Fermi liquid instabilities in the spin channel

0.47 MB·English

by  Congjun Wu

Fermi liquid instabilities in the spin channel
Congjun Wu,1 Kai Sun,2 Eduardo Fradkin,2 and Shou-Cheng Zhang3
1Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030
2Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080
3Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-4045
(Dated: February 6, 2008)
7 WestudytheFermisurfaceinstabilitiesofthePomeranchuktypeinthespintripletchannelwith
0 highorbitalpartialwaves(Fla(l>0)). Theorderedphasesareclassifiedintotwoclasses,dubbedthe
0 αandβ-phasesbyanalogytothesuperfluid3He-AandB-phases. TheFermisurfacesintheα-phases
2 exhibitspontaneousanisotropicdistortions,whilethoseintheβ-phasesremaincircularorspherical
withtopologicallynon-trivialspinconfigurationsinmomentumspace. Intheα-phase,theGoldstone
n
modes in the density channel exhibit anisotropic overdamping. The Goldstone modes in the spin
a
channel have nearly isotropic underdamped dispersion relation at small propagating wavevectors.
J
Due to the coupling to the Goldstone modes, the spin wave spectrum develops resonance peaks in
8
both the α and β-phases, which can be detected in inelastic neutron scattering experiments. In

thep-wavechannelβ-phase,achiralgroundstateinhomogeneityisspontaneouslygenerateddueto
]
l a Lifshitz-like instability in the originally nonchiral systems. Possible experiments to detect these
e
phases are discussed.
-
r
t PACSnumbers: 71.10.Ay,71.10.Ca,05.30.Fk
s
.
t
a
m I. INTRODUCTION the electron fluid behaves, from the point of view of its
symmetriesandoftheirbreaking,verymuchlikeanelec-
-
d tronic analog of liquid crystal phases [5, 6]. The charge
The Landau theory of the Fermi liquid is one of the
n nematic is the simplest example of such electronic liquid
most successful theories of condensed matter physics [1,
o crystal phases, a concept introduced in Ref. [7] to de-
c 2]. It describes a stable phase of dense interacting
scribe the complex phases of strongly correlatedsystems
[ fermionicsystems,aFermi liquid(FL). Fermiliquidthe-
suchasdopedMottinsulators. Thechargenematicphase
ory is the foundation of our understanding of conven-
2 has also been suggested to exist in the high T materi-
tional,weaklycorrelated,metallicsystems. Itscentralas- c
v als near the melting of the (smectic) stripe phases [7, 8],
6 sumptionistheexistenceofwell-definedfermionicquasi-
andinquantumHallsystemsinnearlyhalf-filledLandau
2 particles,singleparticlefermionicexcitationswhichexist
levels [9, 10]. Experimentally, the charge nematic phase
3 as long-livedstates at very low energies, close enough to
has been found in ultra-high mobility two-dimensional
0 the Fermi surface. In the Landau theory, the interac-
1 electron gases (2DEG) in AlAs-GaAs heterostructures
tions among quasi-particles are captured by a few Lan-
6 dauparametersFs,a,wherel denotestheorbitalangular and quantum wells in large magnetic fields, in nearly
0 l half filled Landau levels for N 2 at very low tem-
t/ msinogmleetntaunmd tpriaprlteitalchwaanvneelcsh,arnesnpeel,ctaivnedly.s,Pahydseincoatlequspanin- peratures [10, 11, 12, 13]. Strong≥evidence for a charge
a
nematic phase has been found quite recently near the
m tities, such as the spin susceptibility, and properties of
metamagnetictransitionoftheultra-cleansamplesofthe
collective excitations, such as the dispersion relation of
- bilayer ruthenate Sr Ru O [14, 15].
d zerosoundcollectivemodes,acquiresignificantbutfinite 3 2 7
n renormalizations due to the Landau interactions. In the Electronic liquid crystal phases can also be realized
o
FL phase, except for these finite renormalizations, the as Pomeranchukinstabilities in the particle-hole channel
c
: effects of the interactions become negligible at asymp- with non-zero angular momentum. This point of view
v totically low energies. It has, however, long been known has been the focus of much recent work, both in con-
i
X thatthe stabilityofthe FL requiresthatthe Landaupa- tinuum system [4, 16, 17, 18, 19] as well as in lattice
r rameters cannot be too negative, Fls,a > −(2l + 1), a systems[20,21, 22,23, 24],inwhichcasethesequantum
a resultfirstderivedby Pomeranchuk[3]. The mostfamil- phase transition involvethe spontaneous breaking of the
iar of these Pomeranchuk instabilities are found in the point symmetry groupof the underlying lattice. The 2D
as-nwdapvehacsheasnenpeal:ratthioenSatotnFe0rs <fer−ro1m.agnetism at F0a < −1 qd-uwaanvtuem(ln=em2a)tcichaFnernmelieflxuhidibpithinagseaissapnonintsatnaeboiulistyeilnliptthie-
It has been realized quite recently that when these cal distortion of the Fermi surface [4]. In all cases, these
bounds are violated in a channel with a non-vanishing instabilities typically result in anisotropic Fermi surface
angular momentum, there is a ground state instability distortions, sometimes with a change of the topology of
in the particle-hole, spin singlet, channel leading to a the Fermi surface. Nematic phases also occur in strong
spontaneous distortion of the Fermi surface. This is a coupling regimes of strongly correlated systems such as
quantum phase transition to a uniform but anisotropic the Emery model of the high temperature superconduc-
liquid phase of the fermionic system [4]. In such a phase tors [25]. If lattice effects are ignored, the nematic state
2
hasGoldstonemodescanbeviewedastherotationofthe which was first proposed by Leggett in the context of
distortedFermisurface,i.e.,thesoftFermisurfacefluctu- superfluid 3He-B phase [1]. In fact, it is a particle-hole
ations. Withinarandomphaseapproximation(RPA)ap- channel analog of the superfluid 3He-B phase where the
proach[4], confirmed by a non-perturbative high dimen- pairing gap function is isotropic over the Fermi surface.
