1304.3126 hologdraphic spintronics

7/28/2019 1304.3126 Hologdraphic Spintronics

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OU-HET-783, RIKEN-MP-69, YITP-13-22

Towards Holographic Spintronics

Koji Hashimoto,1,2, Norihiro Iizuka,3, and Taro Kimura2,

1Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, JAPAN2Mathematical Physics Laboratory, RIKEN Nishina Center, Saitama 351-0198, JAPAN3Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, JAPAN

We study transport phenomena of total angular momentum in holography, as a first step towardholographic understanding of spin transport phenomena. Spin current, which has both the localLorentz index for spins and the space-time vector index for current, couples naturally to the bulkspin connection. Therefore the bulk spin connection becomes the source for the boundary spincurrent. This allows us to evaluate the spin current holographically, with a relation to the stresstensor and metric fluctuations in the bulk. We examine the spin transport coefficients and thethermal spin Hall conductivity in a simple holographic setup.

Introduction

Spintronics is a technology where we manipulatethe intrinsic electron spin degrees of freedom insteadof the electric charge [1, 2]. In ferromagnetic/anti-ferromagnetic materials, spin-charge separation can oc-cur and in such a situation, it is useful to consider spinas an independent degree of freedom which carries infor-mation. Because electric charge transport is not involvedthere, spin devices can reduce power consumption com-pared to usual electric ones, and exceed the velocity limitof the electron charge. This spintronics is actually usedwidely, for example, for read-heads of hard-drives, andis based on a recent development of experimental tech-nologies manipulating imbalance between up-spins anddown-spins. For these reasons, spin transport phenom-ena have been attracting special interests recently.

Recent research on the spin transport basically relieson one-body quantum mechanical analyses, especially inthe presence of a spin-orbit interaction. However, instrongly correlated systems, we have to go beyond the

one-body physics by treating the interaction effect se-riously. In this Letter, we propose a method to studythe spin transport phenomena for strongly correlated sys-tems by using the holography, i.e., gauge/gravity corre-spondence [35]. The method of holography is one ofthe most useful tools to study strongly correlated quan-tum field theories. While there are some attempts toinclude effects of spins in holography, e.g., [614], studyof spin transport itself has not yet been performed inthe literature. In order to discuss the spin degrees offreedom, we first show a definition of spin current froma relativistic field theoretical viewpoint as a conservedNothers current. Then with this definition, we show

how to deal with the spin transport coefficients from theholographic viewpoint. The key point is that the spinconnection is naturally regarded as a source for the spincurrent. We demonstrate a holographic treatment of thespin transport, on a boosted Schwarzschild black brane

Electronic address: [email protected] address: [email protected] address: [email protected]

FIG. 1: (a) The charge current is just the total contribution ofup- and down-spin currents J= J + J. (b) The spin current

is given by difference between them, Jz =1

2(J J). This

picture is available if and only ifz-direction spin is conserved.

background in AdS, and we calculate a spin transport co-efficient and a thermal spin Hall conductivity.

Spin current

The spin current is, as the name suggests, a flow of theintrinsic spin degrees of freedom, instead of the electriccharge. If z-spin is conserved, namely a good quantumnumber, we can apply a naive definition of the spin cur-rent,

Jz =1

2

J J

. (1)

This means that the spin current is given by the differencebetween flows of up- and down-spins, J and J, whilethe electric current is the total contribution of them,J = J + J, as shown in Fig. 1. This definition (1)corresponds to the Schwinger representation of the spinoperator, s = 12

.The expression (1) is available if and only if the spin

is conserved, or at least, approximately conserved [29].However generically the electron spin is not conservedby itself, due to the spin-orbit interaction. Therefore thenaive definition of the spin current (1) has to be modifiedin the presence of such an effect.

First we consider how to define the spin current fromthe field theoretical point of view. Lets recall the treat-ment of conserved currents in the context of quantumfield theories. A conserved current is defined as a varia-tion of an action with respect to its corresponding source.For example, the electric current J is derived by differ-entiating an action with respect to a U(1) gauge field,

arXiv:1304.3

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[hep-th]10

Apr2013
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J =S

A. (2)

Conservation of J is guaranteed by the Nothers theo-rem, associated with a U(1) gauge symmetry,

J = 0. (3)

In the weak coupling limit of a U(1) gauge theory, theU(1) local symmetry reduces to a global one. The Abecomes a non-dynamical background gauge potential,which is a source, and the J becomes a global current.In this limit, the global U(1) current J couples to thesource A in the Lagrangian as Lsource = AJ. There-fore, the U(1) current J is obtained by differentiatingthe action with respect to its source A.

