Exact diagonalization of Heisenberg $SU(N)$ chains in the fully symmetric and antisymmetric representations

2.2 MB·

by  Pierre Nataf

Exact diagonalization of Heisenberg SU(N) chains in the fully symmetric and
antisymmetric representations
Pierre Nataf, Fr´ed´eric Mila1
1Institut de Physique Th´eorique, E´cole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
(Dated: January 6, 2016)
Motivatedbyrecentexperimentalprogressinthecontextofultra-coldmulti-colorfermionicatoms
inopticallattices,wehavedevelopedamethodtoexactlydiagonalizetheHeisenbergSU(N)Hamil-
tonian with several particles per site living in a fully symmetric or antisymmetric representation of
SU(N). The method, based on the use of standard Young tableaux, takes advantage of the full
SU(N) symmetry, allowing one to work directly in each irreducible representations of the global
SU(N) group. Since the SU(N) singlet sector is often much smaller than the full Hilbert space,
6 this enables one to reach much larger system sizes than with conventional exact diagonalizations.
1 The method is applied to the study of Heisenberg chains in the symmetric representation with two
0 and three particles per site up to N = 10 and up to 20 sites. For the length scales accessible to
2 this approach, all systems except the Haldane chain (SU(2) with two particles per site) appear to
be gapless, and the central charge and scaling dimensions extracted from the results are consistent
n
with a critical behaviour in the SU(N) level k Wess-Zumino-Witten universality class, where k is
a
J the number of particles per site. These results point to the existence of a cross-over between this
universality class and the asymptotic low-energy behavior with a gapped spectrum or a critical
5
behavior in the SU(N) level 1 WZW universality class.

]
l
e
- I. INTRODUCTION ansatzsolvable7andcanbestudiedbyaQuantumMonte
r Carlo algorithm free from the minus sign problem17–19,
t
s Recent advances in ultracold atoms allow experimen- the study of these systems is in fact often challeng-
.
t taliststoartificiallyengineeradvancedmodelsofstrongly ing. Analytical studies can be made with the help of
a
m correlated systems1. In particular, alkaline-earth atoms quantum field theory in some large N development20,
such a 137Yb or 87Sr loaded in optical lattices can be strongcouplinglimit21,22,ormean-fieldapproach15,16,23,
d- used to realize the Fermi-Hubbard model SU(N) inter- or through flavor-wave theory24,25. There is a crucial
n action symmetry2–6. When the number of particle per need to associate those with numerical methods in order
o site m is an integer, and when the on-site repulsion is to test their validity, or to compensate for them when
c large enough, the system is expected to be in a Mott in- they are unapplicable or inconclusive. Among them,
[
sulatingphase,whichiswelldescribedbytheHeisenberg Quantum Monte Carlo can be used only in very specific

1 SU(N) model. This is a generalization of the familiar cases (to avoid sign problem): as we already said, in 1D
v SU(2)spin1/2Heisenbergmodel. Dependingonthege- for m = 1, and on any bipartite lattice provided that
2 ometry of the lattice (a chain, or the square, triangular, pairs of interacting sites correspond to conjugate irre-
2 honeycomb.. lattices in 2D), the number of colors N, ducible representations (’irrep’)26–30. In the case where
9
the number of particles per site (and in particular the the local wave-function on each site is completely an-
0
0 SU(N) symmetry or - irreducible representation- of the tisymmetric, variational Monte Carlo simulations based
. local wave-function on each site), such a model can lead on Gutzwiller projected wave-functions have been found
1 to a rich variety of quantum phases. For instance, in 1D to lead to remarkably accurate results31–34, but it is not
0
6 and for m = 1, the SU(N) chain, for which a general clearhowgeneralizethisapproachtootherirreps,theto-
1 Bethe ansatz solution exists7, is gapless with algebraic tally symmetric one for instance. Density Matrix Renor-
: decayingcorrelations,whilethesamesystemwithm=2 malization Group (DMRG) methods have also been em-
v can lead to the opening of the gap. The famous Haldane ployedtoinvestigateSU(N)Hamiltoniansin1D35–39,as
i
X gap appears for SU(2) with an even number m of parti- wellasInfiniteProjetcedEntangledPairStates(iPEPS)
r clespersiteinthetotallysymmetricrepresentation(cor- in 2D, in a very efficient way12,14,40,41, but the perfor-
a responding to spin j = m/2)8,9 . In 2D, for m = 1, the mances of both methods significantly decrease when the
groundstatehasbeenshowntobecharacterizedbysome dimensionofthelocalHilbertspaceincreases,asaconse-
N´eel-type ordering for SU(2), SU(3)10,11, SU(4)12 and quenceofthelargenumberofcolorsN (typicallyN ≥6),
SU(5)13 on the square lattice, while the SU(4) model or of the large number of particles per site m. Finally,
on the honeycomb lattice is an algebraic spin liquid14. the Exact Diagonalization (ED) are limited by the size
Moreover,onthesquarelattice,withmparticlespersite of the clusters.
