Non-Abelian adiabatic statistics and Hall viscosity in quantum Hall states and p_x+ ip_y paired ...

44 Pages·2008·0.68 MB·English

Non-Abelian adiabatic statistics and Hall viscosity
in quantum Hall states and p +ip paired superfluids
x y
N. Read
Department of Physics, Yale University, P.O. Box 208120, New Haven, CT 06520-8120, USA
(Dated: October31, 2008)
Many trial wavefunctions for fractional quantum Hall states in a single Landau level are given
by functions called conformal blocks, taken from some conformal field theory. Also, wavefunctions
8 for certain paired states of fermions in two dimensions, such as px +ipy states, reduce to such
0 a form at long distances. Here we investigate the adiabatic transport of such many-particle trial
0 wavefunctionsusingmethodsfromtwo-dimensionalfieldtheory. Onecontextforthisistocalculate
2 thestatisticsofwidely-separatedquasiholes,whichhasbeenpredictedtobenon-Abelianinavariety
ofcases. TheBerryphaseormatrix(holonomy)resultingfrom adiabatictransportaroundaclosed
t
c loop in parameter space is the same as the effect of analytic continuation around the same loop
O withtheparticlecoordinatesheldfixed(monodromy),providedthetrialfunctionsareorthonormal
andholomorphicintheparameterssothattheBerryvectorpotential(orconnection)vanishes. We
1
show that this is the case (up to a simple area term) for paired states (including the Moore-Read
3
quantumHallstate),andpresentgeneralconditionsfor ittoholdforothertrialstates(suchasthe

Read-Rezayi series). We argue that trial states based on a non-unitary conformal field theory do
]
l notdescribeagappedtopological phase, atleast inmanycases. By consideringadiabatic variation
l
a oftheaspectratioofthetorus,wecalculatetheHallviscosity,anon-dissipativeviscositycoefficient
h analogous to Hall conductivity, for paired states, Laughlin states, and more general quantum Hall
- states. Hallviscosityisaninvariantwithinatopological phase,andisgenerally proportionaltothe
s
“conformal spin density”in theground state.
e
m
t. I. INTRODUCTION holes(orvortices)whentheyareexchangedadiabatically,
a
where adiabatic transport is calculated using trial forms
m
of the wavefunctions. The trial wavefunctions are taken
A. Background and motivation
- to be “conformal blocks” obtained from some conformal
d
field theory (CFT), as in MR.
n
There has been renewed interest in the past few years
o
A central idea of MR is that the adiabatic effect of an
in non-Abelian quantum states of matter, both in the
c
exchangeofquasiholesinthetrialwavefunctionsgivenby
[ original setting of quantum Hall states [1], and also in
conformal blocks is the same as the effect inferred from
othersystemsincludingonesinwhichthesymmetriesun-
3 simpleanalyticcontinuationofthewavefunctionsviewed
der time reversal and parity are unbroken (see e.g. Ref.
v
as functions of the quasihole coordinates, with the par-
[2]). Briefly, a non-Abelian phase of matter is a gapped
7
ticle coordinates fixed. In contrast to analytic contin-
0 (topological)phase in whichthere arequasiparticleexci-
uation, adiabatic transport involves integration over the
5 tations over the ground state, the adiabatic exchange of
2 whichproducesamatrixeffectonthestateofthesystem, particlecoordinatesforeachinfinitesimaltimestep,since
5. with the matrices corresponding to distinct exchanges this defines the inner product in the Hilbert space. In
severalexamples with Abelian statistics, this was shown
0 not all commuting (thus the term “non-Abelian”). This
8 requires that there be a degenerate space of states when in MR to give the same result, but the result was not
0 therearequasiparticlesatwell-separatedpositionsinthe demonstratedforthetrialstateswithnon-Abelianstatis-
: tics. Here we provide a detailed explicit calculation for
v system. Given a reasonably local Hamiltonian, such be-
thecasesoftwoorfourquasiholesintheMRpairedstate
i haviorcanonlyoccurwhenthedimensionalityofspaceis
X (withthechargepartofthewavefunctionsremoved);the
two,fortopologicalreasons. Thebasicexampleproposed
r byMooreandRead(MR)[1]isapairedstate,whichcan four quasihole case is the first to exhibit non-Abelian
a
statistics. Finally, we also give a general criterion for
be viewed as a p-wave, or more precisely a p + ip -
x y
other states, and discuss when this may hold, with ex-
wave, Bardeen-Cooper-Schrieffer (BCS) [3] paired state
amples.
of spinless or spin-polarized composite fermions in zero
oralmostzeronetmagneticfield. Itturnsoutthatmuch An additionalissue that we addressis adiabatic varia-
of the physics of this state is also found in such paired tionofthegeometryinaclosedfinitesystem,forexample
states of ordinary fermions [4]. In this paper we address a parallelogram with periodic boundary conditions (i.e.
bothofthese situations together,notonly quantumHall a torus). There is a discrete group of transformations
systems. Much of the current interest in these systems that map the geometry to one equivalent to the original
is driven by their potential use for topological quantum (called the modular group). It is of interest to perform
computing [5]. these adiabatically also. It turns out that in these cases
The major issue that we address in this paper is the the relevant Berry connection (vector potential on the
derivation of the non-Abelian statistics of the quasi- space of inequivalent geometries)has non-zerocurvature
2
(field strength). Because varying the aspect ratio varies has been formulated recently [11], and applied to non-
the strain on the system, this response can be related to Abelian states [12] as we were finishing this paper.)
a non-dissipative viscosity coefficient [6], which we call We follow the framework of their method, but gener-
Hall viscosity (earlier it was termed “odd”, or “antisym- alize to allow for non-Abelian statistics. Thus suppose
metric” viscosity [6], or “Lorentz shear modulus” [7]). that we have a space of degenerate states with basis
Hall viscosity is so-named because it is the natural ana- Ψ (w) , where a runs over some finite set, and we will
a
| i
log in viscosity of the Hall conductivity. We show that use w to stand for the set of n quasihole coordinates w ,
l
this viscosity arises in quantum Hall fluids and in paired l = 1, ..., n. We assume that the basis is orthonor-
superfluids, and its magnitude is always proportional to mal, Ψ (w)Ψ (w) =δ foreachw. As w varies,these
a b ab
h | i
aspindensity. Thisresultshouldbeofwideinterest. For vectors sweep out a subspace of a common multiparticle
trial wavefunctions given by conformal blocks, the spin Hilbertspaceinwhichtheyalllive[13,14]. Thestatesfor
per particle is given by the conformal weight. anyfixedw aresupposedtobedegenerateinenergy,and
In the remainder of this Introduction, we explain the we will compute the adiabatic statistics within this sub-
central issues in more depth, introduce some basic ar- space. Weneglectthe“dynamicalphase”whichdepends
guments that will be crucial later, and also review some on the energy and the total time taken for an exchange.
