Boson Condensation in Topologically Ordered Quantum Liquids

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by  Titus Neupert

Boson Condensation in Topologically Ordered Quantum Liquids
Titus Neupert,1 Huan He,2 Curt von Keyserlingk,1 Germ´an Sierra,3 and B. Andrei Bernevig2
1 Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
2 Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
3 Instituto de F´ısica Teo´rica, UAM-CSIC, Madrid, Spain
(Dated: January 8, 2016)
Boson condensation in topological quantum field theories (TQFT) has been previously investi-
gated through the formalism of Frobenius algebras and the use of vertex lifting coefficients. While
general, this formalism is physically opaque and computationally arduous: analyses of TQFT con-
densationarepracticallyperformedonacasebycasebasisandforverysimpletheoriesonly,mostly
not using the Frobenius algebra formalism. In this paper we provide a new way of treating bo-
son condensation that is computationally efficient. With a minimal set of physical assumptions,
6
such as commutativity of lifting and the definition of confined particles, we can prove a number of
1
theoremslinkingBosoncondensationinTQFTwithchiralalgebraextensions,andwiththefactor-
0
2 ization of completely positive matrices over Z+. We present numerically efficient ways of obtaining
a condensed theory fusion algebra and S matrices; and we then use our formalism to prove sev-
n
eral theorems for the S and T matrices of simple current condensation and of theories which upon
a
condensation result in a low number of confined particles. We also show that our formalism easily
J
reproduces results existent in the mathematical literature such as the noncondensability of 5 and
6
10 layers of the Fibonacci TQFT.

]
l
e I. INTRODUCTION ternal defects49–56.
-
r The universal aspects of topologically ordered phases
st In two spatial dimensions the Pauli principle general- are captured by topological quantum field theories57.
t. izes, allowing for anyonic particles with quantum statis- Among these, the axiomatic approach of category the-
ma tics different from that of fermions and bosons1. While ory58–62,moreconcretelytheformulationofmodularten-
such anyons do not appear as free particles in nature, sor categories (MTCs), is particularly powerful and, to
d- they can occur as emergent excitations in quasi two- our knowledge, provides a complete characterization of
n dimensionalfractionalquantumHallsystems2,3,andthe- topological order in two-dimensional space.4,15 At a ba-
o oretically in other states of quantum matter, such as the siclevel,MTC’sarecharacterizedbythetypesofanyons
c toric code and its generalizations4,5. A state with any- thatappearinthephaseaswellastheirinterrelationsin
[ onic excitations is called topologically ordered6–12 and theformoffusionandbraidinginformation,theso-called
1 represents a new paradigm in condensed matter physics “F moves” and “R moves”.
v with far reaching potential technological applications in In correspondence with the different descriptions of
0 quantum computation14. topological order itself, several formulations of anyon
2
Prior to the discovery of topological order, it was well condensation were developed. In the context of MTCs,
3
1 known that bosons can macroscopically occupy a sin- the phase after condensation is found by studying com-
0 gle quantum state, a fact which allows for the possibil- mutative separable Frobenius algebras63–67 of the ini-
. ity of a Bose-Einstein condensation phase transition. In tialtheory.68,69 BaisandSlingerlandtranslatedthispro-
1 a topologically ordered phase, bosons are more compli- cedure into the language of anyon models16, but their
0
6 cated particles: They can have nontrivial braiding be- formulation did not give a systematic method for de-
1 havior with other anyons4,15, and even more exotically termining properties of the phase after condensation.
: they can carry nonlocal internal degrees of freedom4, in This was later achieved by Eli¨ens et al. in Ref. 17 via
v
i whichcasetheyarecallednon-Abelianbosons. Notwith- a diagrammatic formulation of the condensation prob-
X standing, such bosons can sometimes condense16–27. It lem that makes use of the so-called vertex lifting coef-
r is then natural to ask how this condensation affects the ficients. These allowed them to embed the fusion and
a topological order, namely, what is the fate of the other braiding processes of the condensed phase in the initial
anyons in the phase. The answer is that anyon conden- anyon model. However, all these approaches fall short of
sationinducestransitionsbetweendifferenttopologically providinganalgorithmicformulationofbosoncondensa-
orderedphasesinsuchawaythatuniversalpropertiesof tion in a way that could, for example, be implemented
the anyons of the condensed phase can be inferred from in a computer algebra program allowing for systematic
those of the initial phase, together with a list of con- studies of possible condensations.
densed bosons. This framework of anyon condensation In this paper, we reformulate the problem of boson
transitionsfoundmanyapplicationsinthestudyoftopo- condensation in anyon models axiomatically and purely
logicalorder,28–33 inparticularinsolvingthequestionof algebraically. Theresultingformalismisbasedonasmall
bulk-boundary correspondence34–39, or in deducing the number of natural assumptions such as the commutativ-
universal properties of domain walls40–48 and other ex- ity of fusion and condensation as well as an assumption
2
aboutthetopologicalspinsoftheanyonsaftercondensa- rounding condensate. Projecting out these confined ex-
tion. Our approach puts the modular matrices S and T citations, we are left with a deconfined condensed MTC
of the initial anyon model center stage, instead of focus- that we denote as U. Our goal is to find possible MTCs
ing on the F and R moves, which are the key objects of U givenAandsomebasicinformationabouttheconden-
interest for the diagrammatic approach.17 The F and R sate, such as which bosons condense.
movesareingeneralnotoriouslyhardtocomputeevenfor Inwhatfollows,wewillconsidertheBosecondensation
relativelysimpletheories. Ourgoalistofindthemodular of a collection of bosons in the original theory A. This
matrices S˜ and T˜ of the final theory after condensation. collectionofanyonsiscalledthecondensateandbecomes
Usingouralgebraicformulation,weproposeanalgorithm part of the vacuum in the new intermediate fusion cat-
that determines all possible condensation instabilities of egory T. In condensing these bosons, a generic anyon
an anyon model and can be efficiently implemented on a ∈ A will become (or “restrict to”) a superposition of
a computer. We solve for the condensation via a series particles t∈T
of linear algebra problems, involving the factorization of
a(cid:55)−→r ntt, ∀a∈A (1)
nonnegative integer matrices. a
Besides its utility for computer-aided calculations, our t(cid:88)∈T
algebraic formulation of condensation also facilitates an- with the coefficients nt ∈ Z , where we assume that
a ≥0
alytical derivations. As an example, we discuss layer nt =nt¯ andbarsdenoteantiparticles(seeAppendixA).
