The Study of Goldstone Modes in $ν$=2 Bilayer Quantum Hall Systems

0.24 MB·English

by  Y. Hama

RIKEN-QHP-27
The Study of Goldstone Modes in ν=2 Bilayer Quantum Hall Systems
Yusuke Hama,1,2 Yoshimasa Hidaka,2 G. Tsitsishvili,3 and Z. F. Ezawa4
1Department of Physics, The University of Tokyo, 113-0033, Tokyo, Japan
2Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, Japan
3 Department of Physics, Tbilisi State University, Tbilisi 0179, Georgia
4Advanced Meson Science Laborary, Nishina Center, RIKEN, Wako 351-0198, Japan
Atthefillingfactorν=2,thebilayerquantumHallsystemhasthreephases,thespintripletphase,
2 the spin singlet phase and the canted antiferromagnet (CAF) phase, depending on the relative
1 strength between the Zeeman energy and interlayer tunneling energy. We present a systematic
0 methodtoderivetheeffectiveHamiltonianfortheGoldstonemodesinthesethreephases. Wethen
2 investigate the dispersion relations and the coherence lengths of the Goldstone modes. To explore
a possible emergence of the interlayer phase coherence, we analyze the dispersion relations in the
n
tunneling energy zero limit. We find one gapless mode with the linear dispersion relation in the
u
CAFphase.
J

9
PACSnumbers: 73.43.-f,11.30.Qc,73.43.Qt,64.70.Tg
2

]
l I. INTRODUCTION model has been derived to describe low-energy coherent
l
a phenomena[20]. However, the effective Hamiltonian for
h theGoldstonemodeshasnotbeenderived. Thoughthere
In the bilayer quantum Hall (QH) system, a rich
s- physics emerges by the interplay between the spin and are some results with the use of Grassmannian fields in
e the layer (pseudospin) degrees of freedom[1, 2]. For in- the spin and ppin phases, no attempts have been made
m stance, at the filling factor ν = 1, there arises uniquely in the CAF phase. On the other hand, experimentally, a
. thespin-ferromagnetandpseudospin-ferromagnetphase, role of a Goldstone mode has been suggested by nuclear
t
a showing various intralayer and interlayer coherent phe- magnetic resonance[19] in the CAF phase.
m In this paper we developa generic formalism to deter-
nomena. On the other hand, the phases arising at ν =2
- are quite nontrivial. According to the one-body pic- minethesymmetrybreakingpatternandtoderivetheef-
d fectiveHamiltonianfortheGoldstonemodesinthethree
ture we expect to have two phases depending on the
n
relative strength between the Zeeman gap ∆ and the phases of the ν = 2 bilayer QH system. The symmetry
o Z
breaking pattern reads
c tunneling gap ∆SAS. One is the spin-ferromagnet and
[ pseudospin-singlet phase (abridged as the spin phase)
SU(4) U(1) SU(2) SU(2), (1.1)
for ∆ > ∆ ; the other is the spin-singlet and pseu- → ⊗ ⊗
1 Z SAS
dospin ferromagnet phase (abridged as the ppin phase) and there appear eight Goldstone modes in each phase.
v
3 for∆SAS >∆Z. Instead,anintermediatephase,acanted The corresponding Goldstone modes in the two phases
0 antiferromagnetic phase (abridged as the CAF phase) match smoothly at the phase boundary. All the modes
0 emerges. This is a novel phase where the spin direc- are actually gapped except along the phase boundaries
0 tion is canted and make antiferromagnetic correlations due to explicit symmetry breaking terms. It is im-
7. between the two layers[3, 4]. Das Sarma et al. obtained portant if gapless modes emerge in the limit ∆Z 0
→
0 the phase diagramin the ∆ d plane basedon time- or ∆ 0, where the spin coherence or the inter-
2 dependent Hartree-Fock anSaAlySs−is, where d is the layer layerScAoShe→rence is enhanced. Gapless modes are genuine
1 separation[3, 4]. Later on, an effective spin theory, a Goldstonemodesassociatedwithspontaneoussymmetry
:
v Hartree-Fock-Bogoliubovapproximationandanexactdi- breaking. Naturally we have gapless modes in the spin
i agonalizationstudy wereemployedto improvethe phase phase as ∆ 0 and in the ppin phase as ∆ 0.
X Z → SAS →
diagram[5–8]. Effects of the density imbalance on the It is intriguing that we find one gapless mode with the
r
a CAF were also discussed[9, 10]. lineardispersionrelationintheCAFphaseas∆SAS 0.
→
ThefirstexperimentalindicationoftheCAFphasewas This paper is organized as follows. In Sec. II, we re-
givenbyinelasticlightscatteringspectroscopy[11]. They view the Coulomb interaction of the bilayer QH system
also have observed softening signals indicating second- projectedtothelowestLandaulevel(LLL)andtheSU(4)
order phase transitions[12]. Subsequently, an unambigu- effective Hamiltonianafter making the derivative expan-
ous evidence of the CAF phase was obtained through sion. We also review the ground state structure in the
capacitance spectroscopy as well as magnetotransport three phases. In Sec. III, which is the main part of this
measurements[13–18]. paper, we develop a unified formalism to derive the ef-
The ground state structure of the ν = 2 bi- fective Hamiltonian for the Goldstone modes. Then we
layer QH system has been investigated based on the discuss the SU(4) symmetry breaking pattern and the
SU(4) formalism[20–25]. The expectation values of the Goldstone mode spectrum, such as the dispersion rela-
SU(4) isospin operators are the order parameters, in tions and the coherence length in each phase. In partic-
terms of which an anisotropic SU(4) nonlinear sigma ular, for the investigation of the CAF phase, we find it
2
useful to introduce two convenient coordinates of SU(4) The electron field ψ (x) has four components, and the
α
group space, the s-coordinate and the p-coordinate. We bilayer system possesses the underlying algebra SU(4)
study the dispersions and the coherence length in the with having the subalgebra SU (2) SU (2). We
spin ppin
limit ∆ 0, to explore a possible emergence of the denote the three generators of the SU× (2) by τspin,
SAS → spin a
interlayer phase coherence in the CAF phase. Remark- and those of SU (2) by τppin. There are remaining
ppin a
ably, we find one coherent mode whose coherence length ninegeneratorsτspinτppin. Theirexplicitformisgivenin
a b
diverges. Section IV is devoted to discussion. Apendix A.