earlier work relevant to the problems of adiabatic statis- The desired adiabatic phase or matrix is the holonomy,
tics and Hall viscosity. which can be written [9, 13, 14, 15]
expi (A dw+A dw) (1.2)
w w
P I · ·
B. Adiabatic transport and quasiparticle statistics C
where C is a closed directed path that begins and ends
Here for the time being we are concerned primar- at a base point w , in the configuration space of the
(0)
ily with quantum Hall states. The basic examples are quasiholes. (This configuration space can be thought of
Laughlin’s trial wavefunctions for a ground state with n as Cn where C is the complex plane or Riemann sphere,
quasiholes(eachofthelowestpossiblenon-zerofractional minus the “diagonal” on which wk = wl for some k, l,
charge), which in their original form are [8] k =l,modulotheactionofthepermutationgroupSn for
6
the identical quasiholes; if the quasiparticles are not all
(zk wl) (zi zj)Qe−14 i|zi|2. (1.1) of the same type there are obvious modifications of the
Yk,l − Yi,j − P permutation group.) P is the path ordering operator, in
which the matrix for a later point on the path is to the
Here the exponent Q determines the filling factor of the left of earlier ones, Aw·dw = lAw,ldwl, and Aw,l is a
state in the thermodynamic limit as the particle number matrix. It is possible to changPe to a different basis set
N goes to infinity, zi (i = 1, ..., N) are the complex |Ψ′b(w)i = |Ψa(w)iMab(w) as a function of position on
coordinates of particles, and w are the complex coordi- the path C, where M(w) is a unitary matrix for each
l
nates of the quasiholes. When Q is even, the particles point w on the path C, in order to preserve orthonor-
are bosons, and when Q is odd they are fermions, such mality. This is referred to as a change of gauge. The
aselectrons. Ineithercase,thefunctionsareforparticles expression is correct as written when the vector poten-
in the lowest Landau level (LLL). tial A refers to a single gauge choice for the whole path
Nowweconsidertheadiabaticcalculationofstatistics. C [as in the case of the states in eq. (1.1) above]. In
The adiabatic calculation is meaningful if the excitation general such a choice may not be convenient, and then
spectrum of the system above the possibly degenerate one mustuse patches with a gaugechoice oneachpatch.
quasiparticle state subspace is gapped (with the gap go- The patches overlap, and in the overlap region there is
ing to a constant as the system size goes to infinity). It a transition function (gauge transformation), which is a
canalsobevalidifanygaplessexcitationsaresufficiently unitary matrix M. The holonomy is then the matrix
weakly coupled to the quasiparticles. For example, in a composite of the path-ordered exponentials for the part
quantumHallsystemwithanedge,therearegaplessedge of the path within each patch, with gauge transforma-
excitations. When quasiparticles are present within the tions by the transition matrices inserted in between. It
regionoccupiedbytheparticles,wewillassumethrough- is frequently the case that a single transition function
outthispaperthatthesystem(i.e.N)islargeenoughso M is sufficient, and the transition can be located at the
that during the exchange the distance of the quasiparti- basepoint w(0). Then the adiabatic transport maps the
cles from the edge is large compared with their separa- basis states |Ψb(w(0))i to a|Ψa(w(0))iBab, where the
tion, which is itself large compared with the microscopic holonomy is given by P
correlation length scale (which frequently is of order the
particle spacing). In this case the edge excitations effec- B =M expi (Aw dw+Aw dw). (1.3)
P I · ·
tively decouple from the calculation. C
The pioneering calculation of adiabatic statistics in Defined this way, the holonomy is gauge invariant in all
a QH system was performed by Arovas, Schrieffer and cases, up to conjugation by a unitary matrix that cor-
Wilczek [9, 10]. (An interesting alternative approach responds to a change of orthonormal basis at w . We
(0)
3
note also that the use of a different base-point on the
1 Q
same path C conjugates the holonomy by some unitary +Q2 ln zi zj 2 wk 2 zi 2. (1.7)
| − | − 2 | | − 2 | |
matrix, and has no effect on the structure of the braid- Xi<j Xk Xi

group representation.
The Berry connection is given in components by Given the partition function, we can define the free en-
ergyF asF = ln (orasthistimesthetemperatureQ,
− Z
∂Ψ (w) but this makes no difference for us). Then by screening,
b
Aw,l,ab(w) = i(cid:28)Ψa(w)(cid:12) ∂w (cid:29), (1.4) forsufficientlylargeseparationofthequasiholes,thisfree
(cid:12) l energy goes to a constant at sufficiently large separation
(cid:12)∂Ψ (w)
A (w) = i Ψ (w)(cid:12) b . (1.5) of the quasiholes. This constant has the form
w,l,ab (cid:28) a (cid:12) ∂w (cid:29)
(cid:12) l
(cid:12) F =Af (Q)+nf (Q), (1.8)
(cid:12) 0 qh
ThenA (w)=A (w) , whichensuresthatthe holon-
w,l w,l †
omy is unitary (A is not holomorphic in w in general). consisting ofan extensive backgroundterm, whichis the
w
For functions in the LLL, the inner product is the usual area A occupied by the particles plus quasiholes times
hΨa|Ψbi= d2ziΨ∗aΨb. In the calculation in Ref. [9], a constant f0(Q), plus a “defect free energy” fqh(Q) for
the trial quRaQsihole states used were the wavefunctions each quasihole. (In a more general situation with dif-
(1.1)whicharenotorthonormalwithrespecttothisinner ferent types of quasiparticles, these latter terms would
product (for Laughlin quasiholes, there is only a single be different for each type of quasiparticle, as well as de-
state for n quasiholes,so the labela is dropped). Arovas pendingontheunderlyinggroundstate.) Bothconstants
et al. appeared to neglect this point [10], but arrived at f0(Q) and fqh(Q) are well-defined but Q-dependent,
the correct answer nonetheless. The result is that if two given the definition above, but have no universal signifi-
quasiholesareexchangedaroundacounterclockwisepath cance.