a a¯
constructions of topologically ordered states and easily
Equation (1) defines the “restriction map”. We will also
reproduce the known result that 5 and 10 layers of the usethephrase“arestrictsto ntt”todescribeEq.(1).
t a
Fibonacci anyon model cannot undergo a condensation
It is possible that more than one particle t appears on
transition. (cid:80)
the righthand side of Eq. (1), in which case we say that
Thispaperisstructuredasfollows. InSec.IIweformu- “a splits into ntt”. Condensed particles (bosons b
t a
late the condensation problem along with the axioms re- in the condensate) have the additional special property
latingtofusionrules. InthefollowingSec.III,wepresent that nϕ (cid:54)= 0, w(cid:80)here ϕ is the vacuum particle in T, that
b
the assumptions that allow us to deduce the braiding is, their restriction contains the identity of the new T
properties of the theory after the condensation transi- theory. If nϕ (cid:54)= 0, then nϕ = nϕ, i.e., both the boson
tion, and several implications are derived. In Sec. IV, and its antipbarticle must cobndens¯be at the same time.
we derive central equations which constrain S˜ and T˜. The reverse (or, more precisely, adjoint) operation to
Subsequently, we show in Secs. V and VI that a weaker restrictioniscalled“lifting”. Foraparticlet∈T,allthe
set of axioms suffices, if the condensate consists of so- particles in A which restrict to t are defined to be the
called simple currents and if only one particle is con- lifts of t. The lifting coefficients are the same nt that
a
fined through the condensation transition, respectively. we used in defining the restriction. Formally, lifting is
We formulate an algorithm for solving the condensation defined by
problem in Sec. VII. The final Sec. VIII gives examples
of condensation transitions in multi-layered anyon mod- t→(cid:55)−l nta, ∀t∈A. (2)
a
elsanddiscussesobstructionstobosoncondensationin5 a∈A
(cid:88)
and 10 layers with Fibonacci anyons. We have included
Finally, we define particles in T whose lifts do not share
eight appendices containing brief summaries of MTCs
a common topological spin θ as confined, that is
(Appendix A) and chiral algebras (Appendix C), math- a
ematical proofs of the results explained in the main text t:confined ⇔ ∃a,bsuchthatntnt (cid:54)=0withθ (cid:54)=θ .(3)
a b a b
(Appendices B, E, F) and further examples of condensa-
Conversely, the deconfined particles in T are the parti-
tion (Appendices D, G, H).
cles whose liftings do share a common topological spin,
which becomes identified with the spin of the deconfined
particle, that is
II. DEFINITIONS AND ASSUMPTIONS
t:deconfined ⇔ ∀a,bsuchthatntnt (cid:54)=0thenθ =θ .(4)
a b a b
In this section we present the formalism underpinning Obviously,anyparticlet∈T iseitherdeconfined(t∈U)
anyon condensation, following Refs. 16 and 17 closely. or confined (t ∈ T/U). With these definitions in place,
Our discussion is self-contained with respect to the pre- we now make a fundamental assumption from which we
viousliteratureonanyoncondensation,butassumesthat willderivethestructureofthetheoryaftercondensation.
the reader is familiar with the basic concepts of MTCs5 We assume that the restriction A → T commutes with
(see Appendix A for a short review). fusion. This is represented by the diagram
The input for our approach to anyon condensation is
a MTC A (the uncondensed theory), and a set of re- A⊗A f (cid:47)(cid:47)A
striction and lifting coefficients, which relate the particle
excitations in A to those in T (the condensed theory). r⊗r r
(cid:15)(cid:15) (cid:15)(cid:15)
In general, T is only a fusion category, because it may (cid:47)(cid:47)
T ⊗T T
contain some excitations which are confined by the sur- f
3
in which f represents fusion and r represents restriction. whereaparticletisbraidedarounditself. Thisprocessis
Moreexplicitly,thecommutingdiagramcanbewritten equivalenttobraidingtheliftsaoftheparticlet(namely
in terms of anyon basis nt (cid:54)= 0). Each of these braidings is given by the phase
a
θ , while the loop with particle a is equal to the quan-
nransbN˜rts = Nacbntc, (5) tuam dimension da. The result we obtain is the quantity
r,s∈T c∈A β which we assume vanishes when t ∈ T/U as confined
(cid:88) (cid:88) t
particles cannot form a braided category. While in gen-
whereNc andN˜t arethefusioncoefficientsinAandT,
ab rs eral we assume Eq. (10) in addition to the statements in
respectively. This elementary constraint is surprisingly
Sec. II, we note that it can be derived from the assump-
restrictive. For instance, it immediately leads us to two
tions underpinning the approaches in Kirillov-Ostrik68,
constraintsonthequantumdimensionsofparticlesinthe
Kong69 and Eli¨ens et al.17 In certain special cases we
A and T theories (see Appendix B)
can show that Eq. (10) follows from the assumptions: in
d = nrd , ∀ a∈A, (6a) Sec. II, e.g., we do so for the so-called simple current
a a r condensates (see Sec. V).
r∈T
(cid:88) With these additional assumptions in place, we define
1
d = ntd , ∀ t∈T, (6b) some useful quantities. The vacuum component t=ϕ of
t q a a
Eq. (5) will be a central object in our analysis, the left
a∈A
(cid:88)
hand side of which reads:
where q := nϕd . Diagrammatically, Eq. (6b) is
a a a
(cid:80) 1 Ma(cid:48)c := ntantc = Nacbnϕb , (12a)
t = q nta a (7) t(cid:88)∈T (cid:32) b(cid:88)∈A (cid:33)
a
X as will be
It will also be useful to define the quantity
M := ntnt. (12b)
ac a c
βt := θadanta, (8) (cid:88)t∈U
a(cid:88)∈A Notice how the two above definitions of the matrices M(cid:48)
whereθaisthetopologicalspinofa∈A. Givenaparticle and M with nonnegative integer entries differ subtly but
t ∈ U, it follows from the aforementioned definition of a crucially: The expression for M(cid:48) involves a summation
deconfined particle Eq.(4) over T while that for M involves a summation over the
deconfined condensed theory U.
