Electron stars for holographic metallic criticality

Abstract

We refer to the ground state of a gravitating, charged ideal fluid of fermions held at a finite chemical potential as an 'electron star'. In a holographic setting, electron stars are candidate gravity duals for strongly interacting finite fermion density systems. We show how electron stars develop an emergent Lifshitz scaling at low energies. This IR scaling region is a consequence of the two way interaction between emergent quantum critical bosonic modes and the finite density of fermions. By integrating from the IR region to an asymptotically AdS 4 spacetime, we compute basic properties of the electron stars, including their electrical conductivity. We emphasize the challenge of connecting UV and IR physics in strongly interacting finite density systems.arXiv:1008.2828v2 [hep-th] 6 Jan 20111 The broader context A challenge facing contemporary condensed matter theory is the description of a 2+1 dimensional finite density of fermions interacting with a gapless collective bosonic excitation, such as a spin density wave or emergent gauge field. Such theories arise, for instance, when a Fermi liquid is tuned across a quantum phase transition. The low energy dynamics of the system of fermions interacting with the critical bosonic mode can be characterised as metallic quantum criticality. While in 3+1 dimensions one can proceed to integrate out the fermions and obtain a stable Gaussian theory for the boson [1], this approach does not give correct answers in 2+1 dimensions, see e.g. [2,3,4], as it ignores an infinite number of nonlocal marginal couplings in the effective theory for the boson. One should not integrate out the fermions in this case but rather flow to a scaling regime involving both the boson and fermion fields. The resulting low energy theory is strongly interacting, e.g. [4].One might have hoped to perform a (vector) large N analysis as a perturbative handle on the theory. It has recently been demonstrated [5,6,4] that the vector large N expansion breaks down for 2+1 dimensional metallic quantum critical systems. This occurs because a potential IR divergence at high loop order is cured by a self-energy of order 1/N , leading to extra factors of N in the numerator in certain Feynman graphs. Partially motivated by these difficulties, in this paper we will use the holographic correspondence [7,8,9, 10] to study a strongly interacting system of gapless bosons with a finite density of fermions. Before proceeding we should note that more traditional approaches to this problem have also been proposed [11,12,13] and that our framework does not appear to include ingredients that are likely crucial for applications to the original systems of interest, such as Fermi lines with cold regions as well as hot spots. We will, however, describe the emergence of a low energy scaling regime from the interaction of critical bosons with a finite density of fermions. The essential physics of this process was noted in [14].In the holographic correspondence a charge density is implemented by a bulk Maxwell field, dual...

Fermi surfaces and gauge-gravity duality

“…The challenge for the future is to describe the phases of Sections V and VI using the dual gravity theory, in a manner which captures their expected properties of finite-range models in d = 2 and d = 3 respectively, and there have been recent studies in this direction [10,[12][13][14][15][16][17][18]20]. In particular, the d = 3 model of Gubser and Rocha [10] is a promising model for future study.…”

Holographic quenches and fermionic spectral functions

“…We deliberately choose parameters such that 2ν k F > 1, so that sharp quasiparticles are formed in the finite density system. It is however by now well understood that, when backreaction from bulk fermions is taken into account, the RN black hole is quantum mechanically unstable to the formation of an electron star [29][30][31][32][33] (see [34] for a review). This also corresponds to a thermodynamic instability, in the sense that for fixed temperature and chemical potential, the electron star solution has lower free energy than the RN black brane.…”

Neutral order parameters in metallic criticality ind=2+1from a hairy electron star

“…Interestingly, the case we consider also includes z = 2. A finite value of z in the IR is achieved by replacing the AdS 2 ×R d−1 near-horizon geometry of the RN-AdS d+1 background with one that is of, say, Lifshitz type [13] as in [14]. Our work here then represents a holographic description of quantum criticality in a metallic system in two spatial dimensions with z = 2, and with a neutral bosonic order parameter.…”

“…For simplicity, and with an eye towards condensed matter applications, we assume that the boundary theory is d = 2 + 1 dimensional. A background with such properties was recently constructed by Hartnoll and Tavanfar in [14] following [25,26]. As in [14], we refer to this background as the electron star background, or simply the star background.…”

“…A background with such properties was recently constructed by Hartnoll and Tavanfar in [14] following [25,26]. As in [14], we refer to this background as the electron star background, or simply the star background. Compared to other backgrounds which are asymptotically AdS 4 and Lifshitz in the IR [27], the electron star background has the appealing feature that 2 This mechanism is identical to the one used for the construction of holographic superconductors [15,[17][18][19].…”