that does not enclose any other quasiholes (the others Nowwecanreformulatethestatisticscalculation. Sup-
stay at fixed positions throughout), the adiabatic phase pose that, in addition to being (ortho-)normalized, the
change is eiπ/Q. wavefunctions of the states Ψa(w) are holomorphic in
| i
The central point of the calculation is that the neces- w, as the Halperin functions above are (except for the
saryinnerproductscanbeevaluatedbyusingtheplasma Gaussianfactors,whichwe ignorehere,but commenton
mapping of Laughlin[8], plus the fact that screening oc- afterwards, and except on the diagonal wk = wl, which
curs in the Coulombplasma providedQ is not too large. is to be avoided). Then it follows that
We reformulate this as follows (following a line of ar-
∂
gument begun by Halperin [16]). Consider instead the
A (w)=i Ψ (w) Ψ (w) =0. (1.9)
w,l,ab a b
followingwavefunctions,whichrepresentthe samequan- ∂wl h | i
tum states as (1.1) because they differ only by functions
Thentheholonomyisgivenentirelybythetransitionma-
of the parameters w:
trix or matrices: B = M. For the wavefunctions (1.6),
the transition matrix is just a number M of modulus 1
Ψ(w ,...,w ;z ,...,z )=
1 n 1 N that is required to transform back to the original sheet
(wk wl)1/Q (zi wk) (zi zj)Q (or gauge) after making an exchange, due to the wave-
− · − · − ·
kY<l Yi,k Yi<j functions not being single valued in the wks. For the ex-
×e−41QPk|wk|2−41Pi|zi|2 (1.6) cohthaenrgse(owfhtiwcho qsutaaysihaotlefisxaedlonpgosaitpioanths tnhortoeungchloousitn)g, tahniys
gives the phase M = eiπ/Q. It is important to empha-
(here and below we take the notational liberty of using
size that when applying this argumentto wavefunctions,
some indices more than once in distinct factors, which
they should be functions in a common Hilbert space for
have been separated here by the dots ). The modulus-
· all w, and with the integration measure independent of
squaredistheBoltzmannweightforaclassicalplasmaof
the parameter w being varied.
chargesQ atallz , 1 atallw , with a neutralizing back-
i k ThefactthatthenormalizingGaussianfactorsarenot
groundof density 1/2π,impurities at fixed positions w ,
k holomorphic in w means that the calculation also pro-
with the two-dimensional Coulomb interaction potential
duces a phase factor eiA(C)/Q, where A(C) is the area
betweenunitchargestakentobeminusthenaturalloga-
enclosedbytheloopC [9]. Thiscanbeinterpretedasthe
rithm of the distance-squared, and with temperature Q.
fractionally-charged quasiparticles detecting the back-
Thus the overlap integral is the corresponding partition
ground magnetic field [9] through an Aharonov-Bohm
function:
phase, though in fact it is Q times the particle density
that they detect, which happens to give the same re-
(w ,...,w )= Ψ(w ,...,w ) 2 =
Z 1 n k| 1 n ik sult because the particle density is uniform, with filling
factor 1/Q, outside the quasiholes. (If the particle den-
1
d2zi exp  ln wk wl 2+Q ln zi wk 2 sity were not uniform, the normalizing Gaussian factors
Z Q | − | | − |
Yi Xk<l Xi,k in ws would be modified.) This effect is ubiquitous for

4
QH states. We will comment on it further in connec- ground on CFT, see Ref. [17].
tion with systems that are not QH systems. Other than The trial wavefunctions for QH systems that we will
this effect, the holonomy depends only on the homotopy study in this paper take the form
classofthepathC intheconfigurationspace,notonthe
Ψ (w ,...,w ;z ,...,z )= (1.10)
precise path; that is, it is invariant under small defor- a 1 n 1 N
mations of the path such that C does not pass through Ψ ψ(z ) ψ(z )τ(w ) τ(w ) .
charge 1 N 1 n a,CFT
·h ··· ··· i
the diagonals w = w during the deformation. Then
k l
Here again coordinates z are those of particles (either
the holonomy givesa unitary representationof the braid i
bosons or fermions), so Ψ is single valued and either
group, acting in the space of degenerate states at w , a
(0)
symmetric or antisymmetric in these variables,while co-
times the path-dependent factor eiA(C)/Q which we will
ordinates w are those of quasiholes. The label a again
frequently just ignore. l
runs over a basis for a space of functions. On the right
The transition matrices M result from the behavior
hand side, Ψ is independent of a, and is a func-
of the wavefunctions under analytic continuation of the charge
tion of the same coordinates of similar form to eq. (1.6),
quasiholes, with the particle coordinates held fixed (or
though the exponents (or charges of the particles and
even if they are not fixed). In the context of solutions
impurities in the corresponding plasma) may take other
of differential equations, this is called monodromy, and
rational-fraction values. We note that the exponent in
wefollowthisterminologyhere. Halperin[16]notedthat
the particle-particle factors is the inverse of the filling
themonodromyofhisfunctions(1.6)suggestedfractional
factor, ν = P/Q, and the Gaussian in the particle coor-
statistics, but did not perform the adiabatic calculation.
dinatesisalwaysasin(1.6). Thevaluesoftheexponents
Theuseofthepresentapproachintheadiabaticcalcula-
in the charge part are determined by the requirement
tion gives the statistics without the further calculations
that the whole function Ψ be single valued and (anti-)
performed by Arovas et al. [9], in particular, confirm- a
symmetricintheparticlecoordinates;thismayalwaysbe
ing the sign of the phase (alternatively, their calculation
done consistently with a plasma form, thanks to consis-
[9] can also be used to produce the normalization of the
tency properties of conformalblocks. The exponents are
Laughlin quasihole functions [10]).
determined only up to addition of integers, and we usu-
The statistics calculation for the Laughlin quasiholes
ally consider the smallest possible positive values, which
generalizes easily to wavefunctions that correspond to
givesthehighestvalueofthefillingfactor,andthelowest
multicomponent Coulomb plasmas [1]. Here we wish
valueofthequasiholecharge;furtherthefieldτ isusually
to generalize it to non-Abelian cases, specifically those
alsochosen,giventheCFT,toobtainthelowestpossible
in which the trial wavefunctions are conformal blocks
quasihole charge for a given possible filling factor.
from CFT. These are holomorphic in the quasihole co-
Theexpectationvalue (w ,...;z ,...)=
ordinates, and have non-trivial, sometimes non-Abelian, Fa 1 1 h···ia,CFT
stands for a conformal block in some CFT, in which ψ
monodromy;theHalperinform(1.6)ofAbelianquasihole
andτ standfor fields. The notationis somewhatformal,
wavefunctionscanalsobeviewedasconformalblocks[1].
becausethefunctionisgenerallynotsingle-valued,anda
Thegoalwillbetoseewhethertheeffectnotedabovefor
sheetshouldbespecified. Thefunctionisholomorphicin
thelatterstates,thatholonomyequalsmonodromy,holds
the zs and ws off the diagonal on which some zs and/or
for these trial functions, as conjectured by MR. The re-
ws coincide. The field ψ must have the property that its
sult will hinge on whether the screening property in the
monodromy is Abelian, which means that there is a sin-
plasmamappingfortheLaughlinstates,whichmakesthe
gle primary field (of some chiral algebra) in its operator
Halperin functions (conformal blocks) orthonormal, also
product expansion (ope) with itself (we omit non-zero
generalizeswhenotherconformalblocksareusedastrial
ope coefficients, which play no role here):
wavefunctions. The answer will be yes in some cases.