Similarly a stress tensor is given by a variation of anaction with respect to a metric,

T =1

g

S

g. (4)

The conservation of energy and momentum

T = 0 (5)

comes from the translation invariance in temporal andspatial directions, respectively. In the weak gravity limit(where gravity is decoupled), non-dynamical backgroundmetric g becomes a source for the stress-tensor, andthey couple as Lsource = gT in the Lagrangian.

In this way, in order to obtain a conserved quantity, wehave to introduce a corresponding field (or source) whichcouples to the conserved quantity. For the case of thespin current J

ab

, our claim is that the spin connection

ab is the corresponding field (source). This implies that

they couple as Lsource = ab Jab in the Lagrangian. Bydifferentiating an action with respect to the spin connec-tion, we can obtain the spin current.

To see why it is so, lets recall the nature of spin. Thespin operator sa = a/2 has an index a for the orien-tation of the spin. Here the hatted index a takes onlya spatial coordinate as a = x, y, z and is the Paulimatrix. Spin is conserved only in the sense that the to-tal angular momentum is conserved. The total angularmomentum is associated with the global rotational sym-metry of the system. If we uplift this global rotational

symmetry to a local one, then these become a subgroupof the local Lorentz symmetry. Therefore, it is very nat-ural to associate the conserved spin a to a local Lorentz

generator ab =i4 [a, b] as

a = abcbc, where abc is

anti-symmetric tensor taking 1 defined on the spatialpart of the local Lorentz indices, i.e., a, b, c of abc takes

only x, y, z. Furthermore, since the spin connection abis a gauge field associated with the local Lorentz symme-try, it is very natural to associate it to the conserved spincurrent J

ab, as equation (2).

Therefore, we reach a conclusion that a spin current isgiven by a variation of an action with respect to a spinconnection as

Jab

=S

ab. (6)

From now on, the hatted indices a, b, represent thelocal Lorentz indices, so they stand for

t, x, y, z. Greekindices ,, stand for curved spacetime vector indices.

The spin connection is written in terms of a vielbein e aas

ab = ea

eb = e a eb + e a eb

= e b ea = ba , (7)

where stands for the Christoffel symbol, and the viel-

bein e a satisfies g = ab ea

eb

, with the local Lorentzmetric ab = diag(1, 1, 1, 1).

Usually, we call the following current as a spin current

Ja = 0abcJbc

, (8)

rather than the former one Jab

. Here we use the conven-

tion 0123 = 1. One can easily see that the definition (8)is consistent with, for example, the standard free fermionspin current. To see this, let us consider the generic formof a fermionic Lagrangian on a curved space, which isgiven by

LF =

ieaa

iA i

2 ab ab

m

. (9)

From this, we have the spin current by differentiating it

with the spin connection,

Jab

=1

2

ab Ja =

1

2(a 1 ). (10)

This is regarded as a current carrying a-direction spin.We can see that the zero-th component correctly givesthe spin density

J0a = (sa 1 ). (11)

In this way, we have seen that the definition (8) is con-sistent with the conventional one for the spin current.However it is more convenient to consider J

abas a spin

current, since J

a defined in equation (8) is not localLorentz invariant tensor. This is because 0abc tensortakes explicit index component 0.

The conservation of the spin current Jab

J

ab= 0 (12)

is associated with the local Lorentz invariance, and the

spin current Jab

couples to the source term ab in the

Lagrangian as Lsource = ab Jab.


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Precisely speaking, what we define above is total an-gular momentum current, rather than spin current.Note that only the total contribution of the angular mo-mentum current, coming from both the orbital and thespin angular momentum, is conserved. A difficulty indealing with spin transport phenomena is in the defini-tion of the spin current, because the intrinsic spin is notconserved solely but rather conserved as a whole angular

momentum. Therefore the spin current, by itself, can-not be introduced as a conserved Nother current at leastin the relativistic limit. Thus, in this sense, the spincurrent defined above is slightly different from the con-ventional definition of the spin current often used in thenon-relativistic condensed matter physics, which includesthe contribution of only the intrinsic electron spin.

