Holographic model for antiferromagnetic quantum phase transition induced by magnetic field

0.16 MB·English

by  Rong-Gen Cai

Holographic model for antiferromagnetic quantum phase transition induced by
magnetic field
Rong-Gen Cai,1,∗ Run-Qiu Yang,1,† and F.V. Kusmartsev2,‡
1State Key Laboratory of Theoretical Physics,Institute of Theoretical Physics,
Chinese Academy of Sciences,Beijing 100190, China.
2Department of Physics, Loughborough University,
Loughborough, Leicestershire, LE11 3TU, United Kingdom
We propose a gravity dual of antiferromagnetic quantum phase transition (QPT) induced by
magnetic field and study the critical behavior around the quantum critical point (QCP). It turns
outthattheboundarycriticaltheoryisastrongcouplingtheorywithdynamicexponentz=2and
thatthehyperscalinglaw isviolated and logarithmic corrections appearneartheQCP.Somenovel
5 scaling relations are predicated, which can be tested by experiment data in future. We also make
1 some comparison with experimental data on low-dimensional magnets BiCoPO5 and pyrochlores
0 Er2−2xY2xTi2O7.
2

t
c Introduction. –Quantum phase transition (QPT) and the holographic antiferromagnetic/paramagnetic phase
O the behavior of quantum systems in the vicinity of the transition was studied. We showed that the antiferro-
corresponding quantum critical point (QCP) have at- magnetictransitiontemperatureT isindeedsuppressed
9 N
tracted a lot of attention both in theory and experi- byanexternalmagneticfieldandtendstozerowhenthe

] ment sides recently [1–3]. In contrast to their classical magnetic field reaches its critical value Bc. In this way
h
counterparts induced by thermal fluctuations arising at the antiferromagnetic QPT induced by magnetic field is
t
- finite temperature T > 0, QPTs happen at zero tem- realized. However,itwasshownthatthemodelproposed
p
perature and are governed by quantum fluctuations as- inRef.[21]containsavectorghost,veryrecently,amod-
e
h sociated with the Heisenberg uncertainty and are driven ified model was proposed [23], which is shown not only
[ by a certain control parameter rather than temperature, ghost free, but also causal well-defined, while it keeps

e.g.,composition,magneticfieldorpressure,etc. Incon- themainresultsintheoriginalmodelqualitatively. Here
2
v densedmatterphysics,suchaquantumcriticalityiscon- we will elaborate in some detail this QPT and study the
1 sideredtoplayanimportantroleinsomeinterestingphe- corresponding critical properties in this new model.
8 nomena [4, 5]. Holographicmodel. –Inordertodescribethespontaneous
4
staggered magnetization which breaks the time rever-
4 One of intensively discussed QPTs is ordered-
sal symmetry, we introduce two real antisymmetric ten-
0 disordered QPT in antiferromagnetic materials induced
. sor fields coupled with U(1) Maxwell field strength [22].
1 by magnetic field (see for example, Refs. [6–10]), espe-
Based on the discussions in Refs. [22, 23], we take the
0 cially in the heavy-fermion systems, since they can be
bulk action as follows,
5 tunedcontinuouslyfromanantiferromagnetic(AF)state
1
toaparamagnetic(PM)metallicstatebyvaryingasingle 1 6
: S = d4x√ g[R+ FµνF λ2(L +L +L )],
v parameter[4]. Inthesematerials,QPTnaturallybelongs 2κ2 Z − L2− µν− 1 2 12
Xi to the phenomenon involving strongly correlated many- (1)
body systems [11–13]. However, the complete theoreti- where
r
a cal descriptions valid in all the energy (or temperature)
k
region are still lacking. In order to study and character- L12 = 2M(1)µνMµ(2ν),
ize stronglycoupledquantumcriticalsystems,somenew
1 m2
methods are called for. L = (dM(a))µντ(dM(a)) + M(a)µνM(a)
(a) 12 µντ 4 µν (2)
Thanks to the feature of the weak/strong duality, the 1
+ M(a)µνF +JV(M(a)),
AdS/CFT correspondence provides a powerful approach 2 µν µν
to study such strongly coupled systems. This duality V(M(a))=(∗M(a) M(a)µν)2, a=1,2.