In addition, we apply the same formalism to consider ψ(z)ψ(0) z−2hψ+hψ∗ψ∗(0)+..., (1.11)
adiabatic variation of the geometry of the system. We ∼
willshowlaterthatthecurvatureoftheBerryconnection as z 0, where ψ∗ is another field, and the ... stands
→
is non-zero, and this determines the Hall viscosity. We as usual in CFT for terms smaller by positive integer
also consider the holonomy arounda loop corresponding powersof z asz 0,whicharedescendants ofψ∗ under
→
to a modular transformation. In some cases we can ap- thechiralalgebra. Moreover,ψmustgenerateinthisway
ply a similar argument, that the orthonormalized wave- only fields that are also Abelian. Further, there must be
functionsarealmostholomorphicintherelevantcomplex exactly one term in the ope of ψ with τ:
parameter τ, except for a very simple non-holomorphic
part. ψ(z)τ(0)∼z−hψ−hτ+hτ∗τ∗(0)+..., (1.12)
where again τ is another field, h is the conformal
∗ ψ
weightofψ (andsimilarlyforotherfields), andthe same
C. Conformal blocks as trial wavefunctions is true for τ and so on. These requirements mean that
∗
ψ is what is called a “simple current”: the operation of
Next we discuss how conformal blocks from CFT can taking the ope of ψ with the primary fields in the the-
be used as trial wavefunctions, following MR. For back- ory (including ψ itself) just permutes the primary fields.
5
This has the effect of guaranteeing that the full function many-particlewavefunctions,wewouldinsteadcountthe
Ψ has the stated properties. The ope of τ with itself number of linearly-independent functions of the particle
a
may be nontrivial,containing terms that do not differ in coordinates z for fixed w. In general, the latter number
conformal weight only by integers: might be less than the number of blocks (for example,
when N = 0). However, in examples these numbers do
τ(z)τ(0) z−2hτ+hτ(1)τ(1)+...+z−2hτ+hτ(2)τ(2)+...+..., coincide when the particle number N is large enough,
∼
(1.13) and we will assume this from here on, as large N is the
where, depending on which CFT is used, any number of case of interest anyway. More generally, of course, one
distinct primaries could appear on the right. Those ap- could have more than two types of primary field in the
pearingwith nonzerocoefficients maybe summarizedby correlator, which could represent the particles and more
the fusion rules (analogous to the Clebsch-Gordan for- than one type of quasihole;such could resultfrom use of
mulas for SU(2) tensor products), which generally have the ope starting from one type of “basic” quasihole.
the form In CFT, correlation functions are constructed from
combinations of blocks and their conjugates, for exam-
φ φ = γ φ , (1.14)
α× β Nαβ γ ple
Xγ
ψ(z ,z¯ ) ψ(z ,z¯ )τ(w ,w¯ ) τ(w ,w¯ )
1 1 N N 1 1 1 1 CFT
in which the fields φ run over the full set of primary h ··· ··· i
α
fields in the CFT used (ψ, ψ∗, ..., as well as τ, τ∗, τ(1), = |Fa(w1,...;z1,...)|2, (1.15)
..., will be among these), and the γ = γ are non- Xa
Nαβ Nβα
negative integers (which may be larger than 1 in some
where the sum is over the basis for the space of blocks.
cases). The product here is formal and simply refers to
Such an expression is supposed to represent a single-
terms in an actual ope. We define φ = 1, the identity
0 valued correlation function of local operators in a CFT;
operator. For an Abelian field (one obeying Abelian fu-
the local operators ψ(z,z¯) and so on that appear here
sion rules), β is equal to 1 for one value of β (and
Nαα differfromthechiralversionsψ(z)thatappearedbefore,
zero otherwise), andsimilarly for iteratedproducts; oth-
and are related roughly by ψ(z,z¯) = ψ(z)ψ(z¯), but also
erwisethe fieldisnon-Abelian. (As explainedinRef. [1],
require the sum over the blocks. The “diagonal” form
Ψ can also be viewed as a kind of conformal block
charge given is single-valued if the monodromy of the space is
in the CFT of a single scalarfield ϕ, with the role of the
given by unitary matrices B in the basis used. Such a
ab
fields ψ, τ played by charged fields (exponentials of ϕ),
diagonalformisalwaysavailableinaunitarytheory,and
but which also includes a chiral version of the neutral-
even in non-unitary rational theories, such as the Vira-
izing background charge density [8].) A rational CFT is
sorominimalmodelsofBelavin,Polyakov,andZamolod-
one in which there is a chiral algebra, which is either the
chikov (BPZ) [17, 19]. More generally, when the CFT is
Virasoro algebra or an algebra extension of it, obtained
formulated on a surface of higher genus, the number of
fromope’s ofafinite setofAbelianconformalfields, and
blocks is in most cases larger than one even when no
a finite set of primary fields, defined generally as confor-
fields are inserted in the correlator. For N, n 0, the
mal fields that generate irreducible highest-weight rep- ≥
single label a labels the full space of blocks. A diagonal
resentations of the chiral algebra (by operator products)
theory, in which all correlatorsare given by diagonal ex-
[17, 18]. Examples of non-trivial chiral algebras include
pressionslikethataboveexistsinmostcases,andisvalid
the affine Lie algebras. At many places we will need to
for surfaces of any genus; it is referred to as the diago-
assume that the CFT is rational, as is the combined one
nal modular invariant theory, where modular invariance
that includes the charge part. Irrational examples will
referstothecaseofthetorus(genus1),withnofieldsin-
be discussed at the end of the paper.
serted. At this point we should emphasize that what we
Theconformalblocksaremultisheetedfunctionswhen
call a correlation function (or correlator)here is really a
the exponents such as 2h h are not integers, and
ψ − ψ∗ partitionfunctionfortheCFTonthegivensurface,with
because of non-Abelian fields τ. The multisheetedness
fields ψ and τ inserted at specified points. This differs
due to fractional exponents from ψ is no worse than in
from the more usual use of normalized correlation func-
the case of the Halperin functions (1.6), and produces
tions, in which one divides by the partition function for
only a phasefactor whenthese are exchangedorencircle
the case of no field insertions. For example, when ψ is a
a τ, so not a linearly-independent function of the par-
basicfieldofthe theory,describedbyanactionS[ψ],the
ticle coordinates (as the latter vary over some open set
theory can be described by a functional integral,
that lies on a single sheet). Finally, a labels a basis for
the space of linearly-independent conformal blocks asso-
ciatedwithagivencorrelationfunction; the rangeofa is Z =Z D[ψ]e−S[ψ], (1.16)
the dimension of the space of blocks, which can be cal-
culated for the case of CFT on the sphere by repeated and the unnormalized correlation function is
use of the fusion rules. By definition, this is the number
ψ(z ,z¯ ) ψ(z ,z¯ )τ(w ,w¯ ) τ(w ,w¯ )
of linearly-independent functions of all the variables z 1 1 N N 1 1 1 1 CFT
h ··· ··· i
and w as they vary oversome open set. When viewedas =Z(w ,w¯ ,...;z ,z¯ ,...)