All the physical operators required for the description
of the system are constructed as the bilinear combina-
II. THE SU(4) EFFECTIVE HAMILTONIAN tions of ψ(x) and ψ†(x). They are 16 density operators
AND THE GROUND STATE STRUCTURE
ρ(x)=ψ†(x)ψ(x),
Electrons in a plane perform cyclotron motion under 1
S (x)= ψ†(x)τspinψ(x),
perpendicular magnetic field B⊥ and create Landau lev- a 2 a
els. The number of flux quanta passing through the sys- 1
P (x)= ψ†(x)τppinψ(x),
tem is NΦ B⊥S/ΦD, where S is the area of the sys- a 2 a
≡
tem and ΦD = 2π¯h/e is the flux quantum. There are R (x)= 1ψ†(x)τspinτppinψ(x), (2.6)
NΦ Landau sites per one Landau level, each of which is ab 2 a b
associated with one flux quantum and occupies an area
S/NΦ =2πℓ2B, with the magnetic length ℓB = ¯h/eB⊥. where Sa describes the total spin, 2Pz measures the
electron-density difference between the two layers. The
In the bilayer system an electron has two types of in-
p operator R transforms as a spin under SU (2) and
dices, the spin index ( , ) and the layer index (f,b). ab spin
↑ ↓ as a pseudospin under SU (2).
Theycanbeincorporatedin4typesofisospinindexα= ppin
The kinetic Hamiltonianis quenched,since the kinetic
f ,f ,b ,b . One Landausite may containfour electrons.
↑ ↓ ↑ ↓ energyiscommontoallstatesinthe LLL.TheCoulomb
The filling factor is ν =N/N with N the total number
Φ
interaction is decomposed into the SU(4)-invariant and
of electrons.
SU(4)-noninvariant terms
Weexplorethephysicsofelectronsconfinedtothelow-
est Landau level, where the electron position is specified
1
solely by the guiding center X = (X,Y), whose X and H+ = d2xd2yV+(x y)ρ(x)ρ(y), (2.7)
C 2 −
Y components are noncommutative, Z
H− =2 d2xd2yV−(x y)P (x)P (y), (2.8)
[X,Y]= iℓ2. (2.1) C − z z
− B Z
where
By introducing the ladder operators,
e2 1 1
b= √21ℓB(X −iY), b† = √21ℓB(X +iY), (2.2) V±(x)= 8πǫ |x| ± |x|2+d2!, (2.9)
withthelayerseparationd. Thetpunnelingandbiasterms
obeying [b,b†]=1, we construct the Fock states,
are summarized into the pseudo-Zeeman term. Combin-
1 ing the Zeeman and pseudo-Zeeman terms we have
n = (b†)n 0 , n=0,1,2, , b0 =0. (2.3)
| i √n! | i ··· | i
H = d2x(∆ S +∆ P +∆ P ), (2.10)
ZpZ Z z SAS x bias z
−
ThesestatesaretheLandausitesinthesymmetricgauge. Z
We expand the electron field operator by a complete with the Zeeman gap ∆ , the tunneling gap ∆ , and
Z SAS
set of one-body wave functions ϕn(x) = hx|ni in the the bias voltage ∆bias =eVbias.
LLL, The total Hamiltonian is
NΦ H =HC++HC−+HZpZ. (2.11)
ψ (x) c (n)ϕ (x), (2.4)
α α n
≡
n=1 We investigate the regime where the SU(4) invariant
X CoulombtermH+dominatesallotherinteractions. Note
C
where cα(n) is the annihilation operator at the Landau that the SU(4)-noninvariant terms vanish in the limit
site n withα=f ,f ,b ,b . Theoperatorsc (m),c†(n) d,∆ ,∆ ,∆ 0.
| i ↑ ↓ ↑ ↓ α β Z SAS bias →
satisfy the standard anticommutation relations, We project the density operators (2.6) to the LLL by
substituting the fieldoperator(2.4)intothem. Atypical
c (m),c†(n) =δ δ , density operator reads
{ α β } mn αβ
{cα(m),cβ(n)}={c†α(m),c†β(n)}=0. (2.5) Rab(p)=e−ℓ2Bp2/4Rˆab(p), (2.12)
3
in the momentum space, with Hamiltonian density
Rˆ (p)= 1 ne−ipX m c†(n)τspinτppinc(m), Heff =Jsd (∂kSa)2+(∂kPa)2+(∂kRab)2
ab 4π Xmnh | | i a b (2.13) +2Js−(cid:16)X (∂kSa)2+(∂kPz)2+(∂kRaz)(cid:17)2
where c(m) is the 4-componentvector made of the oper- +ρ [ǫ(cid:16)X( )2 2ǫ− ( )2+( )2 (cid:17)
ators c (m). φ cap Pz − X Sa Raz
α
(ǫ+ ǫ−)( )2(cid:16)+X( )2+( )2 (cid:17)
What are observed experimentally are the classical − X − X Sa Pa Rab
dSenρˆs(itpi)esS, w,hwichheraereSexpreepctraetsieonntsvaalugeesnesuricchsatastρeˆcli(np)th=e −(∆ZSz +∆XSAS(cid:0)Px+∆biasPz)−(ǫ+X −(cid:1)ǫ−X()2],.17)
h | | i | i
LLL. The Coulomb Hamiltonian governing the classical
densities are given by[23] where ρ =ρ /ν is the density of states, and
Φ 0
1
J = E0,
Heff =π d2pV+(p)ρˆcl( p)ρˆcl(p) s 16√2π C
D −
Z
2 d d2
+4πZ d2pVD−(p)Pˆzcl(−p)Pˆzcl(p) Jsd =Js"−rπℓB +(cid:18)1+ ℓ2Bed2/2ℓ2Berfc(cid:16)d/√2ℓB(cid:17)(cid:19)#,
− π2 d2pVXd(p)[Sˆacl(−p)Sˆacl(p)+Pˆacl(−p)Pˆacl(p) Js± = 21(Js±Jsd),
Z
+Rˆaclb(−p)Rˆaclb(p)]−π d2pVX−(p)[Sˆacl(−p)Sˆacl(p) ǫX = 21 π2EC0, ǫ±X = 12 1±ed2/2ℓ2Berfc d/√2ℓB ,
Z r
+Pˆcl( p)Pˆcl(p)+Rˆcl( p)Rˆcl(p)] d h (cid:16) (cid:17)i
z − z az − az ǫ− = E0, ǫ =4ǫ− 2ǫ−, (2.18)
π d2pV (p)ρˆcl( p)ρˆcl(p), (2.14) D 4ℓB C cap D− X
X
− 8 −
Z with
where VD and VX are the direct and exchange Coulomb E0 = e2 . (2.19)
potentials, respectively, C 4πǫℓ
B
This Hamiltonian is valid at ν =1 and 2.