µν µν
relates a weakcoupling gravitationaltheory in a (d+1)-
dimensional asymptotically anti-de Sitter (AdS) space- Here ∗ is the Hodge star dual operator and dM denotes
time to a d-dimensional strong coupling conformal field the exterior derivative of M. L is the radius of AdS
theory (CFT) in the AdS boundary [14–16]. In recent space, 2κ2 = 16πG with G the Newtonian gravitational
years, we have indeed witnessed that the duality has constant, k, m2 and J are all model parameters with
been extensively applied into condensed matter physics J <0, λ2 characterizes the back reaction of the two po-
and some significant progresseshave been made [17–20]. larization fields M(a) to the background geometry, and
µν
In Ref. [21] we realized the ferromagnetic/paramagnetic L describes the interaction between two polarization
12
phase transition in a holographic setup, and in Ref. [22] fields. Note that by rescaling the polarization fields and
2
the parameter J, λ2 can also be viewed as the coupling With this ansatz, it is found that the equations for
strength between the polarization fields and the back- M(a) are algebraic ones and can be solved directly [25].
tr
ground Maxwell field. In the AdS/CFT duality, the Therefore we pay main attention on α and β. At the
modelparametersmandk arerelatedtothedualopera- horizon,theregularinitialconditionsshouldbeimposed.
tordimensionintheboundaryfieldtheory,J isrelatedto The behavior of the solutions of equations in the UV
the self-coupling coefficient ofthe magnetic moment and region(near the AdS boundary) depends onthe value of
kdescribestheinteractionbetweentwokindsofmagnetic m2+k. When m2+k =0,the asymptotic solutions will
moments. The reason to introduce two antisymmetric have a logarithmic term, we will not consider this case
fields for describing the anti-ferromagnetism was elabo- here. Instead when m2+k =0, we have the asymptotic
rated in Ref. [22]. Note that the form of V(M(a)) is not solution as [25] 6
µν
unique,wechoosethisformasitcanleadtospontaneous
B
symmetrybreaking(seeFig.1inRef.[23])andtosimplify α =α r(1+δ1)/2+α r(1−δ1)/2 ,
the equations of motion of the model. Compared with UV + − − m2+k
the original model for antiferromagnetism in Ref. [22], βUV =β+r(1+δ2)/2+β−r(1−δ2)/2, (7)
the key change is to replace the covariant derivative of
δ = 1+4k+4m2, δ = 1 4k+4m2,
the polarization field M by the exterior derivative. This 1 p 2 p −
change can avoid the problems such as ghost and causal where α and β are all finite constants. To make the
± ±
violation, while keep the significant results in the origi- system condense into the antiferromagnetic phase, as in
nal model qualitatively and in addition this model has a Ref. [22], we require that the term associated with the
potential origin in string/M theory [23]. magnetic field B in Eq. (7) should be the leading term.
In the probe limit of λ 0, we can neglect the When B = 0, we require that the condensation for β
→
back reaction of the two polarization fields on the back- appearsspontaneously. Withthose,theparametershave
ground geometry. The background we will consider is a to satisfy m2 >k >0 and
dyonic Reissner-Nordstro¨m-AdS black brane solution of
the Einstein-Maxwell theory with a negative cosmologi- J+(k,m2)<J <J−(k,m2), (8)
c c
cal constant, and the metric reads [24]
with J±(k,m2) = (m2 + k)2(m2 + 3/2 k)/12 and
dr2 α =βc =0 accord−ing to the AdS/CFT di±ctionary (for
ds2 =r2( f(r)dt2+dx2+dy2)+ , + +
− r2f(r) details please see [25]).
(3)
1+µ2+B2 µ2+B2 QCP, energy gap and spectrum.–Let us first consider
f(r)=1 + . theinfluenceoftheexternalmagneticfieldB ontheanti-
− r3 r4
ferromagnetic critical temperature T . Near the critical
N
Here both the black brane horizon rh and AdS radius L temperature, the staggered magnetization is very small,
have been set to be unity. The temperature of the black i.e.,β isasmallquantity. Inthatcasewecanneglectthe
brane is nonlinear terms of β and obtain the equations for α and
β,
T =(3 µ2 B2)/4π. (4)
− −
f′α′ m2 B f′β′ m2
For the solution (3), the corresponding gauge potential α′′+ αeffα= , β′′+ βeffβ =0. (9)
f − r2f r2f f − r2f
is A = µ(1 1/r)dt+Bxdy. Here µ is the chemical
µ
−
potential and B can be viewed as the external magnetic Herem2 andm2 aretwofunctionsofα[25]. Without
αeff βeff
field of the dual boundary field theory.