1 1 1 1
6
= [ψ]ψ(z ,z¯ ) τ(w ,w¯ ) e S[ψ], (1.17) (The coordinates, and the integration measure are writ-
1 1 1 1 −
Z D ··· ··· ten for the plane, though the definition applies to other
geometries with modifications that are hopefully ob-
whereas usually one would define the correlation func-
vious. We do not imply that the overlap matrices
tiontobeZ(w ,w¯ ,...;z ,z¯ ,...)/Z. Evenleavingaside
1 1 1 1
(w ,...,w ) are holomorphic in the ws.) The differ-
a factor of the form exponential of minus a free energy ab 1 n
Z
encebetween andZ aboveshouldbenoted: bothare
proportional to the area of the surface, this makes a dif- Zab
sesquilinear in conformal blocks, but Z(w ,...,z ,...)
ferenceforgenus>1,becausethedenominatorisinmost 1 1
depends on z’s as well as w’s, and is a diagonal sum
CFTs a sum of more than one mod-square conformal
of mod-square conformal blocks, while is integrated
blocks. The exceptions to the latter are “holomorphic” ab
Z
over z’s, but not summed over the indices a, b.
CFTs, in which the only primary field of the chiral alge-
bra is the identity operator 1; an example is the current From the discussion we now see that when conformal
algebraorWess-Zumino-Wittentheoryforthe Liegroup blocksareusedas(orin)trialwavefunctionsasdescribed
E at level 1. above,thenthe holonomyunder adiabatictransportwill
8
As an example of the conformal blocks and their use equal the monodromy provided that the overlap matrix
astrialwavefunctions,wegivetheMRstate,forthecase ab(w1,...,wn) is proportional to δab with proportion-
Z
of no quasiholes,in which ψ is a Majoranafermion field, alityconstantindependent ofthe positionsw, asymptot-
with conformal weight h =1/2. The ground state trial ically for large separations of the w’s, in a basis for the
ψ
wavefunctionforthesphereorinfiniteplanecontainsthe conformal blocks in which the braiding matrices B are
conformal block [1] unitary. This orthonormality of the conformal blocks is
then the desired statement generalizing screening in the
1
Coulomb plasma. (MR noted in Abelian examples such
ψ(z )ψ(z ) =Pf , (1.18)
1 2 Ising
h ···i zi zj as the Laughlin states that when conformal blocks are
−
used as trial wavefunctions, the gauge is such that the
where the Pfaffian is defined for any even-by-even anti-
Berry connection vanishes, and the holonomy is given
symmetric matrix with matrix elements M by
ij entirely by the monodromy.)
PfMij = (M12M34 MN 1,N), (1.19) Let us also point out here that the so-called shift for
A ··· −
the ground state on the sphere or disk geometries can
wheretheantisymmetrizer sumsoverallpermutations
A be obtained from the CFT as well. The shift in the
thatproducedistinctpairings(i,j),timesthesignofthe S
number of flux N piercing the sphere is defined by
permutation. Forquasiholes,oneuses[1]τ =σ,thespin φ
field of the critical Ising model; 1, ψ and σ are the only
N =ν 1N . (1.24)
primary fields in the Ising (or Majorana fermion) CFT. φ −
−S
The scaling dimensions are h = 1/2, and h = 1/16.
ψ σ
Explicitconformalblocksforthiscasewillbequotedlater The flux Nφ can be obtained from the degree of the
in the paper. The fusion rules for this CFT (other than wavefunction in each coordinate zi, which itself can be
for products with the identity 1) are obtained by letting zi and extracting the leading
→ ∞
power of z (neglecting the Gaussian factor). The CFT
i
ψ ψ = 1, (1.20) contributes 2h to this, as particle i and the N 1 re-
× − ψ −
ψ σ = σ, (1.21) mainingparticlesmustfusetogivetheidentity[20]. The
× charge sector contributes ν 1(N 1). Then the shift is
σ σ = 1+ψ. (1.22) −
−
×
The fusion rules imply that the number of conformal =2(ν−1/2+hψ). (1.25)
blocks for n quasiholes on the sphere is 2n/2 1, and n S
−
must be even. Other examples include the RR states
Wewriteitinthisformbecausewhenthechargesectoris
[20], in which the field ψ = ψ , one of the parafermion
1 interpretedasaCFTalso[1],theconformalweightofthe
currentsintheZkparafermionCFT;thesefieldsaregood field contributing to the particle is ν−1/2, so that /2 is
examplesof“simplecurrents”. OnecanalsoconsiderQH S
simplythetotalconformalweightofthefieldrepresenting
systems in which the particles carry SU(2) spin greater
the particle, including the charge sector.
than0,andtheninnerproductsinvolvetheinnerproduct
of spin states as well as spatial integrals.
Wemaynowdefinetheoverlapintegrals,orinnerprod-
ucts of the trial states with wavefunctions Ψ , as D. Bose-Einstein condensates and paired states
a
Ψ (w ,...,w )Ψ (w ,...,w )
h a 1 n | b 1 n i Somesimilartrialwavefunctionsalsohaveapplications
= ab(w1,...,wn) outside of the QH effect. Let us start again with the
Z
N simplestcase,thatofaBose-Einsteincondensate(BEC),
= d2ziΨa(w1,...;z1,...) with n 0 vortices included in infinite space with no
Z ≥
Yi=1 backgroundpotential. A trial function for this is similar
Ψ (w ,...;z ,...). (1.23) totheLaughlinfunctiondividedbyitsmodulus(butwith
b 1 1
×
7
the particle-particle factors completely removed) as for the trial function above, there is no net chargeac-
cumulatedat the vortex,and the net phase for exchange
(z w )
i− l (1.26) of two vortices is zero (in the borderline case of 1/r2 in-
z w
Yi,l | i− l| teraction, a non-zero result is possible).
In this paper we will consider BCS paired states of
Unlike the Laughlin states, in such a condensate, while
fermions in addition to QH states. In these, the vor-
the averagecharge(or number)density is uniform, there
tices carry vorticity in multiples of half the usual unit,
are large (Poissonian) fluctuations in the density, or in
due to the pairing. If the fermions are not coupled to
thenumberinasubregion. Thisisconnectedwiththein-
any gauge field (either the electromagnetic field, or the
finite compressibilityofthis BECinnon-interactingpar-
Chern-Simons field that arisesin composite particle the-
ticles. One may wonder if the vortices possess fractional
ory[23]),thenthefermionwavefunctionsmustbesingle-
statistics. It is simple to perform the adiabatic calcula-
valuedeveninthepresenceofvortices,whilethelocalgap
tion, using expressions similar to those in Arovas et al.
function (or condensate wavefunction, or pairing func-
[9]. There is a part of the holonomy phase factor re-
tion)mustwindinphaseby2πonmakingcircuitaround
lated to the expectation of the charge density times the
aminimum-vorticityvortex. Thechargesectorcontribu-
areaofthe loopenclosedbythe path, whichis relatedto
tion to the adiabatic statistics is expected to come from
the Magnus force on the vortex. When vortices are ex-
viewing the system as made of composite pairs of par-
changed,thereisacorrectionduetothechargedeficiency
ticles, and the pairs behave as bosons, similar to the
at the vortex. However,for this trial function, the latter
BEC. The point we wish to emphasize [4] is that, like
is clearly zero, as the vortices disappear from the den-
the BEC wavefunction above, the “nice” trial wavefunc-
sitycalculationwhenthemod-squareistaken. Thus,the
tions that will be considered here do not include the re-
vorticesarebosons. The screeningeffectof the Coulomb
sult of the self-consistent calculation of the gap. That
plasma that arose on taking the mod-square,which pro-
is, the gap function for the pairing should be calculated
ducedanetdeficitofchargearoundeachquasihole,which
by solving self-consistency conditions from BCS mean-
was so important in the discussion of the QH functions,
fieldtheorythatincorporatesthepresenceofthevortices.
is simply absent here.
