Zero Sound in Strange Metallic Holography

Abstract

One way to model the strange metal phase of certain materials is via a holographic description in terms of probe D-branes in a Lifshitz spacetime, characterised by a dynamical exponent z. The background geometry is dual to a strongly-interacting quantum critical theory while the probe D-branes are dual to a finite density of charge carriers that can exhibit the characteristic properties of strange metals. We compute holographically the low-frequency and low-momentum form of the charge density and current retarded Green's functions in these systems for massless charge carriers. The results reveal a quasi-particle excitation when z<2, which in analogy with Landau Fermi liquids we call zero sound. The real part of the dispersion relation depends on momentum k linearly, while the imaginary part goes as k^2/z. When z is greater than or equal to 2 the zero sound is not a well-defined quasi-particle. We also compute the frequency-dependent conductivity in arbitrary spacetime dimensions. Using that as a measure of the charge current spectral function, we find that the zero sound appears only when the spectral function consists of a single delta function at zero frequency. Comment: 20 pages, v2 minor corrections, extended discussion in sections 5 and 6, added one footnote and four references, version published in JHEP

Full text

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arXiv:1007.0590v1 [hep-th] 4 Jul 2010
Zero Sound in Strange Metallic Holography
Carlos Hoyos1∗, Andy O’Bannon2†, and Jackson M. S. Wu3‡
1Department of Physics, University of Washington, Seattle, WA 98195-1560 USA
2Max-Planck-Institut f¨ ur Physik (Werner-Heisenberg-Institut),
F¨ ohringer Ring 6, 80805 M¨ unchen, Germany
3Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics,
University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
July 6, 2010
Abstract
One way to model the strange metal phase of certain materials is via a holo-
graphic description in terms of probe D-branes in a Lifshitz spacetime, char-
acterised by a dynamical exponent z. The background geometry is dual to a
strongly-interacting quantum critical theory while the probe D-branes are dual
to a finite density of charge carriers that can exhibit the characteristic proper-
ties of strange metals. We compute holographically the low-frequency and low-
momentum form of the charge density and current retarded Green’s functions
in these systems for massless charge carriers. The results reveal a quasi-particle
excitation when z < 2, which in analogy with Landau Fermi liquids we call zero
sound. The real part of the dispersion relation depends on momentum k linearly,
while the imaginary part goes as k2/z. When z ≥ 2 the zero sound is not a well-
defined quasi-particle. We also compute the frequency-dependent conductivity
in arbitrary spacetime dimensions. Using that as a measure of the charge current
spectral function, we find that the zero sound appears only when the spectral
function consists of a single delta function at zero frequency.
MPP-2010-76
∗email: choyos@phys.washington.edu
†email: ahob@mppmu.mpg.de
‡email: jbnwu@itp.unibe.ch
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1Introduction and Summary
Gauge-gravity duality [1–3] provides a novel way to compute observables in strongly-
coupled, scale-invariant systems at finite density, and thus may be useful for studying
condensed matter systems near quantum critical points, such as the “strange metal”
phase of some heavy fermion compounds and possibly also of some high-Tcmaterials.
In particular, gauge-gravity duality may be able to reproduce the simple properties
of strange metals, such as electrical resistivity scaling linearly with temperature T,
which cannot be derived in a Fermi liquid description, which assumes the appropriate
degrees of freedom are weakly-interacting quasi-particles. Of course the ultimate goal
is to understand why some strange metals have a high Tc.
Gauge-gravity duality is holographic: it equates a weakly-coupled theory of gravity
on some spacetime with a strongly-coupled field theory living on the boundary of that
spacetime. The spacetime symmetries of the field theory are dual to the isometries of
the bulk metric. Here we will focus on field theories invariant under the Lifshitz group,
which includes scale transformations of the time coordinate t and spatial coordinates
? x of the form
t → λzt,
where λ is real and positive and z is called the “dynamical exponent.” The dual gravity
theory then lives in Lifshitz spacetime, with metric [4]
? x → λ? x, (1)
ds2= gµνdxµdxν=dr2
r2−dt2
r2z+d? x2
r2, (2)
where r is the holographic radial coordinate: r → ∞ corresponds to the infrared (IR)
of the field theory, while the near-boundary region r → 0 corresponds to the ultraviolet
(UV). Notice that when z = 1, the metric becomes that of anti-de Sitter space (AdS),
and the dual field theory symmetry is enhanced to the relativistic conformal group.
