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arXiv:1003.5913v2 [hep-th] 16 Jun 2010
Preprint typeset in JHEP style - HYPER VERSION
MPP-2010-37
Holographic Flavor Transport in Schr¨ odinger Spacetime
Martin Ammon,1∗Carlos Hoyos,2†Andy O’Bannon,1‡and Jackson M. S. Wu3§
1Max-Planck-Institut f¨ ur Physik (Werner-Heisenberg-Institut)
F¨ ohringer Ring 6, 80805 M¨ unchen, Germany
2Department of Physics, University of Washington
Seattle, WA 98195-1560, United States
3Albert Einstein Center for Fundamental Physics
Institute for Theoretical Physics, University of Bern
Sidlerstrasse 5, 3012 Bern, Switzerland
Abstract: We use gauge-gravity duality to study the transport properties of a finite density
of charge carriers in a strongly-coupled theory with non-relativistic symmetry. The field
theory is N = 4 supersymmetric SU(Nc) Yang-Mills theory in the limit of large Ncand with
large ’t Hooft coupling, deformed by an irrelevant operator, coupled to a number Nfof massive
N = 2 supersymmetric hypermultiplets in the fundamental representation of the gauge group,
i.e. flavor fields. The irrelevant deformation breaks the relativistic conformal group down to
the Schr¨ odinger group, which has non-relativistic scale invariance with dynamical exponent
z = 2. Introducing a finite baryon number density of the flavor fields provides us with charge
carriers. We compute the associated DC and AC conductivities using the dual gravitational
description of probe D7-branes in an asymptotically Schr¨ odinger spacetime. We generically
find that in the infrared the conductivity exhibits scaling with temperature or frequency that
is relativistic, while in the ultraviolet the scalings appear to be non-relativistic with dynamical
exponent z = 2, as expected in the presence of the irrelevant deformation.
Keywords: AdS/CFT correspondence, Gauge/gravity correspondence.
∗E-mail address: ammon@mppmu.mpg.de
†E-mail address: choyos@phys.washington.edu
‡E-mail address: ahob@mppmu.mpg.de
§E-mail address: jbnwu@itp.unibe.ch
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Contents
1. Introduction1
2. Adding Flavor to Schr¨ odinger Spacetime
2.1 Review: D7-Branes in AdS
2.2 The Null Melvin Twist
2.3Twisting with Probe D7-branes
4
4
7
11
3. DC Conductivity of Probe Flavor
3.1In the DLCQ of AdS
3.2In Schr¨ odinger Spacetime
13
14
18
4. AC Conductivity of Probe Flavor
4.1 D7-brane Embeddings
4.2 AC conductivity in the IR
4.3AC conductivity in the UV
23
25
27
28
5. Discussion and Conclusion 28
1. Introduction
Gauge-gravity duality [1, 2, 3], or more generally holography, provides a new tool for studying
strongly-coupled systems at finite density, and in particular may provide novel insights into
low-temperature systems controlled by quantum critical points. Quantum critical theories
are invariant under scale transformations of the form
t → λzt,? x → λ? x, (1.1)
where λ is some real, positive scaling parameter, z is the “dynamical exponent,” and we
have assumed spatial isotropy. If rotations, space translations, and time translations are also
symmetries, then the theory is invariant under the so-called Lifshitz symmetry algebra. When
z = 1 the Lifshitz algebra may be enhanced to the relativistic conformal algebra.
The theory of fermions at unitarity, which can be realized experimentally using cold
atoms, is invariant under the so-called Schr¨ odinger symmetry [4, 5]. The Schr¨ odinger symme-
try includes time translations, spatial translations, spatial rotations, and Galilean boosts, as
well as scale transformations with z = 2, a special conformal transformation, and a number
symmetry with generator N, which is a central element of the Schr¨ odinger algebra. The
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Schr¨ odinger algebra is in fact easy to obtain from the relativistic conformal algebra in one
higher spatial dimension. If we use the extra spatial dimension to form light-cone coordinates,
x±, and then retain only those generators that commute with the translation generator in the
x−direction, P−, the resulting algebra is precisely the Schr¨ odinger algebra, if we make some
identifications, including identifying the relativistic generator P+ with the non-relativistic
Hamiltonian (generator of time translations) and P−with the number operator N. Notice
that if we want the spectrum of eigenvalues of N to be discrete, which should be the case
for a non-relativistic theory, then we must compactify x−. In other words, if we begin with
a relativistic conformal theory, break the symmetry group down to the subgroup that com-
mutes with P−(via some deformation), and then perform a Discrete Light-Cone Quantization
(DLCQ), then we obtain a non-relativistic theory with Schr¨ odinger symmetry, in one lower
spatial dimension.
Some gravitational duals for theories with Schr¨ odinger symmetry were discovered in refs.
[6, 7] (see also refs. [8, 9]). Via the gauge-gravity dictionary, the Schr¨ odinger symmetry
group translates into the isometry group of the metric. We will thus call such spacetimes
“Schr¨ odinger spacetimes.” A direct method to obtain Schr¨ odinger spacetimes, used in refs.
[10, 11, 12], is to apply a solution-generating technique of type II supergravity, the Null Melvin
Twist (NMT), to known solutions. We will review the NMT below. The basic example is to
start with type IIB supergravity on AdS5×S5, where AdS5is (4+1)-dimensional anti-de Sitter
space and S5is a five-sphere. The dual theory is N = 4 supersymmetric SU(Nc) Yang-Mills
(SYM) theory, in the limits of large Ncand large ’t Hooft coupling. Upon applying the NMT
to this type IIB solution, we obtain Sch5×S5, where Sch5is (4+1)-dimensional Schr¨ odinger
spacetime. The solution also includes a non-trivial Neveu-Schwarz (NS) two-form, B. The
dual theory is then N = 4 SYM deformed by a particular dimension-five operator that breaks
the relativistic conformal group down to the Schr¨ odinger group. We may then additionally
compactify x−, although doing so makes the supergravity approximation to string theory
unreliable since x−is a null circle [11]. The generalization to thermal equilibrium states with
temperature T is straightforward [10, 11, 12].
As holographic models of fermions at unitarity, these systems have various advantages
and disadvantages, some of which we review below. We will mention here two of the biggest
disadvantages, however. First, the NMT does not produce a genuinely non-relativistic the-
ory, but rather a deformation of a relativistic theory which then has Schr¨ odinger symmetry.
Second, as emphasized in ref. [13], the U(1) number symmetry generated by N is not sponta-
neously broken in any of the known supergravity solutions, whereas real systems are typically
superfluids.
Our goal is to study the transport properties of some “charge carriers” in such systems.
For simplicity, we will work with the basic example above, type IIB supergravity in Sch5×S5.
In the field theory, we will introduce a number Nfof massive N = 2 supersymmetric hyper-
multiplets in the fundamental representation of the SU(Nc) gauge group, i.e. flavor fields.
We will work in the probe limit, in which Nf ≪ Nc, which amounts to ignoring quantum
effects due to the flavors, such as the running of the coupling. The theory has a flavor symme-
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try analogous to the baryon number symmetry of Quantum Chromodynamics (QCD), with
conserved current Jµ. We will introduce a finite baryon number density, ?Jt?, giving us our
charge carriers. In condensed matter terms, we will study some dilute1massive charge carriers
propagating through some quantum critical heat bath. We will then compute (holographi-
cally) both the DC and AC conductivities associated with baryon number transport. In the
relativistic case these were computed in refs. [14, 15, 16, 17, 18, 19]. The flavor fields appear
in the supergravity description as a number Nfof probe D7-branes in Sch5× S5. The mass
and density are encoded in the D7-branes’ worldvolume fields, as we will review.
Our study is complementary to that of ref. [20], where the DC and AC conductivities of
probe flavor were computed holographically using probe branes in Lifshitz spacetimes, that
is, spacetimes whose isometry group is the Lifshitz group, with general z. Two of the main
results of ref. [20] were that at temperatures low compared to the density and mass the DC
conductivity σ scales as ?Jt?T−2/z(for all z) and the AC conductivity σ(ω) scales as
(here Λ is a dimensionful scale that renders the argument of the logarithm dimensionless).
The authors of ref. [20] then suggested that, by introducing a scalar field, such as a dilaton,
with nontrivial dependence on the holographic radial coordinate, the powers of T and ω in
the DC and AC conductivities, respectively, can be engineered to take essentially any value
we like. In such a fashion we can produce holographic systems with scalings that match any
number of real strongly-coupled electron systems. Moreover, with varying scalars we can even
engineer flows in which the scalings change between the ultraviolet (UV) and infrared (IR).
Examples of such flows, with exponent z = 2 in the UV and z = 1 in the IR, produced by
relevant or marginally relevant deformations of Lifshitz spacetime, appear in refs. [21, 22].
Using Schr¨ odinger instead of Lifshitz spacetime, we find that, in appropriate limits, for
example low-temperature and large mass, the scalings with temperature or frequency in the
IR are relativistic, meaning z = 1, while in the UV the scalings are non-relativistic, meaning
z = 2. That is precisely what we expect in the dual (relativistic) field theory, in the presence
of an irrelevant operator that, roughly speaking, produces z = 2 in the UV. Schr¨ odinger
spacetime is thus a good example of a flow from non-relativistic scaling in the UV to relativistic
scaling in the IR.
Our results suggest that the NMT may be a useful tool for the kind of model-building
proposed in ref. [20]. We can imagine starting with a relativistic bulk system, introducing
a scalar and engineering whatever exponents we like, and then performing a NMT. We will
generically obtain a theory with Schr¨ odinger symmetry, and exponents that flow from UV
values, presumably with z = 2, to the IR values we gave them in the original relativistic
σ(ω) ∝
?Jt?z/2ω−1
?Jt?(ω logωΛ)−1for z = 2
?Jt?ω−2/z
for z < 2,
for z > 2,
(1.2)
1Here dilute refers to energy density at finite temperature: the adjoint fields will have energy density of
order N2
cwhile the flavor fields will have energy density of order NfNc.
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setting. Such an approach of engineering scaling exponents directly in a relativistic system
may be technically easier than engineering them in a non-relativistic system.2
The paper is organized as follows. In section 2 we review how to add probe D7-branes
to AdS5× S5, review the NMT and Sch5solution, and then discuss how the NMT affects
the D7-branes’ action, the Dirac-Born-Infeld (DBI) action. In section 3 we compute the DC
conductivity and in section 4 we compute the AC conductivity. We conclude with some
discussion and suggestions for future research in section 5.
2. Adding Flavor to Schr¨ odinger Spacetime
In this section we will review how to obtain Schr¨ odinger spacetime from a NMT of AdS5×S5,
and review the field theory dual to supergravity on Schr¨ odinger spacetime. Our new ingredient
will be probe D7-branes. We will discuss in general terms what effect the NMT has on the
D7-branes’ worldvolume action.
2.1 Review: D7-Branes in AdS
In type IIB supergravity, we begin with the solution describing the near-horizon geometry of
non-extremal D3-branes, AdS5-Schwarzschild times S5. The metric is
ds2= grrdr2+ gttdt2+ gyydy2+ gxxd? x2+ ds2
1
r2
S5
(2.1)
=
?dr2
f(r)− f(r)dt2+ dy2+ d? x2
?
