Table Of ContentCharge Spin Separation in 3D
M. Cristina Diamantini∗
INFN and Dipartimento di Fisica, University of Perugia, via A. Pascoli, I-06100 Perugia, Italy
Carlo A. Trugenberger†
SwissScientific, chemin Diodati 10, CH-1223 Cologny, Switzerland
(Dated: February 1, 2012)
Electronfractionalization intospinonsandchargeonsplaysacrucialrolein2Dmodelsofstrongly
correlatedelectrons. Inthispaperweshowthatspin-chargeseparationisnotaphenomenonconfined
2 tolowerdimensionsbut,rather,wepresentafield-theoreticmodelinwhichitisrealizedin3D.The
1 modelinvolvestwogauge fields,astandardoneandatwo-form gaugefield. Thephysicalpictureis
0 that of a two-fluid model of chargeons and spinons interacting by the topological BF term. When
2 a Higgs mechanism of the second kind for the two-form gauge field takes place, chargeons and
spinons are bound together into a charge 1 particle with spin 1/2. The mechanism is the same
n
one that gives spin to quarks bound into mesons in non-critical string theories and involves the
a
J self-intersection number of surfaces in 4D space-time. A state with free chargeons and spinons is a
topologicalinsulator. Whenchargeonscondense,thesystembecomesatopologicalsuperconductor;
1
a condensate of spinons, instead realizes U(1) charge confinement.
3
PACSnumbers: 11.10.-z,11.15.Wx,73.43.Nq,74.20.Mn
]
l
e
-
r I. INTRODUCTION sential physics of 2D Josephson junction arrays. In the
t
s same paper it was also pointed out that this two-fluid
. construction can be generalized to 3D, the topological
t The concept of spin-charge separation is one of the
a interaction being encoded in what is known as the BF
m guiding principles of the modern approach to strongly
term [8].
correlated, low-dimensional systems [1]. The idea is
-
d that, in specific ground states, the electron is fractional- Based on this construction we proposed a new, topo-
n ized into two ”constituent” quasi-particles,the chargeon logical mechanism for superconductivity based on the
o (holon) carrying only the charge degree of freedom and condensationoftopologicaldefectsinaBFmodel[9],no
c thespinon,carryingonlythespindegreeoffreedom. The symmetry breaking being involved. A pure BF theory
[
twoquasi-particlesinteractviaemergentgaugefields: the has also been recently shown [10] to represent the long-
2 electron is reconstituted when the gauge interaction be- distance physics of 3D topological insulators [11]. The
v comes strong enough to cause confinement. electric permittivity and magnetic permeability of the
1
materialgovernquantumphasetransitionsfromtopolog-
8 The idea originates from the work of Tomonaga and
ical insulators to topological superconductors and possi-
2 Luttinger and was shown by Haldane [2] to be a generic
bly to a U(1) charge confinement phase [12].
3 feature of 1D metallic systems. Moreover, the idea of
. electronfractionalizationisthoughttoplayacrucialrole Spin-chargeseparationismostlybelievedtobeaphe-
2
1 in the physics of the high-Tc cuprates. Indeed, spin- nomenon confined to lower space dimensions (1D and
1 charge separationseems to be an unavoidable character- 2D), where it is typically derived from fermionic models
1 istics of the 2D tJ model [3, 4] of the doped Mott insu- like, e.g., the Hubbard model. In this paper we show
v: lators, capturing the essential physics of high-Tc super- that this is not so. The topological excitations driving
i conductivity [5]. Thereis anabundance ofevidence that the topologicalinsulator-topologicalsuperconductorand
X
electron fractionalization in these models leads to new topological insulator-confinement quantum phase tran-
r quantumordersnotcharacterizedbysymmetrybreaking sitions in 3D are indeed chargeons and spinons. We
a
[6]. show that the presence of a specific combination of rel-
evant terms in the action leads to a generalized Higgs
A key ingredient of the spin-charge separation idea in
2D is the representation of chargeons and spinons as a mechanism of the second kind for the two-form gauge
two-fluid model with mutual Chern-Simons interactions field. When this happens, chargeons and spinons are
[5],apicturethatcanbeanalyticallyderivedfromthetJ bound together and the resulting excitation describes a
model [3, 4]. Mixed Chern-Simons fluids as representa- charge 1 particle with spin 1/2. The spin arises in a
tionsofcondensedmattersystemswherefirstintroduced subtle way: particle-antiparticle fluctuations about the
in [7], where it was shown that they capture all the es- symmetry-brokengroundstateareconnectedbyastring
whose only action is a topological term measuring the
self-intersectionnumber ofthe world-surfaceitsweepsin
4D (Euclidean) space-time. As was shown in [13], this
∗Electronicaddress: [email protected] factor is equivalent to the spin factor for fermionic par-
†Electronicaddress: [email protected] ticles in the path integral formalism [14]. Indeed, this is
2
the samemechanismgivingspintothe quarksboundto- of the charge and spin field degrees of freedom just in-
gether in mesons in non-critical string theories, the only troduced. The mostgeneralsuchmodel, containing only
difference being that in the present case the string ten- relevantandmarginaltermsandcompatiblewiththetwo
sions and all other curvature terms vanish, so that the separate gauge invariances (1,2) is given by:
particles are not confined and the string carries only the
spin information. This way we derive the fermionization S = ik d4x b ǫ f + iθ d4x f f˜ +
of a bosonic two-fluid model rather then the other way 4π Z µν µναβ αβ 16π2 Z µν µν
around. 1
+ d4x f f , (3)
The idea is to represent chargeons and spinons as 4e2 Z µν µν
topological excitations of charge and spin gauge fields.