VD(p)= 4πeǫ2p e−ℓ2Bp2/2, inIttheisStUo(4b)eirnevmarairaknetdlitmhiatt, walhlepreotpeenrttiaulrbtaetrimves evxacniitsah-
| |
√2πe2ℓ tions are gapless. They are the Goldstone modes associ-
VX(p)= 4πǫ BI0(ℓ2Bp2/4)e−ℓ2Bp2/4, (2.15) ated with spontaneous breaking of the SU(4) symmetry.
There are eight Goldstone modes, as we shall show in
Section III. They get gapped in the actual system, since
and the SU(4) symmetry is explicitly broken. Nevertheless
we call them the Goldstone modes.
e2 The ground state is obtained by minimizing the ef-
VD±(p)= 8πǫp 1±e−|p|d e−ℓ2Bp2/2, fective Hamiltonian (2.17) for homogeneous configura-
| |(cid:16) (cid:17) tions ofthe classicaldensities. The orderparametersare
√2πe2ℓ
VX±(p)= 8πǫ BI0(ℓ2Bp2/4)e−ℓ2Bp2/4 tshheowcnla[2ss0i]caaltdνen=sit2ietshfaotrtthheeygarroeungidvesntaitne.teIrtmhsasofbteweno
± e42πℓ2Bǫ ∞dke−12ℓ2Bk2−kdJ0(ℓ2B|p|k), (2.16) parameters α and β as
Z0 ∆
0 = Z(1 α2) 1 β2,
Sz ∆ − −
0
with I (x) the modified Bessel function, and J (x) the
0 0 ∆ p ∆
Bessel function of the first kind. 0 = SASα2 1 β2, 0 = SASα2β,
Px ∆ − Pz ∆
0 0
Since the exchange interaction V±(p) is short ranged, ∆ p
itisagoodapproximationtomakethederivativeexpan- 0 = SASα 1 α2β,
Rxx − ∆ −
sion,orequivalently,themomentumexpansion. Wemay 0
asentdρˆcRˆl(cpl()p=) ρ=0,ρSˆacl(p)(p=) ρfΦorSat(hpe),sPtˆuacdl(yp)of=GρoΦldPsat(opn)e, R0yy =−∆∆Z0α p1−α2 1−β2,
ab ΦRab ∆ p p
modes. Taking the nontrivial lowest order terms in 0 = SASα 1 α2 1 β2, (2.20)
the derivative expansion, we obtain the SU(4) effective Rxz ∆0 − −
p p
4
with all others being zero. The parameters α and β, that (1 β2)/∆2 = O(1). Up to the order O(∆2 ),
− SAS SAS
satisfying α 1 and β 1, are determined by the (2.22) is reduced to
| | ≤ | | ≤
variational equations as
∆2 4ǫ−(1 α2)
∆2 4ǫ− ∆2 β2∆2 ∆2 SAS 1+ X − =0.
∆bi∆as2Z == 41−SǫA−XβS2+−2α2(Xǫ−D∆(cid:0)−0p0ǫ−−X1)−β+2SAS(cid:1)1, , ((22..2221)) T(cid:18)he sZol−ut1io−nsβa2r(cid:19)e  q∆2Z(1−α2)+ ∆12S−AβSα22(2.27)
β∆SAS (cid:0) ∆0 (cid:1) 1 β2
−
∆ 2
where p β = 1 SAS +O(∆4 ), (2.28)
±s −(cid:18) ∆Z (cid:19) SAS
∆0 = ∆2SASα2+∆2Z(1−α2)(1−β2). (2.23) with
q
As a physical variable it is more convenient to use the ∆ ∆ +O(∆3 ), (2.29)
0 → SAS SAS
imbalance parameter defined by
for (2.23). By using (2.24) we have
∆
σ 0 = SASα2β, (2.24)
0 ≡Pz ∆ ∆2
0 0 =σ = α2 1 SAS +O(∆4 ). (2.30)
instead of the bias voltage ∆ . This is possible in the Pz 0 ± s − ∆2Z SAS
bias
ppin and CAF phases. The bilayer system is balanced
Theparametersαandβ aresimplefunctionsofthephys-
at σ = 0, while all electrons are in the front layer at
0 ical variables ∆ /∆ and σ in the limit ∆ 0.
σ =1, and in the back layer at σ = 1. SAS Z 0 SAS →
0 0 − In particular, one of the layers becomes empty in the
There are three phases in the bilayer QH system at
ppinphaseandalsoneartheppin-phaseboundaryinthe
ν =2. We discuss them in terms of α and β.
CAF phase, since we have σ 1 as α 1. On the
First, when α=0, it follows that 0 =1, 0 = 0 = 0 → ± →
Sz Pa Rab other hand, the bilayer system becomes balanced, since
0a,llsfionrcmeu∆la0s=in∆(Z2.201)−. βT2h.isNiostethtehastpiβndpihsaapsep,eawrhsicfrhomis twheehsapvien-σp0h→ase0baosuαnd→ary0iinntthheesCpiAnFphpahsaesaen.dWaelsomnigehatr
p
characterized by the fact that the isospin is fully polar- expect novel phenomena associated with the interlayer
ized into the spin direction with phase coherence in the CAF phase.
0 =1, (2.25)
Sz
III. EFFECTIVE HAMILTONIAN FOR
and all others being zero. The spins in both layerspoint
GOLDSTONE MODES
to the positive z axis due to the Zeeman effect.