+ (dχ + A)2+ ds2
CP2, (2.2)
where r is the radial coordinate, with the AdS boundary at r = 0, and (t,y,? x) are field
theory directions. We have singled out one field theory spatial direction, y, for use in the
NMT below. Here f(r) = 1 − r4/r4
using units in which the radius of AdS is one, in which case the temperature of the black hole
is T = 1/(πrH). We have written the S5metric as a Hopf fibration over CP2, with χ the
Hopf fiber direction. A gives the K¨ ahler form J of CP2via dA = 2J. To write the metric of
CP2and A explicitly, we introduce CP2coordinates α1, α2, α3, and θ and define the SU(2)
left-invariant forms
H, with rHthe position of the black hole horizon. We are
σ1=1
2(cosα2dα1+ sinα1sinα2dα3),
σ2=1
2(sinα2dα1− sinα1cosα2dα3),
σ3=1
2(dα2+ cosα1dα3), (2.3)
2AdS space is a solution of Einstein gravity plus a negative cosmological constant. Gravity alone cannot
produce Lifshitz or Schr¨ odinger spacetime, however. These require matter fields. In the Schr¨ odinger case, the
NMT takes AdS and generates the needed matter fields, in particular the NS two-form B. Introducing scalars
in a theory of gravity alone and then doing the NMT may be easier than introducing scalars in a theory with
gravity and other matter fields.
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so that the metric of CP2is
ds2
CP2 = dθ2+ cos2θ?σ2
1+ σ2
2+ sin2θ σ2
3
?, (2.4)
and A = cos2θ σ3. The full solution also includes a nontrivial five-form, but as shown in refs.
[10, 11, 12] that will be unaffected by the NMT, so we will ignore it.
When T = 0, type IIB supergravity in the above spacetime is of course dual to N = 4
SYM theory in the limits of large Ncand large ’t Hooft coupling. The symmetries of the
theory include the (3+1)-dimensional relativistic conformal group, dual to the isometry group
of AdS5, and an SO(6) R-symmetry group, dual to the isometry group of the S5. The AdS5-
Schwarzschild geometry is dual to the N = 4 SYM theory in a thermal equilibrium state
with temperature T. In our units, we can convert from supergravity to SYM quantities using
α′−2= 4πgsNc= g2
gY Mis the SYM theory coupling, and λ is the ’t Hooft coupling.
Following ref. [23], we will introduce a number Nf of probe D7-branes into the above
geometry. The D7-branes will be extended along AdS5-Schwarzschild times an S3⊂ S5.
More specifically, the D7-branes will be extended along the three angular directions α1, α2
and α3. The two worldvolume scalars are then θ and χ. As explained in ref. [23], the NfD7-
branes are dual to a number Nfof N = 2 supersymmetric hypermultiplets in the fundamental
representation of the SU(Nc) gauge group. We will work in the probe limit, which consists
of keeping Nffixed as Nc→ ∞, so that Nf≪ Nc. On the gravity side, we may then ignore
the back-reaction of the D7-branes on the supergravity fields. On the field theory side, we
ignore quantum effects due to the flavor fields, such as the running of the coupling, as these
are suppressed by powers of Nf/Nc.
Massless flavor fields break the SO(6) R-symmetry to an SO(4)×U(1) symmetry, where
the U(1) and one SU(2) subgroup of the SO(4) form the R-symmetry of the N = 2 super-
symmetric theory [24]. The SO(4) symmetry is dual to the SO(4) isometry of the S3that
the D7-branes wrap. An N = 2 supersymmetric mass m for the flavor fields preserves the
SO(4) but explicitly breaks the U(1).
To study transport, we need to introduce a finite charge density, which we do as follows,
following ref. [14]. We will introduce an N = 2 supersymmetry-preserving mass identical
for all Nf flavors. The field theory then has a global U(Nf) flavor symmetry. In analogy
with QCD, we will identify the overall diagonal U(1) as baryon number (or more precisely,
quark number). We will denote the associated conserved current as Jµ, and study states
with a finite density, that is, states with nonzero3?Jt?. To study transport we will introduce
a constant external electric field that pushes on anything with U(1) baryon number charge,
pointing in the x direction, i.e. a constant Ftx. We will then compute (holographically) the
resulting current ?Jx? and from that extract a conductivity.
Each field on the worldvolume of the D7-branes is dual to some gauge-invariant operator
built from the flavor fields. The worldvolume field θ is dual to the mass operator [23], hence
Y MNc≡ λ, where α′is the string length squared, gsis the string coupling,
3We will always work in the canonical ensemble, with fixed ?Jt?, rather than in the grand canonical ensemble,
with fixed chemical potential.
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we will study embeddings with nonzero θ(r). The worldvolume scalar χ will be trivial, that
is, the D7-branes will sit at a fixed value of χ. Giving all the flavors the same mass means
that in the bulk we introduce Nfcoincident D7-branes. The U(Nf) gauge invariance of the
D7-branes is dual to the U(Nf) global symmetry of the field theory. The U(1) worldvolume
gauge field Aµis dual to the U(1) current Jµ. To study states with nonzero ?Jt?, Ftx, and
?Jx?, we will introduce worldvolume gauge fields At(r), Ftx, and Ax(r).4
The D7-brane action describing the dynamics of the worldvolume fields is the DBI action
plus Wess-Zumino (WZ) terms. For our ansatz with θ(r) and only the Abelian worldvolume
gauge field, we will only need the Abelian D7-brane action. Additionally, for our ansatz the
WZ terms vanish because the relevant form fields will not saturate the D7-branes’ indices. In
short, for our ansatz we only need the Abelian DBI action
SD7= −NfTD7
?
d8ξ e−Φ?
−det(P [g + B]ab+ (2πα′)Fab), (2.5)
where TD7=α′−4g−1
nates (hence a,b = 0,1,...7), Φ is the dilaton, P[g+B]abdenotes the pull-back of the metric
and NS two-form to the D7-branes’ worldvolume, and Fab is the D7-branes’ worldvolume
U(1) field strength.
Our ansatz for the gauge field involves only the (r,t,x) directions, hence we may write SD7
as a (3+1)-dimensional DBI action times some “extra” factors, with the (3+1)-dimensional
part being the (r,t,x,y) subspace:
s
(2π)7
is the D7-brane tension, the ξaare the D7-branes’ worldvolume coordi-
SD7= −N
?
drcos3θ(r)g1/2
xx
?
−g −1
2g˜F2−1
4
?˜F ∧˜F
?2. (2.6)
The square root factor is the characteristic form of a (3+1)-dimensional DBI action, in our
case in the (r,t,x,y) subspace, so that g is the determinant of the induced metric in that
subspace: g = gD7
rr denotes a component of the induced metric on the
D7-branes, so gD7
rr=
factors of (2πα′), for example˜Fab= (2πα′)Fab. We have also performed the trivial integration
over the S3, producing a factor of the S3volume, 2π2, and defined
rrgttgxxgyy, where gD7
1
r2f(r)+θ′(r)2. Starting now, primes will denote
∂
∂rand tildes will denote
N ≡ NfTD72π2=
1
(2π)4λNfNc,(2.7)
(not to be confused with the N of supersymmetry), where we have also written N in terms
of SYM theory quantities. We also performed the trivial integration over the field theory
directions and divided both sides by this (infinite) volume, so now SD7is actually an action
density. We will use that convention in what follows. More explicitly, for our ansatz the DBI
action is
SD7= −N
?
drcos3θ(r)gxx
?
|gtt|gxxgD7
rr− (2πα′)2?
gxxA′2
t+ gD7
rr
˙A2
x− |gtt|A′2
x
?
, (2.8)
4Throughout the paper we use a gauge in which Ar = 0.
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where dots denote
The equations of motion for the gauge fields are trivial, since the action depends only on
the derivatives A′t(r) and A′x(r). We thus obtain two constants of motion, which in terms of
field theory quantities are simply ?Jt? and ?Jx?. Explicitly, we have5
∂
∂t. We define a Lagrangian L via SD7= −?drL.
?Jµ? =
δL
δA′µ
. (2.9)
These are two equations (for ?Jt? and ?Jx?) for two unknowns (A′t(r) and A′x(r)), hence we
can solve just algebraically for A′t(r) and A′x(r). These solutions appear explicitly in ref. [14].
We obtain θ(r)’s equation of motion by varying the DBI action eq. (2.8) and then inserting
the solutions for A′t(r) and A′x(r). When the density is zero, ?Jt? = 0, but the temperature is
finite, two topologically-distinct classes of embedding are possible [25, 26]. In the first class,
called “Minkowski embeddings,” at the boundary r → 0 the D7-brane wraps the equatorial
S3⊂ S5, but as the D7-brane extends into the bulk the S3shrinks and eventually “slips off”
the S5, collapsing to a point at some value of r, which we will call rΛ, outside of the horizon
(so rΛ< rH). From the AdS point of view, the D7-brane simply ends at rΛ. In the other
class of embeddings, called “black hole embeddings,” the S3shrinks but does not collapse,
and the D7-brane intersects the horizon. As shown in ref. [27], with a finite density ?Jt?, such
that At(r) is non-trivial, only black hole embeddings are allowed.
We can extract the mass m of the flavor fields from the coefficient of the leading term
in θ(r)’s asymptotic expansion, as explained in ref. [23]. Generically, solving the equation of
motion for θ(r) and hence extracting the value of m, when the temperature and density are
finite, requires numerics. Fortunately, we know the result for θ(r) in the two limits of zero
and large mass. Zero mass corresponds to the trivial embedding, θ(r) = 0, in which case the
D7-brane wraps the equatorial S3for all values of r. If we take the mass to be much larger
than any other scale in the problem, then θ(r) →π
such that At(r) is zero on the D7-brane worldvolume, then the D7-brane ends very close to
the boundary, i.e. rΛ→ 0. At finite density, the D7-brane still has to reach the horizon.
In that case the D7-brane forms a “spike” [27]: the D7-brane almost ends at some rΛ, but
then wraps a small S3of approximately constant volume all the way down to the horizon. In
other words, θ(r) is nearly constant, almost but not quite equal to π/2, along the spike. In
what follows we will either use the analytic solutions θ(r) = 0 or θ(r) → π/2, or we will find
approximate solutions for θ(r) in certain limits, as we do in section 4.
2. Here, if the density is zero, ?Jt? = 0,
2.2 The Null Melvin Twist
Following refs. [10, 11, 12], we now apply the NMT to the supergravity solution in eq. (2.1).
The NMT is a species of TsT (T-duality, shift, T-duality) transformation that produces
new supergravity solutions from old ones. The input is some solution with two commuting
U(1) isometries. The output is a new solution with different asymptotics. In our case, we
5For a rigorous derivation of ?Jµ?, see ref. [14].
– 7 –
Page 9
will begin with AdS5-Schwarzschild times S5, using the Hopf fiber direction χ and the field
theory spatial direction y as isometry directions, and find a new solution for which the metric
is asymptotically Schr¨ odinger. The steps of the NMT are:
1. Boost by an amount γ in y
2. T-dualize in y
3. Shift in the χ direction dχ → dχ + αdy
4. T-dualize in y
5. Boost by −γ in y
6. Take a limit: γ → ∞ and α → 0 keeping β =1
Steps 2, 3, and 4, are the TsT part of the NMT. We will not write the explicit result
of each step (for that, see ref. [12]), but we will make some generic comments about each
step. The first step changes gttand gyyand generates a dy dt term in the metric. The second
step produces a nontrivial NS B-field, with dy ∧dt component, and dilaton, and also changes
gttand gyy. The third step changes gyyand also introduces a dχ + A and dy cross-term in
metric, but leaves the B-field and dilaton unchanged. The fourth step changes gtt, gyy, gty,
and the (dχ + A)2term in the metric, and generates a (dχ + A) ∧ dy term in B. In the end,
the metric that results from the NMT is asymptotically Sch5. Explicitly, the final result for
the metric is
2αeγfixed
ds2=
1
r2
?dr2
f(r)− f(r)1 + β2r−2
K(r)
dt2+1 − β2r−2f(r)
K(r)
dy2−2β2r−2f(r)
K(r)
dtdy + d? x2
?