This is difficult, and in general these details should not
However, for a BEC, the trial wavefunction above is
be relevant to the topological properties of a topological
not very physical. For one, it has an unpleasant sin-
phase. The result of such a calculation should be simi-
gularity at the locations of the vortices. Even for non-
lar to the hydrodynamic or Gross-Pitaevskii calculation
interacting particles, it does not solve the Schr¨odinger
for the Bose superfluid, and (for short range interaction
equation (a function that does is (z w ), which
i,l i − l of the fermions) the density deficiency at the vortices,
is not normalizable; the situation iQs better for trapped
and consequently the charge sector contribution to the
atoms in e.g. a harmonic potential, when the LLL can
adiabatic statistics, will not be well defined. Again, the
again be used). Further, in an interacting Bose super-
neutralpaired fermionsuperfluid is a gaplessphase with
fluid, the circulation of the fluid around the vortex pro-
a Goldstone mode, if there is no long-range interaction.
duces a “centrifugal” force effect, and due to the finite
Thus, while there may be a non-Abelian contribution to
compressibility of the fluid there is a long-range tail in
statistics from the paired wavefunctions, as discussed in
the deficiency ofdensity comparedwith the background,
going as ξ2/r2, where r is the distance from a vor- Ref. [4] and the present paper, the Abelian contribution
∼ is not well-defined; in this sense, these systems are not
tex, and ξ is the healing length (this result may also be
in a topological phase. The same applies to other gap-
obtained from a Gross-Pitaevskii equation analysis—see
less degrees of freedom, for example when spin-rotation
Ref. [21]—and shows that the simple form above is not
symmetry is broken. This should be kept in mind when
evenvalidasympticallyoutsidethevortices). Aspointed
considering the use of non-QH paired systems for topo-
out by Haldane and Wu [22], this leads to a charge
logical quantum computation, such as the half-flux vor-
deficit in a circle of radius R centered at the vortex that
ticesinHe3[24]. Thesewillmostlikelyonlybesuccessful
increases logarithmically with R. Thus, the fractional
if the Abelian phases drop out of computations. (Even
statistics phase is not path independent; it depends on
in QH systems, the Aharonov-Bohm phase is an incon-
the separation of the vortices. (Compared with the QH
venience.)
case,italsodependsdifferentlyonthesignsofthevortic-
ities,sincethedeficiencyofparticlenumberatavortexis For interactions falling slower than 1/r2, with a neu-
independent of the sign of its vorticity; the above wave- tralizingbackgroundadded,thecontributiontoadiabatic
function represents vortices all with positive vorticity.) statistics from the charge sector is well-defined, but is
Clearly,thisisconnectedwithbeinginaphaseofmatter zero. Thus for interacting electrons with 1/r Coulomb
that is not “topological”,due to the existence of gapless interactions, only the effects other than the charge sec-
Goldstone (density) modes of the superfluid, as required tor will be left. These can be calculated using the trial
bythebrokensymmetry. Ref.[22]alsopointedoutthatif wavefunctions, if there are no other Goldstone modes in
the interactionbetweenthe particles fallsoffslowerthan the system apart from the charge mode.
1/r2 (in which case a neutralizing background potential Returning to trialpaired states in the presence of vor-
will be required), then the fluid exhibits screening, and tices but without the self-consistent calculation of the
8
gapfunction,onecanmakeasingulargaugetransforma- which amounts to using the originalgauge. Then we ex-
tion as a function of the fermion coordinates that turns pect still to obtain the charge sector contribution that
the trial state into one with a gap function that does is the area enclosed by the loop (because the particle
not wind in phase, with the many-particle wavefunction density is uniform), in addition to the contribution of
for the fermions changing in sign on making a circuit theconformalblocks. Anothersimplesolution,whichwe
around the vortex. If all the vortices have positive vor- adopthere,isthatwhentheexchangeofvorticesismade
ticity,the transformationismultiplicationbythe inverse alongapathC thatiscontractibletoalimitpointonthe
square root of the above BEC trial function. Then for intersection of some diagonals, then we may define it as
thissecondgaugechoice,itwasshowninRef.[4]thatthe madealongadifferentpath,homotopictotheoriginalin
long-distancebehaviorofa p+ip pairedstate ofspinless the vortex (or quasihole) configuration space, that does
or spin-polarizedfermions is the same as that of the MR notencloseanyparticles. Thisisacceptableforthemon-
trialstate,asgivenbytheIsingconformalblock(orPfaf- odromyas it does not requirewell-separatedvortices (or
fian) above. More generally, we might view conformal quasiholes). This approach is not available in the case
blocks, now without the charge sector factor Ψ , as of exchange by non-contractible paths, for example on a
charge
trial wavefunctions for Abelian anyons in zero magnetic surfaceofnon-trivialtopology,butwewillnotenterinto
field. Theserepresentsuperfluidstatesofanyons,andwe this in this paper.
will include this possibility in the following discussion. The preceding applies to fermions that either are not
We should point out that this relation of QH wavefunc- coupledtoagaugefield,orarebutthepenetrationdepth
tions to those for particles (possibly of different, though forthegaugefieldislarge. Ifthepenetrationdepthisin-
stillAbelian,statistics)inzeromagneticfieldisprecisely steadsmall,andweconsiderexchangeofvorticesatsep-
the ideaof“compositeparticles”[23],herespecializedto arationslarger than the penetration depth, then there is
composite particles in zero net magnetic field. Later we a circulating pure-gauge vector potential outside a pen-
will argue when performing the calculations of overlaps etration depth from the vortices, and the gap function
for the QH trial functions that the charge part may as is covariantly constant. This corresponds to the use of
well be removed, implying that the results also have ap- the second gauge choice above. The calculation of adi-
plication to particle systems in zero magnetic field, con- abatic statistics may be made well defined by the gauge
sistent with the composite particle point of view. How- transformation technique as described above.
ever, as we have just seen for the boson case (which is
relatedtocompositebosons[23]),inpracticetherecanbe
importantdifferencesinthebehaviorinthechargesector
E. Earlier work
that distinguishes these two types of physical systems.