Second,whenα=1,itfollowsthat 0 =0and( 0)2+
Sz Px Havingreviewedthethreephasesinthebilayersystem
( 0)2 =1. Thisistheppinphase,whichischaracterized
Pz at ν = 2, we proceed to discuss the symmetry breaking
by the fact that the isospin is fully polarized into the
pattern and construct the effective Hamiltonian for the
pseudospin direction with
Goldstone modes in each phase. There is a systematic
methodforthispurpose,whichwasdevelopedinparticle
0 = 1 β2, 0 =β =σ , (2.26)
Px − Pz 0 and nuclear physics[26, 27].
p We analyze excitations around the classical ground
and all the others being zero.
state (2.20). It is convenient to introduce the SU(4)
For intermediate values of α (0 < α < 1), not only
isospin notation such that
thespinandpseudospinbutalsosomecomponentsofthe
residualspinarenonvanishing,wherewemaycontrolthe
(0) = 0, (0) = 0, (0) = 0 . (3.1)
density imbalance by applying a bias voltage as in the Ia0 Sa I0a Pa Iab Rab
ppin phase. It follows from (2.20) that, as the system
We set all of them into one 15-dimensional vector (0)
goesawayfromthespinphase(α=0),thespinsbeginto Iµν
cant coherently and make antiferromagnetic correlations withtheindexµν: NotethatthereisnocomponentI0(00).
between the two layers. Hence it is called the canted Most general excitations are described by the operator
antiferromagnetic phase.
µ′ν′
The interlayer phase coherence is an intriguing phe-
nomenon in the bilayer QH system[1]. Since it is en- Iµν(x)=I(x)expi πγδTγδ Iµ0′ν′, (3.2)
hanced in the limit ∆SAS →0, it is worthwhile to inves- Xγδ µν
tigatetheeffectiveHamiltonianinthislimit. Weneedto   
knowhowtheparametersαandβ areexpressedinterms where T are the matrices of the broken SU(4) genera-
γδ
of the physical variables. Form (2.21) it is trivial to see torsintheadjointrepresentationofSU(4),eachofwhich
5
is a 15 15 matrix. The greek indices run over 0,x,y,z. in eight directions, xµ and yµ, the broken generators
×
The phase field π (x) are the Goldstone modes associ- are T and T . Consequently, the symmetry breaking
γδ xµ yµ
atedwiththebrokengenerators,andthecoefficient (x) pattern reads
I
is the amplitude function corresponding to the “sigma”
field in the linear sigma model. SU(4)→U(1)Iz0 ⊗SU(2)I0a ⊗SU(2)Iza, (3.10)
It has been argued[20] that there are nine indepen-
implying that the unbroken generators are T , T and
dent real physical fields. They are the amplitude fluctu- z0 0a
T .
ationfield (x)satisfying 2(x) 1,andeightGoldstone za
I I ≤ Werequire(3.2)tosatisfytheSU(4)algebraicrelation
modesπ (x). Hence,onlyeightgeneratingmatricesT
γδ γδ
are involved in the formula (3.2). We shall explicitly de- [ (x,t), (y,t)]=iǫ ρ−1 (x,t)δ(x y), (3.11)
termine them in each phase in the following subsections. Iaµ Ibµ abc Φ Ic0 −
Since we are only interested in an effective low energy so that the field describes the SU(4) isospin. From
µν
theory of the Goldstone bosons, we set (x) = 1. Then (3.11), we obtainIthe equal-time commutation relations
I
we may identify for the Goldstone modes,
Sa =Ia0, Pa =I0a, Rab =Iab, (3.3) [π˜xµ(x,t),π˜yµ(y,t)]=iδ(x−y), (3.12)
and express various physical variables in terms of the with π˜ = ρ1/2π . Equivalently, we may construct a
γδ Φ γδ
Goldstone modes π (x).
γδ Lagrangianformalismsothat(3.12)isthecanonicalcom-
We expand the formula (3.2) in π ,
γδ mutation relation.
Itfollowsfrom(3.3)and(3.9)thattheeightGoldstone
Iµν(x)=Iµ0ν +Iµ(1ν)(x)+Iµ(2ν)(x)+··· , (3.4) modes are explicitly given by
mwhoedreeπIµ(nν.)(Uxp) tios tthheesnetchonodrdoerdretre,rtmheyinatrhee Goldstone Sx =−πy0, Sy =πx0, Rxa =−πya, Rya =(π3x.a1.3)
γδ
Substituting them into (2.17), we obtain the effective
Iµ(1ν)(x)=−fµν,γδ,µ′ν′πγδIµ0′ν′, (3.5) Hcaanmonilitcoanliasentsoofftπ˜he Ganodldπs˜tonaesmodes in terms of the
1 xµ yµ
Iµ(2ν)(x)= 2!fµν,γδ,µ′′ν′′fµ′′ν′′,γ′δ′,µνπγδπγ′δ′Iµ0ν, (3.6) 2J
spin = s (∂ π˜ )2+(∂ π˜ )2
k xµ k yµ
H ρ
where fµν,γδ,µ′ν′ are the structure constant of SU(4), 0 µ=0,z
X (cid:2) (cid:3)
(Tγδ)µµν′ν′ =ifµν,γδ,µ′ν′, (3.7) + 2ρJsd (∂kπ˜xa)2+(∂kπ˜ya)2
0
a=x,y
X (cid:2) (cid:3)
about which we explain in Appendix A (A7). 4ǫ− (2) 2ǫ− (π˜ )2+(π˜ )2
Each phase is characterized by the order parameter − XIz0 − X xµ yµ
µ=0,z
0 ,whicharenothingbut(2.20). Thekeyobservationis X (cid:2) (cid:3)
Iµν (2) (2) (2)
thatthefirstorderterm µ(1ν)(x)containsallinformations −ρφ ∆ZIz0 +∆SASI0x +∆biasI0z , (3.14)
I
aboutthesymmetrybreakingpatternandtheassociated h i
Goldstonemodes,yieldingtheirkinematicterms. Onthe where (2) are given by (3.6), and read
I0a
otherhand,the secondorderterm (2)(x) providesthem
with gaps. Iµν (2) =1 1 (π2 +π2 ),
Iz0 − 2 xµ yµ
µ=0,x,y,z
X
π π +π π π π π π
(2) = xy yz yz xy − yy xz− xz yy,
A. Spin Phase I0x 2
π π +π π π π π π
(2) xx yy yy xx xy yx yx xy
= − − . (3.15)
First we analyze the spin phase. Setting α = 0 in the I0z 2
order parameters (2.20), we obtain
The annihilation operators are defined by
0 =δ δ . (3.8)
Iµν µz ν0 π˜x0+iπ˜y0 π˜xx+iπ˜yx
η = , η = ,
1 2
With the use of this, itis straighforwardto calculate the √2 √2
first order term (1)(x) in (3.5), π˜xy+iπ˜yy π˜xz +iπ˜yz
µν η = , η = , (3.16)
I 3 4
√2 √2
(1) = π , (1) =π . (3.9)
Ixµ − yµ Iyµ xµ and satisfy the commutation relations,
There are eight fields π and π with µ = 0,x,y and
z, which are the Goldstyoµne modxeµs. Since they emerge ηi(x,t),ηj†(y,t) =δijδ(x−y), (3.17)
h i
6
with i,j =1,2,3,4. They depend not only ∆ but also on the exchange
Z
The effective Hamiltonian (3.14) reads in terms of the Coulomb energy ǫ− and the interlayer stiffness originat-
X
creation and annihilation variables (3.16) as ing in the interlayer Coulomb interaction.