These topological excitations represent quasi-particles where fµν = ∂µaν ∂νaµ is the field strength associ-
−
that arise due to the compactness of the correspond- ated with a , f˜ = (1/2)ǫ f its dual and k (the
µ µν µναβ αβ
ing gauge groups, the charge and spin fields mediating BF coupling), θ and e are dimensionless couplings. We
the emergent gauge interactions between these quasi- use relativistic notation in Euclidean space-time: possi-
particles. ble non-relativistic effects would not alter the main con-
A massive spin 1/2 particle in 3D is characterized by clusions.
three degrees of freedom, a scalar charge degree of free- The first two terms in this action are purely topolog-
dom and two degrees of freedom for the spin. Repre- ical terms: the first is called generically the BF term
senting the spin as a separate bosonic entity, the spin [8] and represents a generalization to 3D of the mutual
field, requires thus a massive vector with two degrees of Chern-Simons terms in 2D. It preserves the P amd T
freedom. It is well known that, in 3D, massless vectors symmetriesifthetwo-formgaugefieldisapseudotensor.
(photons) carry two degrees of freedom, called helicities, Thesecondisthefamedθ-termofaxionelectrodynamics
while massive vectors obtained from spontaneous sym- [17]. The parameter θ is an angle variable with period-
metry breaking and described by the Proca Lagrangian icity2π, the partitionfunction being invariantunder the
carrythree degreesoffreedom. This is the reasonwhy it shift θ θ+2π. The θ-term breaks generically the P
→
is mostly believed that spin-charge separation is impos- and T symmetries: these are however restoredwhen θ is
sible in 3D. It is, however not widely appreciated that quantized: θ =nπ, n Z. Thus there are only two pos-
∈
there is a way to describe a vector particle in 3D with sible θ values compatible with the P and T symmetries:
a gauge invariant mass. Exactly as its Chern- Simons θ =0andθ =π. Thethirdtermintheaction(3)hasthe
counterpart in 2D [15], the gauge invariant mass arises form of a standard Maxwell term for the effective gauge
from a topological term in the action [16]: such vector field a .
µ
particles in 3D carry thus only two helicity degrees of The physical interpretation is that
freedom and are ideal candidates to describe spin fields.
k k
Spin fields are thus described by a vector particle a
µ j = ǫ ∂ b ,φ = ǫ ∂ a , (4)
µ µναβ ν αβ µν µναβ α β
with gauge invariance under the transformations: 2π 2π
are the conserved charge and spin currents representing
a a +∂ λ . (1)
µ → µ µ thelow-energyfluctuationsaboutatopologicallyordered
state. WhenthetwoAbeliangaugesymmetries(1,2)are
Sincea playstheroleofanon-dynamicalLagrangemul-
0
U(1) compact symmetries, the dual field strengths (4)
tiplier inthe action,this gaugeinvarianceeliminates one
contain singularities [18]
degree of freedom from the remaining three, leaving the
two helicities as the only surviving degrees of freedom.
dx (τ)
Charge fields must be described by a single scalar de- J = dτ µ δ4(x x(τ)) ,
µ
Z dτ −
greeoffreedom. Therearetwowaystodescribeamassive C
scalarin3D,either directlyorbyembedding it inanan- Φ = 1 d2σ X (σ) δ4(x x(σ))
µν µν
tisymmetric gauge potential of the second kind b with 2Z −
µν S
gauge invariance under the transformations: ∂x ∂x
X =ǫab µ ν , (5)
µν ∂σa ∂σb
b b +∂ η ∂ η . (2)
µν µν µ ν ν µ
→ −
where C and S are closed curves and compact surfaces
The mixed components b = b will also play the role parametrized by x(τ) and x(σ) respectively. These have
0i i0
−
ofnon-dynamicalLagrangemultipliers,leavingthreedy- standard couplings to the charge and spin gauge fields:
namical degrees of freedom. The above gauge invariance ika J andikb Φ respectivelyanddescribechargeon
µ µ µν µν
eliminates, however, two of these, since there are two and spinon quasi-particle fluctuations about the ground
independent gauge parameters η (the other one being state. Sincein3Dthespinonisavectorwithtwodegrees
i
eliminatedbytheequivalenceη η +∂ ρ),leavingthus offreedom,theallowedpolarizationsaretransversewhen
i i i
≡
one overall degree of freedom. itis moving,exactlyasinthe caseofastandardphoton.