sional bosonizationtreatment [17], the Goldstone modes The phases with anisotropic Fermi surfaces distortions
wereshowntobeoverdampedinalmostallthepropagat- are the analog of the 3He-A phase where the gap func-
ing directions, except along the high symmetry axes of tion is anisotropic. The two possibilities of keeping or
thedistortedFermisurface. TheGoldstonemodecouples breaking the shape of Fermi surfaces are dubbed β and
stronglytoelectrons,givingrisetoanon-Fermiliquidbe- α-phases respectively, by analogy with B and A super-
havior throughout the nematic Fermi fluid phase [4, 26]: fluid phases in 3He systems. An important difference is,
in perturbation theory, the imaginary part of the elec- however, that while in the A and B phases of superfluid
tron self-energy is found to be proportional to ω32 on 3He all the Fermi surface is gapped, up to a set of mea-
most of the Fermi surface, except along four “nodal” di- sure zero of nodal points in the A phase, in the β and
rections, leading to the breakdown of the quasi-particle α-phases of spin triplet Pomeranchuk systems no gap in
picture. Away from the quantum critical point, this ef- the fermionic spectrum ever develops.
fect is suppressed by lattice effects but it is recovered at
An important common feature of the α and β-phases
quantum criticality[21] and athigh temperatures (if the
is the dynamical appearance of effective spin-orbit (SO)
lattice pinning effects are weak). Beyond perturbation
couplings,reflectingthefactthatinthesephasesspinand
theory [26] this effect leads to a form of ‘local quantum
orbit degrees of freedom become entangled [37]. Con-
criticality’.
ventionally, in atomic physics, the SO coupling origi-
Richer behaviors still can be found in Pomeranchuk nates from leading order relativistic corrections to the
instabilitiesinthe spintripletandl>0angularmomen- Schr¨odinger-Pauli equation. As such, the standard SO
tum channels [4, 8, 27, 28, 29, 30, 31, 32, 33, 34]. Typ- effects in many-body systems have an inherently single-
ically, these class of instabilities break both the spacial particle origin, and are unrelated to many-body correla-
(orbital) and spin SU(2) rotationsymmetries. A p-wave tioneffects. ThePomeranchukinstabilitiesinvolvingspin
channel instability was studied by Hirsch [27, 28]. The we are discussing here thus provide a new mechanism
Fermi surfaces of spin up and down components in such to generate effective SO couplings through phase tran-
a stateshift alongoppositedirections. Thep-wavechan- sitions in a many-body non-relativistic systems. In the
nelinstabilitywasalsoproposedbyVarmaasacandidate 2D β-phase, both Rashba and Dresselhaus-like SO cou-
forthehiddenorderappearingintheheavyfermioncom- plingscanbegenerated. Intheα-phase,theresultingSO
pound URu Si [31, 32] below 17 K. Gor’kov and Sokol coupling can be considered as a mixture of Rashba and
2 2
studied the non-N´eelspin orderingsin itinerantsystems, Dresselhaus with equal coupling strength. Such SO cou-
and showed their relation to the Pomeranchuk instabil- pling systems could in principle be realized in 2D semi-
ity[29]. Oganesyanandcoworkers[4,8]proposedtheex- conductor materials leading to interesting new effects.
istence of a nematic-spin-nematic phase, a nematic state For instance, a hidden SU(2) symmetry in such systems
in both real space and in the internal spin space, where was found to give rise to a long lived spin spiral excita-
the two Fermi surfaces of up and down spins are sponta- tion with the characteristic wavevector relating the two
neously distorted into two orthogonal ellipses. Podolsky Fermi surfaces together [38]. Recently, many proposals
and Demler[35] considered a spin-nematic phase as aris- have been suggested to employ SO coupling in semicon-
ing from the melting of a stripe phase; this phase can ductormaterialstogeneratespincurrentthroughelectric
also be a nematic-spin-nematic if it retained a broken fields. The theoretical prediction of this “intrinsic spin
rotational symmetry, as expected from the melting of a Hall effect” [39, 40] has stimulated tremendous research
stripe (or smectic) phase[7, 36]. Kee and Kim [33] have activityboththeoreticalandexperimental[41,42]. Thus,
suggested a state to explain the behavior of Sr Ru O Pomeranchuk instabilities in the spin channel may have
3 2 7
which can be shown to be a lattice version of a partially a potential application to the field of spintronics.
polarized nematic-spin-nematic state.
Asystematicdescriptionofthehighpartial-wavechan-
Interestingly, the spin triplet Pomeranchuk instabili- nel Pomeranchuk instabilities involving spin is still lack-
tiescanalsooccurwithoutbreakingrotationalinvariance ing in the literature. In this paper, we investigate this
inrealspace,andkeepthesymmetriesoftheundistorted problemforarbitraryorbitalpartialwavechannelsintwo
Fermi surface. Wu et al. [30] showed that a state of dimensions (2D), and for simplicity only in the p-wave
this sort can exist in the p-wave channel, with the dis- channel in three dimensions (3D). We use a microscopic
tortions affecting only the spin channel. In this state, modeltoconstructageneralGinzburg-Landau(GL)free
there are two Fermi surfaces with different volumes, as energy to describe these instabilities showing that the
inferromagneticsystems. Althoughboththespatialand structure of the α and β-phases are generalfor arbitrary
spin rotation symmetries are broken, a combined spin- valuesofl. Theα-phasesexhibitanisotropicrelativedis-
orbit rotation keeps the system invariant, i.e., the total tortions for two Fermi surfaces as presented in previous
angular momentum is conserved. The broken symme- publications. We also investigatethe allowedtopological
try in such a state is the relative spin-orbit symmetry excitations(textures)ofthesephasesandfindthatahalf-
3
quantum vortex-like defect in real space, combined with rentcanbeinducedwhenachargecurrentflowsthrough
spin-orbit distortions. The β-phases at l 2 also have thesystem. InSectionXI,wediscussthe possibleexper-
≥
a vortex configuration in momentum space with wind- imental evidence for Pomeranchukinstabilities involving
ing numbers l which are equivalent to each other by a spins. We summarizethe results ofthis paper atSection
±
symmetry transformation. XII. Details of our calculations are presented in two ap-
We study the collective modes in critical regime and pendices. In Appendix A we specify our conventions for
in the orderedphases at zero temperature. At the quan- LandauparametersandinAppendix Bwegivedetailsof
tumcriticalpoint,asinthe casespreviouslystudied, the the Goldstonemodes forspinoscillationsintheα-phase.