+
1
K(r)(dχ + A)2+ ds2
CP2,
=
1
r2
?
dr2
f(r)−
f(r)
r2K(r)dx+2+
2
K(r)dx+dx−+1 − f(r)
2K(r)
?dx+
√2β−√2βdx−
?2
+ d? x2
?
+
1
K(r)(dχ + A)2+ ds2
CP2, (2.10)
where
f(r) = 1 −r4
r4
H
,K(r) = 1 +β2r2
r4
H
, (2.11)
and in the second equality of eq. (2.10) we have introduced light-cone coordinates X±,
X+= t + y,X−=1
2(−t + y), (2.12)
which we then rescaled by factors of β to produce the light-cone coordinates x±,
x+= β (t + y),x−=
1
2β(−t + y). (2.13)
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Page 10
We discuss the utility of this rescaling at the end of this subsection. The solution also includes
the NS two-form field
B = −
β
r2K(r)(dχ + A) ∧ (f(r)dt + dy),
1
2r2K(r)(dχ + A) ∧?(1 + f(r)) dx++ (1 − f(r))2β2dx−?
Φ = −1
Notice that if we take β → 0 then we recover the pre-NMT solution.
The metric in eq. (2.10) has a horizon at r = rH, with some associated Hawking temper-
ature T. We obtain T = 0 by sending rH→ ∞, which sends f(r) → 1 and K(r) → 1. The
resulting metric is then the metric of Sch5(not just asymptotically Sch5), which is essentially
the metric of AdS5× S5with an extra term −dx+2/r4,
1
r2
1
r2
= −
(2.14)
and a dilaton
2logK(r). (2.15)
ds2=
?
?
dr2−β2
dr2−1
r2dX+2+ 2dX+dX−+ d? x2
?
+ ds2
+ ds2
S5
=
r2dx+2+ 2dx+dx−+ d? x2
?
S5
(2.16)
Notice that the dx+2term diverges faster as r → 0 than the metric of AdS5. When T = 0
the geometry includes an S5, however, the NS B-field breaks the SO(6) isometry of the S5
down to SU(3) × U(1), which is the isometry group of CP2. The T = 0 solution also breaks
all supersymmetry6[11] and has a singularity at r = ∞ [12]. When T is finite the geometry
is only asymptotically Sch5, the S5is deformed, the singularity is hidden behind a horizon,
and the dilaton becomes non-trivial.
What is the field theory dual to type IIB supergravity on Sch5? Equivalently, we can ask
what field theory operation is dual to the NMT? Put briefly, the NMT is dual to adding an
irrelevant operator to the N = 4 SYM theory Lagrangian. We can easily see this as follows.
Given the solution above, if we perform a Kaluza-Klein reduction7on the S5the NS B-field
gives rise to a massive vector in Sch5whose dual operator is a vector of dimension five, in the
antisymmetric tensor representation of the SU(4) R-symmetry. The dual operator, which we
will denote Oµ, is a linear combination of operators of the form [10]
?
K
OIJ
µ = TrFν
µΦ[IDνΦJ]+
?
DµΦKΦ[KΦIΦJ]
?
+ fermion terms, (2.17)
where ΦIare the adjoint scalars of N = 4 SYM transforming in the 6 of SU(4), Fµν is
the field strength, and Dµis the covariant derivative. To be precise, recall that the 15 of
6Supersymmetric Sch5 solutions do exist [28, 29], obtained by using different directions of the S5(besides
the Hopf fiber) in the TsT transformation, which are very similar in form to the solution above. We will leave
a thorough analysis of probe branes in those backgrounds for the future.
7For details of the reduction, which is in fact consistent, see ref. [11].
– 9 –
Page 11
SU(4) decomposes into representations of SU(3) as 15 = 8 + 3 +¯3 + 1, so that we can
write Oµ = MIJOIJ
the 1 of SU(3). In short, the NMT generates an NS B-field whose presence indicates an
irrelevant deformation of N = 4 SYM: we have added O+to the N = 4 SYM Lagrangian.
Indeed, adding O+breaks the relativistic conformal group down to the algebra of generators
that commute with P+∝ P−, producing the Schr¨ odinger group, and breaks the SU(4) R-
symmetry down to SU(3)×U(1). As an irrelevant deformation, we also expect the geometry
to be deformed near the boundary, which is indeed the case: we see explicitly in eq. (2.16)
that the effect of the deformation (the β2term) grows near the boundary r → 0.
The number generator N of the Schr¨ odinger algebra is dual to the isometry of the x−
direction. If we want the eigenvalues of N to be discrete, we must compactify x−, that is,
we must perform a DLCQ. When T = 0 a DLCQ of the above geometry produces a null
circle. Any closed strings that wrap the null circle will be massless, hence when T = 0 and
we compactify x−the supergravity approximation becomes unreliable [11]. As emphasized in
ref. [11], if the spacetime has momentum in the x−direction then the x−circle is no longer
null, and supergravity is reliable in most of the spacetime, although x−becomes null again
near the boundary r → 0.
The finite-T solution in fact has x−momentum, since the NMT involves boosts, so with
finite T the x−direction is no longer null (as is obvious from eq. (2.10)). The dual field
theory is in a state with a finite number density N, or equivalently a finite chemical potential
[10, 12]. As shown in refs. [10, 12], the field theory temperature T and chemical potential8µ
are
T =
πrH
The dual field theory has the correct equation of state for a scale-invariant, non-relativistic
theory with z = 2 in two spatial dimensions (here we are performing a DLCQ), ǫ = P, with ǫ
the energy density and P the pressure. The NMT does not change the area of horizons [30],
so the metrics in eqs. (2.1) and (2.10) have the same horizon area, although the conversion
to field theory quantities and interpretation differs in the two cases. Of central importance
is the fact that the entropy, and other thermodynamic quantities, such as the free energy
density, scale with negative powers of µ/T, and hence diverge in the limit µ/T → 0. Such
odd singular behavior appears to be a direct consequence of the DLCQ: exactly the same
scalings occur in a gas of non-interacting, non-relativistic Kaluza-Klein particles [31]. Type
IIB supergravity in Schr¨ odinger spacetime is apparently not dual to a theory of fermions at
unitarity.9
µ
where MIJ is an SU(4) matrix that, after the decomposition, is in
1
1
β,
µ = −
1
2β2. (2.18)
8For us µ will always denote the chemical potential associated with the U(1) along the compact x−, not
the chemical potential associated with the U(1) baryon number of the flavor fields.
9Curiously, however, at T = 0 and µ = 0, the three-point functions computed holographically from Sch5
agree exactly, up to normalization, with the three-point functions of fermions at unitarity [32, 33]. As in a
relativistic conformal theory, the Schr¨ odinger symmetry fixes the form of two-point functions but the three-
point functions are only partially fixed, and so contain dynamical information. The fact that the Sch5 result
agrees with fermions at unitarity is thus a non-trivial statement about the dynamics of the dual theory.
– 10 –
Page 12
Notice that the bulk theory is relativistic, so that under a scale transformation the coor-
dinates transform as r → λr, t → λt, ? x → λ? x for some real positive number λ. The parameter
β has units of length and hence scales as β → λβ. Rescaling the X±in eq. (2.12) by powers of
β produces the light-cone coordinates x±in eq. (2.13), such that under scalings x+→ λ2x+
while x−is invariant. Once we perform the DLCQ and interpret x+as the time coordinate,
the resulting theory indeed exhibits the scaling of eq. (1.1) with z = 2. The fact that x−
is invariant indicates that the conjugate momentum P−is also invariant, which makes sense:
after DLCQ we identify P−with the number operator N, which is a central element of the
algebra, and in particular must commute with the dilation generator.
As reviewed for example in ref. [20], for a theory with d spatial dimensions, if we assign
momentum to have scaling dimension one, then for a given value of z we have the following
scaling dimensions for a density ?Jt?, current ?Jx?, electric field E, magnetic field B, chemical
potential µ and temperature T:
??Jt??= d,[?Jx?] = d + z − 1,[E] = z + 1,[B] = 2,[µ] = [T] = z. (2.19)
From Ohm’s law, ?Jx? = σE, we find that the conductivity has dimension [σ] = d − 2. Our
system has z = 2 and, after DLCQ, d = 2. From eq. (2.18) we see that the factors of β are
essential to produce a T and µ with scaling dimension two.
In the field theory we will also have background gauge fields, dual to the gauge fields
on the D7-branes, such as the (relativistic) electric field Ftx=˙Axin eq. (2.8). Here again,
appropriate factors of β will produce gauge fields with the correct scaling dimensions. From
eq. (2.18) we have β ∝ (−µ)−1/2, so we may interpret all rescalings by powers of β as
rescalings by appropriate powers of the chemical potential µ. Recalling that the gauge field
is a one-form, we have
Atdt + Aydy =
1
2β(At+ Ay)dx++ β(−At+ Ay)dx−≡ A+dx++ A−dx−, (2.20)
so that A+ has scaling dimension two while, after DLCQ, A− is a dimensionless scalar.
An electric field F+x = ∂+Ax− ∂xA+will then indeed have scaling dimension z + 1 = 3.
Recalling the relativistic coupling AµJµ, from the coupling A+J+we see that J+will have
scaling dimension two, so that A+and J+have the correct scaling dimensions of a chemical
potential and charge density, respectively, for z = 2 and d = 2. The coupling A−J−indicates
that after DLCQ J−will be a scalar with scaling dimension four.
2.3 Twisting with Probe D7-branes
We now ask what happens to our probe flavor when we perform the NMT. The field theory
side is easy, so we start there: we simply write the Lagrangian of N = 4 SYM theory coupled
to massive N = 2 supersymmetric hypermultiplets in the fundamental representation of the
gauge group, and then add the operator O+. The flavors break the SU(3) × U(1) symmetry
to the same SO(4) × U(1) as in the relativistic case. In the probe limit, massless flavors will
preserve the Schr¨ odinger symmetry, while a finite mass will explicitly break scale invariance.
– 11 –
Page 13
Now we ask what happens on the gravity side, that is, we ask what happens to probe D7-
branes when we perform the NMT. The effect of the boost in steps 1 and 5 is straightforward.
In the T-dualities of steps 2 and 4, the D7-branes are converted into D6-branes and then back
to D7-branes. The component Ayof the worldvolume gauge field is converted into a scalar,
Φy, which is then converted back into Ay. Crucially, however, the DBI action is consistent
with T-duality [34]. That means that when we T-dualize, the metric, NS B-field, and dilaton
may change, and we replace Ay→ Φy, but the quantity
L ≡ e−Φ?
evaluates to the same function of r, though now with Φyreplacing Ay. If Ayis non-trivial,
so that after T-duality Φyis non-trivial, then the shift in step 3 may change the pullback of
the metric to the D6-branes, and hence potentially change L. If Ayis trivial, however, then
the TsT part of the NMT transformation leaves L unchanged.