Forsuchtrialfunctions(conformalblocks)thatarenot There have been various earlier steps towards demon-
single-valued in the particle coordinates, some technical strating that non-Abelian adiabatic statistics occurs in
issues must be dealt with in order to discuss the adia- trial QH wavefunctions based on conformal blocks, and
batic statistics of the vortices. The monodromy of the in certain BCS paired states. The idea that in the MR
functions is well-defined if one keeps track of all the zs wavefunction, the holonomy equals the monodromy was
and ws. But for our purposes we wish to move w only, re-emphasized (though not using this terminology) by
withz’sfixed,andtheresultdependsontheprecisepath NayakandWilczek(NW)[25],whoalsoemphasizedthat
taken relative to the z’s, due to the square roots in the this generalizes screening in the plasma mapping for the
trial function (this generalizes directly to particles that Laughlin states. They also found explicitly the two con-
areAbeliananyons). As wewishtocomparewithholon- formal blocks corresponding to any even number N of
omy calculated by integrating out the particles in each particles and n = 4 quasiholes. Even though they did
infinitesimal time step, this dependence on particle po- notfindtheseintheformofMajoranafermionzeromode
sitions is unacceptable, even though it leads only to an states on the quasiholes (which was found at around the
ambiguity in sign in the present case (more generally, same time in Ref. [26]), they guessed that this interpre-
to some root of unity). This effect was not present for tation was correct for any number of quasiholes. This
the QH trial wavefunctions above, which included the ledthemtoconjecturetheformofthebraidgrouprepre-
charge sector and were single valued in the particle co- sentation in the monodromy, which apart from Abelian
ordinates (and the particles were bosons or fermions). factors (i.e. tensor product with an Abelian representa-
[We note that the Berry connection is well-defined even tionofthebraidgroup)canbeviewedasanimageofthe
for the non-single valued wavefunctions, because the de- braid group in the spinor representation of the rotation
pendence on the particle coordinates is just a square (or group in n dimensions [25]. This representation of the
other) root, and the phase change cancels in the overlap braid group was known [27], and was also known to oc-
(thus relies on the particles obeying Abelian statistics).] cur in the Ising CFT (its structure is described in more
Onesolutionwhenthechargesectorisremovedfromthe detailinRef.[28]). Asimilarargumentwasspelledoutin
QHwavefunctionstoobtaintheblocksconsideredhereis greaterdetailinRef.[24],aftertheworkofRef.[4]onthe
to retain the gauge-transformation phase factor so that Majorana zero modes in p+ip paired states. However,
thefunctionsaresinglevaluedintheparticlecoordinates, the argument gives only the monodromy of the states
9
(modulo Abelian factors), and it is not clear if adiabatic others,forexamples,Refs.[29,30]. Theresultsforquasi-
transport is actually considered in Ref. [24] (no expres- holes rest on an assumption, that screening occurs in a
sion for the Berry connection appears there). Another certain very conventional two-component plasma, which
argument of NW is somewhat similar to the one we will willbeacceptedbymostphysicists. Theargumentsgiven
give in Sec. V below. inSec.Vapplytoanytrialstategivenintermsofconfor-
Adiabatic transport of quasiholes or vortices in the mal blocks as explained here, and show that holonomy
pairedstatewasconsideredfurthermorerecentlyinRef. equals monodromy under some general conditions that
[29], and especially clearly in Ref. [30] (Appendix A), can be checked for each particular trial state as a well-
where it is shown that the Berry connection is propor- posed physical question in two-dimensional field theory.
tionaltotheidentitymatrix,thusprovingthattheholon- Further, there is a simple easily-checked criterion (rele-
omy is given by the monodromy found by NW, up to vance or irrelevance of a perturbation)that may provide
some Abelian factor. important clues as to whether or not the general condi-
tions hold.
OtherapproachestotheproblemfortheMRQHstate
should also be mentioned. Gurarie and Nayak [31] used Now we turn to earlier work on adiabatic variation
anotherCoulombgasmethodfromCFTtorepresentthe of the aspect ratio of a QH system on a torus. In the
overlap integrals. For the case of only two quasiholes, complex plane, the torus is defined by identifying points
they succeeded in obtaining the vanishing of the Berry under z z + L and z z + Lτ, where τ is in the
→ →
connection,andhencethattheholonomyequalsthemon- upper half plane Imτ > 0; thus the upper half plane is
odromy in this case. For four quasiholes their result de- the parameter space on which we may study adiabatic
pended on some assumptions, the validity of which does transport. This was considered for the filled LLL in an
not appear to be obvious. Other groups [32, 33] for- elegant paper by Avron et al. [6]. They showed that
mulated field-theoretic arguments, but seem to assume there is a contribution that is not holonomy, but curva-
that the edge theory is the expected CFT. Tserkovnyak ture (anholonomy orfield strength) of the Berryconnec-
and Simon [34] evaluated the holonomy numerically for tion (vector potential) on the upper half plane. (This is
two and four quasiholes by Monte Carlo methods, find- somewhat analogous to the Aharonov-Bohm phase that
ing agreement with the expected result, at some degree is proportional to the area enclosed by a loop, in the
of accuracy. Arovas et al. [9] calculation for moving one quasihole.)
For most other states, such as those of Read-Rezayi For the case of the filled LLL, the result is proportional
[20], much less has been shown. But there is a series of to the total flux through the torus [6]. Physically, for
spin-singlettrialstatesduetoBlokandWen[35]forpar- a homogeneous many-particle fluid state, this adiabatic
ticlesofSU(2)spink/2(k =1,2,...;k =1isHalperin’s curvaturedividedby the systemarearepresentstheHall
Abelian spin-singlet state, see e.g. Ref. [1]), in which the viscosity, here denoted η(A), a non-dissipative transport
CFT is SU(2) level k, which have many nice properties. coefficient [38, 39] that is known in plasma physics but
These authors were able to show that the Berry con- often overlookedelsewhere in fluid dynamics; η(A) is the
nection vanishes, and so holonomy equals monodromy, only coefficient of viscosity that can be non-zero in a
for these states by using the Knizhnik-Zamolodchikov two-dimensionalisotropicincompressiblefluid(butmust
equation from CFT [17], and making an assumption vanish if time-reversal symmetry is present) [6]. The re-
that screening holds in an SU(2) generalization of the sult for the filled LLL can also be extracted from the
Coulombplasma,inamannercloselyparalleltothework detailed single-particle results of L´evay [40]; this paper
of Arovas et al. [9]. The screening assumption implies clarifies many aspects of this problem. More generally,
that the trial ground state has short-range spin correla- forthestateinwhichthelowestν Landaulevelsarefilled
tions. We will comment on this further in Sec. V below. (ν integer), the result η(A) = h¯νn/4 is quoted for inte-
Forrecentfurtherresultsonmonodromyofblocksinthe ger ν, ν 1 in Ref. [38], again using results from Ref.