theoryhas dynamic criticalexponentz =3for allvalues
of the orbital angular momentum l. In the anisotropic
α-phase,the Goldstonemodes canbe classifiedinto den- II. MODEL LANDAU HAMILTONIAN
sity and spin channel modes, respectively. The density
channel Goldstone mode exhibits anisotropic overdamp- We begin with the model Hamiltonian describing the
ing in almostall the propagatingdirections. In contrast, PomeranchukinstabilityintheFa (l 1)channelin2D.
l ≥
the spin channel Goldstone modes show nearly isotropic Lateroninthepaperwewilladaptthisschemetodiscuss
underdamped dispersion relation at small propagating the 3D case which is more complex. This model, and
wavevectors. In the β-phase, the Goldstone modes are the related model for the spin-singlet sector of Ref. [4]
relative spin-orbit rotations which have linear dispersion on which it is inspired, has the same structure as the
relations at l 2, in contrastto the quadratic spin-wave effectiveHamiltonianfortheLandautheoryofaFL.The
≥
dispersionintheferromagnet. BoththeGoldstonemodes corresponding order parameters can be defined through
(spinchannel)intheα-phasesandtherelativespin-orbit the matrix form as
Goldstone modes in the β-phases couple to spin excita-
tions in the ordered phases. Thus the spin wave spec- Qµb(r) = ψα†(r)σαµβgl,b(−i∇ˆ)ψβ(r) (2.1)
tra develop characteristic resonance peaks observable in
neutron scattering experiments, which are absent in the where the Greek indices µ denote the x,y,z directions
normal phase. in the spin space, and in 2D the latin indices b = 1,2
denote the two orbital components, and α,β = , label
Thep-wavechannelPomeranchukinstabilityinvolving ↑ ↓
thetwospinprojections. (Hereafter,repeatedindicesare
spin is special because the Ginzburg-Landau (GL) free
summed over.) In 2D, the operators g ig , which
energy contains a cubic term of order parameters with a l,1 ± l,2
carry the azimuthal angular momentum quantum num-
linear spatial derivative satisfying all the symmetry re-
ber L = l, are given by
quirement. Such a term is not allowed in other chan- z ±
nanelds twhiethPlom6=er1a,nicnhculukdiinnsgtatbhielitfieersroimnatghneedtiecnisnitsytacbhilaitny- gl,1(−i∇ˆ)±igl,2(−i∇ˆ)=(−i)l(∇ˆx±i∇ˆy)l, (2.2)
nel. This term does not play an important role in the α-
where the operator ˆa is defined as ~a/ . The 3D
phase. But in the β-phase, it induces a chiral inhomoge- ∇ ∇ |∇|
counterpartof these expressions can be written in terms
neousgroundstateconfigurationleadingtoaLifshitz-like
of spherical harmonic functions. Thus, in 3D the latin
instability in this originally nonchiral system. In other
labels take 2l+1 values. For the moment, and for sim-
words, a Dzyaloshinskii-Moriya type interaction for the
plicity, we will discuss first the 2D case.
Goldstone modes is generatedin the β-phase. The effect
In momentum space (i.e. a Fourier transform) we can
bears some similarity to the helimagnet [43] and chiral
write the operators of Eq. (2.2) in the form g (~k) =
liquid crystal [6] except that the parity is explicitly bro- l,1
kenthere but nothere. The spiralpattern ofthe ground coslθk and gl,2(~k) = sinlθk, where θk is the azimuthal
state order parameter is determined. angle of~k in the 2D plane. In momentum space, Qµb(~q)
The paper is organized as follows: In Section II, we is defined as
construct the model Hamiltonian for the Pomeranchuk
~q ~q
instabilities with spin. In Section III, we present the Qµb(~q)= ψ†(~k+ ) σµ g (~k) ψ (~k ). (2.3)
α 2 αβ l,b β − 2
Ginzburg-Landau free energy analysis to determine the
X~k
allowed ground states. In Section IV, we discuss the re-
sults of the mean field theory. In Section V we discuss It satisfies Qµb( ~q)=Qµ,b,∗(~q), thus Qµb(~r) is real.
−
the topology of the broken symmetry states and classify We generalize the Hamiltonian studied in Ref. [4, 30]
the topological defects. In Section VI, we calculate the to the Fa channel with arbitrary values of l as
l
collective modes at the critical point at zero tempera-
ture. In Section VII and Section VIII, we investigate H = d2~r ψ†(~r)(ǫ(~) µ)ψ (~r)
the Goldstone modes in the α andβ-phases respectively, α ∇ − α
Z
and study the spontaneous Lifshitz instability in the β- 1
phase at l = 1. In Section IX, we present the magnetic + 2 dd~rdd~r′ fla(~r−~r′) Qˆµb(~r)Qˆµb(~r′),
field effects. In Section X, we show that in the α and Z µb
X
β-phases in the quadrupolar channel (l =2), a spin cur- (2.4)
4
where µis the chemicalpotential. For later convenience, Taking into account that fa 1/N(0) around the
| 1| ∼
we include the non-linear momentum dependence in the transitionpoint,weintroduceadimensionlessparameter
single particle spectrum up to the cubic level as λ to denote the above criterion as
ǫ(~k)=vF∆k(1+a(∆k/kF)+b(∆k/kF)2+...) (2.5) λ= κkF2 1. (2.11)
N(0) ≫
with∆k =k k . Herev andk aretheFermivelocity
F F F
−
and the magnitude of the Fermi wave vector in the FL. Finally, notice that, in the p-wave channel, the Hamilto-
The Fouriertransformofthe Landauinteractionfunc- nian Eq. (2.10) can be formally represented through an
tion f(r) is SU(2)non-Abelian gauge field minimally-coupled to the
fermions
fa
f(~q)= d~reiq~~rfa(r)= l , (2.6) 1
l 1+κfa q2 H = d2~r ψ†(~r)( i~ a mAµ(~r)σµ)2ψ(~r)
Z | l | mf 2m − ∇ − a
Z
m
and the dimensionless Landau parametersare defined as ψ†(~r)ψ(~r)Aµ(~r)Aµ(~r). (2.12)
− 2 a a
Fa =N(0)fa(q =0) (2.7)
l l where the gauge field is defined as
withN(0)thedensityofstatesattheFermienergy. This
Aµ(~r)σµ =nµa(~r)σµ. (2.13)
Hamiltonian possesses the symmetry of the direct prod- a
uct of SO (2) SO (3) in the orbit and spin channels.