β =qd θ , ∀t∈U, (9)
t t t 2 The matrices, which can be factorized as in Eq. (12a)
as a useful corollary to Eq. (6b). and (12b), are called completely positive matrices over
the ring of positive integers. We will discuss completely
positive matrix factorization later in the paper. In the
III. THE CONDENSATION MATRIX M following sections, we will demonstrate two important
ab
properties of M, namely [M,S] = [M,T] = 0, where
S and T are modular matrices of the A theory. For a
So far, our formalism does not differ appreciably from
discussion of the role of the matrix M in CFTs, we refer
thatofRefs.16and17. However, inwhatfollowsweopt
the reader to Appendix C.
tonotintroducetheso-called“vertexliftingcoefficients”
on which the approach of 17 is based. Instead, we find
that we can extract a surprising amount of information
A. Proof that M commutes with T matrix of the A
fromsupplementingthealgebraicrelationsinSec.IIwith
theory
two additional assumptions. First, by assumption, we
are only interested in cases where U is a TQFT, so that
its anyons form a braided fusion category. Second, we In the following, we will prove that the M matrix we
assume that defined in Eq. (12b) commutes with the modular T ma-
trix of the A theory. The T matrix of the A theory is
βt =0, ∀t∈T/U, (10) Tab =θaδab. Note that
where t ∈ T/U runs over all confined anyons. To moti-
[M,T] = (M T −T M )
ac ab bc ab bc
vatethisequationletuspictoriallyrepresentthelefthand
b∈A
side of Eq. (10) as (cid:88)
=M θ −M θ (13)
ac c ac a
1
tt= q nta a = ntantc(θc−θa).
Xa (11) (cid:88)t∈U
1
= nt✓ a Sincet∈U,thespinsofalltheliftsoftinAarethesame,
q a a hence θ = θ and each term in the final line vanishes
a a c
X
4
identically. It follows that We start from the expression of the S matrix for a
braided fusion category A (e.g., see Ref. 70)
[M,T]=0. (14)
Note that this is not valid if the sum in the last line of
Eq.(13)wasnotrestrictedtotheU theory,i.e.,[M(cid:48),T](cid:54)=
0, with M(cid:48) defined in Eq. (12a).
1 θ
S = Nx x d , (16)
cb D c¯bθ θ x
A c b
x∈A
(cid:88)
B. Proof that M commutes with S matrix of the A
theory
In this section, we will prove that the M matrix com-
mutes with the modular S matrix of A theory whereDAisthetotalquantumdimensionoftheAtheory
and ¯b denotes the antiparticle of b. From the definition
[M,S]=0. (15) of M, we express the commutator [M,S] as
1
[M,S] = θ d ntntNx −ntntNx
ab D θ θ x x a c c¯b b c ac¯
A a b
c,x∈At∈U
(cid:88) (cid:88) (cid:0) (cid:1)
1
= D θ θ θxdx nta ntcNcx¯b −ntb ntcNaxc¯ (17)
A a b (cid:34) (cid:32) (cid:33) (cid:32) (cid:33)(cid:35)
x∈At∈U c∈A c∈A
(cid:88)(cid:88) (cid:88) (cid:88)
1
= θ d nt ntNc −nt ntNc .
D θ θ x x a c bx b c ax¯
A a b (cid:34) (cid:32) (cid:33) (cid:32) (cid:33)(cid:35)
x∈At∈U c∈A c∈A
(cid:88)(cid:88) (cid:88) (cid:88)
Inthefirstlinewehaveusedthatifntnt (cid:54)=0witht∈U, thenθ =θ andifntnt (cid:54)=0witht∈U, thenθ =θ which
a c c a c b c b
yields the term θ θ in the denominator. To obtain the last line, we have used the equalities Nc = N¯b = Nc¯ and
a b ab ac¯ a¯¯b
Nc =Nc . We now use Eq. (5) to replace the terms in the round brackets and find
ab ba
1
[M,S] = θ d ns ntnrN˜t −ntnrN˜t , (18)
ab D θ θ x x x a b rs b a rs¯
A a b
s(cid:88)∈T x(cid:88)∈A t∈U(cid:88);r∈T (cid:16) (cid:17)
wherewehaveusedtheequalityns =ns¯ (theassumption These equations are essential to the theory of conden-
x x¯
thattherestrictionofx’santiparticlex¯istheantiparticle sation, as they establish that the condensation matrix
of the restriction of x) to transfer the antiparticle on M is a particular symmetry of the S and T modular
ab
the N˜t . We can now split up the r sum in Eq. (18) matrices. While there exist other such symmetries, for
r,s¯
into a sum over U and a sum over T/U. For the first example automorphisms that are represented by permu-
contribution, we have tation matrices, these matrices are not ‘completely pos-
itive’ integer matrices i.e., they cannot be factorized as
ntnrN˜t −ntnrN˜t = ntnr N˜t −N˜r nnT in terms of a nonnegative integer matrix n.
a b rs b a rs¯ a b rs ts¯
r(cid:88),t∈U(cid:16) (cid:17) r(cid:88),t∈U (cid:16) (cid:17)
(19)
IV. THE MODULAR TENSOR CATEGORY
by exchanging the labels r and t in the second term.
