Holographic metals at finite temperature

Abstract

A holographic dual description of a 2+1 dimensional system of strongly
interacting fermions at low temperature and finite charge density is given in
terms of an electron cloud suspended over the horizon of a charged black hole
in asymptotically AdS spacetime. The electron star of Hartnoll and Tavanfar is
recovered in the limit of zero temperature, while at higher temperatures the
fraction of charge carried by the electron cloud is reduced and at a critical
temperature there is a second order phase transition to a configuration with
only a charged black hole. The geometric structure implies that finite
temperature transport coefficients, including the AC electrical conductivity,
only receive contributions from bulk fermions within a finite band in the
radial direction.

Full text

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Preprint typeset in JHEP style - PAPER VERSION
NORDITA-2010-102, RH-27-2010
Holographic metals at finite temperature
V. Giangreco M. Puletti,1)S. Nowling,1)L. Thorlacius,1),2)and T. Zingg1),2)
1) NORDITA, Roslagstullsbacken 23
SE-106 91 Stockholm, Sweden
2) University of Iceland, Science Institute
Dunhaga 3, IS-107 Reykjavik, Iceland
E-mail: valentina, nowling, larus, zingg @nordita.org
Abstract: A holographic dual description of a 2+1 dimensional system of strongly
interacting fermions at low temperature and finite charge density is given in terms of
an electron cloud suspended over the horizon of a charged black hole in asymptotically
AdS spacetime. The electron star of Hartnoll and Tavanfar is recovered in the limit
of zero temperature, while at higher temperatures the fraction of charge carried by
the electron cloud is reduced and at a critical temperature there is a third order
phase transition to a configuration with only a charged black hole. The geometric
structure implies that finite temperature transport coefficients, including the AC
electrical conductivity, only receive contributions from bulk fermions within a finite
band in the radial direction.
Keywords: Field Theories in Lower Dimensions, Quantum Critical Points,
Gauge-gravity correspondence, Black Holes.
arXiv:1011.6261v2 [hep-th] 3 Dec 2010