We will also point out that it is possible that the or-bital contribution gives only a sub-leading contribution,when we take the non-relativistic limit. This is becausethe orbital angular momentum includes the spatial mo-

mentum as L = x p. Thus, by taking an appropriatelimit, the spin current, defined as a conserved one, can

provide a good description of the spin transport. We willdiscuss how we take the non-relativistic limit a bit morein detail in the discussion later.

There are a number of attempts to define the spin cur-rent in the literature. The original idea of using the spinconnection as a source to obtain a spin current is foundin [15, 16], especially in the 2 + 1 dimensional theory. Inthe paper [15] they treated the space and time separatelyand broke the Lorentz invariance explicitly. Another at-tempt to define a spin current is performed by introduc-ing an SU(2)-valued gauge field, coupled to a spin de-grees of freedom, in addition to a U(1) electromagnetic

field [1719]. This SU(2) symmetry can be seen as a rem-nant of the local Lorentz symmetry, which is decomposedas SO(1, 3) = SU(2) SU(2) in 3 + 1 dimensions. How-ever, since these SU(2) are not decoupled except for themassless case, it is difficult to define the spin current asa conserved current only with the SU(2) gauge field. Ac-tually, this SU(2) symmetry is broken in the presence ofthe spin-orbit interaction.

Holography

Given the spin current definition in terms of spinconnection, in order to study the spin current by thegauge/gravity duality scheme, we will evaluate the fluc-tuation mode of the spin connection. Note that holog-raphy induces one extra coordinate, i.e., a radial direc-tion. So in the gravity side, the local Lorentz index runsas a = t, x, y, z and r. Similarly the vector index runs = t ,x,y,z,r.

Before studying a component of the spin connectioncorresponding to a spin current in a spatial direction, we

analyze a temporal component of a spin current J xyt ,

as an example. This term couples to xyt . When thebackground metric is diagonal, the static contribution is

calculated as

xyt =1

2exxeyy

ygtx xgty

. (13)

Here we apply a gauge choice e a=rr = gr=r = 0. Fromthe indices, it is clear that this represents a rotationof a metric fluctuation in the xy-plane. In terms ofthe gauge/gravity duality, the non-normalizable mode

of this component is regarded as a chemical potentialfor z-component of the total angular momentum, i.e.,xyt (NN) =

12

z, where the index (NN) represents the

non-normalizable mode [30]. This chemical potential isnaively interpreted as the difference between those forup- and down-spins, z = 12( ). The z-componentspin density J zt corresponds to the normalizable mode

of xyt (N) in the holographic viewpoint, where the index

(N) represents the normalizable mode.Similarly, let us study a fluctuation of the spin con-

nection along the x-spatial direction, xyx . This corre-sponds to a spin current J xyx =

12

J zx , i.e., z-orientedspin flows along x direction. Here we can see thatwe need to turn on some of the off-diagonal elements ofthe background metric, in particular gtx and gty, whichcorrespond to non-vanishing off-diagonal contributions ofvielbeins, e xt and e

yt . To see this, assuming that the

fluctuation depends only on r and t directions, we obtain

xyx = 1

2etxeyytgxy +

1

2exxetytgxx. (14)

From this expression one can see that the off-diagonal

components of the metric, e xt and ey

t , or equivalentlygtx and gty, are required in order to give the spin currentJ zx . A physical meaning of this condition is discussedlater.

Example: boosted black brane

So far we have considered a boundary theory in 3 + 1(x, y, z and t) dimensions. However, even if the boundarytheory is 2 + 1 dimensional, none of our argument so faris modified since 2+1 dimensional theories still admit aspin along the z-direction; Here z-direction is simplythe (a, b) = (x, y) component, J xy . We will conduct acalculation of the spin current in a holographic setting,but for simplicity of the calculation in the bulk, we con-sider a bulk theory in 3+1 dimensions, which correspondsto a boundary theory in 2+1 dimensions.

We demonstrate a calculation of the transport coeffi-

cients for spin with the simplest holographic setup,i.e.,

pure gravity in 3+1 dimensions,

S = Sbulk + Sboundary, (15)

Sbulk =

d4x

g (R[g] 2) , (16)

Sboundary =

d3x

, (17)


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where the cosmological constant is = 3, and isthe boundary metric, defined by the metric componentsalong the boundary dimensions. is a scalar definedwith the extrinsic curvature = 12 (n + n),as =

. n is outward unit vector pointing alongthe radial direction. This boundary action is requiredfor dealing with a surface term, and giving the Einsteinequation appropriately.