inanantisymmetricrepresentation,thegroundstatehas Recently,wehavedevelopedamethodtoexactlydiag-
been predicted by mean-field theory to be a chiral spin onalizetheHamiltonianforoneparticlepersiteindepen-
liquid provided that m/N >515,16. dently in each global irrep of SU(N) by using standard
From a theoretical point of view, apart from 1D with Young tableaux13. Since, for antiferromagnetic interac-
one particle per site, where the system is both Bethe tions, the ground state is in general a singlet, and since
2
II. THE METHOD
In the most general case, a SU(N) Heisenberg-like in-
teraction between two sites i and j can be written as:
H =(cid:88)Sˆi Sˆj , (1)
(i,j) µν νµ
µ,ν
where the SU(N) generators satisfy on each site i the
following commutation relation:
(cid:2)Sˆi ,Sˆi (cid:3)=δ Sˆi −δ Sˆi .
αβ µν µβ αν αν µβ
FIG. 1. a) Young tableau of shape [3,2,2]; b)examples of
standardtableauxrankedaccordingtothelastlettersequence;
c) |Φ[13,2,2](cid:105)=|AAABBCC(cid:105); d) left: integers di,N that enter A. Brief review for one particle per site
into the calculation of the dimension dα; right: hook lengths
N
l ; d is the product of the numbers of the left box divided
i i,N When there is one particle per site, the local states
by the numbers of the right box.
belongtothefundamentalrepresentationofSU(N). The
local Hilbert space is N−dimensional and spanned by
N states, one for each color, that we can call A,B,C,
etc. The interaction in Eq.(1) then takes the form of a
permutation operator P :
i,j
H =P , (2)
(i,j) i,j
the SU(N) singlet sector has a dimension much smaller
which switches the state between site i and j: P |γ(cid:105) ⊗
than that of the full Hilbert space, this enables us to i,j i
|β(cid:105) =|β(cid:105) ⊗|γ(cid:105) , for any γ,β =A,B,C... In that case,
reachessentiallythesamesizesforalargeN asforsmall j i j
an efficient method has been devised to work directly
N: typically, if we call N the number of sites, N ∼ 30
s s in the irreps of the global SU(N) symmetry in Ref. 13.
sites. A natural question was the generalization of the
Here,wejustsummarizethemostimportantresults,and
method to larger number of particles per site: m>1.
we introduce the basic definitions needed to understand
the rest of the section.