We will now consider low-energy effective theories for Since,however,itisamassivevectorparticle,itcanpoint
condensed matter system which are formulated in terms inanyspacedirectioninitsrestframe. So,inthiscase,it
3
isthedirectionofmovementratherthanthepolarization It is well known that adding marginal terms to an ac-
that is restricted: a spinon is a massive vector particle tioncandrivethesystemtoanewfixedpoint,describing
that moves always perpendicularly to the direction in anentirelydifferentphysics. Inthepresentcasethereare
which it is pointing. It has thus ”quantum Hall-type” indeedthreeadditionalmarginaltermsthatcanbeadded
responses to external fields. to the model (3): b b , b f and b ǫ b . All
µν µν µν µν µν µναβ αβ
Inorderto makecomputationstractablewe shalladd, these terms, taken one by one, break the gauge invari-
as a regulator,aninfrared-irrelevantbut gauge invariant ance (2) and introduce thus new, unwanted degrees of
kinetic term for the charge degree of freedom, freedom. There is, however one particular combination
of these three terms that can be added to the effective
S S+S , S = 1 d4x h h , (6) action and that preserves both gauge invariances, albeit
→ reg reg 12Λ2 Z µνα µνα (2) is realized in a different, more subtle way:
whereh =∂ b +∂ b +∂ b isthefieldstrength
µνα µ να ν αµ α µν ik2 θ θ
associated with the two-form gauge field b and Λ is S = d4x b + f ǫ b + f
µν µν µν µναβ αβ αβ
2θ Z (cid:18) 4πk (cid:19) (cid:18) 4πk (cid:19)
a mass parameter of the order of the ultraviolet cutoff
Λ . Thisregulatortermmakesthequadratickernelswell 4π2k2 θ θ
0 + d4x b + f b + f ,(9)
defined by inducing a mass m = eΛk/π for all fields. e2θ2 Z (cid:18) µν 4πk µν(cid:19)(cid:18) µν 4πk µν(cid:19)
This is the anticipatedtopological,gauge invariantmass
[16] that is the 3D analogue of the famed Chern-Simons
Something very interesting happens when the addi-
topological mass [15]. This mass represents the gap for
tional marginal terms in the effective action combine
the topologically-ordered state: it sets the energy scale
with the original ones to give (9). Only the combina-
for charge- and spin-wave excitations and the thickness
tion b + θ f appearsinthe actionandthe tensor
scale for chargeon and spinon quasi-particle excitations. µν 4πk µν
gaug(cid:0)e invariance (2(cid:1)) is preserved if it is combined with
Itthusplaysthesameroleastheinversemagneticlength
a corresponding shift a a 4πkη . This combined
in the quantum Hall effect. The mass can be removed µ → µ− θ µ
transformation can be exploited to entirely absorb f
again after integrations by letting Λ : in this case µν
the gap becomes infinite and only po→int∞-like chargeons into bµν, giving the effective action
and spinons quasi-particles survive.
It is easy to establish that the model (3) describes a S = i d4x b ǫ b + d4x π2 b b
topologicalinsulator [11]. Firstofall,the chargedegrees 32θ Z µν µναβ αβ Z 4e2θ2 µν µν
offreedom,carriedbythe two-formgaugefieldb , have i
µν + d4x b ǫ F +i d4x b Φ , (10)
no dynamics in the bulk of the material since the BF 16π Z µν µναβ αβ Z µν µν
termisatopologicalterm. Theonlydynamicmatterde-
greesoffreedomareedgemodesdescribingsurfaceDirac
wherewehaverescaled,fornotationalsimplicity,theten-
fermions [10]. Secondly, let us consider the coupling of
sor gauge field by a factor 4 and we have included the
the matter currents (4) to an external electromagnetic
couplingstoexternalelectromagneticfieldsandtoquasi-
field A dictated by the form of the effective action (3),
µ particle excitations. Note that the original BF coupling
kfallscompletelyoutoftheactionandisreplacedbythe
S S+i d4xj A . (7) factor θ/π, that takes over the role of charge unit.