loss of generality, we can set β(r )=1. With increasing
h
Weconsideraself-consistentansatzforthetensorfields
the magneticfieldB fromzero,the effective masssquare
with nonvanishing components M(a),M(a) (a=1,2) and
tr xy of β increases, so that the critical temperature TN de-
define
creases. Thecriticaltemperatureisplottedasafunction
of the external magnetic field in Fig. 1. When T is
α=(M(1)+M(2))/2, β =(M(1) M(2))/2. (5) N
xy xy xy − xy decreased to zero, an AdS2 geometry emerges near the
By this definition, the antiferromagnetic order parame- horizon [25]. The existence of a stable IR fixed point in
ter, i.e., the staggered magnetization, can be expressed the emergent AdS2 region demands
as [22, 23]
B =B m2 , m2 =0 (10)
c ≡− αeff|α=αc βeff|α=αc
N†/λ2 = drβ/r2. (6) at the horizon r = r = 1. Then we can see that in
−Z h
the case of T = 0, when B < B , β is unstable near
c
| | | |
Then the antiferromagnetism phase corresponds to the the horizon and the condensation happens so that the
case when N† = 0. In our model, it just corresponds to staggered magnetization is no longer vanishing. When
6
the case of β =0, while α=0. B > B , however, β is stable at the horizon and the
c
6 | | | |
3
staggered magnetization is zero. Therefore, a QPT oc-
Holographic Holographic
curs at B = B and the system is quantum disorder 1 Data 1 x=0
c
| | | | x=0.031
when |B|>|Bc|. TeN PM TeN x=0.085
Inordertoinvestigatethemagneticfluctuationsinthe 0.5 0.5 PM
AF AF
vicinity of QCP, we need to consider the perturbations
of two polarization fields. To make the system be self- 00 0.5 1 00 0.5 1
consistent at the linear order, the perturbations for all B/Bc B/Bc
components of the polarization fields have to be consid-
ered, FIG.1. TheantiferromagneticcriticaltemperatureTN versus
the external magnetic field B compared with experimental
δM(a) =ǫC(a)e−i(ωt+qx), (µ,ν)=(r,y),(t,x) data. ThecriticaltemperatureTN iscalculatedwiththeaidof
µν µν 6 (11) thesolutionofEqs.(9). Ateach fixedvalueofmagneticfield
δM(a) =iǫC(a)e−i(ωt+qx), (µ,ν)=(r,y),(t,x). BthetemperatureisdeterminedatwhichtheAFcondensate
µν µν
starts to appear. Left: Comparion with experimental data
Put this perturbations into the equation of motions and fromBiCoPO5Right: Comparionwithexperimentaldatafor
compute to the 1st order for ǫ, we can get their equa- pyrochlore compounds: Er2−2xY2xTi2O7. The experimental
data are from [26, 28] and rescaled.
tions of the perturbations (the details for perturbational
equations can be found in Ref. [25]). In general, be-
cause of the nonlinear potential, all the components
and QPT occurs. After rescaled, different parameters
of two polarization field couple with each other. Let
give similar behaviors with some slight differences. The
β =(Cx(1y) Cx(2y))/2. Intheparamagneticmagneticphase physical picture of this QPT can be understood as fol-
−
(eT > TN or B > Bc), when q,ω → 0, the equations for lows. Whenmagneticfieldreachesitscriticalvalue,there
β decouple from others [25]. By imposing the ingoing magnetic spins become partially aligned along the direc-
condition at the horizon, it has the following asymptotic tion of the magnetic field. Therefore the system requires
e
solution in the UV region, less thermal energy to destroy the remaining magnetic
spins order.