We study metric fluctuations around a boostedSchwarzschild black brane solution in AdS4,

ds2 = U(r)dt2 + 1U(r)

dr2 + r2dy2

+(r2 a2U(r))dx2 2aU(r)dtdx (18)

with U(r) = (r3 r30)/r. r = r0 is the horizon whiler = is the boundary. r0 is related to the tem-perature T as T = 3r0/4 [31]. This metric was ob-tained by a coordinate transformation t t + ax onthe AdS-Schwarzschild solution, and it suffices our pur-pose since it includes the off-diagonal metric element gtx.

We can check that this satisfies the Einstein equationR 12gR + g = 0, and is not singular for |a| < 1,and we can consider a > 0 without loss of generality.

Let us perform a fluctuation analysis around the back-ground solution. Fluctuations we consider are gty andgxy, and we assume the following form for AC fluctua-tions,

gty = gyt = eitr2f(r) , (19)

gxy = gyx = eitr2h(r) . (20)

Then, nontrivial components of the Einstein equationsto linear order in these fluctuations, O(), are found tobe just the ty-component, the ry-component and the xy-component. The other components of the Einstein equa-tions turn out to be trivially satisfied. Among the threeequations, the ry-component provides a constraint

f(r) =

a +

r3

a(r30 r3)1

h(r) . (21)

where is for the r-derivative. With this relation, the ty-component reduces to a simple equation solely for h(r),

h(r) +r3 r30

2r3d

dr

(r3 r30)r4

(1 a2)r3 + a2r30d

drh(r)

= 0. (22)

Furthermore, the remaining xy-component of the Ein-stein equations also reduces to the same equation (22).So, we just need to solve the equation (22) for h(r), andrelate it to f(r) via the constraint equation (21). Thisequation (22), in the limit a = 0, coincides with the equa-tion for the shear viscosity calculation [20] [32].

(22) can be written by a new coordinate x r0/r as

2

r20h(x) = x2(x3 1) d

dx

1 x3

x2(1 a2 + a2x3)dh(x)

dx

.(23)

The new coordinate x ranging 0 x 1 can make theboundary analysis easier.

Near the horizon x = 1, we can solve (23) as

h exp

i3

r0log(1 x)

, (24)

which amounts to the in-going boundary condition atthe horizon. Note that the equation of motion (23) andthe in-going boundary condition (24) depend on r0 onlythrough the combination /r0. Since T r0, the tem-perature dependence is the same as the 1/ dependence.This is because the background is a finite temperaturesystem of an AdS space, a scale invariant system, andtherefore, any non-trivial dependence comes from onlythe dimensionless ratio, /r0 [33].

Near the boundary x = 0, we have two independentsolutions of (23),

h = h0

1 1

2x2 1

82x4 +

, (25)

h = h3

x3

+ , (26)with (, T) (1 a2)2/r20. Here h0 and h3 are in-tegration constants. We can find that h0 is the non-normalizable mode while h3 is the normalizable mode.Consider the bulk action, equation (16), and expand thataround r in the background equation (18), withthe fluctuation h(r) and f(r). After using the constraint(21), we find, to the quadratic order in h(r), the leadingr behavior of the Einstein action is

g [R[g] 2] |r= (background)

2 e2itr4

2(1 a2)h(r)2 , (27)

neglecting the boundary terms. From this expression, weconfirm that h const. is the non-normalizable mode[34] while h r3 is the normalizable mode.

We can also specify the boundary condition for theother fluctuation f(r). From (21),

f(x) =

x1

a(s3 1)a2s3 a2 + 1

dh(s)

dsds + c, (28)

where c is an integration constant. Near the horizonx = 1, h approximated as (24) can give an in-going wavefor f only if c = 0. So we need to put c = 0, and f(r) is

uniquely determined once h(r) is given. The magnitudef0 of the non-normalizable mode of f(r) can be read by(28) with c = 0, while the magnitude h3 of the normaliz-able mode of f(r) is proportional to that of h(r) (whichis h3), through (28).