The most general statement we can make is: if L is initially invariant under boosts in
the y direction, then the entire NMT has no effect on L. In such cases the boosts in steps
1 and 5 and the TsT transformation each individually leave L unchanged. For boosts to
be a symmetry requires T = 0, and all worldvolume fields must be invariant under boosts
in y.10For example, we may introduce the worldvolume scalar θ(r), as well as the gauge
field Ax(r), both of which are clearly invariant under boosts in y. In the field theory we will
have flavor fields with a finite mass and some current in the x direction, ?Jx?.11Using the
asymptotically Schr¨ odinger background and these worldvolume fields, we find that the action
SD7is identical to the asymptotically AdS case in eq. (2.8), with A′t=˙Ax= 0. We may also
add field strengths describing electric and magnetic fields pointing in the y direction, such as
Fty, which are invariant under boosts12. The NMT leaves L invariant in all such cases.
A number of conclusions follow from the invariance of SD7for y-boost-invariant config-
urations. For example, suppose that, at T = 0, we introduce only the worldvolume scalar
θ(r). After the NMT we will find exactly the AdS result, eq. (2.8), with all gauge fields
set to zero. The equation of motion for θ(r) is then identical to the AdS case, hence the
solution is also identical: θ(r) = arcsin(cr), where c is a constant that determines the mass m
of the flavor fields via m = c/(2πα′) [23].13The counterterms written in ref. [38], needed to
det(P [G + B]ab+ (2πα′)Fab), (2.21)
10In the Sakai-Sugimoto model [35], which is a system of intersecting D4-branes and D8-branes, the NMT
seems to have no effect on the probe D8-branes’ action, even in the black hole background and with nonzero
At(r) [36].
11As mentioned in ref. [37] in the field theory at T = 0 such a current will not dissipate. We may introduce
it simply as an external parameter.
12If we introduce an electric field Fty in the field theory, then we expect a resulting current in the y direction,
?Jy?, which breaks the boost invariance. A bulk solution with Fty and no ?Jy? would probably be pathological
(exhibiting an instability of the kind that we will discuss in section 3.1, for example).
13In AdS the θ(r) = arcsin(cr) solution is supersymmetric, but here the background breaks all supersym-
metry already, so we need not bother checking the supersymmetry of the D7-branes’ embedding. A good
question, though, is whether supersymmetric embeddings could be found for the supersymmetric Schr¨ odinger
solutions of refs. [28, 29].
– 12 –
Page 14
render the on-shell action finite, are then also identical to the relativistic case. Furthermore,
in the T = 0 AdS case, solutions with nonzero θ(r) and Ax(r) were found in ref. [37]. These
solutions will also be identical for D7-branes in Sch5.
Given that the embedding of the D7-branes, θ(r), is identical in the T = 0 AdS and
Schr¨ odinger cases, a natural question is whether the spectra of linearized fluctuations of
worldvolume fields are also the same. These spectra are dual to the spectra of mesons in
the field theory. In general, the spectrum of mesons will not be the same. The simplest
way to see that is to consider the fermionic mesons, dual to fermionic fluctuations of the
D7-branes [39]. The linearized equation of motion for these fermionic fluctuations is simply
the Dirac equation. The Dirac operator is different in AdS and Schr¨ odinger spacetimes [40].
More generally, the differential operators appearing in the fluctuations’ equations of motions,
for example the scalar Laplacian, will differ from their AdS counterparts, so the spectrum
will generically be different. Some subsector of the meson spectrum may be unchanged,
for example the sector with zero momentum in x−and zero charge under the R-symmetry
that corresponds to the Hopf fiber isometry. We leave a detailed investigation of the meson
spectrum for the future.
When T is nonzero the NMT changes L, even when all the worldvolume fields are trivial.
Introducing non-trivial worldvolume fields will then obviously not restore L to its AdS form.
In what follows, we will be interested in finite T solutions with worldvolume fields that are
not invariant under boosts in y, such as electric fields Ftx, so we will not be able to exploit
the T = 0 invariance of L.
In what follows we will study transport. We should, however, first study thermodynamics,
to determine the ground state of the system for all values of the parameters (mass, density,
etc.). In the relativistic case, a variety of phase transitions do indeed occur as the parameters
change (see refs. [25, 27, 41, 42, 43, 44, 45, 46] and references therein). We will leave a detailed
analysis of the non-relativistic case for the future. In the following, in the field theory we will
always assume that the D7-branes intersect the horizon, hence our results for the conductivity
will only be valid when such D7-branes describe the ground state of the field theory.
3. DC Conductivity of Probe Flavor
In this section we will compute (holographically) a DC conductivity associated with transport
of baryon number charge. We will use the method of ref. [14], which captures effects beyond
those of linear response. Our background spacetime will be asymptotically Sch5rather than
asymptotically AdS5. One of the major differences between these is that in Sch5we want
to use light-cone coordinates x±, compactify x−, and in the dual non-relativistic theory
interpret x+as the time coordinate. To understand how these operations affect the result for
the conductivity, we first repeat the calculation of ref. [14] in the DLCQ of AdS5× S5and
then turn to the Schr¨ odinger case.
– 13 –
Page 15
3.1 In the DLCQ of AdS
Consider the trivial solution of type IIB supergravity: the metric is simply (9+1)-dimensional
Minkowski space. Now introduce Nfcoincident probe D7-branes,14and consider a solution
in which the only nontrivial worldvolume field is a constant U(1) electric field Ftx= −E. As
in eq. (2.6), the DBI action assumes the characteristic (3+1)-dimensional form,
SD7= −NfTD7
?
?
d8x
?
?
1 −1
2
˜F2−1
4
?˜F ∧˜F
?2
= −NfTD7
d8x
1 − (2πα′)E2.(3.1)
Clearly when the electric field is greater than the string tension, E > 1/(2πα′), the DBI action
becomes imaginary. That signals the well-known tachyonic instability of open strings in an
electric field [47, 48, 49]. The electric field pulls the endpoints of an open string in opposite
directions. When the electric field is big enough to overcome the tension of the string, it rips
the string apart. Another way to say the same thing is that the electric field reduces the
effective tension of open strings. The instability appears when that effective tension becomes
negative. Notice that if we additionally introduce a magnetic field Fxy= B orthogonal to the
electric field, then the DBI action becomes
SD7= −NfTD7
?
d8x
?
1 − (2πα′)(E2− B2). (3.2)
If E > B we can boost to a frame in which the magnetic field vanishes, and the arguments
above still apply. If E ≤ B then the instability never appears. Indeed, in that case we can
boost to a frame where the electric field is zero.
Now instead of flat space consider AdS5-Schwarzschild times S5, as in eq. (2.1). Here the
effective tension of strings already decreases as a function of r, going to zero at the horizon.
Probe D7-branes with a constant worldvolume electric field will reduce the effective tension
by the same amount at every value of r. We thus expect that for any nonzero E the effective
tension will go to zero at some radial position r∗ outside the horizon, and to be negative
between r∗and the horizon. With an asymptotically AdS space, however, we have a dual
field theory, so we can use our field theory intuition to guess the endpoint of the instability.
The endpoint of a string looks like a quark. The electric field ripping a string apart should
look like a Schwinger pair-production process. We should thus see a current. These arguments
lead us to the ansatz of section 2.1, for which the DBI action appears in eq. (2.8).
The observation of ref. [14] was that, with˙Ax= −E, the action in eq. (2.8) depends
only on r derivatives of Atand Ax, hence the equations of motion for the gauge fields are
trivial. We obtain two “constants of motion,” that is, r-independent quantities, the currents
?Jt? and ?Jx? in eq. (2.9). If we think of eq. (2.9) as two equations for two unknowns, A′tand
14Strictly speaking, to avoid constraints on the number of D7-branes we should set the string coupling to
be precisely zero. Many of the arguments that follow rely only on the form of the DBI action, rather than any
properties unique to D7-branes, however.
– 14 –
Page 16
A′x, then we may solve algebraically for the unknowns. We thus obtain solutions for A′tand
A′xin terms of E, ?Jt?, ?Jx?, metric components, and θ(r). These solutions appear explicitly
in ref. [14]. Plugging these back into the action, we obtain
SD7∝ −
?
drcos6θ(r)g5/2
xx|gtt|1/2g1/2
rr
?
|gtt|gxx− (2πα′)2E2
xxcos6θ(r) + |gtt|?Jt?2− gxx?Jx?2,
N2(2πα′)2|gtt|g3
(3.3)
where we use the notation of section 2.1 and drop some constant prefactors. We are assuming
that the D7-brane intersects the AdS5-Schwarzschild horizon, as must be the case when ?Jt?
is nonzero, hence the r integration runs from the horizon to the boundary. We can see the
instability as follows. Consider the fraction under the square root. At the horizon |gtt| = 0,
so the numerator is negative. At the boundary r → ∞, |gtt|gxxdiverges as r4, hence for any
finite E the numerator becomes positive. The numerator must have a zero in between, which
in fact occurs precisely at r∗,
?|gtt|gxx−?2πα′?E2?
r∗= 0. (3.4)
Notice that when E = 0, the above equation implies r∗= rH, and that as E increases, r∗
moves closer to the boundary. In other words, with a larger E we can probe UV physics (r∗is
close to the boundary) while with a small E we probe IR physics (r∗is close to the horizon).
We also included a current in our ansatz, which allows the denominator under the square
root to have a similar sign change: at the horizon the denominator is negative, but at the
boundary it is positive. We are thus able to avoid an imaginary DBI action (to avoid the
instability) if we demand that the denominator and numerator change sign at the same place,
r∗. We thus require
?
N2?2πα′?2|gtt|g3
xxcos6θ(r∗) + |gtt|?Jt?2− gxx?Jx?2?
r∗= 0. (3.5)
The value of E determines r∗via eq. (3.4), and eq. (3.5) then determines the unique value of
?Jx? that prevents the instability.15Converting to field theory quantities, we find ?Jx? = σE,
where
?
16π2
σ =
N2
fN2
cT2
?
e2+ 1cos6θ(r∗) +
d2
e2+ 1, (3.6)
where
e =
E
π
2
√λT2,d =
?Jt?
√λT2.
π
2
(3.7)
To complete our solution, we must solve numerically for the final worldvolume field, θ(r).
That has been done in refs. [50, 51]. When we compute the DC conductivity we will always
use analytic solutions in the limits of zero and large mass explained at the end of section 2.1.
15From a field theory point of view, we choose the mass, temperature, charge density, and the electric field,
and the dynamics of the theory then determines the system’s response, i.e. the resulting current.
– 15 –
Page 17
The result in eq.
identify the physical origin of each term, following refs. [14, 15, 16, 52]. The second term,
proportional to d2, describes the contribution to the current from the charge carriers we
introduced explicitly via a nonzero ?Jt?. The first term, proportional to cos6θ(r∗), describes
the contribution to the current from charge-neutral pairs. The microscopic process producing
these pairs is not immediately clear. As in ref. [52], however, we can take a limit with ?Jt? = 0,
T = 0, m = 0 (which means cosθ(r∗) = 1), in which case we find a finite conductivity σ ∝√E.
In that limit, the electric field is the only scale in the problem, hence the pair production
must occur via a Schwinger process. When T is finite, however, thermal pair production may
also occur. The first term preumably knows about both kinds of pair production.
Notice also that the result for the conductivity depends explicitly on the electric field,
and may also have implicit dependence through cosθ(r∗). In the regime of linear response we
expect, essentially by definition of linear response, the current to be linear in E and hence the
conductivity to be independent of E. Here we are capturing effects beyond linear response.
Ultimately we can do so because the DBI action sums all orders in (2πα′)Fab. Translating to
field theory quantities, that means the result for σ actually accounts for all orders in E/√λ.