L S
The LP insta⊗bility occurs at Fa < 2 at l 1 in two Notice, however,thatthis is an approximateeffective lo-
l − ≥ calgaugeinvariancewhichonlyholdsforatheorywitha
dimensions. For the generalvalues of l, we representthe
linear dispersion relation, and that it is manifestly bro-
order parameter by a 3 2 matrix
× kenbynon-linearcorrections,suchasthequadraticterm
d2~k of Eq. (2.12), and the cubic terms included in the dis-
nµ,b = |f1a| (2π)2hψα†(k)σαµβgl,b(~k)ψβ(k)i. (2.8) persion ǫ(~k) of Eq. (2.5).
Z
It is more convenient to represent each column of the
matrix form nµ,b (b=1,2) as a 3-vectorin spin space as III. GINZBURG-LANDAU FREE ENERGY
~n =nµ,1, ~n =nµ,2. (2.9)
1 2 A. The 2D systems
For l = 1, ~n are just the spin currents along the x,y
1,2
In order to analyze the possible ground state configu-
directions respectively. When l 2,~n denote the spin
1,2
≥ ration discussed in Section I, we construct the G-L free
multipolecomponentsatthelevellontheFermisurface.
energy in 2D in the arbitrary l-wave channel. The sym-
~n i~n carry the orbital angular momenta L = l
1 2 z
± ± metryconstrainttotheG-Lfreeenergyisasfollows. Un-
respectively. In other words, ~n are the counterpart
1,2
der time-reversal (TR) and parity transformations, ~n
of the spin-moment in the l-th partial wave channel in 1,2
transform, respectively, as
momentum space.
The mean field Hamiltonian, i.e. for a state with a
T~n T−1 =( )l+1~n , P~n P−1 =( )l~n . (3.1)
uniform order parameter, can be decoupled as 1,2 − 1,2 1,2 − 1,2
d2~k Under the SOS(3)rotationRµν inthe spinchannel,~n1,2
H = ψ†(~k) ǫ(~k) µ ~n cos(lθ ) transform as
MF (2π)2 α − − 1 k
Z n (cid:16)
+ ~n2sin(lθk) ·~σ ψβ(~k)+ |n1|22+fa|n2|2. nµ,1 →Rµνnν,1, nµ,2 →Rµνnν,2, (3.2)
(cid:17) o | l | On the other hand, under a uniform rotation by an an-
(2.10)
gle θ about the z-axis, in the orbital channel, the order
Thismeanfieldtheoryisvalidwhentheinteractionrange parameters~na transform as
ξ κfa is much larger than the inter-particle dis-
tan≈cepd≈| 1l/|kF. The actualvalidity ofmeanfieldtheory ~n1 → cos(lθ) ~n1+sin(lθ) ~n2,
at quantum criticality requires an analysis of the effects ~n sin(lθ) ~n +cos(lθ) ~n . (3.3)
2 1 2
→ −
of quantum fluctuations which are not included in mean
field theory [44, 45]. In this theory, just as the case of Thus, the order parameter fields~n and~n are invariant
1 2
Ref. [4], the dynamic critical exponent turns out to be under spatial rotations by 2π/l, and change sign under
z = 3 and mean field theory appears to hold even at a rotation by π/l. In the α-phase this change change be
quantum criticality. compensated by flipping the spins.
5
In order to maintain the SO (2) SO (3) symmetry, be written as
L S
⊗
up to quartic terms in the orderparameter fields~n , the
a 2
uniform part of the GL free energy has the form F (n) = γ (∂ δ igAλ(Tλ) ) nνb
grad 1 a µν − a µν
v v + γ ng2(ǫ nλanνb)2 o
F(n) = rtr(nTn)+(v + 2)[tr(nTn)]2 2tr[(nTn)2] 1 λµν
1
2 − 2 = γ (∂ ~n +g ~n ~n )2
= r(~n 2+ ~n 2)+v (~n 2+ ~n 2)2 1 a b a× b
+ v |~n1| ~n| 22.| 1 | 1| | 2| (3.4) Xab n
2| 1× 2| γ1g2~na ~nb 2 , (3.8)
− | × |
The coefficients r,v ,v will be presented in Eq. (4.12) o
1 2 with g =γ /(2γ ).
by evaluating the ground state energy of the mean field 2 1
HamiltoniansEq. (4.1)andEq. (4.7). ThePomeranchuk
instability occurs at r < 0, i.e., Fa < 2 (l 1). Fur-
l − ≥ B. The 3D systems
thermore, for v > 0, the ground state is the α-phase
2
which favors ~n ~n , while leaving the ratio of ~n /~n
1 k 2 | 1| | 2| In 3D, the order parameter in the Fa channel Pomer-
arbitrary. On the other hand, for v2 < 0 we find a β- l
anchuk instabilities can be similarly represented by a
phase, which favors~n ~n and ~n = ~n .
1 2 1 2
⊥ | | | | 3 (2l+1) matrix. Here we only consider the simplest
The gradient terms are more subtle. We present the ×
case of the p-wave channel instability (l =1), which has
gradient terms of the GL free energy for l=1 as follows
been studied in Ref. [4, 27, 30, 32] under different con-
texts. In the Fa channel, the order parameter nµ,i is a
F (n) = γ tr[∂ nT∂ n]+γ ǫ nµanνb∂ nλb 1
grad 1 a a 2 µνλ a 3 3 real matrix defined as
×
= γ (∂ ~n ∂ ~n )+γ (∂ ~n ∂ ~n )
1 a b· a b 2 x 2− y 1 d3~k
(~n ~n ) . n (3.5) nµ,b =|f1a| (2π)3hψα†(k)σαµβkbψβ(k)i. (3.9)
· 1× 2 Z
o Thedifferencebetweennµ,iandthetripletp-wavepairing
For simplicity, as in Ref. [4], we have neglected the dif- order parameter in the 3He system [1, 46] is that the
ference between two Frank constants and only present
formerisdefinedintheparticle-holechannel,andthusis
one stiffness coefficient γ . (This approximation is accu-
1 real. Bycontrast,thelatteroneisdefinedintheparticle-
rateonlynearthePomeranchukquantumcriticalpoint.)