We finally analyze those of η and η , which are
3,k 4,k
4J 4Jd
spin = s ∂ η†∂ η + s ∂ η†∂ η coupled. Hamiltonian(3.23)canbewritteninthematrix
H ρ0 k a k a ρ0 k a k a form,
a=1,4 a=2,3
X X
+∆Z ηa†ηa+[∆Z+4ǫ−X] ηa†ηa spin = η3,k † Ak −i∆SAS η3,k ,
a=1,4 a=2,3 H3 η4,k i∆SAS Bk η4,k
X X (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
∆ ∆ (3.28)
bias[η†η η†η ] SAS[η†η η†η ].
− i 2 3− 3 2 − i 3 4− 4 3 where
(3.18)
4Jd 4J
A = sk2+∆ +4ǫ−, B = sk2+∆ . (3.29)
The variables η2, η3 and η4 are mixing. k ρ Z X k ρ Z
0 0
In the momentum space the annihilation and creation
operators are η and η† together with the commuta- Hamiltonian (3.28) can be diagonalized as
i,k i,k
tion relations, spin = η˜3,k † Eη˜3 0 η˜3,k , (3.30)
ηi,k,ηj†,k′ =δijδ(k−k′). (3.19) H3 (cid:18)η˜4,k (cid:19) (cid:18) 0 Eη˜4 (cid:19)(cid:18)η˜4,k (cid:19)
where
h i
For the sake of the simplicity we consider the balanced
1
configuration with ∆bias = 0 in the rest of this subsec- Eη˜3 = 2 Ak+Bk+ (Ak−Bk)2+4∆2SAS ,
tion. Then the Hamiltonian density is given by (cid:20) q (cid:21)
1
Hspin = d2k spin, Eη˜4 = 2 Ak+Bk− (Ak−Bk)2+4∆2SAS , (3.31)
H (cid:20) q (cid:21)
Z and the annihilationoperatorη˜ (i=3,4)givenby the
spin = spin+ spin+ spin, (3.20) i,k
H H1 H2 H3 form
where
i C2+4∆2 +C η 2∆ η
− k SAS k 3,k− SAS 4,k
spin = 4Jsk2+∆ η† η , (3.21) η˜3,k = (cid:16)p (cid:17) ,
H1 (cid:20) ρ0 Z(cid:21) 1,k 1,k 2 Ck2+4∆2SAS+Ck Ck2+4∆2SAS
H2spin =(cid:20)44ρJJ0sddk2+∆Z+4ǫ−X(cid:21)η2†,kη2,k, 4J (3.22) η˜4,k = −ir(cid:16)p(cid:16)Ck2+4∆2SAS−Ck(cid:17)pη3,k+2∆SASη(cid:17)4,k,
H3spin = ρsk2+∆Z+4ǫ−X η3†,kη3,k+ ρsk2+∆Z 2 Ck2+4∆2SAS−Ck Ck2+4∆2SAS
(cid:20) 0 ∆ (cid:21) (cid:20) 0 (cid:21) r (cid:16) p (cid:17)(3.32)
η† η SAS η† η η† η . (3.23)
× 4,k 4,k− i 3,k 4,k− 4,k 3,k with C = A B . The annihilation operators (3.32)
k k k
h i −
We firstanalyzethe dispersionrelationandthe coher- satisfy the commutation relations
ence length of η . From (3.21), we have
1,k η˜ ,η˜† =δ δ(k k′), (3.33)
i,k j,k′ ij −
4J
Eη1(k)= ρsk2+∆Z, (3.24) with i,j =3,4.hWe obtaiin the dispersions for the modes
0
η˜ (i=3,4) from (3.29) and (3.31).
i,k
πJ
s By taking the limit k 0 in (3.31), we have two gaps
ξ =2l . (3.25)
η1 B ∆ →
r Z Eη˜3 =∆ +2ǫ− + 4(ǫ−)2+∆2 12 ,
The coherent length diverges in the limit ∆ 0. This k=0 Z X X SAS
Z
modeisapurespinwavesinceitdescribesthefl→uctuation Eη˜4 =∆ +2ǫ− (cid:2)4(ǫ−)2+∆2 (cid:3)21 . (3.34)
of and as in (3.13). Indeed, the energy (3.24) as k=0 Z X − X SAS
x y
welSl as theScoherent length (3.25) depend only on the The gapless condition (Eη˜(cid:2)4 =0) implies(cid:3)
k=0
Zeeman gap ∆ and the intralayer stiffness J .
Z s ∆ (∆ +4ǫ−) ∆2 =0, (3.35)
We next analyze those of η2,k, Z Z X − SAS
which holds only along the boundary of the spin and
4Jd
E (k)= sk2+∆ +4ǫ−, (3.26) CAF phases: See (4.17) in Ref.[20]. In the interior of
η2 ρ0 Z X the spin phase we have ∆Z(∆Z +4ǫ−X)−∆2SAS > 0, as
impliesthattherearisenogaplessmodesfromη˜ andη˜ .
πJd 3 4
ξ =2l s . (3.27) These excitation modes are residual spin waves coupled
η2 Bs∆Z+4ǫ−X with the layer degree of freedom.