Now suppose we want to perform a DLCQ both in the bulk and in the field theory. To
do so, we first write the AdS part of the metric in eq. (2.1) in the light-cone coordinates X±
of eq. (2.12),
(3.6) consists of two terms adding in quadrature. We can easily
ds2= grrdr2+ g++dX+2+ g−−dX−2+ 2g+−dX+dX−+ gxxd? x2
1
r2
=
?dr2
f(r)+14(1 − f(r))dX+2+ (1 − f(r))dX−2+ (1 + f(r))dX+dX−+ d? x2
?
,
Notice that when T = 0 and f(r) = 1, the metric, and its inverse, in the light-cone directions
is strictly off-diagonal, g++= g−−= 0 and g++= g−−= 0.
After the DLCQ, we interpret X+as the new time coordinate. The boundary value of
the D7-brane worldvolume field A+ acts as a source for the field theory operator J+. In
the DLCQ we assume all physical quantities are independent of X−, that is, that ∂−acting
on any quantity gives zero. After the DLCQ, the relativistic equation for conservation of
the current, ∂µ?Jµ? = 0, reduces to ∂+?J+? + ∂i?Ji? = 0, with i the index for the spatial
directions. We thus interpret ?J+? as the charge density after DLCQ.
To study states in the field theory with finite ?J+? our ansatz for the worldvolume fields
will always include A+(r), or equivalently F+r(r) = −A′+(r). Suppose for the moment we
also introduce A−(r). The DBI action will then involve terms of the form
1
2g˜F2⊃ grr
D7g++A′
+(r)2+ grr
D7g−−A′
−(r)2+ 2grr
D7g+−A′
+(r)A′
−(r). (3.8)
Taking variational derivatives and using eq. (2.9), we see that a nontrivial A′+(r) not only
produces a finite ?J+? in the field theory but also a finite ?J−?. In other words, in the field
theory, if we introduce ?J+?, we must introduce ?J−?. We will discuss the meaning of this,
from the field theory point of view, shortly. Furthermore, when T = 0 and g++= 0, a
– 16 –
Page 18
nontrivial A′+(r) produces only a nonzero ?J−?. To obtain a nonzero ?J+? at T = 0, we will
thus also introduce A−(r). Notice that if we return to the original coordinates, with just
At(r), then we obtain both A+(r) and A−(r).
We also want a constant electric field, which after DLCQ should be F+x= −E. Introduc-
ing F+xalone doing does not produce an instability of the DBI action, however. When T = 0,
for example, the inverse metric component g++= 0, hence if we introduce only F+xthen the
DBI action does not depend on the electric field at all, since g˜F2∝ gxxg++F2
ing back to the original coordinates reveals what is happening: F+xdescribes perpendicular
electric and magnetic fields Ftxand Fyxof equal magnitude, such that g˜F2∝ E2− B2= 0.
We will thus also introduce F−x, in which case the DBI action depends on both F+x and
F−x, and exhibits the expected instability. Introducing both F+x and F−x is the same as
introducing Ftxand Fxy. Similarly to the story with A+(r) and A−(r), in what follows we
will begin with Ftxand then switch to light-cone coordinates.
In summary, our ansatz for the worldvolume gauge field will be identical to the relativistic
case, with Frt= A′t(r), Frx= A′x(r) and constant Ftx, but converted to light-cone coordinates.
The bulk field A−is dual to the field theory operator J−. Given that we will be working
with a nontrivial A−(r) in the bulk and states with nonzero ?J−? in the field theory, a natural
question is, from the field theory point of view, what is J−?
After the DLCQ, A−is a bulk scalar and J−is a scalar operator. To gain some intuition
for the physical meaning of A−and J−after DLCQ, consider a complex scalar field in the
relativistic theory that carries the U(1) charge associated with the current Jµ, which in our
case means the scalars in the N = 2 hypermultiplet. Consider in particular the kinetic terms,
written in light-cone coordinates and with a covariant derivative Dµ= ∂µ− iqAµinvolving
the background gauge field Aµ, with q the charge of the scalar under the U(1). Explicitly, we
have (here ∂±are derivatives with respect to X±)
+x= 0. Convert-
gµν|DµΦ|†DνΦ = g+−|D+Φ|†D−Φ + g−+|D−Φ|†D+Φ + ...
= ∂+Φ†(∂−− iqA−)Φ + (∂−+ iqA−)Φ†∂+Φ + ....
If we work with fixed X−momentum N,
Φ?X−,X+,? x?= e−iNX−φ?X+,? x?, (3.9)
then we obtain
gµν|DµΦ|†DνΦ = (N + qA−) i
?
φ†∂+φ −
?
∂+φ†?
φ
?
+ .... (3.10)
Recalling that after DLCQ we interpret N as the “particle number” quantum number,16, we
see that a nonzero A−looks like a shift in the particle number N. Indeed, that will be true
for any U(1): if we introduce a nonzero A−, any fields charged under the U(1) will appear to
have a shifted N. That makes sense since, after DLCQ, a nonzero A−will produce a Wilson
16Here N denotes the eigenvalue of the number operator, which we also called N above.
– 17 –
Page 19
loop in the x−direction, effectively shifting the momentum P−, and hence shifting N. Both
N and qA−couple to the operator representing x+momentum,
P+= i
?
φ†∂+φ −
?
∂+φ†?
φ
?
. (3.11)
As mentioned in the introduction (and ref. [6]), after the DLCQ, P+plays the role of the
Hamiltonian in the Schr¨ odinger algebra.We thus see that A− couples to the operator
J−= qP+, or q times the Hamiltonian. The statement above that a nonzero ?J+? must
be accompanied by a nonzero ?J−? is thus easy to understand: from the perspective of the
non-relativistic theory, a finite density of particles must be accompanied by some energy.
3.2 In Schr¨ odinger Spacetime
Having explained the method for computing the DC conductivity, and some of the subtleties
of working in light-cone coordinates, we proceed to the case where the background spacetime
is the asymptotically Sch5metric of eq. (2.10).
First, we must be careful with factors of β (see the end of section 2.2). As explained
above, our ansatz is the same as in the relativistic case of section 2.1, with At(r), Ftx= −E,
and Ax(r), but converted to the (rescaled) light-cone coordinates x±of eq. (2.13). We also
rescale the gauge field components as in eq. (2.20), so that our A± obey non-relativistic
scaling. Explicitly, our ansatz for the worldvolume gauge field is
A+(x,r) = Eβx + h+(r),A−(x,r) = −2β2Eβx + h−(r),Ax= Ax(r), (3.12)
where we have redefined the electric field to be Eβ = E/(2β) such that Eβ scales non-
relativistically, i.e. [Eβ] = 3, and h±(r) are functions for which we most solve. We also recall
the other scaling dimensions of eq. (2.19), with d = z = 2,
[A+] = 2,[J+] = 2,[A−] = 0,[J−] = 4,[Ax] = 1,[Jx] = 3, (3.13)
and the conductivity is dimensionless, [σ] = d − 2 = 0.
We now want to insert our ansatz for the worldvolume fields into the DBI action, eq.
(2.5), using the background metric, B-field, and dilaton of eqs. (2.10), (2.14), and (2.15),
respectively. To write the action succinctly, let us introduce some notation. We define
Gi1...in≡
1
sin2α1
det(P[g + B]ab), a,b = i1,...,in, (3.14)
i.e. Gi1...inis the determinant of the n×n submatrix of P[g +B]abcontaining only rows and
columns indexed by i1,...,in. All such sub-determinants are functions of r times a factor of
sin2α1, hence we divide by sin2α1to make Gi1...ina function of r only. Similarly we define
the 3 × 3 submatrix determinant (divided by sin2α1):
GB≡
1
sin2α1
det
g+−
−B−α2gα2α2gα3α2
−B−α3gα2α3gα3α3
B+α2B+α3
. (3.15)
– 18 –
Page 20
Explicitly, the submatrix determinants we will need are
Gα2α3=cos2θ(r) + K(r)sin2θ(r)
16K(r)
cos4θ(r),
G+α2α3=r6− 4r4
Hβ2f(r)sin2θ(r)
64r4r4
Hβ2K(r)
1 + f(r)
32r2K(r)cos4θ(r),
cos4θ(r),G−α2α3=K(r) − 1
16K(r)
cos4θ(r),
GB=
G+−α2α3= −f(r)cos4θ(r)
16r4K(r)
. (3.16)
We note for later use that G+−α2α3vanishes at the horizon r = rhand otherwise is strictly
negative, while Gα2α3is strictly positive. We also define the shorthand notation
G3≡ G−α2α3+ 4β4G+α2α3+ 4β2GB=β2[r2− β2f(r)sin2θ(r)]
4r4K(r)
cos4θ(r). (3.17)
Plugging our gauge field ansatz eq. (3.12) into the D7-brane action, we obtain
SD7= −NfTD7
?
?
drd3αe−Φ?
−K(r)detMab,
−det(P [g + B]ab+ (2πα′)Fab)
= −N
?
dr
(3.18)
where the matrix Mabhas determinant
detMab≡ gxxgα1α1
?
+gxx
grr
?E2
?G−α2α3(A′
βG3+ gxxG+−α2α3
?+ G+−α2α3(A′
−)2− 2GBA′
x)2+ E2
βGα2α3(2β2A′
++ A′
−)2
+)2+ G+α2α3(A′
+A′
−
??
. (3.19)
Using eq. (2.9) we obtain the current components
?J+? =NKgxxgα1α1
√−KdetMab
?(2β2E2
βGα2α3− gxxGB)A′
−,+(4β4E2
βGα2α3+ gxxG−α2α3)A′
+
?
(3.20a)
?J−? =NKgxxgα1α1
?Jx? =NKgxxgα1α1
√−KdetMab
√−KdetMab
?(2β2E2
G+−α2α3A′
βGα2α3− gxxGB)A′
++ (E2
βGα2α3+ gxxG+α2α3)A′
−
?, (3.20b)
(3.20c)
x.
Solving eqs. (3.20) for A′±and A′xand plugging the solutions back into the action, we obtain
the on-shell action
SD7= −N2
?
drK(r)gxxg1/2
rrgα1α1
?
gxx|G+−α2α3| − E2
U(r) − V (r)
βG3
(3.21)
– 19 –
Page 21
where
U(r) =
?Jx?2
G+−α2α3
E2
+ N2K(r)gxxgα1α1,
βGα2α3(?J+? − 2β2?J−?)2+ gxx
(3.22a)
V (r) =
?G+α2α3?J+?2+ G−α2α3?J−?2+ 2GB?J+??J−??
gxxGα2α3(gxx|G+−α2α3| − E2
βG3)
.
(3.22b)
We now focus on the square root factor in the on-shell action in eq. (3.21), and demand
that the action remain real for all r, as in the relativistic case of section 3.1. First, notice
that as a function of r, the factor in the numerator, gxx|G+−α2α3|−E2
horizon, r = rH, and positive near the boundary r → 0, and hence must have a zero at some
r = r∗,
βG3, is negative at the
?gxx|G+−α2α3| − E2
Similarly, the denominator U(r)−V (r) is negative at the horizon and positive at the boundary,
and so must have a zero also at some value of r. As in the relativistic AdS case reviewed above,
the two zeros must coincide to avoid an imaginary action, so we require U(r∗) − V (r∗) = 0.