Each irrep of SU(N) is labeled by a Young tableau
Inthepresentpaper,weproceedtothisgeneralization
α = [α ,α ,...,α ] (1 ≤ k ≤ N) where the lengths of
1 2 k
in the cases where the local Hilbert space is a totally
the rows α satisfy α ≥ α ≥ ... ≥ α ≥ 1 (see Fig.1
j 1 2 k
symmetric or antisymmetric irrep. In the first part, we
a). The construction relies on the concept of standard
explain our method which is also based on the use of
Young tableaux (SYT) associated to a given shape α,
standard Young tableaux. We build an orthonormal ba-
i.e. tableaux filled with numbers from 1 to N (equal to
s
sis of states belonging to each irrep of SU(N), and show
thenumberofboxes)inascendingorderfromlefttoright
how to write the SU(N) two-sites interaction in such andfromtoptobottom. Theirnumberisdenotedbyfα,
a basis. Then, we apply this method to the study of and they can be ranked from 1 to fα according to the
the Heisenberg SU(N) symmetric chain with m=2 and
last letter sequence: two SYTs S and S are such that
r s
m=3particlespersite,inordertoinvestigatetheprob-
S < S if the number N appears in S in a row below
lemoftheHaldanegapinthecontextofSU(N)chain35, r s s r
the one it appears in S . If those rows are the same, one
s
a problem still open for most values of N (N ≥ 4) in
looks at the rows of N −1, etc (see Fig.1 b).
s
spite of the efficiency of DMRG algorithm to treat 1D
Then, for a given Young tableau, it has been shown
short range interactions Hamiltonian. The results turn
in Ref. 13 that one can construct an orthonormal ba-
out to be quite surprising: for all systems except the
sis with the help of linear superposition of permutations
Haldane chain (SU(2) with two particles per site), the {oα} called orthogonal units which satisfy the
excitation gap seems to tend to zero with the system rs r,s=1..fα
property:
size, consistent with a gapless behavior. In addition, the
central charge and the scaling dimension that could be oαrsoβuv =δαβδsuoαrv ∀ r,s=1...fα,∀u,v =1...fβ, (3)
extractedfromthefinite-sizeenergiesareconsistentwith
which allow one to write the projector on the irrep α as:
theSU(N)levelkWZWuniversalityclass,wherekisthe
(cid:88)
number of particles per site (then k =m). These results Tα = oα , (4)
rr
contradict the DMRG results of Ref. 35 for N = 3 and r=1...fα
the field theory expectation that, if the system is criti-
andwhich, moregenerally, allowonetouniquelyexpress
cal,theuniversalityclassshouldbeSU(N)level1WZW.
any linear superposition of permutations η:
We propose an explanation in terms of a cross-over be-
(cid:88)
tweenSU(N)levelkWZWatintermediateenergies,and η = µβ(η)oβ, (5)
tq tq
agappedbehaviororSU(N)level1WZWatlowenergy.
β,t,q
3
where µβ(η) are the coefficients of the decomposition. IfwesolvethisHamiltonianinthefullHilbertspace,the
tq
Indeed, attaching a site to each integer of the SYTs and spectrum will include all the spectra obtained with all
interpreting the permutations as operators acting in the possible combinations of local irreps.
Hilbert space, the family of states To work in a specific irrep at each site, one needs to
construct an appropriate basis of the projected Hilbert
(cid:110) (cid:111)
|Ψα(cid:105)=||oα |Φα(cid:105)||−1oα |Φα(cid:105) (6) space. Since the Hamiltonian with one particle per site
r 11 1 r1 1
r=1...fα takes a very simple form in the basis of SYTs, the natu-
ral idea is to try and express the projectors in this basis
where |Φα(cid:105) is a product state with A on the first line, B
1 to get a basis as linear combinations of SYTs. In the
on the second line, etc.,(see Fig. 1 c) ) can be proven to
following, we show that for the fully symmetric or fully
beanorthonormalbasisofoneofthesectorsoftheirrep
antisymmetric representations, on which we want to fo-
α (if the quadratic Casimir of α is not equal to zero, α
cus in this paper, this can be achieved quite easily.