Spin current and stress tensor

Let us pose and understand the physical meaning ofthe modes we consider above. The spin connection canbe written with the metric, or the vielbein as eq. (7).This means that the spin current, which is dual to the


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spin connection, should be associated with the stress ten-sor, which is dual to the metric. Therefore we have toevaluate the spin current by taking into account its rela-tion to the stress tensor. In other words, the spin currentcan be determined by comparing the coefficients appear-ing in the following relation,

Jab

ab = T = L . (29)

Here L is the Lagrangian of the quantum field theory inthe boundary 2 + 1 dimensions. Note that these metricand spin connection are defined on the boundary, there-fore all the indices run without the radial direction. Wehave omitted the volume factor

for simplicity.To obtain an explicit relation between the spin cur-

rent and the stress tensor, we first need to choose a localLorentz frame. Any spin current is dependent on thechoice of the frame. The boundary metric is

gtt = 1 , gtx = gxy = a , gxx = 1 a2 , gyy = 1 . (30)These are given by subtracting the scale factor r of thebulk metric in the limit r

. A natural choice of the

local Lorentz frame for the background vielbein consis-tent with this metric is given by [35]

e tt = 1 , et

x = a , ex

x = 1 , ey

y = 1 . (31)

We turned on the AC fluctuation of the metric givenby equation (19) and (20), and the most generic viel-bein fluctuations consistent with (19) and (20) is a set

{e yt , e yx , e ty , e xy }, which satisfies the two following rela-tions

e yt e ty = eit+ikxx+ikyyf0 , (32)e yx + e

xy

ae ty = e

it+ikxx+ikyyh0 . (33)

coming from the constraint = ea

eb

ab. Here we

used Fourier modes as eit+ikxx+ikyy, and (, kx, ky)is the frequency/momentum for the fluctuations. Theother components of the vielbein are consistently put tozero in our case.

With this at hand, all nontrivial components of thespin connection are

tyt = iet

y ,

xyt = i

2kx(e

yt e ty ) +

i

2(e xy e yx + ae ty ) ,

ty

x = i

2 kx(e

y

t + et

y ) +

i

2 (ex

y ey

x + aet

y ) ,

xyx = i

2kx(2e

xy + ae

yt ae ty )

+i

2a(e xy e yx + ae ty ) ,

txy = i

2kx(e

yt e ty ) +

i

2(e xy e yx + ae ty ) ,

tyy = ikye yt , xyy = iky(ae yt e yx ) . (34)

Keeping the two relations (32) and (33) satisfied, wecan make a gauge choice of the Local Lorentz frame,e ty = e

xy = 0, and restrict ourselves to homogeneous

fluctuation, kx = ky = 0. In this local Lorentz frame,the above spin connections are simplified, and all thenonzero components are

xyx =

ia

2

eith0 , (35)

xyt = ty

x = tx

y = i

2 eith0 , (36)

Since h0 is the constant mode of the boundary metricgxy, it is a source for the boundary stress tensor T

xy,therefore we obtain the spin current coupled to the spinconnection from this expression as

J xyx = 1

a

1

2iTxy , (37)

J xyt = Jty

x = Jtx

y = 1

2iTxy . (38)

All the other components, other than each anti-

symmetric partner Jba

= Jab

, are zero. These com-bined with (35) and (36) clearly satisfy (29). Jxyx is the

spin current along x direction, and J xyt (= Jty

x = Jtx

y ) isthe temporal component of the spin current, correspond-ing to the spin density.

Here we have employed a choice of the local Lorentz

frame e ty = ex

y = 0. However other local Lorentz framechoices are also possible. Actually, for a certain otherchoice of the local Lorentz frame, one can show that thespin current determined in this way is equivalent to apopular definition of the angular momentum current Mmade by the stress-energy tensor,

M

xT

xT

. (39)

Note that this current is with the target spacetime in-dices, so in order for this to be equivalent to our spincurrent J, a certain local Lorentz frame should be ap-propriately chosen.

To check this explicitly, we consider our case of nonzeroTty and Txy . We consider a = 0 for simplicity. From thedefinition (39), one obtains

Mtty = tTty , Mtxy = xTty, Mxty = tTxy ,Mxxy = xT

xy , Mytx = xTty tTxy ,

Myty = yTty , Myxy = yTxy . (40)

One can show that all of these are consistent with thespin connections (34) only when we choose a local Lorentzframe at which

e yt = e ty , e yx = e xy (41)are satisfied. To see this, in this case, (32), (33) become

e yt = e ty =1

2 eit+ikxx+ikyyf0 =

1

2gty , (42)

e yx = ex

y =1

2 eit+ikxx+ikyyh0 =

1

2gxy , (43)


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same as our definition of the spin current (6). We willnow discuss that the spin current evaluated by the defi-nition (6) yields zero value. Therefore we need to relatethe spin current to the stress tensor as (29) which wehave used in this paper.