RR series, see Refs. [36, 37]. [40] (n|is|t≥he particle density n = ν /(2πℓ2); h¯ and the
| | B
magnetic length ℓ are set to 1 elsewhere in this paper).
Weconcludefromthissurveythatwithfewexceptions B
[Note that following comments from the authors of Ref.
existing results in the literature are either only partial
[7], we have corrected the coefficient to 1/4 to account
ones for the MR state (not calculating the Abelian fac-
for apparent typos in Ref. [38].]
tors), or else depend on unproven assumptions, or apply
only to particular states. By contrast, the results pre- A recent paper [7] has argued that the result of Ref.
sented below for the MR state are complete in that they [6] generalizes to an arbitrary QH state in the LLL to
yield the full holonomy for up to four quasiholes, or for give η(A) = h¯ν/(8πℓ2) for arbitrary (possibly negative)
B
thegroundstatesonasurfaceofanygenus. Anargument values of ν, independent of the state (this paper [7], and
that we invoke frequently, which is supported by renor- references therein, use the Hall viscosity in an interest-
malization group arguments herein, is that the charge ing hydrodynamic approach to collective modes in QH
sector factor of the QH wavefunctions can be dropped fluids). Unfortunately, the end of the calculation uses
withoutjeopardizingtheresultsforholonomyintheCFT the incorrect argument that all particles are close to the
sector;this isrelatedto conventionalloreaboutcompos- x-axis. The single-particle results of L´evay [40] can be
iteparticlemethods,andhasofcoursealsobeenusedby applied to obtain a seemingly similar result. He finds,
10
fortheBerrycalculationintheN -folddegeneratespace pairing phases. In Sec. III, we examine similar questions
φ
ofsingle-particlestatesinasingleLandaulevel(N isthe for the Laughlin QH states, relate the Hall viscosity for
φ
number of flux piercing the torus), that the Berry con- trial states given by conformal blocks to the conformal
nectionanditscurvatureareproportionaltotheidentity weightofthe fieldforthe particle,andconcludethe gen-
matrix in this space, with a coefficient related to the ki- eral discussion for Hall viscosity. In Sec. IV, we present
netic energy of the Landau level. This implies (e.g. by directargumentsforthe non-Abelianadiabaticstatistics
using the Slater determinant basis) that for the space of oftwoandfourquasiholesinthe MRstateonthe sphere
all many-particle states with all the particles in a sin- or plane in the thermodynamic limit. The calculations
gleLandaulevel,theBerryconnectionandcurvatureare work by “doubling” (taking two copies) of the system
again proportional to the identity matrix, and so in this with the charge part removed, and using an argument
senseareindependentofthe state. (This generalizesfur- that a plasma is in a screening phase. In Sec. V, we
ther if we consider a space consisting of all the many- presentgeneralargumentsthatamountto necessaryand
particle states with a given number of particles in each sufficientconditionsfortrialwavefunctionsgivenbycon-
Landau level.) However, this adiabatic transport is not formal blocks to describe a topological phase with adi-
what should be considered for a quantum fluid state, in abatic statistics given by the monodromy of the blocks.
which because of the presence of interactions we take a Theconditionisthatrelatedcorrelationfunctions intwo
family consisting of a single ground state (as considered dimensionsshouldgotoaconstant,asinaorderedphase.
inRef.[7])foreachvalueofτ,ormoregenerallyaspaceof We discuss numerous examples in this light. Cases not
“degenerate”states for eachτ, but generallynot the full obeying the condition arearguedto be gaplessphases or
spaceofmany-particlesingleLandaulevelstates. Thisis critical points. We argue that use of non-unitary ratio-
anexampleofthe generalset-upforadiabatictransport, nal CFTs in our way cannot produce such a topological
in which a family of subspaces within a common Hilbert phase if there are any negative quantum dimensions in
space is considered, as discussed above in Sec. IB. (The the theory (which follows if there are any negative con-
τ-dependent Landau level states themselves arise in this formalweights). Theargumentassumesthatthetwistin
way,asallaresubspacesofthespaceofsquare-integrable the theory, defined adiabatically, is also the same as in
functions on the torus with given boundary conditions.) theCFT,whichhasnotbeenshown. InanAppendix we
The curvature of the Berry connection definitely does discuss this, which is the last step in deriving a modular
in general depend on the choice of a subspace for each tensorcategoryfromtheconstruction,andshowthatthe
value of τ (though not on the gauge, that is the basis consistent twists are very limited, so that the argument
for the subspace), and cannot be reduced to a calcula- does go through in at least one family of examples.
tion that ignores restriction to the subspace. Hence the
resultthatisindependentofthestatechosen,asclaimed
inRef. [7], cannotbe asgeneralas wasstated. Later,we
will show that the correct result depends on the form of
the ground state, not only on the density, though it is
universal within a topological phase. (We will also find
II. PAIRED STATES IN A CLOSED FINITE
related results for paired superfluids. We are not aware
SYSTEM
of any earlier results for Hall viscosity of paired states.)
FromL´evay[40],theresultofRef.[7]canbeviewedas
the correct one for non-interacting particles in the LLL.
In this section, we consider the p+ip pairedstates on
Also, using L´evay’s results for non-interacting particles
compact two-dimensional surfaces (no boundary). The
but at non-zero temperature T, and going to high tem-
set-up was already described in Ref. [4] (see also Ref.
perature using classical equipartition of energy, we find
[30]), but we review some essential steps, and add a few
that the result agrees with that in Ref. [39], which is
points. The basic case, other than the sphere, is the
η(A) =k Tn¯/(2ω ),whereω isthe cyclotronfrequency.
B c c torus, or equivalently periodic boundary conditions on
a parallelogram in the plane. The other cases, surfaces
of genus higher than one, require more work to set up,
F. Structure of paper
and we will be more brief. For all these problems, we
canshowstartingfromthegeneralpairingproblem,that
InSec.II,wefirstreviewtheessentialsofpairedstates at long wavelengths the orthonormalized wavefunctions
at the BCS mean-field level, then specialize to ground (within the BCS mean-field formulation) are the confor-
states on the torus. We calculate the normalization mal blocks of the Majorana fermion (Ising) CFT. This
factors, then examine the monodromy under modular allows us to calculate explicitly the adiabatic transport
transformations, and then the curvature and holonomy ofthestatesastheaspectratioofthetorusisvaried;this
of the Berry connection for changes in the aspect ratio leads to the Hall viscosity and the modular transforma-
τ; this determines the Hall viscosity of BCS paired sys- tion group. We also consider the generalization to other
tems, for which we also give a simple direct calculation. paired phases of fermions, including the strong-pairing
We also discuss higher genus surfaces, and the strong- phases.