Since N˜r = N˜t¯ = N˜t , Eq. (19) vanishes identically. AFTER CONDENSATION
ts¯ r¯s¯ rs
Thus, r in Eq. (18) can only take values in T/U. By
assumption, U is a closed fusion category. This implies Intheprevioussection,weidentifiedamatrixM which
thatnotrivalentvertexwithasingleleginT/U existsin commutes with the modular matrices S and T of the A
T. As a result, the s-sum in Eq. (18) may only run over theory. Inthissectionweproveastrongerpairofresults,
T/U. However, for s ∈ T/U we can use the assumption namely that
Eq. (10) to find θ d ns = 0 for the remaining
x∈A x x x nS˜ = Sn, (20a)
terms in Eq. (18). We conclude that Eq. (18) vanishes
identically and thu(cid:80)s [S,M]=0. nT˜ = Tn, (20b)
5
whereS˜andT˜ arethemodularmatricesoftheU theory We have thus proven Eq. (20a) and Eq. (20b) within
and n is the matrix of coefficients that enter the restric- our algebraic formulation of the condensation transition.
tion and lifting maps, (n) = nt, ∀ a ∈ A, t ∈ U. Our These two equations have a well known parallel in the
at a
assumptionEq.(10)willbecrucialfortheseproofs. The studyofchiralalgebraextensions,whichwedetailinAp-
second equality Eq. (20b) is the statement that, compo- pendix C. As a side remark, let us derive a consequence
nent by component, whenever nt (cid:54)= 0, θ = θ ; this is of Eq. (22), namely
a t a
true by recalling our definition of deconfined particles of
D >D , (28)
the U (⊂T) theory, Eq. (4). A U
BeforestartingtheproofofthefirstequalityEq.(20a), which follows from the fact that the embedding dimen-
we note the following equalities derived in Appendix E sion q = d nϕ > 1 and q = 1 if no condensa-
a∈A a a
tion is happening. This is always true even if the as-
βtβt∗ =q2DU2 (21a) sumption E(cid:80)q. (10) is not used. If Eq. (10) is used, then
t∈U Eqs. (21a) and (21b) imply D = qD . By analogy
(cid:88) A U
with the Zamolodchikov c-theorem of Ref. 71 one can
and
call this result the D-theorem which can be interpreted
β β∗ =D2. (21b) asthedisappearanceofsomeanyonsuponcondensation.
t t A
There is a stronger connection between this result and
t∈T
(cid:88) the g-theorem, according to which the Affleck-Ludwig
It then follows from assumption Eq. (10), that boundaryentropyofanopenconformalsystemdecreases
β β∗ = β β∗ and thus q2D2 =D2 or undertherenormalizationgroupflowoftheboundaryas
t∈U t t t∈T t t U A
long as the bulk theory remains critical throughout the
(cid:80) (cid:80) q =DA/DU. (22) flow.72–74 (This situation is, however, distinct from the
case of a condensation transition in which the bulk is
To prove Eq. (20a), we multiply Eq. (5) by θ d and
a a only critical at the transition.) The boundary entropy is
sum both sides over a∈A to obtain
inturnrelatedtothequantumdimensionoftheprimary
field that characterizes the boundary condition, which
N˜t nsβ = Nc d θ nt
rs b r ab a a c suggests a relation between the g-theorem and Eq. (28).
r,s∈T a,c∈A
(cid:88) (cid:88) (23)
=θ D S θ nt,
b A bc c c
V. SIMPLE CURRENTS
c∈A
(cid:88)
wherewehaveusedthedefinitionofS fromEq.(16)and
Simplecurrentsareabeliananyonsthat,whenraisedto
that θ =θ . For particles t∈U, we furthermore have
a a¯ acertainpowerbyfusion,equaltheidentity,seeRefs.75–
81. The precise definition follows below. In this section,
N˜t nsβ = N˜t nsqθ d , ∀t∈U, (24)
rs b r rs b r r weconsideracondensatethatiscomposedofsimplecur-
r,s∈T r,s∈U rents only. In this situation, we can prove that Eq. (10),
(cid:88) (cid:88)
i.e., β = 0, ∀t ∈ T/U, follows from the assumptions in
because (i) only r ∈U contributes to the sum (as β =0 t
r Sec. II.
if r ∈ T/U) and (ii) only s ∈ U contributes since U is
closedunderfusionbyassumption. Wefurthermoreused
Eq. (9) to rewrite the righthand side of Eq. (24). Since
A. Introduction to simple currents
we assumed that U forms a braided fusion category with
its S˜ matrix, we use the usual definition of the S˜ matrix
There are several equivalent definitions of simple cur-
to write
rents in the context of rational conformal field theory
N˜t θ d ns = S˜ D θ θ ns. (25) (RCFT).First, asimplecurrentisaprimaryfieldJ that
rs r r b st U t s b has a unique fusion channel with any other primary field
r(cid:88),s∈U(cid:16) (cid:17) s(cid:88)∈U of the RCFT
Since s,t ∈ U, we furthermore have θ = θ if ns (cid:54)= 0
s b b J ×φ=φ(cid:48), ∀φ. (29a)
which allows us to combine Eq. (23) and Eq. (25) into
A second definition is that a simple current is a primary
1D
A S θ nt = S˜ nsθ . (26) fieldJ thatwhenfusedwithitsantiparticleorconjugate
q DU bc c c st b t field J¯only fuses to the identity (see Ref. 77)
c∈A s∈U
(cid:88) (cid:88)
Since for all nt (cid:54)= 0, we have θ = θ (t ∈ U) and using J ×J¯=1. (29b)
c c t
Eq. (22), this expression reduces to A third definition is that the quantum dimension of J is
1,
S nt = S˜ ns. (27)
bc c st b
d =1. (29c)
c∈A s∈U J
(cid:88) (cid:88)
6
One can show that all these definitions are equivalent.76 powers of all the bosonic simple currents, which is some-
GiventwosimplecurrentsJ andJ ,theirfusionprod- times called the bosonic center C of the RCFT. Powers
1 2
uctJ J isalsoasimplecurrent. Thenumberofprimary of a condensed bosonic simple current, or the products
1 2
fields of a RCFT is finite, therefore each simple current of different condensed bosonic simple currents are also
J hasanassociatedintegerN suchthatJN =1byusing bosonic and condensed. To see this, examine the J ,J
i j
Eq.(29a). ThesmallestintegerN >0withthisproperty componentofEq.(5),whereJ ,J areassumedtobecon-
i j
is called the order of J. A simple current J generates a densed. Recalling that J := J ×J is automatically
i+j i j
setofsimplecurrents{Jm|m=0,1,...,N−1},whichis a simple current (see above), and making use of the fact
isomorphic to the abelian group Z . A RCFT may con- that condensed simple currents like J have quantum di-
N i
tainsimplecurrentsgeneratedbymorethatoneprimary mension 1 so that nt =δt, we find
Ji ϕ
field. ThecollectionofallofthemformanAbeliangroup
cwahnicchhoisoisseoma obrapshisicoftositmhpelpercoudrurecnttZsNs1u×ch·t·h·×atZNNr.arOenoef 1=nϕJinϕJj =NJJii,+JjjnϕJi+j =nϕJi+j. (32)
i
tthhee ffuonrmdampniein,tnail t∈heZo,rewmithofpfiinaitepraibmeeliannumgrboeurp.s.This is AsaresultnϕJi+j =1,indicatingthatJi+j restrictssolely
tothevacuum,soitmustbeaboson. Thereforetheprod-
As an example, consider the RCFT constructed from
uct of any two condensed simple current is a condensed
the Kac-Moody algebra SU(2) . The primary fields are
k (hence bosonic) simple current.