In order to obtain the spin current following the def-inition (6) in holography, note that (6) means that wehave to differentiate the action (15) with the boundary

spin connection, which is defined by the spin connectionsalong the boundary directions. The contribution comingfrom a variation of the bulk action (16) by the boundaryspin connection, vanishes by using the bulk equations ofmotion. Thus the contribution to the spin current comesfrom a variation of the boundary action (17) only. How-ever we will see that this contribution also vanishes.

Whenever we take a variation, we have to fix all theother quantities. In this case, we regard each of theboundary spin connection component as an independentdegree of freedom, and then, we take a variation of the ac-tion by that, while keeping all the other quantities, whichinclude metric, fixed. In this formulation, each spin con-nection component is an independent degree of freedomfrom the metric; The independent degrees of freedom aremetric and spin connection. In fact, we can formulategeneral relativity in such a way, by i.e., so-called Pala-tini formulation of gravity. However this procedure turnsout to give a vanishing spin current.

To see this, let us conduct a variation of the bound-ary action (17) by the boundary spin connection. Theextrinsic curvature is written with the normal vec-tor n as = n. In the Palatini formalism,the boundary metric and the boundary spin connec-tion are independent, therefore, the contribution formthe boundary action variation yields

Jab

=

ab(n)

= eaebn . (48)

Since n = 0 only when = r and erb = 0 only when b =r, there is no spin current on the boundary. This showsthat the spin current evaluated by the Palatini formalismvanishes [38]. In order to obtain a non-vanishing spincurrent, we should not regard the metric and the spinconnection as independent degrees of freedom. We needto modify our definition of the spin current (6) slightly.

Therefore in this paper, we do not regard the spin con-nection as an independent variable, but associate it withthe metric. This further implies that our spin current,

which is dual to the spin connection, should be associ-ated with the stress tensor, which is dual to the metric.In the Palatini formalism, the relation (7) comes from theequation of motion for the spin connection. Therefore wehave evaluated the spin current by taking into accountits relation to the stress tensor as (29) in this paper.

Discussions : Spin vs angular momentum

In this paper, we have investigated the spin transportphenomena from the view point of gauge/gravity corre-spondence. We have introduced the proper definition of

FIG. 3: When the off-diagonal background metric gtx, namelya constant energy flow in x-direction, is turned on, the angu-lar momentum current as a spin current J zx is induced byapplying the fluctuation gty.

the spin current, as a conserved Nothers current, whichcouples naturally to the spin connection.

We have analyzed the AdS Schwarzschild black branegeometry as a simple example to demonstrate how tostudy the spin transport in the context of the hologra-phy. We have calculated the spin transport coefficient

and the thermal spin Hall conductivity sH by studyingthe fluctuations of the metric components. We have ob-tained the corresponding transport coefficient from thenon-normalizable and normalizable modes propagatingin the bulk gravity.

Let us comment on a physical meaning of the holo-graphic analysis done in this paper. We have seen thatthe off-diagonal metric component for the background,i.e., gtx(= gxy), is required for giving the spin current.Note that if there is such a component in the backgroundgeometry, that leads to a constant energy flow coupled togtx. By applying the fluctuation gty in addition to thebackground flow, we should have an angular momentum

current in x-direction as shown in Fig. 3. It seems thatour spin current almost corresponds to the orbital partof angular momentum.

However, at least from the relativistic theoretical view-point, we cannot split the total angular momentum intocontributions from orbital and intrinsic spin; Spin isoriginally defined in the non-relativistic system, wherethe Lorentz invariance is broken and we should treatspace and time separately. Since in this paper we haveconsidered the total angular momentum current de-fined in relativistic field theory, in order to really discussthe spin-current, we need to take an appropriate non-relativistic limit of our system. Only after taking that,

we can extrapolate the spin contribution from the totalangular momentum current.The non-relativistic limit of relativistic conformal field

theories is obtained by taking the DLCQ, discreet light-cone quantization. This limit reduces the boundary met-ric from AdS into the following form [2125]

ds2 = r2z(dx+)2 + dr2

r2+ 2r2dx+dx + r2dx2 , (49)

where x+ is the light front time, r is again the holographic


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radial direction. x is a new direction associated withthe boost direction and we compactly x x + R, andhas an interpretation as dual to the conserved particlenumber since P is quantized as N/R, where N is particlenumber. z is called dynamical exponent and representsthe difference of the scaling between time x+ and spatialcoordinate x.

