a beam of high energy photons which kick out electrons. The energy and momentum that is missing in the detected electrons as compared to the initial photons allow to draw conclusions about the properties of the electronic structure of the material.
Photoemission experiments essentially measure the spectral density function A(ω, k) = 1
πImGR. (3.8)