Now consider V (r), which has a factor of gxx|G+−α2α3| − E2
numerator of V (r) is finite at r∗, then V (r) will diverge at r∗. Notice, however, that U(r) is
not divergent at r∗. The only way to achieve U(r∗) − V (r∗) = 0 is thus to demand that the
numerator of V (r) vanish at r∗(at least as quickly as gxx|G+−α2α3| − E2
numerator of V (r) to zero at r∗, we obtain (after some algebra)
βG3
?
r∗= 0 =⇒
1
E2
β
=4r2
f(r∗)
∗β2
?r2
∗− β2f(r∗)sin2θ(r∗)?. (3.23)
βG3in its denominator. If the
βG3). Setting the
?J−? = −GB+ 2β2G+α2α3
2β2GB+ G−α2α3
????r∗
?J+?.(3.24)
Notice that eq. (3.24) has no explicit dependence on the current ?Jx?, and depends on the
electric field only implicitly through r∗. In the absence of the electric field, eq. (3.24) becomes
?J−? = −?J+?/(2β2), which is independent of the temperature. Recalling the statements at
the end of the last subsection, here we see that, indeed, we cannot introduce ?J+? without
also introducing ?J−?.
Now the condition U(r∗) = V (r∗), combined with eq. (3.24), fixes the current to be
?Jx?2= E2
β
?
N2Kgα1α1G3+
?
G2
3?J+?2
g2
xx(G−α2α3+ 2β2GB)2
?
r∗
?
(3.25)
= E2
β
f(r∗)
64E2
βr6
∗
N2cos6θ(r∗) +16?J+?2r2
∗f(r∗)
β4E2
β
, (3.26)
which gives the DC conductivity, σ, through Ohm’s law ?Jx? = σEβ,
?
64E2
σ =
N2f(r∗)
βr6
∗
cos6θ(r∗) +
f(r∗)2
4β4r4
∗E4
β
?J+?2.(3.27)
– 20 –
Page 22
Note that σ is dimensionless, as it should be. Notice that once we fix T, Eβand ?J+?, reality
of the action determines ?J−? and ?Jx?.
As in the relativistic case, the result for σ consists of two terms adding in quadrature.
Once again, the second term, proportional to ?J+?2, describes the contribution to the current
from the charge carriers we introduced explicitly via the net density ?J+?. The first term,
again proportional to cos6θ(r∗), appears to describe the contribution from charge-neutral
pairs. We suspect that, as in the relativistic case, these pairs come from Schwinger and/or
thermal pair production. From the field theory point of view, such pair production at first
seems counter-intuitive, since in a non-relativistic theory, the number of particles should not
change. Recall, however, that we are actually studying a relativistic theory which we deform
in two ways, first by introducing an irrelevant operator and then by performing a DLCQ.
From that perspective, nothing is wrong with pair production. Notice also that what plays
the role of the “number of particles” is the eigenvalue of N ∼ P−, the momentum in the x−
direction, which is indeed fixed.
We now take two different limits to explore the scaling of the conductivity. The two
limits depend on the relative strengths of Eβ and T. In the limit of a very weak electric
field, E2
conductivity takes the form
ββ2r4
H≪ 1, or equivalently Eβ≪ βT2, we have from eq. (3.23) r∗→ rH, and the
σ ≈
?
π2N2cos2θ(r∗)
16
T2β4+4?J+?2
π4β4T4=
?
π2N2cos2θ(r∗)
64
T2
µ2+16
π4
?J+?2
T2
µ2
T2, (3.28)
where in the second equality we have replaced β with µ using eq. (2.18). At low temperatures
or large masses, i.e. T/µ → 0 or cosθ(r∗) → 0, the pair-production term will be suppressed,
in which case the conductivity approaches
σ ≈
4
π2
?J+?
T
(−µ)
T
. (3.29)
In a scale-invariant theory with dynamical exponent z, the conductivity should behave as
σ ∼ ?J+?T−2/z. Here we see that, if we fix the chemical potential µ, then we obtain the
dynamical exponent of a relativistic theory, z = 1. On the other hand, if we hold fixed the
ratio µ/T as we vary the temperature, then we find non-relativistic scaling, with z = 2.
The opposite limit is strong electric field, E2
the mass is small, so that θ(r∗) ≈ 0 and hence sinθ(r∗) ≈ 0, then eq. (3.23) implies that
r∗∼ (2Eββ)−1/2, and we find
?
8
E2
ββ2r4
H≫ 1, or equivalently Eβ≫ βT2. If
σ ≈
N2cos6θ(r∗)
Eββ3+?J+?2
ββ2=
?
N2cos6θ(r∗)
83/2
Eβ
(−µ)3/2+ 2?J+?2(−µ)
E2
β
.(3.30)
If we set the mass and density to zero, so cosθ(r∗) = 1 and ?J+? = 0, then we obtain a
finite conductivity,
N
83/4
σ ≈
?
Eβ
(−µ)3/2, (3.31)
– 21 –
Page 23
We indeed find a nonzero current, which must come from Schwinger pair production. The
bulk mechanism is exactly the same as in the relativistic case: the worldvolume electric field is
ripping strings apart. If we fix the value of µ, then σ ∝?Eβ, which is the same scaling with
Eβ, however, then the conductivity is a constant, which we expect for a (2+1)-dimensional
theory with non-relativistic scale invariance, if the only scale is the electric field.
the electric field as in the T = 0 relativistic case. If we fix the ratio Eβ/(−µ)3/2as we vary
If the electric field is small, then the second term under the square root in eq. (3.30)
dominates. We then find
σ ≈?J+?(−2µ)1/2
Eβ
, (3.32)
Switching back to relativistic coordinates,
?J+? = β(?Jt? + ?Jy?) =
1
(−2µ)1/2(?Jt? + ?Jy?), (3.33)
we find that, for a fixed µ, the equation for the current becomes
?Jx? = ?Jt? + ?Jy?, (3.34)
which is similar to the AdS result at zero temperature (see Appendix A of ref. [52]). The
physics here is simply that at zero temperature the charge carriers are accelarated to the
speed of light, so the system is not really stationary. If we fix the ratio Eβ/(−µ)3/2, so that
we can use (−µ)1/2∼ E1/3
σ ∝?J+?
E2/3
β, then
β
, (3.35)
which is the appropriate dependence on the electric field for a two-dimensional non-relativistic
conformal theory with a finite density.
To summarize: of the two scales Eβand T, we can take one to be large relative to the
other. If we hold the larger scale fixed relative to the scale set by µ, then we obtain non-
relativistic scaling. If on the other hand we hold µ fixed and vary the larger scale, we obtain
relativistic scaling.
We can understand our results in terms of the geometry as follows. Consider for example
the case in which we can neglect the electric field, so the temperature is the larger scale.
The conductivity is then evaluated at r∗≃ rH. Fixing µ/T is the same as fixing rH/β. We
are thus probing the geometry on the scale of the deformation β, so intuitively we expect to
obtain non-relativistic behavior. Indeed, in this case the conductivity in (3.29) behaves in a
non-relativistic way. If instead we fix µ as we take T/µ → 0 (see the text above eq. (3.29)),
then the ratio rH/β → ∞ and the horizon enters the region where the geometry is similar to
AdS-Schwarzschild. In that case (3.29) indeed exhibits relativistic scaling.
– 22 –
Page 24
4. AC Conductivity of Probe Flavor
We now proceed to compute the frequency-dependent conductivity in the linear response
approximation. In the field theory we consider thermal equilibrium states with temperature
T and massive flavor fields with a finite density ?J+? and finite ?J−?, but now, unlike the last
section, no constant electric field.
In the holographic dual the system at equilibrium is described by probe D7-branes in the
asymptoticaly Schr¨ odinger black hole geometry of eq. (2.10), with nontrivial worldvolume
fields A+(r) and A−(r). We can obtain the solution for these in exactly the same way as the
last section: each has an associated constant of motion, ?J+? and ?J−? from eq. (2.9), so we
obtain equations similar to those in eqs. (3.20), which we then algebraically invert to find
A+(r) and A−(r) in terms of ?J+? and ?J−?. We will not present these solutions explicitly,
but we will record that, in the absence of a constant worldvolume electric field, we have from
the last section that r∗= rH and ?J−? = −?J+?/(2β2) (see eqs. (3.23) and (3.24)). Notice
also that, as always with a nontrivial A+(r) (or At(r)), the D7-branes must extend all the
way to the black hole horizon [27]. We will also consider a nontrivial embedding θ(r), whose
form we will discuss in detail below.
To obtain the conductivity in the regime of linear response, we consider a small, frequency-
dependent perturbation of the worldvolume electric field about the background solution
(which has A+(r), A−(r), and θ(r)),
Ax(x+,x−,r) = Re
?
e−iω(x++2β2x−)ax(r,ω)
?
. (4.1)
For simplicity we will work with zero spatial momentum. Here we note that, in fact, re-
moving all x−dependence does not qualitatively change our results for the behavior of the
conductivity with frequency.
To quadratic order, the Lagrangian density for the perturbation is
L2= α++f2
+x+ α−−f2
−x+ 2α+−f+xf−x− αrrf2
xr, (4.2)
where fαβ= ∂αAβ− ∂βAαis the field strength associated with the fluctuation in eq. (4.1).
The equation of motion for the fluctuation is then
a′′
x+α′rr
αrr
a′
x+ ω2α+++ 4β4α−−+ 4β2α+−
αrr
ax= 0.(4.3)
Here the α are r-dependent coefficients that depend on the background solutions for A+(r)
and A−(r) about which we are perturbing. To write these succinctly, let us introduce some
notation. We define
ks(r) ≡ 1 +sin2θ(r)β2r2
r4
H
,ρ(r) ≡ 1 + r2f(r)θ′2, (4.4)
Q2≡64?J+?2
β2
,γ2(r) ≡ cos6θ(r)ks(r) + Q2r4(r2+ β2sin2θ(r)). (4.5)
– 23 –
Page 25
The coefficients in the quadratic Lagrangian density are then
αrr(r) =
N
16r?ks(r)ρ(r)f(r)γ(r),
16r4
N
64β2r3f(r)γ(r)
α++(r) =
N
Hf(r)γ(r)β2r3?Q2r2r4
H+ cos6θ(r)??
?r6
?r4
H
ks(r)ρ(r),
α−−(r) =
?
Q2r8+ cos6θ(r)
r4
H
− 4sin2θ(r)β2f(r)
???
???
ks(r)ρ(r),
α+−(r) =
N
32rf(r)γ(r)
?
Q2r6− cos6θ(r)
r4
− 2
ks(r)ρ(r). (4.6)
We follow the now-standard procedure to compute transport coefficients, in the regime
of linear response, holographically (for a review see refs. [53, 54, 55]). We must first solve
the linearized equation of motion for the fluctuation ax(r,ω) with the boundary condition
that near the horizon the solution has the form of a traveling wave moving into the black
hole, i.e. an in-going wave. We then insert that solution into the action, which then acts as
a generating functional for field theory correlators. Taking two functional derivatives of the
on-shell action gives us the retarded Green’s function. We then extract the conductivity from
the retarded Green’s function via a Kubo formula. Ultimately, we find
σ(ω) ∝ lim
r→0
?αrra′x(r,ω)
ωax(r,ω)
?
, (4.7)
where, because we are primarily interested in σ(ω)’s scaling behavior with ω, we omit the
overall prefactor.
Crucially, notice that the result will depend on the background solution θ(r) describing
the embedding of the D7-brane. In general, with finite temperature T and density ?J+?, we
can only solve for θ(r) numerically. We leave a complete numerical solution for future work.
Here we will focus on regimes of physical interest, applying some approximations to obtain
analytic results for σ(ω)’s scaling with ω.