denoted by φ where (cid:96) = 0,1,...,k is twice the topo-
(cid:96) As an aside, we note that for general bosonic currents
logical spin. The field φ is a simple current because its
k which are not necessarily condensed: (i) as before, any
fusion rule is φ ×φ =φ . Indeed φ is the only non-
k (cid:96) k−(cid:96) k power of such a simple current is a bosonic simple cur-
trivial simple current, and satisfies φ ×φ = φ = 1.
k k 0 rent;(ii)however,theproductoftwosuchbosonicsimple
The simple currents form a Z ={φ ,φ } subcategory.
2 0 k currentsdoesnothavetobebosonic. Forexample,inthe
Whenactingonaprimaryfieldφ,J generatesanorbit
toriccodethatwewilldiscussindetailinSec.VII,eand
formed by the fields Jnφ
m are bosonic simple currents while their product f is
actually a fermion (which cannot condense). To prove
[φ]={φ,Jφ,J2φ,...,Jd−1φ}, Jdφ=φ. (30)
(i), one can use the symmetry S = S∗ of the S ma-
ab a¯b
Here,disthesmallestpositiveintegersuchthatJdφ=φ. trix and the fact that for any anyon θa = θa¯. Choosing
The orbit Eq. (30) is denoted by a representative field φ a=b=J with J a simple current gives
but one can choose another field belonging to the orbit.
θ θ
In general d need not equal N, the order of the current J2 = 1 . (33)
θ θ θ∗θ∗
J, but d must divide N. In the example of SU(2)k, if k J J J J¯
is odd, all the orbits have two elements, (d = N = 2),
If J is bosonic, then so is J¯ and the above implies J2
while for k even, φ ×φ =φ for the action of φ and
k k k k
sotheorbithasonlyone2eleme2nt, φ . Generally, wewill is also a bosonic simple current, i.e., θJ2 = 1. This ar-
k2 gument can be iterated by assuming that up to some n0
simplycalltheanyona“fixedpoint”whenitisinvariant all Jn, n=1,...,n , are bosonic (then so are all JN−n,
0
under fusion with a simple current, or equivalently, if its
n=1,...,n ,withN theorderofJ). Solvingtheequal-
0
oevrbenit)ceoxnatmaipnlse,oφnlyisthaefiaxnedyopnoiintsteulfn.deInr tthheefuSsUio(n2)wkit(hk itySJ,Jn0 =SJ∗,JN−n0 forθJn0+1 yieldsthatJn0+1 isalso
k bosonic.
2
φ . As we shall see below, the existence of fixed points
k
is crucial for the construction of the condensed theory.
1. Vafa’s theorem
B. Simple current condensation
We first aim to find information about the topological
spins of some of the particles in the theory by analyzing
We consider a condensation transition, where the con- the implications of Vafa’s theorem82
densate consists only of the set of bosonic simple cur-
rents, generated by n simple currents J1,...,Jn with or- θ NxpyNpuz θ NyuqNxqz
p q
ders N ,...,N . Any anyon in the condensate can thus
1 n θ θ θ θ
be represented as J1i1...Jnin, where il = 0,...,Nl − 1 (cid:89)p (cid:18) x y(cid:19) (cid:89)q (cid:18) x z(cid:19) (34)
and l = 1,...,n (note that the fusion product of sim- Nr Nu
θ yz xr
u
ple currents is unique.) We use the shorthand notation =
θ θ
i=(i1,...,in) and (cid:89)r (cid:18) x r(cid:19)
J :=Ji1···Jin. (31) ∀x,y,z,u. Forthecaseofthesimplecurrentcondensate,
i 1 n
we pick a particle x = a, a particle y = J and a par-
i
The initial theory might have simple currents which ticle z = J . Note that a can be any particle in the A
j
are not bosons. We do not consider these, as they can- theory, not necessarily a simple current. This choice of
not condense. We consider the group generated by the theanyonsuniquelyfixesallotheranyonsintheequation
7
(p=a×J , u=a×J , q =a×J , r =J ). Using for any a,b∈A. To prove (ii), note that:
i i+j j i+j
thefactthatthesimplecurrentsandtheirpowersareall
• If b (cid:54)= a×J for all j, then ntnt = 0 and
bosons, Vafa’s theorem gives j t∈T a b
particles a,b do not have any common restrictions.