particle channel and is complex. Each column of the
Moreimportantly,becausenµ,bisoddunderparitytrans- matrixformnµ,b(b=x,y,z)canbeviewedasa3-vectors
formation for l = 1, a new γ term appears, which is
2 in the spin space as
of cubic order in the order parameter field nµ,b, and it
is linear in derivatives. This term is allowed by all the ~n =nµ,1, ~n =nµ,2, ~n =nµ,3, (3.10)
1 2 3
symmetry requirements, including time reversal, parity,
and rotation symmetries. This term has no important whichrepresentsthe spin currentinthe x,y andz direc-
effects in the disordered phase and in the α-phase, but tions respectively. The G-L free energy in Ref. [30] can
it leads to a Lifshitz-like inhomogeneous ground state be reorganizedas
with spontaneous chirality in the β-phase in which par-
ity is spontaneously broken. We will discuss this effect F(n) = r′tr(nTn)+(v′ + v2′ )[tr(nTn)]2 v2′ tr[(nTn)2]
in detail in Section VIII. The coefficient of γ will be 1 2 − 2
1,2
presented in Eq. (8.19). Similarly, for all the odd values = r′(~n 2+ ~n 2+ ~n 2)+v′(~n 2
| 1| | 2| | 3| 1 | 1|
of l, we can write a real cubic γ term satisfying all the
2 + ~n 2+ ~n 2)2+v′ ~n ~n 2
symmetry constraints as | 2| | 3| 2 | 1× 2|
n
+ ~n ~n 2+ ~n ~n 2 , (3.11)
γ ǫ nµanνb(i)lg (ˆ)nλb. (3.6) | 2× 3| | 3× 1|
2 µνλ a
∇ o
where the coefficients r′,v′ will be presented in Eq.
However,thistermcorrespondstohighordercorrections, 1,2
(4.19). Similarly, the α-phase appears at v > 0 which
2
and is negligible (irrelevant) for l 3.
≥ favors that ~n1 ~n2 ~n3, and leaves their ratios arbi-
Similarly to the approximate gauge symmetry for the trary. The β-phkase apkpears at v < 0 which favors that
2
fermions in Eq. (2.12), the γ2 term can also be repro- vectors~n1,2,3 are perpendicular to eachother with equal
duced by a non-Abelian gauge potential defined as amplitudes ~n = ~n = ~n .
1 2 3
| | | | | |
Similarly,wepresentthegradienttermsintheG-Lfree
iAλ(x)(Tλ) =ǫ nλa(x), (3.7) energy as
a µν λµν
where Tµλν =−iǫλµν is the generatorof the SU(2) gauge Fgrad(n) = γ1′tr[∂anT∂an]+γ2′ǫµνλnµinνj∂jnλi,
group in the vector representation. Then Eq. (3.5) can (3.12)
6
δk↑ The value of n¯ can be obtained by solving the self-
~s
consistent equation in the α-phase
n¯ d~k
= n (ξ (~k)) n (ξ (~k)) cos(lθ ).
fa(0) (2π)2{ f ↑ − f ↓ } k
| l | Z
(4.3)
where n (ξ (k)) and n (ξ (k)) are the Fermi functions
f ↑ f ↓
for the up and down electrons respectively. The dis-
~s
δk↓ tortions of the Fermi surfaces of the spin up and spin
l=1 α-phase l=2 downbandsaregivenbyanangle-dependentpartoftheir
Fermi wave vectors:
FIG.1: Theα-phasesintheF1a andF2a channels. TheFermi δkF↑,↓(θ) = 2a−1x2 (x+ a−2a2x3)coslθ
surfaces exhibit thep and d-wavedistortions, respectively. kF 4 ± 2
ax2cos2lθ (2a2 b)x3cos3lθ+O(x4),
− ± −
wherethecoefficientγ′ willbe presentedatEq. (8.36). (4.4)
1,2
Again, we neglectthe difference among three Frank con-
where we introduced the dimensionless parameter x =
stants. The γ′ term can be represented as
2 n¯/(v k ), where v and k are the Fermi velocity and
F F F F
themagnitudeoftheFermiwavevectorinthe FLphase.
γ′ (∂ ~n ∂ ~n ) (~n ~n )
2 x 2− y 1 · 1× 2 Thissolutionholdsforsmalldistortionsanditisaccurate
n +(∂ ~n ∂ ~n ) (~n ~n ) only close to the quantum phase transition. By inspec-
y 3 z 2 2 3
− · ×
tion we see that in the α-phases the total spin polariza-
+(∂ ~n ∂ ~n ) (~n ~n ) . (3.13)
z 1− x 3 · 3× 1 tionvanishes. Moreimportantly,underarotationbyπ/l,
o the charge and spin components of the order parameter
It can also be represented in terms of the non-Abelian
both change sign, i.e. a rotation by π/l is equivalent to
gauge potential as in Eq. (3.8).
a reversalof the spin polarization.
The single particle fermion Green function in the α-
phase at wave vector~k and Matsubara frequency ω is
IV. MEAN FIELD PHASES IN THE Fa n
l
CHANNEL
1 1+σ 1 σ
G(~k,iω )= z + − z . (4.5)
n 2 iω ξ (~k) iω ξ (~k)
Inthissection,wediscussthesolutiontothemeanfield n n− ↑ n− ↓ o
Hamiltonian Eq. (2.10), for the ordered α and β-phases
in both 2D and 3D.