β β+r(1+δ2)/2+β−r(1−δ2)/2. (12) The holographic model can give some interesting scal-
≃
e e e ing relations near the QCP. For small B, numerical re-
AccordingtothedictionaryofAdS/CFT,uptoapositive
sults show that T T B2, where T denotes the
N N0 N0
constant, the retarded Green’s function for β reads − ∝
critical temperature in the case without external mag-
G =β /β . e (13) netic field. When magnetic field is close to Bc, we find
ββ − + that N´eeltemperature is fitted well by following relation
e e
Using the retardedGreen’s function, we candefine spec-
T /lnT (1 B/B ), (15)
N N c
trum function as P(ω,−→q) = ImG(ω,−→q)/π. When we ∝ −
turn on a small momentum −→q, the energy of long-life where TN = TNe/TN0.eWe will analytically present the
quasi-particle,whichcorrespondstothepeakofP(ω,−→q), relatione(15)byconsideringtheemergentgeometryAdS2
can be given by following dispersion relation, in the IR limit [25].
WhenmagneticfieldB islargerthanthecriticalvalue
ω∗ =∆+ǫ→−q, ǫ→−q=0 =0. (14) Bc, the antiferromagnetic phase disappears even at zero
temperature. In this case, the system comes into quan-
Here ∆ is the energy gap of quasi-particle excitation. In
tumdisorderedphaseatzerotemperature,inwhichthere
the vicinity of QCP, for the case of ω = 0, the retarded
isagappedmagneticexcitation. IntheleftplotofFig.2,
Green’s function usually has the form of G 1/(q2 +
∼ we show ImG with respect to the frequency of antiferro-
1/ξ2), where ξ is called correlation length. At QCP, in
magnetic excitation in the case with different magnetic
general,theenergygapvanishes. Thuswehaveω∗ =ǫ→−q. field. In the case of 0 < B/B 1 1, there is a
c
In addition, for small frequency andwave vector,we can − ≪
distinct peak which gives the energy gap for the exci-
define the dynamic exponent z in the way as ω qz.
∗ tation. With increasing magnetic field, the peak moves
∼
Numerical Results. As the equations involved here
towardshigherenergyandbecomesmoreandmoreindis-
are nonlinear, we have to solve them numerically. The
tinct. This meansthatthe gapincreasesbutthe lifetime
different parameters satisfying restrictions (8) give sim-
decreases when magnetic field increases. At the critical
ilar results, we here therefore just take parameters as
magneticfieldB =B ,weseeω =0,whichcorresponds
m2 = 1,k = 7/8 and J = 0.67 as a typical example in c ∗
− toagaplesslong-lifetimeantiferromagneticexcitation. In
theleftplotofFig.1,andFigs. 2and3andasJ = 0.71 the region of B/B 1 0+, we find the energy gap is
− c
in the right plot of Fig. 1 − →
fitted well by following equation (see the right plot of
WecanseefromFig.1thattheN´eeltemperatureT is
N Fig. 2)
suppressedby externalmagneticfield. There is acritical
magnetic field for given parameters, at which T is zero ∆ (B/B 1),with ∆=∆/T . (16)
N c N0
∝ −
e e
4
150 B=1.001Bc 0.3 magneticorderedisdominatedbyitinerantelectrons,dy-
B=1.003Bc namical exponent z is 2 near the QCP. Since our dual
Im1G00 B=1.006Bc ∆e0.2 boundary theory is a 2-D theory,the effective dimension
is thus d = d+z = 4, which is just the upper critical
50 0.1 eff
dimension of the Hertz field theory [2, 30]. In this case,
Numerical results
0 −8 −6 −4 −2 0 00 0.02 Fittin0g.0 c4urve 0.06 the hyperscaling is violated and logarithmic correction
ln(ω/TN0) B/Bc−1 behavior appears. In fact, the d = z = 2 quantum crit-
ical theory is in general not a weak coupling theory at
FIG.2. Left: Theantiferromagneticspectrumfunctioninthe
any T >0. Instead, a strongly coupled effective classical
casewith differentmagnetic fieldwhen B>Bc. Right: The
modelemergesthatcanbe usedtodeterminethecritical
gap energy versus the external magnetic field when B/Bc−
1→0+. dynamics[31]. Our resultsshowthatit canbe described
well by AdS/CFT correspondence and this provides a
.
new example of the applicability of the gravity/gauge
1.485 0.08 duality in condensed matter theory.