7
B. Ppin Phase The SU(4) isospin density fields p satisfy the SU(4)
Iµν
algebraic relations
We next analyze the ppin phase. Setting α=1 in the
order parameters (2.20), we obtain p (x,t), p (y,t) =iǫ ρ−1 p(x,t)δ(x y), (3.46)
Iµa Iµb abc Φ I0c −
h i
0 = 1 β2δ δ +βδ δ . (3.36)
Iµν − µ0 νx µ0 νz from which we obtain the canonical commutation rela-
In order to determpine the symmetry breaking pattern, tions for the Goldstone modes,
werotatethis vectoraroundthe 0y axissothatonly one
π˜p (x,t),π˜p (y,t) =iδ(x y), (3.47)
component becomes nonzero. We can show that µy µz −
p(0) [V (θ )]µ′ν′ (0) =δ δ , (3.37) with π˜p (cid:2)=ρ1/2πp . (cid:3)
Iµν ≡ β β µν Iµ′ν′ µ0 νx µν Φ µν
WegoontoderivetheeffectiveHamiltoniangoverning
by choosing these Goldstone modes. The first step is to convert the
relation (3.41) to express the original fields in terms of
V (θ )=exp(iθ T ), (3.38)
β β β 0y those in the rotated system. Explicitly we have
with cosθ = 1 β2 and sinθ = β.
In the rβotated b−asis the orderβpar−ameter has a single Iµx =cθβIµpx+sθβIµpz,
nonzero compponent just as (3.8) in the case of the spin Iµz =−sθβIµpx+cθβIµpz,
phase. Therefore the further analysis goes in parallel = p , = p . (3.48)
withthatgivenintheprevioussubsection. Namely,there Ia0 Ia0 Iyµ Iyµ
are eight Goldstone fields, The second step is to expand (3.42) in terms of πp ,
γδ
p(1) = πp , p(1) =πp , (3.39)
Iµy − µz Iµz µy p = πp + (π2), p =πp + (π2),
and the symmetry breaking pattern reads Iµy − µz O Iµz µy O
πp πp +πp πp πp πp πp πp
p = zz yy yy zz − zy yz− yz zy + (π3),
SU(4)→SU(2)Iap0 ⊗U(1)I0px ⊗SU(2)Iapx, (3.40) Ix0 πp πp +πp πp 2 πp πp πp πp O
p = zy xz xz zy − zz xy− xy zz + (π3),
precisely as in the spin phase. Iy0 2 O
Letusrelatethevariablesintherotatedsystemtothe πp πp +πp πp πp πp πp πp
originalvariablesintheformula(3.2). TheSU(4)isospin p = xy yz yz xy− yy xz − xz yy + (π3),
Iz0 2 O
operator after the rotation is given by
πp πp +πp πp +πp πp +πp πp
p = xz 0z 0z xz xy 0y 0y xy + (π3),
Iµpν(x)=[Vβ(θβ)]µµν′ν′Iµ′ν′(x), (3.41) Ixx −πp πp +πp πp +2πp πp +πp πp O
p = yz 0z 0z yz yy 0y 0y yy + (π3),
with the use of (3.38). We substitute (3.2) into this for- Iyx − 2 O
mula to find πp πp +πp πp +πp πp +πp πp
p = zz 0z 0z zz zy 0y 0y zy + (π3),
µ′ν′ Izx − 2 O
Iµpν(x)=expi πγpδTγδ Iµp′(ν0′), (3.42) I0px =1− (πµpy)2+2 (πµpz)2 +O(π3). (3.49)
γδ
X µν µ=0,x,y,z
X
  
with (3.37), where πp is defined by
γδ Now, using (3.3) we obtain the expressionof a, a, ab
in terms of πp , which we substitute into thSe ePffecRtive
πγpδ = [Vβ(θβ)]γγδ′δ′πγ′δ′, (3.43) Hamiltonian (γ2δ.17).
Iγpδ(0) = [Vβ(θβ)]γγδ′δ′Iγ0′δ′, (3.44) GoIlndstthoinsewmayodweesdienritveermthseoeffftehcteivceanHoanmicialtlosneitasnooffπ˜thpe
µy
and π˜p . In the momentum space it reads
while p(0) has been used by (3.37). Here, we have used µz
Iγδ
the formula of the SU(N) group, p = p+ p+ p, (3.50)
H H1 H2 H3
T Φ′ = T [expiθ Ad(T )]cΦ
b b b a a b c where
b b
X X
=exp[iθaTa]ΦbTbexp[ iθaTa], (3.45) p =Cpπ˜p† π˜p +Bpπ˜p† π˜p , (3.51)
− H1 k 0y,k 0y,k k 0z,k 0z,k
w1,h.e.r.e,dΦimb iSsUa(nN)a,rbaintrdaryexapd[jioθinTt ]veicstotrhweitehlema,ebn,tc o=f H2p =Apkπ˜zpy†,kπ˜zpy,k+Bkpπ˜zpz†,kπ˜zpz,k, (3.52)
a a p =(~π˜p)† p~π˜p, (3.53)
SU(N). Here we have N=4 and Φb corresponds to πµν. H3 M
8
with Their coherence lengths are
Apk = 2ρJ01βk2+ 2 ∆1S−ASβ2 −2ǫ−X(1−β2), ξπ˜zpy =2lBvu√∆1S−AβS2 −π4Jǫ1β−X(1−β2), (3.62)
Bp = 2Jsdk2+ p∆SAS , ut
Ckp = 2ρJ01βk2+2p∆1S−ASβ2 +ǫ (1 β2), ξπ˜zpz =2lBsπJsd∆pS1A−S β2. (3.63)
k ρ0 2 1 β2 cap − Itappearsthatξπ˜zpy isill-definedfor∆SAS 0in(3.60).
− →
Thisisnotthe caseduetotherelation(3.65)inthe ppin
J1β =(1−β2)Js+pβ2Jsd, phase, which we mention soon.