We are interested in a regime dominated by the physics of a zero-temperature “critical
point.” Remember that the Schr¨ odinger geometry interpolates between a UV critical point
with z = 2 and an IR critical point with z = 1. The scale that separates the two regimes is
the chemical potential (β in the metric). In order to eliminate thermal effects, we will work
in a limit where the temperature is much smaller than any other scale, which in particular
means ω ≫ T. We then expect that we can explore both regions by taking µ ≫ ω for the IR
regime and µ ≪ ω for the UV.
In the probe approximation the critical behavior is not spoiled by a nonzero mass or
charge density for the flavors. The sector described by the probe D7-branes is sensitive to
these quantities, however, so we expect deviations from scale invariance whenever ω ∼ ?J+?z/d
or ω ∼ m2. In order to avoid such deviations we will only explore frequencies below these
scales, which means ?J+?z/d≫ µ and m2≫ µ.
– 24 –
Page 26
Instead of computing the exact current-current correlator in the holographic description,
we will use the radial coordinate as an approximation to the frequency scale, following ref. [20].
We will call r0the reference scale around which we will give an estimate of the conductivity.
More precisely, we will define a “local conductivity” σ(ω,r0) to be the quantity in brackets
in eq. (4.7), evaluated at r = r0,
σ(ω,r0) ≡
?αrra′x(r,ω)
ωax(r,ω)
?
r0
. (4.8)
If we think about the charge carriers as strings attached to the brane, the length of the string
from r0to the horizon17(times the string tension) is, roughly speaking, the energy of the
charge carriers we are exciting when we apply an oscillating electric field, which is on the
order of 1/r2
0. The holographic conductivity evaluated at r0should thus give us a rough idea
of the response of the system to an external field with a fixed frequency of the order 1/r2
Notice that in such a picture we would expect to produce pairs if the energy is of the order
of, or larger than, the mass. That is another good reason why we only explore scales much
below the mass. Notice also that in the Lifshitz case of ref. [20], both σ(ω,r0) and σ(ω) had
the same scaling with frequency in the limit that ωrz≪ 1.
In terms of the quantities in our formulas, for frequencies small relative to the chemical
potential (the IR) we should set r0≫ β while for large frequencies (the UV) we should set
r0≪ β. In the low temperature limit we should send rH→ ∞ relative to any other scale,
and by large mass and density we mean rΛ≪ r0and ?J+? ≫ 1/r2
We will show in the following subsection that, with some assumptions, the D7-branes have
a very simple embedding that we can compute analytically in our regimes of interest. Using
our analytic results for θ(r), we will analytically compute the ω-dependence of σ(ω) in the
subsequent subsections and compare the ω scaling with the results from Lifshitz backgrounds,
eq. (1.2).
0.
0.
4.1 D7-brane Embeddings
To determine analytic forms for θ(r), we return to the DBI action of eq. (2.5) and insert our
ansatz for the worldvolume fields: A+(r), A−(r), and θ(r), using the background metric, NS
B-field and dilaton of eqs. (2.10), (2.14), and (2.15), respectively. The result will be precisely
eq. (3.18), if in eq. (3.18) we take Eβ= A′x(r) = 0. Writing SD7= −?drL (recall the text
below eq. (2.8)), we find
−L =
N
8r5cos3θ(r)
?
a0− b0cos2θ(r) + r2f(r)θ′2≡
N
8r5cos3θ(r)ˆL,(4.9)
17We are using black hole embeddings only, in which case the endpoint of a string at r0 would be free to
move along the D7-brane and into the horizon. Our statements about the physical meaning of r0 are meant
only to provide intuition.
– 25 –
Page 27
where for later convenience we defined a “reduced Lagrangian”ˆL. Here a0(r) and b0(r) are
functions of the U(1) field strength, not yet evaluated on any solution for the field strength.18
a0(r) = 1 + [1 + f(r)]r4A′
+(r)A′
−(r) + r2A′
−(r)2
?
f(r) −
r6
4β2r4
H
?
−r8β2
r4
H
A′
+(r)2
b0(r) = r2f(r)A′
−(r)2
(4.10)
The equation of motion for θ(r) is
?cos3θ(r)r3f(r)θ′
ˆL
?′
−sinθ(r)
r5
?
3cos2θ(r)ˆL −b0cos4θ(r)
ˆL
?
= 0. (4.11)
Solving for the gauge fields by inverting eq. (3.20), with the charge density fixed, and plugging
the solutions into a0(r) and b0(r), we find the on-shell values of a0(r) and b0(r),
a0(r) =cos6θ(r)ks(r)2
γ2(r)ks(r)
+Q2r4f(r)
γ2(r)ks(r)
?
β2cos2θ(r)
− r2θ′2
?
b0(r) = Q2β2r4f(r)
β2r4
r4
H
− β2cos(2θ(r)) + r2
?
1 +β4sin4θ(r)
r4
H
???
, (4.12)
γ2(r)
ρ(r)
ks(r),(4.13)
and the action evaluated on the solution is proportional toˆL = cos3θ(r)?ks(r)ρ(r)/γ(r).
As we reviewed in section 2.1, when At(r), or A+(r), is zero, the D7-brane can end at
some value of the radial coordinate rΛ, but when At(r) or A+(r) is nontrivial, the D7-brane
must extend all the way to the horizon, and has a spike when rΛ≪ rH. Along the spike
and far from the horizon, rΛ< r ≪ rH, θ(r) is approximately constant. Such a region in
r corresponds to scales below the mass gap of the charge carriers, where scale invariance is
approximately restored.
Let us consider the limits r ≪ rHand Qr3≫ 1, corresponding to low temperature and
large density, as explained in the last subsection. If we take the IR limit β ≪ r, we find
?
H
γ2(r) = cos6θ(r)
1 +β2sin2θ(r)r2
r4
?
+ Q2r4(r2+ β2sin2θ(r)) ≈ cos6θ(r) + Q2r6≈ Q2r6.
(4.14)
In the UV limit β ≫ r we find
γ2(r) ≈ cos6θ(r) + Q2r4β2sin2θ(r) ≃ Q2β2sin2θ(r)r4. (4.15)
The IR and UV limits of all the quantities that appear in the equation of motion eq. (4.11)
are similarly straightforward to determine. In addition, we can assume that r2θ′2≪ 1 as long
18Alternatively, we can derive θ(r)’s equation of motion by first solving for the gauge fields, plugging the
results into L, performing a Legendre transform with respect to the gauge fields, and then finding the Euler-
Lagrange equations of motion [27].
– 26 –
Page 28
as rΛ≪ r, so we will take 1 + r2fθ′2≈ 1, sinθ(r) ≈ sinθ0and cosθ(r) ≈ cosθ0with θ0a
constant.
We then find that in the IR limit, to leading order, the equation of motion for θ(r) eq.
(4.11) becomes
θ′′+β2sinθ0cosθ0
r4
while in the UV limit, the equation of motion becomes
= 0, (4.16)
?θ′
r
?′
+cotθ0
r3
= 0. (4.17)
The solutions in the IR and UV limits are
IRθ(r) = θ0−β2
θ(r) = θ0+1
6r2sinθ0cosθ0,
2cotθ0logr
(4.18)
UV
r0, (4.19)
where, following ref. [20], and as discussed above, we have introduced the reference scale
r0 obeying rΛ < r0 ≪ β, which makes the argument of the logarithm in the UV solution
dimensionless. Notice that in order to satisfy the condition r2θ′2≪ 1 in the UV solution,
we need |cotθ0| ≪ 1, meaning a narrow spike and probably a large mass gap rΛ≪ r0. The
expansion is also limited to a region around the reference scale r0, such that |log(r/r0)| ≪ 1.
Otherwise these results are consistent with all the approximations we have made.
4.2 AC conductivity in the IR
In the IR limit the equation of motion for the gauge field fluctuation, eq. (4.3), becomes
simply
a′′
ra′
with solutions
ax(r,ω) = a0
x+2
x+ 4β2ω2ax= 0, (4.20)
x
e±2iβωr
r
,(4.21)
with a0
xa constant. The in-going solution corresponds to the positive sign in the exponential.
Next we follow the procedure described in ref. [20] for probe branes in Lifshitz geometries.
As mentioned above, we will not directly apply the formula eq. (4.7) in our case because the
r → 0 limit takes us out of the regime of our approximations. Instead we compute the local
conductivity σ(ω,r0) at a reference scale r0such that βωr0≪ 1. Plugging the solution into
eq. (4.8) and expanding the result with βωr0≪ 1, we find that at leading order the local
conductivity is
σ(ω,r0) ∝N?J+?r0
2β
ω−1. (4.22)
The ω−1scaling is consistent with the result of ref. [20], our eq. (1.2), for the relativistic case
with dynamical exponent z = 1.
– 27 –
Page 29
4.3 AC conductivity in the UV
In the UV the equation of motion for the gauge field fluctuation eq. (4.3) also takes a simple
form. Using sinθ0≃ 1, we find
x+1
ra′
a′′
x+ 4r2ω2ax= 0. (4.23)
The general solutions are Bessel functions. The solution describing an in-going traveling wave
at the horizon is a Hankel function,
ax(r,ω) = a0
xH(1)
0
?ωr2?,(4.24)
where again a0
in ref. [20] for a fluctuation of a probe brane’s worldvolume gauge field in Lifshitz spacetime
with dynamical exponent z = 2. As before, we choose a cutoff r0such that ωr2
the solution into eq. (4.7) and expanding the result in powers of ωr2
xis a constant. Notice that eqs. (4.23) and (4.24) coincide with the equations
0≪ 1. Plugging
0, we obtain
σ(ω,r0) ∝N?J+?
16
?ω log?ωr2
0
??−1, (4.25)
which is indeed the same scaling with ω as obtained from a probe brane in a Lifshitz spacetime
with z = 2, see eq. (1.2).
In summary: taking the temperature scale to be very low and the mass and density
scales to be very high, we find that in the IR, meaning ω ≪ µ, the AC conductivity exhibits
relativistic scaling with frequency, while in the UV, meaning ω ≫ µ, we find that the AC
conductivity exhibits non-relativistic scaling with dynamical exponent z = 2. These results
clearly confirm our intuition on the bulk side, where the space is similar to AdS deep in
the interior but not asymptotically, and in the field theory, where we have introduced an
irrelevant deformation to N = 4 SYM that breaks the relativistic conformal group down to
the Schr¨ odinger group.
5. Discussion and Conclusion
Using gauge-gravity duality, we computed both DC and AC conductivities associated with a
finite density of charge carriers in a strongly-coupled theory with non-relativistic symmetry.
The theory was N = 4 SYM theory deformed by a dimension-five operator that breaks the
relativistic conformal group down to the Schr¨ odinger group, with dynamical scaling exponent
z = 2, and the charge carriers were comprised of a finite baryon density of massive N = 2
supersymmeric hypermultiplets. We found that, generally speaking, both the DC and AC
conductivities exhibited relativistic scaling (with temperature or frequency, respectively) in
the IR and non-relativistic scaling, with z = 2, in the UV. These results are in accord with our
expectations, given the origin of the non-relativistic symmetry via an irrelevant deformation
of the theory. For the future, we can think of many questions that deserve further research.
– 28 –
Page 30
First, a more detailed analysis of probe D-branes in Schr¨ odinger spacetime would be
extremely useful. Given background Sch5 geometries that preserve some supersymmetry
[28, 29], can we find supersymmetric embeddings of probe D-branes? What happens to the
many phase transitions that occur in the relativistic setting when the probe D-branes are
instead in Sch5?
Introducing a magnetic field on the worldvolume of a probe D-brane is straightforward,
and allows us to compute not only the Hall conductivity but indeed the entire conductiv-
ity tensor, in the DC limit [15, 16]. What scaling does the Hall conductivity have in the
Schr¨ odinger case?