1.48 0.06 It is quite interesting to apply this holographic model
−×2110G11.4.4775 ξ−02.04 tinoderepaelnisdteicntmoaftetrhiaelsm. icSrionscceoptihcedheotaloilgsraopfhtichemmodaetleriis-
0.02 als and their interactions, it should be suitable for a
11.4.46650 500 10q002/T1N25000NFiuttmi2ne0gr0 icc0uarlv rees2u5lt0s0 00 0.01 B0.0/2Bc−0NF.0iut13tminegr icc0ua.0rlv 4reesul0ts.05 calraessBioCfomPaOt5erwiailtsh. cTriwticoalpomtaengtnieatlicAfiFe-lQdPBTc mat1e5r.i3aTls
≃
(which is obtained by fitting a power-law relation [26])
FIG.3. Left: G−1 asafunctionofq2 whenω=0inthecase
and Er Y Ti O with B 1.5T for x= 0 [27, 28].
2−2x 2x 2 7 c
of |1−B/Bc| = 0.01. The solid line is the fitting curve by ≃
In Fig. 1 we present the experimentaldata for these two
G−1 ∝q2+1/ξ2. Right: Thecorrelation lengthξ versusthe
materials and the holographic results, where we choose
magnetic field when |B/Bc−1|→0.
the model parametersso that they can give out the best
.
fitting. The holographic model gives B 16.2T and
c
≃
1.45T, respectively. While the experiment data show
In the left plot of Fig. 3, we plot the inverse Green’s
that the energy gap for Er Y Ti O when B B
function G−1(q) in the case of ω = 0 and 1 B/B = 2−2x 2x 2 7 ≥ c
ββ | − c| obeysthelinearrelationship(16)withslope4.2(seeFig.8
0.01. WecanseethatitobeysthebehaviorofG−1 q2+ in Ref. [28]), our holographic model gives the slope 5.0
∼
1/ξ2 as we expected before. Thus the Green’s function with the chosen model parameters. These two slopes
cangivethecorrelationlengthbyfittingthecurveofG−1 are in the same order. Note that these two kinds of
as a function of q2, which is shown in the right plot of material have different microscopic structures and com-
Fig.3. Weseethatthecorrelationlengthξ asafunction plex interactions, it is remarkable that the simple holo-
of the tuning parameter B obeys the following relation graphic model can give a self-consistent description for
those two materials. In addition, it is also worth men-
ξ B/B 1−ν, with ν 1/2. (17)
c
∝| − | ≃ tioning here that the doping at the magnetic site (x up
As to the dynamical exponent z, in antiferromagnetic to0.085)has averylittle influence onthe criticalbehav-
metals, z = 2 [5]. In our holographic model, the dy- ior of Er2−2xY2xTi2O7. This indicates that an emergent
namical exponent can be calculated by using similar nu- universal behavior appears in these materials from very
meric method. The results indeed show that z 2. different microscopic details and could be described by
≃
This numerical result can be confirmed by the emergent the holographic model.
AdS geometry in the IR region [25] and agrees with As the critical behavior of a QPT induced by a mag-
2
the result from a different holographic model proposed netic field, the three scaling relations (15), (16) and (17)
in Ref. [29]. Furthermore, from the energy gap (16), we nearthecriticalpointareourmainresultsfromtheholo-
seethatthis energygapsatisfiesthe universalscalingre- graphic model. Besides the energy gap (16), our predi-
lation ∆ B B zν, which further indicates z = 2 in cations on the scaling relations (15) and (17) can also
c
∼ | − |
this model. be confirmed by experiments. Unfortunately at the mo-
Discussion. -The relation we found in this paper of menttheycannotbecheckedbytheexistingexperiments
the N´eel temperature with respect to the magnetic field because the experiment data are absent when the N´eel
in the vicinity of QCP is quite non-trivial. Note that temperature is very close to zero. Of course whether
the relation (15) is not the usual power-law behavior or the holographicmodel is suitable for these twomaterials
square-rootform. Butit is anexpected resultinthe 2-D needs more evidence. It is also very interesting to find
QPTs in strong coupling case [5]. This non-trivial coin- more materials satisfying the conditions of this model
cidence strongly implies a connection between these two andto check ourpredictions. We expect that this model
different theories. In the antiferromagnetic metal where can be confirmed or falsified experimentally soon.
5
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