~π˜p =(π˜p ,π˜p ,π˜p ,π˜p ), Finally, making an analysis of the Hamiltonian (3.53)
yy,k xz,k xy,k yz,k
Ap ∆ /2 0 0 as in the case of the spin phase, we obtain the condition
∆ k/2 BZp 0 0 for the existence of a gapless mode,
Mp = Z00 00k −∆AZpk/2 −∆BZkp/2. (3.54) ∆1SASβ2 " ∆1SASβ2 −4ǫ−X(1−β2)#−∆2Z =0. (3.64)
− −
The canonical commutation relations are
Itpoccurs alonpg the ppin-canted boundary: See (5.3) and
(5.4) in Ref.[20]. Inside the ppin phase, since we have
π˜p ,π˜p =iδ(k+k′), (3.55)
µy,k µz,k′
h i ∆SAS ∆SAS 4ǫ−(1 β2) ∆2 >0, (3.65)
for each µ=0,x,y,z. 1 β2 " 1 β2 − X − #− Z
− −
We first analyze the dispersions and the coherence
lengths of the canonical sets of the modes π˜p and π˜p thpere are no gpapless modes.
0y 0z
from (3.51). Since the ground state is a squeezed co-
herent state due to the capacitance energy ǫ , it is
cap
C. CAF phase
more convenient[1] to use the dispersion and the coher-
ence lengths of π˜p and π˜p separately. The dispersion
0y 0z
relations are given by FinallyweanalyzetheCAFphase. Thisphaseischar-
acterized by the order parameters (2.20), which we may
Eπ˜0py = 2J1βk2+ ∆SAS +ǫ (1 β2), (3.56) rewrite as
k ρ0 2 1−β2 cap − Iµ(0ν) =cθδcθαδµzδν0+sθδsθα cθβδµ0δνx−sθβδµ0δνz
Eπ˜0pz = 2Jsdk2+ p∆SAS , (3.57) +sθδcθαsθβδµxδνx−cθδs(cid:0)θαδµyδνy +sθδcθαcθβδµ(cid:1)xδνz,
k ρ0 2 1 β2 (3.66)
−
p where
and their coherence lengths are
c cosθ = 1 α2, s sinθ =α,
θα ≡ α − θα ≡ α
ξπ˜0py =2lBvuut√∆1S−AβS2 +π2Jǫc1βap(1−β2), (3.58) ccθθβδ ≡≡ccoossθθβδ ==p∆pZ1−∆1β−2,β2sθβ1≡−sαin2,θβ =sθδ−≡β,sinθδ = ∆∆SASα.
p 0 0
ξπ˜0pz =2lB πJsd 1−β2. (3.59) p (3.67)
s ∆pSAS
. The order parameter (0) is quite complicated. Never-
µν
I
They describe a ppin wave. theless, the problem is just to find an appropriate rota-
The similar analysis can be adopted for the canonical tion in the SU(4) space so that the order parameter has
sets of π˜p and π˜p in (3.52). The dispersion relations only a single nonzero component after the rotation.
zy zz
are given by There aretwoways. One is by choosingthe rotational
transformation as
Eπ˜zpy = 2J1βk2+ ∆SAS 2ǫ−(1 β2), (3.60) Uαs,β =exp[iθδTyz]exp[iθαTxy]Vβ(θβ), (3.68)
k ρ0 2 1 β2 − X −
− with V given by (3.38), and we obtain
Ekπ˜zpz = 2ρJ0sdk2+ 2p∆1S−ASβ2. (3.61) β Iµscν(0) ≡ Uαs,β µµ′νν′Iµ(0′ν)′ =δµzδν0. (3.69)
p (cid:2) (cid:3)
9
Inthis rotatedbasis,the further analysisgoesin parallel for the fields in the ppin phase.
with that given in the spin phase. Another choice of the
We require (3.72) to satisfy the SU(4) algebraic rela-
rotational transformation is given by
tion,
π π
Up =exp i θ T exp i θ T V (θ ),
α,β δ− 2 yz α− 2 xy β β
h (cid:16)π (cid:17) i π h (cid:16) (cid:17) i sc(x,t), sc(y,t) =iρ−1 sc(x,t)δ(x y), (3.78)
=exp i T exp i T Us , (3.70) Ixµ Iyµ Φ Iz0 −
− 2 yz − 2 xy α,β
(cid:2) (cid:3)
h i h i
obtaining from which we obtain the canonical commutation rela-
µ′ν′ tion,
pc(0) Up (0) =δ δ . (3.71)
Iµν ≡ α,β µν Iµ′ν′ µ0 νx
h i
π˜sc(x,t),π˜sc(y,t) =iδ(x y), (3.79)
Inthis rotatedbasis,the further analysisgoesin parallel xµ yµ −
with that given in the ppin phase. We call the rotated
(cid:2) (cid:3)
basisoftheSU(4)groupgivenby(3.68),thes-coordinate,
and the rotated basis given by (3.70), the p-coordinate. with π˜sc =ρ1/2πsc.
µν Φ µν
They give the identical results.
We are able to derive the effective Hamiltonian for
We make an analysis by employing the s-coordinate.
the Goldstone modes precisely as we did for the ppin
Namely, we define the SU(4) isospin operator in the s-
phase. Namely,weobtaintherelationsbetweentheorig-
coordinate by inal fields and the fields πsc from (3.72). We give
Iµν γδ
Iµscν(x)= Uαs,β µµν′ν′Iµ′ν′(x) tThheuesxwpleicditerrievleattiohneseiffnecAtpivpeenHdaimx:iltSoeneia(nB2o)f, tahnedG(Bo1ld)-.
(cid:2) (cid:3) µ′ν′ stonemodesintermsofthecanonicalsetsofπ˜xscµandπ˜yscµ.
= exp i πscT sc(0), Working in the momentum space, the effective Hamilto-
  γδ γδ Iµ′ν′ nian reads,
γδ
X µν
  
(3.72)
sc = sc+ sc, (3.80)
H H1 H2
where
πγscδ = Uαs,β γγδ′δ′πγ′δ′ (3.73) where
(cid:2) (cid:3)
with (3.2) and (3.45).