B. The 2D β-phases
The β-phase appears for v <0, which favors~n ~n
A. The 2D α-phases 2 1 2
⊥
and n = n . Like the case of ferromagnetism, the
1 2
| | | |
Fermi surfaces split into two parts with different vol-
The α-phase is characterizedby an anisotropic distor-
umes, while each one still keeps the round shape undis-
tion of the Fermi surfaces of up and down spins. In
torted. However, an important difference exists between
this phase s is as a good quantum number. It is a
z the β-phase at l 1 and the ferromagnetic phase. In
straightforward generalization of the nematic Fermi liq- ≥
the ferromagnet, the spin is polarized along a fixed uni-
uid for the case of the spin channel [4]. For example,
form direction, which gives rise to a net spin moment.
the Fermi surface structures at l = 1,2 are depicted in
On the other hand, in the β-phase with orbital angu-
Fig. 1. The quadrupolar, l = 2, case is the nematic-
lar momentum l 1, the spin winds around the Fermi
spin-nematic phase [4, 8]. Without loss of generality, we ≥
surface exhibiting a vortex-like structure in momentum
choose ~n =n¯ zˆ, and n =0. The mean field Hamilto-
1 | 2| space. Consequentlyintheβ-phasethenetspinmoment
nian H for the α-phase becomes
α is zero, just as it is in the α-phase. (Naturally, partially
d2k polarizedversionsoftheαandβ-phasesarepossible but
Hα,l = (2π)2ψ†(~k)[ǫ(~k)−µ−n¯cos(lθk)σz)]ψ(~k). will not be discussed here.) In other words, it is a spin
Z nematic, high partial wave channel generalization of fer-
(4.1)
romagnetism. In the previously studied cases of the Fa
1
channel in Ref. [30], it was shown that in this phase
The dispersion relations for the spin up and down elec-
effective Rashba and Dresselhaus terms are dynamically
trons become
generated in single-particle Hamiltonians. The ground
ξ (~k) = ǫ(k) µ n¯cos(lθ ), (4.2) state spin configuration exhibits, in momentum space, a
↑,↓ k
− ∓
7
δk↑ δk↑
δk↓ ~s ~s
δk↓
~s
(a)w=2 (b)w=−2
FIG.3: Theβ-phasesintheFa channel. Spinconfigurations
2
(a)Rashba(w=1) exhibit the vortex structure with winding number w = ±2.
Thesetwoconfigurationscanbetransformedtoeachotherby
performing a rotation around thex-axiswith theangle of π.
S2
~s
~n2
~n1
(b)Dresselhaus(w=−1)
FIG.4: ThespinconfigurationsonbothFermisurfacesinthe
β-phase (Fa channel) map to a large circle on an S2 sphere
l
FIG.2: Theβ-phasesintheFa channel. Spinconfigurations with thewinding number l.
1
exhibit the vortex structures in the momentum space with
winding number w = ±1, which correspond to Rashba and
Dresselhaus SO coupling respectively. by solving the self-consistent equation
2n¯ d2~k
vortex structure with winding number w = 1 depicted = n (ξ (~k)) n (ξ (~k) , (4.8)
± fa(0) (2π)2 f ↑ − f ↓
in Fig. 2. | l | Z (cid:16) (cid:17)
Here we generalize the vortex picture in momentum
where the single particle spectra read
space in the Fa channel to a general Fa channel. We
1 l
assume n = n = n¯. Without loss of generality, we
| 1| | 2| ξ (~k)=ǫ(~k) µ n¯, (4.9)
can always perform an SO(3) rotation in spin space to ↑,↓ − ±
set ~n xˆ, and ~n yˆ. Then, much as in the B phase of
3He,1thke mean fie2ldkHamiltonian H for the β-phase in which is clearly invariant under spacial rotations. The
β,l Fermi surface splits into two parts with
angular momentum channel l can be expressed through
a d-vector, defined by
δk x2
F,↑,↓ = x (a b)x3+O(x4). (4.10)
d~(~k)=(cos(lθ ),sin(lθ ),0), (4.6) kF ± − 2 ± −
k k
The single particle Green function in the β-phase reads
as follows
d2k 1 1+~σ dˆ 1 ~σ dˆ
H = ψ†(~k) ǫ(~k) µ n¯d~(θ ) ~σ) ψ(~k), G (~k,iω )= · + − · , (4.11)
β,l (2π)2 − − k · l n 2 iω ξ (~k) iω ξ (~k)
Z h i n n− ↑ n− ↓ o
(4.7)
where ~σ d~(kˆ) is the l-th order helicity operator. Each
whered~(θ )isthe spinquantizationaxisforsingleparti- Fermi su·rface is characterized by the eigenvalues 1 of
k
±
clestateat~k. Thesaddlepointvalueofn¯canbeobtained the helicity operators~σ d~(kˆ).
·
8
From the mean field theory of α and β-phases, we can Now we discuss the general configuration of the d-
calculate the coefficients of the G-L theory, Eq.(3.4), as vectorintheβ-phaseintheFachannel. ~n and~n canbe
l 1 2
any two orthogonal unit vectors on the S2 sphere. The
N(0) 1 1 plane spanned by ~n intersects the S2 sphere at any
r = ( ), 1,2
2 Fa − 2 large circle as depicted in Fig. 4, which can always be
| l |
N(0) N′(0) N′′(0) obtained by performing a suitable SO(3) rotation from
v = [ ]2
1 the large circle in the xy plane. The spin configuration
32 N(0) − 2N(0)
n o around the Fermi surface maps to this large circle with
N(0)
= (1 a 2a2+3b) , thewinding numberofl. Furthermore,the configuration
− − 32v2k2
F F ofwinding number l are equivalent to eachother up to
N′′(0) N(0) rotation of π aroun±d a diameter of the large circle. For
v = =( a+2a2 b) , (4.12)
2 48 − − 8v2k2 example, the case of w = 2 is depicted in Fig. 3 B,
F F −
whichcanbe obtainedfromthatofw =2 byperforming
where v1,2 do not depend on the value of l at the mean such a rotation around the xˆ-axis. Similarly, with the
field level. N′(0) and N′′(0) are the first and second SO (3) symmetry in the spin space, the configurations
S
orderderivatives ofdensity of states atthe Fermi energy with w = l are topologically equivalent to each other.
Ef respectively. They are defined as However, i±f the SOS(3) symmetry is reduced to SOS(2)
because of the existence of an explicit easy plane mag-
dN N(0)
N′(0) = =(1 2a) , netic anisotropy (which is an effect of SO interactions at
dǫ |ǫ=Ef − vfkf thesingleparticlelevel),orbyanexternalmagneticfield
d2N N(0) B~, then the two configurations with w = l belong to
N′′(0) = dǫ2 |ǫ=Ef =6(−a+2a2−b)v2k2. (4.13) two distinct topological sectors. ±
f f
Both of them only depend on the non-linear dispersion
relation up to the cubic order as kept in Eq. 2.5.
