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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

2.1Review: D7-Branes in AdS

2.2The 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

4.1D7-brane Embeddings

4.2AC conductivity in the IR

4.3AC conductivity in the UV

23

25

27

28

5. Discussion and Conclusion28

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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