One proposal for holographic model building of strange metals is to use probe D-
branes in a Lifshitz spacetime [5]. In this approach, an appropriate configuration of
fields on the probe D-brane represents a finite density of massive charge carriers [6–8],
while the background spacetime represents a strongly-coupled neutral quantum critical
theory. Holographic calculations then reveal that such systems can indeed reproduce
the simple properties of strange metals, including electrical resistivity linear in tem-
perature [5]. However, ingredients beyond Lifshitz spacetime, for example a nontrivial
dilaton, are needed to reproduce all the measured strange metal properties [5].
When z = 1 and T = 0, holographic calculations also reveal the existence of a prop-
agating quasi-particle excitation producing a pole in the charge density and current
retarded two point functions, which we denote Gtt
frequency and k is momentum [9–11]. In a weakly-coupled Fermi liquid, fluctuations
in the shape of the Fermi surface produce a collective excitation at zero temperature,
Landau’s so-called “zero sound.” The dispersion relation of the mode in these holo-
graphic systems was identical in form to that of Landau’s zero sound, with real part
R(ω,k) and Gxx
R(ω,k), where ω is
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linear in k and imaginary part going as k2, hence the mode was dubbed zero sound
by analogy. We will call this mode “holographic zero sound.” The physical origin
of this mode remains mysterious since it appears in systems with both fermions and
scalars—systems that do not fit neatly into either Fermi liquid or quantum Bose liquid
theory. Indeed, given the existence of the holographic zero sound, as well as unusual
scaling of the heat capacity with temperature, these holographic systems may be new
kinds of (strongly-coupled) quantum liquids [9].
In this paper we explore the fate of holographic zero sound in these systems when
z > 1. To do so, we compute holographically – and completely analytically, without
numerics – the low-frequency and low-momentum forms of Gtt
massless charge carriers. When z ?= 2 we find
R(ω,k) =ω2
k2Gtt
R(ω,k) and Gxx
R(ω,k), for
Gxx
R(ω,k) ∝
ω2
k2−
1
v2ω2− cω2/z+1, (3)
where v and c are dimensionful constants that depend on the density.1We obtain the
holographic zero sound dispersion relation by setting the denominator of Gtt(ω,k) to
zero. Whether the mode is a quasi-particle depends on whether the ω2or ω2/z+1term
is larger at low frequency. When z < 2, the ω2term dominates, and the mode remains
a quasi-particle, although with a dispersion relation modified relative to the z = 1
case: the real part remains linear, but the imaginary part goes as k2/z. When z > 2
the real and imaginary parts are of the same order, and hence the mode is no longer a
sharply-defined quasi-particle.
We also compute the AC conductivity associated with the charge carriers, generaliz-
ing the results of ref. [5], for two spatial field theory directions, to an arbitrary number
of spatial directions. When z < 2, the AC conductivity’s dependence on ω is a pole
formally identical to the high-frequency or “collisionless” limit of the standard Drude
conductivity, iω−1[5], indicating that the charge current spectral function, which is a
measure of the density of charged states, consists only of a delta-function at ω = 0.
When z > 2, the ω-dependence is instead a power law of the form ω−2/z, indicating a
nontrivial density of states at low frequency.
In short, we find that holographic zero sound appears in the absence of low-frequency
charged states. That alone is perhaps not surprising: a continuum of states can “smear
out” a pole. Indeed, in interacting systems with a Fermi surface, the zero sound pole
is most easily identified if the zero sound velocity is greater than the Fermi velocity, so
that the pole is outside a continuum of particle-hole states. The change in behavior of
the dispersion relation at z = 2 is the novel feature of the holographic systems.
This paper is organized as follows. We review the thermodynamics of these systems
in section 2. We compute the Green’s functions in section 3, and use the results to
study the holographic zero sound and AC conductivity in sections 4 and 5, respectively.
1When z = 2 the last term in the denominator becomes cω2log(αω2), with a scale α depending
on the density.
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We conclude with some discussion and suggestions for future research in section 6.
2Probe Thermodynamics
In this section we review the results of refs. [8,9,12–14] for the thermodynamics of the
charge carriers in these holographic systems.
In gauge-gravity duality, a theory of gravity is dual to some large-N, strongly-coupled
non-Abelian gauge theory. Nfprobe D-branes are dual to Nffields in the fundamental
representation of the gauge group(s) in the probe limit Nf≪ N [15]. We will call these
fields quarks or flavor fields, in analogy with Quantum Chromodynamics.