θa×Ji θa×Jj = θa×Ji+j. (35) Let us write this result as (cid:80)
θ θ θ
a a a
If b∈/ [a] =⇒ ntnt =0, ∀t, (41)
This equation implies that the fractions θ /θ are ir- a b
reduciblecharactersofthegroupZ ⊗Za×J⊗i..a.⊗Z ,
N1 N2 Nn where [a] = {Jja,Jj ∈ C} is the orbit obtained
the bosonic center of RCFT which condenses. The one-
acting on a with all the bosonic simple currents.
dimensional characters of this group can be written as
• If b=a×J for some j then
θ j
a×Ji =ωi1ωi2...ωin, (36)
θ 1 2 n
a N1,N2,...,Nn
where ωi’s satisfy ωiNi = 1. Also note that the ωi’s se- ntanta×Jj = δa×Jj,a×Ji (42)
cretly depend on the subindex a. There are two cases: t(cid:88)∈T i1,i(cid:88)2,...in
=R ∈Z .
a +
Case 1: θ /θ =1, (37a)
a×Ji a But from (i), nt =nt, so the LHS of this equa-
Case 2: θ /θ (cid:54)=1. (37b) a×Jj a
a×Ji a tion is positive. Hence Ra > 0. For example, for
In the latter case, if particles a and a×Ji restrict to n=1, Ra =N1/d, with d defined in Eq. (30).
the same particle t ∈ T then this particle is confined
Hence we have proved (ii). From (ii) we know that if
(as θ (cid:54)= θ ). Moreover, from the orthogonality of
a×Ji a a and b are in the lift of t then b = a×J for some j.
characters Eq. (36) we know that in this case j
On the other hand from (i), if a is in the lift of t, so is
N1,N2,...,Nn θa×Ji =N1−1ωi1N2−1ωi2...Nn−1ωin wah×erJej.a iHseanncye ptairstidcelecoinnfitnheedliiffft θoaf t=. Iθna×oJtjheforrwaolrldjs,,
i1,i(cid:88)2,...,in θa i(cid:88)1=0 1 i(cid:88)2=0 2 i(cid:88)n=0 n (38) given an a ∈ A, the character θa×Jj/θa (cid:54)= 1 for some j
=0. iff a restricts only to confined particles.
Let us now prove the assumption (10). We first mul-
This happens when at least one ωil is not equal to 1. tiply Eq. (40) by dbθb and sum over all particles b in the
A theory to obtain
2. Condensation N1,N2,...,Nn
β nt = d θ
t a a×Ji a×Ji
Without loss of generality, we will assume t(cid:88)∈T i1,i(cid:88)2,...,in (43)
J1,J2,...,Jn condense. If only a subset of the simple N1,N2,...,Nn θ
currents condense then the same analysis applies to just =d θ a×Ji,
a a
θ
the bosons that condense (the others factor out). Since a
i1,i(cid:88)2,...,in
d =...=d =1, the bosons restrict only to the new
J1 Jn
vacuum ϕ with coefficients unity wherewehaveusedthefactthatthequantumdimension
ofanyproductofaparticlewithsimplecurrentsremains
nϕ =...=nϕ =1 (39)
J1 Jn the same. Now if θa×Ji/θa is not the identity character
and do not split. Using the reasoning in Sec. VB it fol- of the trivial representation, then the particle a restricts
lows that all products of these simple currents also con- only to confined particles and we have from Eq. (38)
dense – indeed, all bosonic simple currents J condense.
i
Wewillnowproceedtoproveafewcruciallemmasfor β nt = β nt =0. (44)
t a t a
any a,b ∈ A: (i) nta = nta×Ji, ∀i, for all t ∈ T and (ii) t(cid:88)∈T t∈(cid:88)T/U
ntnt (cid:54)=0 if and only if b=a×J for some j.
t a b j
(i)iseasilyprovedbyexaminingtheb=Ji component In fact, the second equality holds even if θa×Ji/θa = 1
(cid:80)of Eq. (5). To show (ii) we examine the t = ϕ compo- for all j, because in that case nta = 0∀t ∈ T/U – as a
nent of Eq. (5) and note that all bosonic simple currents result,thesecondequalityholdsforalla. Multiplyingthe
condense giving second equality by θa∗da and summing over all particles
a we obtain
N1,N2,...,Nn
ntantb = Nab,JinϕJi 0= βt θa∗danta = βtβt∗. (45)
t(cid:88)∈T i1,i(cid:88)2,...in (40) t∈(cid:88)T/U a(cid:88)∈A t∈(cid:88)T/U
N1,N2,...,Nn
= δb,a×Ji. The unique solution is βt = 0 for t confined, coinciding
with our assumption (10).
i1,i(cid:88)2,...,in
8
VI. ONE CONFINED PARTICLE THEORIES anyons. The algorithm provides all condensation solu-
tions of theory A. Another algorithm which does not
In this section we study a simple boson condensation sidestepT isto(1)buildthematrixM(cid:48) asinthebracket
with just one confined particle t0 in the T theory. Fur- of Eq. (12a); (2) factorize it in nta; (3) keep only the
thermore, assume that the confined particle t has only deconfined particle t’s; and then apply step (3) and
0
two lifts a and a with lifting coefficients both 1, i.e., Eq. (49). Whether the two theories are identical hinges
1 2
onEq.(10),whichweassumetobetrue. Wenowaddress
nt0 =nt0 =1, (46) the above steps one by one.
a1 a2
otherwise nt0 =0,∀a(cid:54)=a ,a . (47)
a 1 2
A. Solutions for M
With these assumptions, we can prove that the con-
densate has only one boson besides vacuum, and this
SinceT isdiagonal,theequation[M,T]=0issatisfied
condensed boson has quantum dimension 1. This im-
if and only if M is a block diagonal matrix with nonzero
plies that the boson is a simple current, so the results of
off-diagonalentriesonlybetweenparticleswiththesame
the previous section imply β = 0. However, we choose
t0 topological spin. Imposing this block structure, we can
to prove this equation through another method which
solve [M,S]=0, imposing that
gives more information about bosonic condensation the-
ories with only one confined particle. Further we find
1. M =M ≥0, M ∈Z,
rthesattridctt0t=o odnae1p=ardtaic2l,ewt0hiicnhtmheeaTnsththeoarty,a1wiatnhdnao2oothnelyr 2. M1a1b =1.ba (50)
particles in T. Finally, in this special one-confined par-
ThesecondconditionensuresthattheAvacuumrestricts
ticle case, we prove that βt0 := a∈Anta0daθa =0 which to the vacuum ϕ of U. In this case, the first row (or col-
clearly support the assumption we used in previous sec-
(cid:80) umn)ofM isequaltothefirstcolumnofnanddescribes
tions. We give the detailed proof in Appendix F.
the particles that condense into the vacuum nϕ. (From
a
this, it is also clear that only solutions with M ≤ d
1a a
can lead to a valid theory.)