Moreover, with probe D-branes that do not fill all of AdS5, we have more options about
how to perform the NMT. Consider for example a D5-brane extended along AdS4×S2inside
AdS5×S5. Here we must make a choice: the D5-brane may be extended along the T-duality
direction y or not. If the D5-brane is along y, we expect the results to be similar those of
section 2.3 for the D7-brane. What happens when the D5-brane is transverse to y, however?
Perhaps the most exciting direction for future research would be model-building of the
kind advocated in ref. [20]. Our results indicate that a straightforward way to engineer
scaling exponents would be to start in the relativistic case, i.e. probe D-branes in AdS5×S5,
introduce a scalar to produce the desired IR exponents, and then perform the NMT. We
generically expect that the result will be DC and AC conductivities that in the UV have
non-relativistic scalings, with z = 2, and in the IR have whatever scalings we initially gave
them.
Acknowledgements
We would like to thank J. Erdmenger, A. Karch, and D. Tong for reading and commenting on
the manuscript, and M. Rangamani for useful discussions. A.O’B. also thanks K. Landsteiner,
D. Mateos, and K. Peeters for useful discussions. C.H. additionally thanks C. Herzog, K.
Jensen, and P. Meessen for useful discussions. The work of M.A. and A.O’B. is supported
in part by the Cluster of Excellence “Origin and Structure of the Universe.” M.A. would
also like to thank the Studienstiftung des deutschen Volkes for financial support. The work
of C.H is supported in part by the U.S. Department of Energy under Grant No. DE-FG02-
96ER40956. J.W. thanks the Galileo Galilei Institute for Theoretical Physics for hospitality
and the INFN for partial support while this work was being completed. The work of J.W. is
supported in part by the “Innovations- und Kooperationsprojekt C-13” of the “Schweizerische
Universit¨ atskonferenz SUK/CRUS” and the Swiss National Science Foundation.
References
[1] J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv.
Theor. Math. Phys. 2 (1998) 231–252, [hep-th/9711200].
[2] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from non-critical
string theory, Phys. Lett. B428 (1998) 105–114, [hep-th/9802109].
– 29 –
Page 31
[3] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253–291,
[hep-th/9802150].
[4] T. Mehen, I. W. Stewart, and M. B. Wise, Conformal invariance for non-relativistic field
theory, Phys. Lett. B474 (2000) 145–152, [hep-th/9910025].
[5] Y. Nishida and D. T. Son, Nonrelativistic conformal field theories, Phys. Rev. D76 (2007)
086004, [arXiv:0706.3746].
[6] D. T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the
Schroedinger symmetry, Phys. Rev. D78 (2008) 046003, [arXiv:0804.3972].
[7] K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett.
101 (2008) 061601, [arXiv:0804.4053].
[8] C. Duval, G. W. Gibbons, and P. Horvathy, Celestial Mechanics, Conformal Structures, and
Gravitational Waves, Phys. Rev. D43 (1991) 3907–3922, [hep-th/0512188].
[9] C. Duval, M. Hassaine, and P. A. Horvathy, The geometry of Schr¨ odinger symmetry in gravity
background/non-relativistic CFT, Annals Phys. 324 (2009) 1158–1167, [arXiv:0809.3128].
[10] C. P. Herzog, M. Rangamani, and S. F. Ross, Heating up Galilean holography, JHEP 11 (2008)
080, [arXiv:0807.1099].
[11] J. Maldacena, D. Martelli, and Y. Tachikawa, Comments on string theory backgrounds with non-
relativistic conformal symmetry, JHEP 10 (2008) 072, [arXiv:0807.1100].
[12] A. Adams, K. Balasubramanian, and J. McGreevy, Hot Spacetimes for Cold Atoms, JHEP 11
(2008) 059, [arXiv:0807.1111].
[13] A. Adams, C. M. Brown, O. DeWolfe, and C. Rosen, Charged Schrodinger Black Holes, Phys.
Rev. D80 (2009) 125018, [arXiv:0907.1920].
[14] A. Karch and A. O’Bannon, Metallic AdS/CFT, JHEP 09 (2007) 024, [arXiv:0705.3870].
[15] A. O’Bannon, Hall Conductivity of Flavor Fields from AdS/CFT, Phys. Rev. D76 (2007)
086007, [arXiv:0708.1994].
[16] M. Ammon, T. H. Ngo, and A. O’Bannon, Holographic Flavor Transport in Arbitrary Constant
Background Fields, JHEP 10 (2009) 027, [arXiv:0908.2625].
[17] R. C. Myers, A. O. Starinets, and R. M. Thomson, Holographic spectral functions and diffusion
constants for fundamental matter, JHEP 11 (2007) 091, [arXiv:0706.0162].
[18] D. Mateos and L. Patino, Bright branes for strongly coupled plasmas, JHEP 11 (2007) 025,
[arXiv:0709.2168].
[19] J. Mas, J. P. Shock, J. Tarrio, and D. Zoakos, Holographic Spectral Functions at Finite Baryon
Density, JHEP 09 (2008) 009, [arXiv:0805.2601].
[20] S. A. Hartnoll, J. Polchinski, E. Silverstein, and D. Tong, Towards strange metallic holography,
arXiv:0912.1061.
[21] S. Kachru, X. Liu, and M. Mulligan, Gravity Duals of Lifshitz-like Fixed Points, Phys. Rev.
D78 (2008) 106005, [arXiv:0808.1725].
– 30 –
Page 32
[22] M. C. N. Cheng, S. A. Hartnoll, and C. A. Keeler, Deformations of Lifshitz holography,
arXiv:0912.2784.
[23] A. Karch and E. Katz, Adding flavor to AdS/CFT, JHEP 06 (2002) 043, [hep-th/0205236].
[24] M. Kruczenski, D. Mateos, R. C. Myers, and D. J. Winters, Meson spectroscopy in AdS/CFT
with flavour, JHEP 07 (2003) 049, [hep-th/0304032].
[25] J. Babington, J. Erdmenger, N. J. Evans, Z. Guralnik, and I. Kirsch, Chiral symmetry breaking
and pions in non-supersymmetric gauge / gravity duals, Phys. Rev. D69 (2004) 066007,
[hep-th/0306018].
[26] I. Kirsch, Generalizations of the AdS/CFT correspondence, Fortsch. Phys. 52 (2004) 727–826,
[hep-th/0406274].
[27] S. Kobayashi, D. Mateos, S. Matsuura, R. C. Myers, and R. M. Thomson, Holographic phase
transitions at finite baryon density, JHEP 02 (2007) 016, [hep-th/0611099].
[28] N. Bobev, A. Kundu, and K. Pilch, Supersymmetric IIB Solutions with Schr´ odinger Symmetry,
JHEP 07 (2009) 107, [arXiv:0905.0673].
[29] A. Donos and J. P. Gauntlett, Solutions of type IIB and D=11 supergravity with Schrodinger(z)
symmetry, JHEP 07 (2009) 042, [arXiv:0905.1098].
[30] E. G. Gimon, A. Hashimoto, V. E. Hubeny, O. Lunin, and M. Rangamani, Black strings in
asymptotically plane wave geometries, JHEP 08 (2003) 035, [hep-th/0306131].
[31] J. L. F. Barbon and C. A. Fuertes, Ideal gas matching for thermal Galilean holography, Phys.
Rev. D80 (2009) 026006, [arXiv:0903.4452].
[32] C. A. Fuertes and S. Moroz, Correlation functions in the non-relativistic AdS/CFT
correspondence, Phys. Rev. D79 (2009) 106004, [arXiv:0903.1844].
[33] A. Volovich and C. Wen, Correlation Functions in Non-Relativistic Holography, JHEP 05
(2009) 087, [arXiv:0903.2455].
[34] R. C. Myers, Dielectric-branes, JHEP 12 (1999) 022, [hep-th/9910053].
[35] T. Sakai and S. Sugimoto, Low energy hadron physics in holographic QCD, Prog. Theor. Phys.
113 (2005) 843–882, [hep-th/0412141].
[36] S. S. Pal, Null Melvin Twist to Sakai-Sugimoto model, arXiv:0808.3042.
[37] A. Karch and A. O’Bannon, Holographic Thermodynamics at Finite Baryon Density: Some
Exact Results, JHEP 11 (2007) 074, [arXiv:0709.0570].
[38] A. Karch, A. O’Bannon, and K. Skenderis, Holographic renormalization of probe D-branes in
AdS/CFT, JHEP 04 (2006) 015, [hep-th/0512125].
[39] I. Kirsch, Spectroscopy of fermionic operators in AdS/CFT, JHEP 09 (2006) 052,
[hep-th/0607205].
[40] R. G. Leigh and N. N. Hoang, Fermions and the Sch/nrCFT Correspondence,
arXiv:0909.1883.
[41] D. Mateos, R. C. Myers, and R. M. Thomson, Holographic phase transitions with fundamental
matter, Phys. Rev. Lett. 97 (2006) 091601, [hep-th/0605046].
– 31 –
Page 33
[42] T. Albash, V. G. Filev, C. V. Johnson, and A. Kundu, Quarks in an External Electric Field in
Finite Temperature Large N Gauge Theory, JHEP 08 (2008) 092, [arXiv:0709.1554].
[43] J. Erdmenger, R. Meyer, and J. P. Shock, AdS/CFT with Flavour in Electric and Magnetic
Kalb-Ramond Fields, JHEP 12 (2007) 091, [arXiv:0709.1551].
[44] J. Erdmenger, M. Kaminski, and F. Rust, Holographic vector mesons from spectral functions at
finite baryon or isospin density, Phys. Rev. D77 (2008) 046005, [arXiv:0710.0334].
[45] J. Erdmenger, N. Evans, I. Kirsch, and E. Threlfall, Mesons in Gauge/Gravity Duals - A
Review, Eur. Phys. J. A35 (2008) 81–133, [arXiv:0711.4467].
[46] J. Erdmenger, M. Kaminski, P. Kerner, and F. Rust, Finite baryon and isospin chemical
potential in AdS/CFT with flavor, JHEP 11 (2008) 031, [arXiv:0807.2663].
[47] C. P. Burgess, Open string instability in background electric fields, Nucl. Phys. B294 (1987) 427.
[48] C. Bachas and M. Porrati, Pair creation of open strings in an electric field, Phys. Lett. B296
(1992) 77–84, [hep-th/9209032].
[49] N. Seiberg, A note on background independence in noncommutative gauge theories, matrix
model and tachyon condensation, JHEP 09 (2000) 003, [hep-th/0008013].
[50] J. Mas, J. P. Shock, and J. Tarrio, On the Einstein Relation in holographic systems at finite
baryon density, Nucl. Phys. Proc. Suppl. 192-193 (2009) 184–186, [arXiv:0812.4432].
[51] J. Mas, J. P. Shock, and J. Tarrio, Holographic Spectral Functions in Metallic AdS/CFT, JHEP
09 (2009) 032, [arXiv:0904.3905].
[52] A. Karch, A. O’Bannon, and E. Thompson, The Stress-Energy Tensor of Flavor Fields from
AdS/CFT, JHEP 04 (2009) 021, [arXiv:0812.3629].
[53] S. A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant.
Grav. 26 (2009) 224002, [arXiv:0903.3246].
[54] C. P. Herzog, Lectures on Holographic Superfluidity and Superconductivity, J. Phys. A42 (2009)
343001, [arXiv:0904.1975].
[55] J. McGreevy, Holographic duality with a view toward many-body physics, arXiv:0909.0518.
– 32 –