C. The 3D instabilities of the p-wave spin triplet
It is worth to stress that the coefficients of Eq. (4.12) channel
were calculated (within this mean field theory) at fixed
density. Similarcoefficientswereobtainedinthespinless
ThemeanfieldtheoryforthePomeranchukinstability
system analyzed in Ref. [4] at fixed chemical potential.
in the Fa channel has been studied in Ref. [30]. To
The is a subtle difference between these two settings in 1
make the paper self-contained, here we summarize the
the behavior of the quartic terms. At fixed chemical po-
main results.
tentialthesignofb,thecoefficientofthecubicterminthe
In the α-phase, taking the special case nµa =n¯δ δ ,
freefermiondispersionrelation,iscrucialforthenematic µz az
the mean field Hamiltonian reads
phasetobestable. However,ascanbe seeninEq.(4.12),
the sign of the coefficient of the quartic term v1 is deter- d3k
minedbyseveraleffects: thecoefficientsaandb,andthat H2D,α = (2π)3ψ†(~k) ǫ(~k)−µ−n¯σzcosθk) ψ(~k),
ofanextra(additive)contributionwhichoriginatesfrom Z h i
(4.15)
the curvature of the Fermi surface and hence scales as
N(0)/k2. IthasbeennotedinRefs. [23,47,48]thatthe
nematicFinstability for lattice systems may be a continu- where θ is the angle between ~k and z-axis. The Fermi
ous quantum phase transition or a first order transition, surfaces for the two spin components are distorted in an
in which case it involves a change in the topology of the opposite way as
Fermisurface. Asshownabove,this dichotomyisthere-
sult of the interplay of the single particle dispersion and ∆kF↑,↓(θ) = 1(1 a)x2 [x+ 2a(1 a)3x3]cosθ
effects due to the curvature of the Fermi surface. The kF 3 − ± 3 −
sameconsiderationsapplytothecoefficientsthatwewill ax2cos2θ (2a2 b)x3cos3θ+O(x4)
present in the following subsection. − ± −
(4.16)
Thegroundstatespinconfigurationintheβ-phaseex-
hibits a vortex structure with winding number w = l
In the β-phase, rotational symmetry is preserved and
in momentum space. The case of w = 2 is depicted in
a SO interaction is dynamically generated. With the
Fig. 3 A. Interestingly, in the case of l =3, after setting
ansatz nµa =n¯δ , the MF Hamiltonian reduces to
d~= (cos(3θ+π/2),sin(3θ+π/2),0), the effective single µa
particle Hamiltonian becomes
H = ψ†(k)(ǫ(k) µ n¯~σ kˆ)ψ(k). (4.17)
3D,β
− − ·
HMF(k)=ǫ(k)+n¯[−sin(3θk)σx+cos(3θk)σy]. (4.14) Xk
This single particle Hamiltonian has the same form as is The single particle states can be classified according to
thatoftheheavyholebandofthe2-dimensionaln-doped the eigenvalues 1 of the helicity operator ~σ kˆ, with
GaAs system[49, 50], whichis results fromSO coupling. dispersion relatio±ns ξB(k) = ǫ(k) µ n¯. Th·e Fermi
↑,↓
− ±
9
surfaces split into two parts, but still keep the round z
shape for two helicity bands with
x
y
∆k
F↑,↓ = x x2 (2a b)x3+O(x4). (4.18)
k ± − ± −
F
The β-phase is essentially isotropic. The orbital angular
momentum L~ and spin S~ are no longer separately con-
served, but the total angular momentum J~=L~ +S~ =0
remainsconservedinstead. Forthegeneralcaseofn =
µa
n¯D , it is equivalent to a redefinition of spin operators
µa
asS′ =S D δ ,thusFermisurfacedistortionsremain
µ ν νa aµ
isotropic and J~′ =L~ +S~′ is conserved.
FIG. 5: The half-quantum vortex with the combined spin-
Fromtheabovemeanfieldtheory,wecancalculatethe
orbit distortion for the α-phase at l = 1. The triad denotes
coefficients in Eq. (3.11) as
thedirection in thespin space.
N(0) 1 1
r′ = ( ),
2 Fa − 3 A. Topology of the α-phases
| 1|
N(0) 1 N′(0) N′′(0)
v′ = [ ]2
1 24 3 N(0) − 5N(0) 1. 2D α-phases
n o
N(0)
= 180v2k2 (7−2a−8a2+9b), For the 2D α-phases, for which we can set nµa =
F F n¯δ δ , the system is invariant under SO (2)hrotatiion
v2′ = N92(00) = 45Nv(F20k)F2 (1−6a+6a2−3b), iinnµtthhzeeaso1prbinitcahlacnhnaneln,etlh,eanZdl raoZta2tiroontawtiiothnwthiethSanthgeleaonfg2leπ/olf
(4.19) πaroundthex-axisinthespinchannelcombinedwithan
orbital rotation at the angle of π/l. Thus the Goldstone
where manifold is
N′(0) = dN = 2−2aN(0), [SOL(2)⊗SOS(3)]/[SOS(2)⋉Z2⊗Zl]
dǫ |ǫ=Ef v k = [SO (2)/Z ] [S2/Z ], (5.1)
f f L l 2
⊗
d2N 2(1 6a+6a2 3b)
N′′(0) = = − − N(0). giving rise to three Goldstone modes: one describes the
dǫ2 |ǫ=Ef v2k2
f f oscillation of the Fermi surface, the other two describe
(4.20) the spinprecession. Theirdispersionrelationwillbe cal-
culated in Section VIIA.