VII. FORMALISM AND IMPLEMENTATION
With conditions 1 and 2 in Eq. (50) in place, we ob-
taintwotypesofsolutionsforM,whichwecallautomor-
Wenowpresentanalgorithmicprescription,whichcan
phismsandcondensations,asidefromthetrivialsolution
be implemented on a computer, and which strongly con-
M =11.
trainsthepossiblecondensationtransitionsstartingfrom
Automorphisms are defined by a fully-ranked matrix
aTQFTwithgivenmodularmatricesS andT. Wethen
M satisfying
apply this procedure to several example TQFTs. The
algorithm is performed in 3 steps:
M =1 ∀b. (51)
ab
1. SearchforthesymmetricmatricesM withnonneg- (cid:88)a
ative integer entries and M1,1 =1 satisfying They satisfy M2 = 11 because of the following reasons:
Since M = 1 and all entries of M can only be
[M,S]=[M,T]=0. (48) a ab
nonnegative integers, for any b ∈ A, there is only one
corresp(cid:80)onding particle b(cid:48), such as Mb(cid:48)b = 1. Further,
2. For each M, find all nonnegative integer rectangu- M = 0, ∀a (cid:54)= b(cid:48) and M is fully-ranked. As a result,
ab
lar matrices n such that M =nnT. if a (cid:54)= b then a(cid:48) (cid:54)= b(cid:48). Hence (M2) = M M =
ab c ac cb
M M = δ . An automorphism M is thus a per-
3. For each M and n, find the putative modular ma- c ac bc ab (cid:80)
mutation matrix of order two – it is a symmetry of the
trices S˜ and T˜ of the TQFT after condensation by (cid:80)
S,T data under relabeling of particles. All automor-
solving
phisms of A form a group under matrix multiplication,
Sn=nS˜ and Tn=nT˜. (49) whichisusedtoconstruct“topologicalsymmetrygroup”
in the presence of a global symmetry.83 Automorphism,
however,stillexistsevenwhenanyothersymmetries(e.g.
Onesubtletyisthatweneedtomakesurethatthe
S˜,T˜matricesweobtainarevalid. Inthispaper,we U(1) charge conservation), are broken.
On the other hand, solutions M that correspond to a
use the necessary conditions for a valid S matrix:
condensation have M (cid:54)=δ for some a, implying that
itshouldbesymmetric, unitary, anditshouldgen- 1a a,1
at least one other boson besides the vacuum restricts
erate non-negative fusion coefficients by Verlinde
formula. These are always satisfied if U is a MTC. to the new vacuum. All the condensations can be su-
perimposed with any of the automorphisms, yielding a
This algorithm sidesteps the discussion of the theory T potentially different condensation. In other words, two
that contains confined anyons and directly yields the condensations can be related via a permutation of A by
resulting MTC U formed by the remaining deconfined multiplyingtheM matrixofonecondensationfromboth
9
sides with the M matrix of the automorphism – we will that exchanges the e and the m particles and two con-
see an example of this below for the toric code TQFT. densations
We can prove that any M that satisfies Eq. (48) and
conditions1and2inEq.(50)iseitheranautomorphism 1 1 0 0 1 0 1 0
or a condensation as follows. We first assume that M is M(2) = 1 1 0 0 , M(3) = 0 0 0 0 , (59)
notacondensationsolution,thatis,thefirstrowandcol- 0 0 0 0 1 0 1 0
umnofM areallzeros(M1a =Ma1 =0, ∀a(cid:54)=1)except 0 0 0 0 0 0 0 0
M = 1 . We show that M must be an automorphism    
11
in this case. From of either the e or the m boson. They are related by
the automorphism M(2) = M(1)M(3)M(1) [note that
M S = S M (52) (M(1))−1 =M(1)].
ab bc ab bc
b b
(cid:88) (cid:88)
we have for c=1
B. Solutions for n
M S =S ⇒ M d =d . (53)
ab b1 a1 ab b a
Nextwesolvefortheintegermatrixnt ≥0,wheretla-
(cid:88)b (cid:88)b a
belsthedeconfined particlesintheMTCU. Itispossible
Thus, da is a strictly positive eigenvector of M with that multiple solutions n exist for a given M. However,
eigenvalue 1. Since every Mab is integer and larger or for some solutions, it still might not be possible to find
equal to zero, Eq. (53) can only hold if a valid condensed MTC: please refer to our Appendix G
for an example of 4-layer Ising model condensation. In
fa ≡ Mab ≥1. (54) that example, we obtain unitary S and T matrices, but
b theydonotcorrespond,viaVerlinde’sformula,tointeger
(cid:88)
fusion coefficients.
On the other hand, summing Eq. (53) over a, and using
An efficient first step in solving for n is to realize that
M =MT, gives
any column of M that only contains zeros and ones is
equaltoacolumninn. WhilethematrixM maycontain
f d = d . (55)
b b a several columns with only zeros and ones that are equal,
(cid:88)b (cid:88)a they all correspond to only a single column in n (there
Again, since f ≥ 1 and the d are strictly positive this are no duplicate columns in n). After removing from M
a a
equation can only be satisfied if allrowsandcolumnsthatcontainonlyzerosandones,an
actualfactorizationroutinecanbeusedontheremaining
f ≡ M =1, (56) sub-block of the M matrix. (As we will discuss for an
a ab
example below, the factorization does not always yield a
b
(cid:88) unique solution for n in this case.) In the situations we
which, together with the fact that M is symmetric, im- haveencountered,thispartofthealgorithmisnotlimited
plies that M has to be an automorphism (a permutation bycomputationalpower. Intheparticularlysimpletoric
matrix). code example, deleting duplicate columns directly yields
Let us illustrate how automorphism and condensation the solution
solutionsforM arisefromcondition(48)fortheexample
of the toric code (TC) TQFT. It contains the anyons 1, M(2) =nnT, nT =(1,1,0,0). (60)
e, m, f and has the modular matrices
Thereisonlyoneparticleinthenewtheory,thevacuum.