We will consider Nf coincident D-branes, whose action is the non-Abelian Dirac-
Born-Infeld (DBI) action, describing the dynamics of the worldvolume scalars and
U(Nf) gauge fields.2The field theory then has a global U(Nf) flavor symmetry. We
produce a finite density of charge carriers by introducing a chemical potential for
the diagonal U(1) ⊂ U(Nf). The density operator Jtis the time component of the
conserved U(1) current Jµ, and is holographically dual to At, the time component
of the U(1) gauge field living on the D-branes. We will thus introduce in the bulk
D-branes with only At(r) present. The action then reduces to the Abelian DBI action.
We will only consider massless flavor fields. A flavor field mass operator would be
dual to a scalar field on the D-branes. We omit any such scalar from our calculations,
although we comment on massive flavors in section 6.
We assume our D-branes are extended along q spatial dimensions of the Lifshitz
spacetime.3The Abelian DBI action for our D-branes is then
?
where TD is the tension, gab the induced metric, and Fab the U(1) field strength of
the D-branes. The factor V is the volume of any internal space that the D-brane may
be wrapping, and 2πα′is the inverse string tension, which will typically be present
in an actual string theory system. Inserting our ansatz At(r), performing the trivial
integrations over dt and dqx, and dividing out the subsequent (infinite) volume factors,
we obtain the action density S (which for brevity we will call the action henceforth),
?
2In string theory D-brane actions also include Wess-Zumino terms, describing the coupling of D-
brane fields to background Ramond-Ramond fields. To keep the discussion general without committing
to a specific string theory realization of Lifshitz spacetime [5], we will omit Wess-Zumino terms.
3If q is less than the total number of spatial dimensions, then in the field theory the flavor fields
propagate only along some q-dimensional defect.
ˆS = −NfTDVdrdtdqx
?
−det[gab+ (2πα′)Fab], (4)
S = −N
drgq/2
xx
?
|gtt|grr−˜A
′2
t,
N ≡ NfTDV ,(5)
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where a tilde denotes a factor of 2πα′, so˜At≡ (2πα′)At, and the prime denotes ∂r.
The charge density in the field theory is
?Jt? =δS
δA′t
= Ngq/2
xx
(2πα′)˜A′
t
?
|gtt|grr−˜A
′2
t
. (6)
Solving for A′
t(r), we find
A′
t(r) =
d
2πα′
?
|gtt|grr
gq
xx+ d2, (7)
where d ≡ ?Jt?/? N. Inserting the solution for A′
t(r) back into the action, we find
S = −N
?
dr|gtt|1/2g1/2
rr
gq
xx+ d2.
xx
?gq
(8)
We obtain the field theory chemical potential and free energy by performing the
integrals for A′
t(r) and S. The chemical potential is
µ0=
?∞
0
drA′
t(r) =
d
2πα′
?∞
0
dr
r−(z+1)
√r−2q+ d2=
dz/q
4πα′q
Γ
?
z
2q
?
Γ
?
1
2−
?
z
2q
?
Γ?1
2
.(9)
The grand canonical (Gibbs) free energy density is
Ω0= −S = N
?∞
0
drr−(2q+z+1)
√r−2q+ d2= −? N
z d
z + qµ0.(10)
These results are deceptively simple. We have actually removed various power-law
divergences at the r = 0 endpoint of the integrations via analytic continuation. To see
the divergences explicitly, we expand the integrand of eq. (10) for small r,
?
The first term gives a density-independent divergence for any z, coming simply from the
volume of Lifshitz spacetime. As we increase z to q and beyond, new density-dependent
UV divergences appear. These are proportional to even powers of d, and appear to
be associated with operators (Jt)2nwith n ∈ N that become relevant for larger values
of z.4We could have cancelled all of these divergences by introducing a regulator by
integrating only to a cutoff r = ǫ and then adding counterterms5at r = ǫ [16–18],
dr
r−(q+z+1)
√1 + d2r2q=
?
drr−(q+z+1)
?
1 −1
2d2r2q+ O?d4r4q??
. (11)
4The scaling dimension of the current operator is [Jt] = q, while the volume element has dimension
[dtdqx] = −z − q, so as we increase z to q and beyond, the operator JtJtgoes from irrelevant to
marginal (at z = q) to relevant. Similarly, when z = 3q, JtJtJtJtbecomes marginal, and so on. Only
even powers of Jtcan appear due to charge conjugation invariance Jt→ −Jt.
5We write the counterterms explicitly in section 3.
5