µ µ
→ Z
In this gauge-fixed form, the original gauge symme-
try (2) appears as broken. This is nothing else than a
Inpresenceofanon-vanishingangleθ,the spinfieldcar-
Higgsmechanismofthesecondkindforthetensorgauge
ries vorticity. The BF coupling (contained in the defi-
symmetry (2). This is also known as the Stu¨ckelberg
nition of the current j ) is the charge unit of chargeons
µ
mechanism [19], in which a scalar longitudinal polariza-
whereas θ is the flux unit of spinons. Integrating out
tion ”eats up” two transverse polarizations to become a
all matter degrees of freedom one obtains, in the limit
massive vector. The Stu¨ckelberg mechanism is the dual
Λ , the effective electromagnetic action
→∞ ofthe standardHiggsmechanismandis thusresponsible
1 iθ for confinement [19], soldering in this case chargeons to
Seff =Z d4x 4e2 FµνFµν + 16π2FµνF˜µν . (8) spinons.
Indeed, corresponding to the merger of the spin and
This describes a bulk dielectric material with an axion charge gauge fields a and b into a single tensor field
µ µν
electrodynamics term that characterizes strong topolog- with3massivedegreesoffreedom,there is alsoa merger
ical insulators when θ =π. of chargeons and spinons into a unique string-like quasi-
In[12]wehaveshownthatacondensationofchargeons particle excitation with open world-sheets, describing
turns the topological insulator into a topological super- magnetic fluxes with charged dyons at their ends. The
conductor, while the condensation of spinons leads to a closed boundaries of the open surfaces represent the
U(1)chargeconfinementregime. In whatfollowswe will world-linesofdyon-antidyonfluctuationswithchargeθ/π
concentrate on a different quantum phase transition. and current J = (1/2π)∂ Φ as can be inferred from
µ ν µν
4
the induced electromagnetic action represents the (signed) self-intersection number of the
world-surface.
1 iθ
S = d4x F F + F F˜µν
eff Z 4e2 µν µν 16π2 µν The important point is that, to avoid infinities and
(θ/π) 4π obtain a well-defined boundary term when we remove
+iZ d4x 2π Aµ∂νΦµν + e2F˜µνΦµν . (11) the UV cutoff, the renormalization flow implies e → ∞
as Λ (and Λ=const.Λ ). As a consequence
0 0
→∞ →∞
This is the same induced actionas in (9) but the matter ofthisrunningcouplingconstant,boththestringtension
degrees of freedom have now dynamics in the bulk. andthe curvaturetermsin(13)vanishatlargedistances
In order to gain more insight into the character of the and the induced quasi-particle action becomes
resulting soldered quasi-particle let us compute its in-
duced actionby using the explicit form(5) for Φ . The
µν
relevant and marginal terms in this action are
dx dx θ
µ µ
S =m dτ i πν , (14)
Λ2 m Λ2 QP sZ∂S r dτ dτ − π
S = K d2σ√g+ d2σ√gR
QP 4π 0(cid:18)Λ0(cid:19)ZS 16πm2 ZS
Λ2
−16πm2 Z d2σ√ggab∂atµν∂btµν where ms is the renormalized mass of the particle. Note
S that the only remnant of the string at large distances
θ π e2m m dxµdxµ is the topological self-intersection number. It has been
i ν+ f dτ , (12)
− π1+ 4eπ2 8π2 (cid:18)Λ0(cid:19)Z∂S r dτ dτ shown [13] that at θ/π = 1 this topological term is just
anotherrepresentationofthespinfactorofapointparti-
where m = eΛ/4π, f(x) = ∞dzK (z)/z and K and clewithspin1/2. Thisshowsthat,intheHiggsphase(of
x 1 0
K1 are Bessel functions ofRimaginary argument, with the second kind) chargeons and spinons recombine into
asymptoticbehavioursK (x) exp( x)/√xandf(x) a single particle with charge 1 and spin 1/2.
0
≃ − ≃
exp( x)/√x3 for large x. The geometric quantities in
− We thus conclude that spin-charge separation can oc-
this expression are defined in terms of the induced sur-
curin3Dwhenanontrivialaxiontermwithθ-angleπ is
face metric g = ∂xµ∂xµ as g = det g = X X /2
ab ∂σa ∂σb ab µν µν generatedinthe electromagneticaction. The phase with
andtµν =Xµν/√g. The quantityRis the scalar(intrin- free separated chargeons and spinons is a strong topo-
sic) curvature of the world-surface while
logical insulator. Topological superconductivity occurs
when chargeons condense and an exotic U(1) confine-
1
ν = d2σ √gǫ gab∂ t ∂ t , (13) mentwouldberealizedinpresenceofspinoncondensate.
µναβ a µν b αβ
4π Z
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