For example, starting from a boundary theory which

is 3+1 dimensional, we can obtain a 2+1 dimensionalnon-relativistic theory where we can identify x+ = t + x3

and x = t x3. This metric possesses the Schrodingersymmetry for the z = 2 case.

Taking this DLCQ limit, or simply replacing theboundary metric from AdS into the above, is not enoughfor extracting the spin information, since spin is not aconserved quantity by itself even here, and only the totalangular momentum is a conserved one. In order to elim-inate the contribution of the orbital angular momentum,it is best to consider a setting where the momentum ofthe particle is suppressed, namely an insulator. The in-sulator is realized as a system which has an energy gap.The energy gap is reflected in a holographic setting inthe bulk as a system which has an IR cut-off, like the

confinement in holographic QCD. The hard wall model isthe simplest setting to realize the mass gap and thereforethis would lead one to a system which has an asymptoticmetric as (49) and has an IR cut-off. Such a bulk set-upis good for us to study the spin-transport phenomena.

In this paper, we considered only the spin-current in-duction by the spin-current potential and also thermo-

potential, but not the one induced by an electric field.In real experiments, it is more often to consider the spincurrent induced by some external electric field, so thisforces us to consider a bulk action coupled to the electro-magnetic field. Adding impurity effects [7, 13, 26, 27] isalso important. We hope to return to these analyses inthe near future.

Acknowledgements

N.I. would like to thank RIKEN Mathematical PhysicsLaboratory for kind hospitality where this projectstarted. The research of K.H. is supported in part byJSPS Grants-in-Aid for Scientific Research No. 23105716,23654096, 22340069. The research of T.K. is supportedin part by Grant-in-Aid for JSPS Fellows (No. 23-593).

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[29] For the spin to be approximately conserved, its coherencetime must b e sufficiently larger than its characteristictime scale.

[30] The factor 1/2 is for a convenience due to the definition,equation (8).

[31] Our boost is simply a coordinate transformation. Sinceit is different from the Lorentz boost, it does not involvethe factor for a Lorentz transformation, therefore thetemperature does not change by this boost.

[32] Note that since these two equations solve all the Einsteinequations, these two modes, gty and gxy, decouple fromthe other components of the fluctuation. Therefore, thisis a consistent truncation of the whole Einstein equations.
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[33] The large r/r0 is equivalent to the r0 0 with rfixed, where the ratio /r0 by fixing . This impliesthat the bulk large (small) r region corresponds to thelarge (small) in the boundary theory as is usual UV/IRcorrespondence [28].

[34] Note that terms like r4h(r)2 are equivalent to terms liker2h(r)2 through the integrate by parts.

[35] When a = 0, the vielbein is simply a unit matrix. Theboost t t + ax in the target space changes only the

target space index , resulting in this form of the vielbein.[36] This can be derived by scaling time in the unit of tem-

perature as gboundarytt = 1/T2, and by using a gauge

transformation, which transform xgbulktt to 2tg

bulktx .

The extra r2 is because ofgbulk = r2gboundary .

[37] One might wonder if the Onsagers reciprocal relationholds in this case. Since we have two thermodynami-cal quantities represented by f(r) and h(r) in the holo-

graphic language, it is natural to argue the reciprocalrelation. There are two points concerning the relation.First, since we have introduced the background AC ex-ternal source for gty and also the background metric gtx,it is expected that we explicitly break the time-reversalsymmetry. So there is no good reason for the reciprocalrelation to hold in our case. Second, as we have noticed,once h(r) is given, then h(r) is completely determined.So, we cannot turn on the external source for gxy and gty

independently. This means that the Onsager reciprocalrelation is not directly measured by our external sources.

[38] In this evaluation, we have used the bulk equation of

motion (7) to define (ea ,

ab ). Instead, if we regard

the extrinsic curvature as being solely written by thevielbein, the variation (48) vanishes in the Palatini for-malism.

1304.3126 hologdraphic spintronics

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