1 1 1 1 Thus, condensation of either the e or the m particle in
1 1 1 −1 −1 the toric code yields the trivial TQFT.
S = (57a)
TC 21 −1 1 −1 As a less trivial example, consider a bilayer of Ising
1 −1 −1 1
  TQFTs. Each layer contains the anyon types 1, σ, ψ
 
with modular matrices
and
√
1 2 1
T =diag(1,1,1,−1). (57b) 1 √ √
TC S = 2 0 − 2 , T =diag(1,eiπ/8,−1).
I 2 √  I
1 − 2 1
It admits three nontrivial solutions to Eq. (48), one au-
tomorphism   (61)
The bilayer S and T matrices are direct prod-
1 0 0 0 ucts SI(2) = SI ⊗ SI, TI(2) = TI ⊗ TI, and the
0 0 1 0 theory supports 9 particle types which we denote
M(1) = (58)
0 1 0 0 11, 1σ, 1ψ, σ1, σσ, σψ, ψ1, ψσ, ψψ, where 11 is the
0 0 0 1 vacuum. There is only one nontrivial solution for M,
 
 
10
which reads in this basis The decomposition M = nnT is generally not unique.
For example, if M =4 for some particle with quantum
aa
1 0 0 0 0 0 0 0 1
dimension 4 or larger, it can either split in 4 particles
0 0 0 0 0 0 0 0 0 with nt = 1 for each or restrict to one particle with
0 0 1 0 0 0 1 0 0 nt = 2a(this issue was discussed previously in Sec. V).
a
0 0 0 0 0 0 0 0 0
  However, in all examples we studied, at most one of all
M =0 0 0 0 2 0 0 0 0. (62)
  possibledecompositionsofM leadtoaconsistentTQFT
0 0 0 0 0 0 0 0 0 with valid solutions for S˜ and T˜. Thus, the uniqueness
0 0 1 0 0 0 1 0 0
  of this step in the condensation is an open question. We
0 0 0 0 0 0 0 0 0 mentioned that factorizing M = nnT is a well-known
1 0 0 0 0 0 0 0 1
  problem in the field of completely positive matrices. In
  ourcases, thefactorizationhappensovertheringofpos-
Itisstraightforwardtoobtaintheuniquesolutionnthat
itive integers. This problem is known to be NP-hard.
yields M =nnT
With the exception of small dimension matrices, it has
not yet been solved. Some outstanding questions are the
1 0 0 0 0 0 0 0 1
characterization of when a matrix M is completely posi-
0 0 0 0 1 0 0 0 0
nT = , (63) tive (sufficient and necessary condition), as well as what
0 0 0 0 1 0 0 0 0
is the minimal number of rows in n (called CP rank),
0 0 1 0 0 0 1 0 0
  whichistranslatedinourcasetotheminimalnumberof
 
particles in U that can be obtained.
which shows that this describes the condensation of the
ψψ particle. In this process, the σσ particle (which has
quantum dimension 2) splits into two particles of quan-
tum dimension 1 and both 1ψ, ψ1 restrict to the same C. The modular matrices of the new theory
particle. All other particles, except for the vacuum, be-
come confined. Having obtained the matrix n, we now solve the equa-
There exists M that solve Eq. (48), but cannot be de- tions
composedasM =nnTwithanonnegativeintegermatrix
n. Someofthemstilladmitaninterpretationintermsof Sn=nS˜, Tn=nT˜ (64)
acondensationinthefollowingsense. IftheMTCU that
is obtained from a condensation with matrix M = nnT forS˜andT˜. Theseequationscanhavespurioussolutions
has an automorphism symmetry P˜, that is equal to its unless we impose a list of additional constraints. For
transpose P˜ =P˜T, then M˜ =nP˜nT is also a symmetric modular theories, these constraints are
matrix that solves Eq. (48). For instance, one necessary
condition for a decomposition M = nnT to be possible • S˜† =S˜−1,
is that M +M ≥2M . If the matrix elements of M
do not sataiasfy thbebtrianglaebequation M +M ≥2M , • S˜2 = Θ(S˜T˜)3 = C˜, where C˜ is a permutation ma-
then M =nP˜nT might be possible insataead. bb ab trix that squares to the identity and Θ = e−iπc/4
with c the chiral central charge of A, which we can
If the TQFT corresponds to a CFT, the possible
prove remains unchanged (mod 8) during conden-
forms of matrices M that solve Eq. (48) are understood
sation.
with the help of the “naturality theorem” by Moore
and Seiberg84,85. This theorem implies that all M that
• T˜ is a diagonal matrix with complex phases on the
solve Eq. (48) in a CFT are either automorphisms of
diagonal,
A, condensations of the form M = nnT, or of the form
M = nP˜nT, with P˜ an automorphism of U. As a corol-
• the fusion coefficients obtained from the Verlinde
lary, we then conclude that for any solution to Eq. (48)
formula
of the from M(2) = nP˜nT, there is another solution
M(1) = nnT, with the same n, since the identity mass N˜ =S˜D˜ S˜−1, (65)
t t
matrix of U always exists. For the purpose of studying
condensations,wethusfocusedonmatricesM thatadmit with (D˜ ) = δ S˜ /S˜ have to be nonnegative
the decomposition M =nnT throughout our analysis. If t rs r,s tr 1r
integers.
we relaxed this constraint to also include M = nP˜nT,
assumptions such as nt = nt¯ would not be justified We do not prove that any solution that obeys the above
a a¯
anymore. We discuss the interpretation of condensation listofconditionsisindeedavalidMTCU. However,any
transitions for CFTs in Appendix C and relate it to the allowed condensation will be a solution to these condi-
“naturality theorem”. Subsequently, in Appendix D we tions. Therefore, if we do not find a solution for a given
give an example of condensation transitions in SU(2)16, MTCA,wecanconcludethatnocondensationtransition
for which two solutions M = nP˜nT and M˜ = nnT to to a modular U theory out of A exists (we will discuss a
Eq. (48) exist. nontrivial example for this situation in Sec. VIIIB).