Onceagain,thecaveatsoftheprevioussubsectiononthe Due to the Z2 structureofthe combinedspin-orbitro-
signofthecoefficientsofthequartictermsapplyheretoo. tation,thevorticesinthe2D-α-phasecanbedividedinto
two classes. The first class is the 1/l-vortices purely in
the orbital channel without distortions in the spin chan-
nel. Thisclassofvorticeshavethesamestructureasthat
V. GOLDSTONE MANIFOLDS AND in the Pomeranchuk instabilities in the density channel
TOPOLOGICAL DEFECTS
dubbed the integer quantumvortex. On the other hand,
anotherclassofvorticesascombinedspin-orbitaldefects
Inthis Sectionwe willdiscuss the topologyofthe bro- exist. This class of vortices bears a similar structure to
kensymmetryαandβ-phases,andtheirassociatedGold- that of the half-quantum vortex (HQV) in a superfluid
stone manifolds in 2D and 3D. We also discuss and clas- with internal spin degrees of freedom [51, 52, 53]. An
sify their topologicaldefects. We shouldwarnthe reader example vortex at l = 1 of this class is depicted in Fig.
that the analysis we present here is based only on the 5 where the π-disclination in the orbit channel is offset
static properties of the broken symmetry phases and ig- by the rotation of π around the x-axis in the spin chan-
nores potentially important physical effects due to the nel. To describe the vortex configuration for the case of
fact that in these systems the fermions remain gapless the p wave channel, we set up a local reference frame in
(although quite anomalous). In contrast, in anisotropic spin space, and assume that the electron spin is either
superconductors the fermion spectrum is gapped (up to parallel or anti-parallel to the z-axis of this frame, at a
possiblyasetofmeasurezeroofnodalpointsoftheFermi point ~x in real space. Let φ = 0 be angular polar coor-
surface). Thephysicsoftheseeffectswillnotbediscussed dinate of~x with respect to the core of the vortex. As we
here. traceapathinrealspacearoundthevortex,theframein
10
spinspacerotatessothatthespinflipsitsdirectionfrom L +σ /2 is still conserved. Generally speaking, the 2D
z z
up to down (or vice versa) as we rotate by an angle of β-phases with the momentum space winding number w
π. (In the d-wave channel the vortex involves a rotation is invariant under the combined rotation generated by
by π/2.) Such behavior is a condensed matter example L +wσ /2. The corresponding Goldstone manifold is
z z
of the Alice-string behavior in the high energy physics
[54, 55]. Another interesting behavior of HQV is that [SO (2) SO (3)]/SO (2)=SO(3). (5.3)
L S L+S
⊗
a pair of half-quantum vortex and anti-vortex can carry
spin quantum number. This is an global example of the Three Goldstone modes exist as relative spin-orbit ro-
Cheshire charge in the gauge theory [51, 53]. The elec- tations around x,y and z-axes on the mean field ground
tron can exchange spin with the Cheshire charged HQV state. Similarly,forthe3Dβ-phasewithl =1,theGold-
pairs when it passes in between the HQV pairs. stone mode manifold is
Due to the SO(2) SO (2) symmetry in the Hamil-
L S
×
tonian,the fluctuationsinthe orbitalchannelarelessse- [SO (3) SO (3)]/SO(3) =SO(3), (5.4)
L S L+S
verethan those inthe spinchannel. In the groundstate, ⊗
the spin stiffness should be softer than that in orbital with three branchesofGoldstone modes. The dispersion
channel. As a result, an integer-valued vortex should relation of these Goldstone modes will be calculated in
be energetically favorable to fractionalize into a pair of Section VIII. The vortex-like point defect in 2D and the
HQV. However, at finite temperatures, in the absence line defect in 3D are determined by the fundamental ho-
of magnetic anisotropy (an effect that ultimately is due motopy group π (SO(3)) = Z . On the other hand, the
1 2
to spin orbit effects at the atomic level) the spin chan- second homotopy group π (SO(3)) = 0, thus, as usual,
2
nel is disorderedwith exponentially decaying correlation no topologically stable point defect exists in 3D.
functions, and thus without long range order in the spin
channel. However, the orbital channel still exhibits the
Kosterlitz-Thoulessbehaviorat2Dwherethelowenergy
VI. THE RPA ANALYSIS IN THE CRITICAL
vortex configurations should be of HQV.
REGION AT ZERO TEMPERATURE
2. 3D α-phases In this section, we study the collective modes in the
Landau FL phase as the Pomeranchuk QCP is ap-
proached, 0 > Fa > 2 in 2D and 0 > Fa > 3 in 3D.
Similarly, the Goldstone manifold in the 3D α-phase l − l −
These collective modes are the high partial wave chan-
for l =1 can be written as
nel counterparts of the paramagnon modes in the 3He
[SO (3) SO (3)]/[SO (2) SO (2)⋉Z ] system. The picture of collective modes in the p-wave
L S L S 2
⊗ ⊗ channel at 2D is depicted in Fig. 6.
= [S2 S2]/Z , (5.2)
L⊗ S 2 For this analysis, it is more convenient to em-
where again the Z operation is a combined spin-orbit ploy the path integral formalism, and perform the
2
rotation at the angle of π to reverse the spin polariza- Hubbard-Stratonovichtransformationto decouple the 4-
fermioninteractiontermoftheHamiltonianpresentedin
tion and orbital distortion simultaneously. This Gold-
stone mode manifold gives rise to two Goldstone modes Eq.(2.4). After integrating out the fermionic fields, we
arrive at the effective action
in the density channel responsible for the oscillations of
the distorted Fermisurfaces,and another two Goldstone
1 β
modes for the spin precessions. The fundamental homo- S (~n )= dτ d~rd~r′ (fa)−1(~r ~r′)~n (~r) ~n (~r′)
topygroupofEq. (5.2)readsπ1[SL2⊗SS2]/Z2 =Z2. This eff b −2Z0 Z l − b · b
means that the π-disclinationexists as a stable topologi- ∂
+trln +ǫ(~) ~n (~r) ~σg ( i~) .
callinedefect. Ontheotherhand,forthepointdefectin ∂τ ∇ − b · l,b − ∇
3D space, the second homotopy groupof Eq. (5.2) reads n o (6.1)
π ([S2 S2]/Z ) = Z Z. This means both the orbit
2 L ⊗ S 2 ⊗
and spin channels can exhibit monopole (or hedgehog) Inthe normalFL state,we set n¯ =0, the fluctuations at
structures characterized by a pair of winding numbers the quadratic level are given by the effective action
(m,n). As a result of the Z symmetry, (m,n) denotes
2
the same monopoles as that of ( m, n). 1
− − S(2)(n)= nµa(~q,iω )L(FL)(~q,iω )
FL 2Vβ n µa,νb n
q~X,iωn
B. Topology of the β-phases nνb( ~q, iω ) (6.2)
n
× − −
In the 2D β-phase with l = 1 and w = 1 in the xy- where we have introduced the fluctuation kernel
plane, the ground state is rotationally invariant, thus L (~q,iω ) which is given by
µa,νb n