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Contents
1. Introduction1
2. Field equations and electron cloud solutions3
3. Free energy9
4. Electric conductivity11
5. Discussion13
1. Introduction
There has been considerable recent interest in developing holographic models of
strongly coupled physics in low dimensions with a view towards condensed mat-
ter systems (see [1, 2, 3, 4] for reviews). This can be motivated both from the point
of view of extending the gauge theory/gravity correspondence to include a variety of
interesting field theoretic systems without supersymmetry, and also from the point of
view of gaining a new theoretical handle on materials containing strongly correlated
electrons.
In a recent paper Hartnoll and Tavanfar [5] considered a simple holographic
model for strongly interacting fermions in 2+1 dimensions at zero temperature and
finite charge density. In their approach, which builds on earlier work in [6, 7], the
bulk Maxwell field, which is dual to the field theory current, is sourced by an ideal
fluid consisting of charged free fermions. The combined Einstein, Maxwell, and fluid
field equations in the bulk spacetime have planar solutions, referred to as electron
stars in [5], where the charge and energy densities of the bulk fermion fluid have a
non-trivial radial profile. The geometry is asymptotically AdS but deep in the elec-
tron star interior the metric exhibits Lifshitz scaling with a non-universal dynamical
critical exponent that depends on the couplings of the model. Similar constructions
were considered in [8, 9] for neutral and charged free fermion fluids supported by a
degeneracy pressure.
In the present paper we extend the gravity dual description of [5] to include
finite temperature configurations in the boundary field theory. We construct static
solutions of the bulk field equations where an ‘electron cloud’ is suspended above
the horizon of a charged black hole, or more precisely a black brane with a planar
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horizon. The electron cloud has both an outer and an inner edge. The outer edge is
also found in the electron stars of [5] but the inner edge is a new feature, found only
at finite temperature. At the inner edge the gravitational pull of the black brane on
the electron fluid is balanced by electrostatic repulsion.
In the fluid description, observables in the boundary theory, such as the electric
conductivity, only receive contributions from bulk fermions located within a band
of finite width in the radial direction. The sharpness of the edges is presumably
an artifact of the classical perfect fluid description and we expect both quantum
corrections and fluid interactions to give rise to tails in the bulk fermion profile that
fall off towards the boundary and the black brane horizon, respectively.1
At the level of classical geometry, a finite temperature in the boundary theory
is introduced by including a black hole in the bulk spacetime, with a non-vanishing
Hawking temperature. Quantum effects in the bulk include thermal Hawking radi-
ation, which comes to equilibrium with the bulk fermion fluid, but at weak gravi-
tational coupling this thermalization in the bulk is suppressed and will not be con-
sidered here. This simplifies our analysis considerably as it allows us to use a zero
temperature equation of state for the free fermions in the bulk to capture finite
temperature effects in the boundary field theory.
At low temperatures in the boundary theory, the electric charge in the bulk
geometry is partly carried by the electron cloud and partly by the charged black
brane inside it. As the temperature is raised, the two edges of the electron cloud
move towards each other and an ever larger fraction of the total electric charge in
the bulk geometry resides inside the black brane. The two edges of the electron
fluid meet at a finite critical temperature, above which there is only a black brane
solution with no electron cloud present. We find that the system undergoes a third
order phase transition at the critical point.2
As the temperature is lowered, on the other hand, the inner edge of the electron
cloud approaches the black brane horizon, which in turn recedes towards vanishing
area. In the zero temperature limit, one recovers the electron star geometry where
there is no longer any black hole and all the charge is carried by the electron fluid. We
confirm this by showing that the horizon recedes from the boundary, the geometry at
low temperature encodes the dynamical exponent z, and by comparing free energy
densities. We find that at low temperatures the free energy density of an electron
cloud geometry smoothly goes over to that of an electron star. Furthermore, we
compare the free energy densities between an electron cloud solution and an AdS-
RN black brane solution at finite temperature. We find that, whenever a solution
with an electron cloud exists, it is favored over an AdS-RN black brane.
1Such tails are, for instance, found in an alternative approach to including bulk fermions based
on a single fermion wave equation [11].
2In an earlier version of the paper it was incorrectly stated that the phase transition was second
order. The correct behavior was identified in [10].
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The electrical conductivity at finite temperature can be obtained for this system
using, by now, standard holographic techniques (see for instance [1, 2]). We find that
the finite temperature conductivity smoothly interpolates between the AdS-RN and
electron star results [5].
The same finite temperature solutions were found independently by Hartnoll and
Petrov in [10]. Initially there was a discrepancy between their work and ours in the
analysis of the phase transition, but after correcting an error in our expression for
the free energy density, we now also find a third order phase transition.
2. Field equations and electron cloud solutions
The Einstein-Maxwell equations with a negative cosmological constant and a charged
perfect fluid are
Rµν−1
2gµνR −
3
L2gµν= κ2(TMaxwell
µν
+ Tfluid
µν ),
∇νFµν= e2Jfluid
µ
.(2.1)
We adopt units where the characteristic AdS length scale is L = 1. The source terms
are given by
TMaxwell
µν
=
1
e2(FµλF
= (ρ + p)uµuν+ pgµν,
λ
ν −1
4gµνFλσFλσ
ν),(2.2)
Tfluid
µν
Jfluid
µ
(2.3)
= σuµ, (2.4)
where σ is the charge density of the fluid, ρ is its energy density, p the pressure, and
uµthe four velocity, uµuµ= −1. The justification and limitations of the perfect fluid
description are discussed in detail in [5] and the same considerations apply here.
We look for static black brane solutions with planar symmetry,
ds2= −f(v)dt2+ g(v)dv2+1
v2(dx2+ dy2),A =e
κh(v)dt,(2.5)
where the radial coordinate goes from v → 0 at the asymptotic boundary to a
constant value v = v0at the black brane horizon. We find it convenient to introduce
a scale invariant variable u = −log(v/v0), such that u = 0 at the horizon and u → ∞
at the boundary, and work with rescaled fields,
ˆf = v2
0f,ˆ g = v2
0g,
ˆh = v0h,ˆ p = κ2p,ˆ ρ = κ2ρ,ˆ σ = eκσ .(2.6)
The equations of motion (2.1) can then be expressed in a first order form, convenient
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for numerical evaluation,
dˆf
du+
ˆk2
2+ˆf(1 − 3e−2uˆ g) = e−2uˆfˆ gˆ p,
dˆk
du+ˆk = e−2u
(2.7)
?1
2
ˆhˆk +ˆf
?
ˆ gˆ σ
?
ˆf
,(2.8)
1
ˆf
dˆf
du+1
ˆ g
dˆ g
du− 4 = e−2uˆ gˆhˆ σ
?
ˆf
,(2.9)
whereˆk ≡ dˆh/du. Following [5], we assume a free fermion equation of state defined
via
?ˆ µ
whereˆβ is a coupling dependent dimensionless constant, ˆ m is proportional to the
electron mass, ˆ m2=κ2
ˆ σ =ˆβ
ˆ m
dεε√ε2− ˆ m2,ˆ ρ =ˆβ
?ˆ µ
ˆ m
dεε2√ε2− ˆ m2,
−ˆ p = ˆ ρ − ˆ µˆ σ ,(2.10)
e2m2, and the (rescaled) local chemical potential ˆ µ is given by
the background Maxwell gauge field in the tangent frame, ˆ µ ≡ˆh/
As discussed in [5], there is a range of parameters,
?
ˆf.
e2∼κ
L? 1, (2.11)
for which we can assume a classical bulk geometry with a non-trivial back-reaction
due to the fermion fluid. If, at the same time, the Compton wavelength of the
fermions is small compared to the AdS length scale,
mL ? 1, (2.12)
then we are also justified in taking spacetime to be locally flat in the fermion equation
of state. These conditions amount to the dimensionless parameters in the equation
of state (2.10) taking order one values [5],
ˆβ ∼ 1,ˆ m2∼ 1. (2.13)
The construction of the electron cloud geometry proceeds in a few steps. First
we solve the vacuum equations, with ˆ σ = ˆ ρ = ˆ p = 0, to find the charged AdS-RN
black brane solution inside the cloud,
ˆf = e2u+ˆ q2
2e−2u− (1 +ˆ q2
2)e−u,ˆ g =e4u
ˆf
,
ˆh = ˆ q(1 − e−u).(2.14)
The dimensionless constant ˆ q is proportional to the charge carried by the black
brane and we have used the freedom to rescale the time coordinate t to fix the
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