The eight Goldstone fields are, sc =Gc (π˜sc )†π˜sc +Gc (π˜sc )†π˜sc , (3.81)
H1 1,k x0,k x0,k 2,k y0,k y0,k
Ixscµ(1) =−πyscµ, Iyscµ(1) =πxscµ, (3.74) H2sc =~πksc†Ms2c~πksc, (3.82)
and the symmetry breaking pattern reads
with
SU(4) SU(2)Isc U(1)Isc SU(2)Isc, (3.75)
→ z0 ⊗ 0a ⊗ za
jusHtearseiwnethreemcaarskeshoofwththeespGionl/dpsptoinnephmaosde.es inthe CAF Gc = 2 Jαk2+ ∆0c−θβ1,
1,k ρ 1 2
phase are transformed into those in spin/ppin phase at 0
the phase boundary. On one hand, the field πsc shift 2 M 4(s2 c2 +c2 )ǫ−
µν Gc = (c2 J +s2 Jβ)k2+ − θδ θβ θδ X,
smoothly to the field (3.13), by the inverse transforma- 2,k ρ θδ s θδ 1 2
0
tion of (3.68), or by taking the limit α,β 0, as
→ J1α =c2θαJs+s2θαJsd, M =4c2θαǫ−X +∆0c−θβ1, (3.83)
πsc π , (3.76)
µν → µν
so that subscript of πµscν perfectly matches with πµν for and
each µν in the spin phase. On the other hand, πsc
µν
shift smoothly to (3.43), by the inverse transformation
of exp(iθδTyz)exp(iθαTxy), or taking the limit α 1 as π˜xscx,k Ac cc ec 0 0 0
→ π˜sc cc Cc fc 0 0 0
πxsc0 →−πzpz, πysc0 →πzpy, ~πsc = π˜yxscyz,,kk , sc = ec fc Fc 0 0 0 .
πππxxxssscccxyz →→→ππ−xyppπyy0p,,z, ππyyssπcczyyscx→→→ππypxpπzz0p,,y, (3.77) k  πππ˜˜˜xyyssscccxzy,,,kkk  M2  000 000 000 Babccc Ddaccc (E3db.ccc84)
10
The Matrix elements in (3.84) are given by D. CAF phase in ∆SAS →0
Ac = 2k2 c2 Jβ +s2 Jd + M 2s2 c2 ǫ−, The effective Hamiltonian in the CAF phase is too
ρ θδ 3 θδ s 2 − θβ θδ X complicated to make a further analysis. We take the
0
Bc = 2k2 hc2 Jβ +s2 Jβi + ∆0 + c2θβǫα, lairmeiotb∆taSiAnSed→. In0ptaorteixcaumlairnweeifwsooumldelsiikmeptolifiseedekfoforrmgualaps-
ρ θα 3 θα 1 2c 2
0 θβ less modes. Such gapless modes will play an important
h i
2k2 M role to drive the interlayer coherence in the CAF phase.
Cc = Jβ + 2c2 ǫ−,
ρ 1 2 − θβ X In this limit we have
0
DEcc == 22ρkk022 hcs2θ2δ(cid:16)sc2θ2αJJ3ββ ++cs2θ2αJJ1ββ(cid:17)++sc2θ2δJJ1ααi++ M2∆cθ0β + c2θδs22θβǫα, cθβ = ∆∆SAZS, sθβ =±s1−(cid:18)∆∆SAZS(cid:19)2
ρ0 h θδ(cid:16) θα 3 θα 1(cid:17) θδ 3i 2 cθδ =cθα, sθδ =sθα, ∆0c−θβ1 =∆Z,
+s2θβs2θδc2θαǫcap−2(c2θβs2θδ +c2θδ)s2θαǫ−X, ac =bc =cc =ec =L=0. (3.88)
2k2 M
Fc = Jd+ , (3.85)
ρ s 2 By using the above equations, (3.81) become
0
and sc = 4 Jαk2+∆ ηsc†ηsc , (3.89)
H1 ρ 1 Z 1,k 1,k
(cid:20) 0 (cid:21)
ac = 2k2c c Jβ + s2θβcθδǫ ,
ρ θδ 2θα 2 4 α with
0
2k2 ∆
bc = s s Jβ +L+ SASc s ǫ , π˜sc +iπ˜sc
− ρ0 θδ 2θα 2 4∆0 θα 2θβ α η1sc,k = x0,k√2 y0,k. (3.90)
2k2
cc = c Jβ +s c ǫ−,
ρ θδ 2 2θβ θδ X
0 From (3.89) we have the dispersion and the coherence
dc =−s2θα4s2θδ[2ρk2 J1β +Jsd−J3β −Js length for mode η1sc
0
(cid:16) (cid:17)
+s2θβ(2ǫ−X −ǫcap)]− N2 , Eη1sc = ρ4 J1αk2+∆Z, ξη1sc =2lB π∆J1α. (3.91)
L N 0 r Z
ec = , fc = , (3.86)
−2 2 This mode is reminiscent of the spin wave (3.24) in the
spin phase.
with
We next investigate sc in (3.82). It yields
H2
s
J1β =c2θβJs+s2θβJsd, J2β = 22θβ(Jsd−Js), H2sc =H2sc,1+H2sc,2, (3.92)
Jβ =c2 Jd+s2 J , Jα =c2 Jd+s2 J , 4
L3 = θβs2θsβ sθβs s (23ǫ− θǫα s)+cθα∆sSASǫ , H2sc,1 =(cid:20)ρ0J1αk2+∆Z(cid:21)η2sc,k†η2sc,k, (3.93)
− 2 θδ 2θα X − cap θα ∆ α sc =~πsc† sc ~πsc , (3.94)
(cid:20) 0 (cid:21) H2,2 2,kM2,2 2,k
N = s2θδs22θαs2θβ(2ǫ−X −ǫcap)+ ∆∆SAS(cθδcθαs2θβǫα+∆Z), where
0
ǫα =4c2θαǫ−X +2s2θαǫcap, (3.87) ηsc = π˜xscx,k+iπ˜yscx,k, (3.95)
2,k √2
where we denote s = sin2θ , s = sin2θ , and
2θα α 2θβ β
s =sin2θ .
2θδ δ and
It canbe verifiedthat the effective Hamiltonian(3.81)
and(3.82)reproducetheeffectiveHamiltonianinthespin π˜sc C˜c f˜c 0 0
xz,k
apghoanseal(i3ze.2t0h)i,sbHyatmakilitnogntiahnelwimitihttαhe→tr0anfisrfsotr,manadtiothneVn−d1i-, ~πsc = π˜yscy,k , sc = f˜c F˜c 0 0 ,
β 2,k π˜sc M2,2 0 0 D˜c d˜c
ortakingα,β 0. Ontheotherhand,wereproducethe yz,k
effective Hami→ltonianinthe ppin phase(3.50), bytaking  π˜xscy,k   0 0 d˜c D˜c 
   (3.96)
the limit α 1, in (3.81) and (3.82).
→