canbedecomposedintoequivalentsectors). Mostimpor-
tantly,thematrix{µα(P )} describingP ,the
tq k,k+1 t,q k,k+1
permutationbetweenneighboringsitesk andk+1,takes
C. Symmetric and antisymmetric local irreps
a very simple form in this basis: if k+1 and k are in the
same row (resp. column) in S , then µα(P ) = +1
t tt k,k+1
First,letusgivepracticalexamplesofthekindofstates
(resp. −1), and all other matrix elements involving t
that live in such local Hilbert spaces. If m = 2, the
vanish. If k+1 and k are not in the same column or the
fully antisymmetric irrep labeled by the Young tableau
same line, and if S is the tableau obtained from S by
u t
λ=[1,1], is represented as:
interchanging k and k+1, then the only non-vanishing
matrix elements involving t or u are given by:
(cid:18)µµαuαttt((PPkk,,kk++11)) µµαuαtuu((PPkk,,kk++11))(cid:19)=(cid:18)(cid:112)1−−ρρ2 (cid:112)1ρ−ρ2(cid:19) (8)
IfN =4,andifwecallA,B,C andDthefourcolors,the
where ρ is the inverse of the axial distance from k to 6 orthogonal basis states of the irrep [1,1] can be chosen
k+1 in S defined by counting +1 (resp. −1) for each as :
t
step made downwards or to the left (resp. upwards or
1 1
to the right) to reach k+1 from k. Since any permuta- {√ {|AB(cid:105)−|BA(cid:105)},√ {|AC(cid:105)−|CA(cid:105)}
tion can be written as product of permutations between 2 2
neighboringsites,thisallowsonetowritedownverysim- 1 1
√ {|AD(cid:105)−|DA(cid:105)},√ {|BC(cid:105)−|CB(cid:105)}
ply the matrix of the Hamiltonian of Eq. (2), which can 2 2
then be diagonalized using Lanczos algorithm. 1 1
√ {|BD(cid:105)−|DB(cid:105)},√ {|CD(cid:105)−|DC(cid:105)}}. (9)
2 2
B. The general Hamiltonian as a sum of The fully symmetric irrep λ=[2] represented as
permutations
(10)
Before entering an explicit construction for symmetric
andantisymmetricirreps,letussummarizethemainidea
is spanned, for N =4 by the 10 states:
of the extension to the general case. If we have m parti-
clespersite,theHilbertspacethatcorrespondstoaspe- {|AA(cid:105),|BB(cid:105),|CC(cid:105),|DD(cid:105)
cific irrep at each given site is the subspace obtained by
1 1
applyingtheappropriateprojectorateachsitetoamuch √ {|AB(cid:105)+|BA(cid:105)},√ {|AC(cid:105)+|CA(cid:105)}
2 2
largerHilbertspace,whereeachparticlewouldbeinany
1 1
of the N states. This latter is the same as the Hilbert √ {|AD(cid:105)+|DA(cid:105)},√ {|BC(cid:105)+|CB(cid:105)}
space for mN sites and the fundamental representation 2 2
s
at each site. In that Hilbert space, the Hamiltonian of 1 1
√ {|BD(cid:105)+|DB(cid:105)},√ {|CD(cid:105)+|DC(cid:105)}} (11)
Eq. ( 1) just corresponds to coupling all particles at site 2 2
i to all particles at site j. So, by numbering each of the
m particles located in each site, it is possible to express The dimension dα of a (local) SU(N) irrep of shape
N
this Hamiltonian as a sum of m2 permutations. Assign- α can be calculated very simply from the shape α as
ing number m(j−1)+l to the lth particle of site j, for dα =(cid:81)n (d /l ), with d =N +γ , where γ is the
N i=1 i,N i i,N i i
l=1...m, it takes the form (see also Appendix VA): algebraicdistancefromtheith boxtothemaindiagonal,
countedpositively(resp. negatively)foraboxabove(be-
l(cid:48)=1...m low) the diagonal (see Fig.1 d). The hook lengths l of
(cid:88) i
H(i,j) = Pm(i−1)+l,m(j−1)+l(cid:48). (7) a box are defined as the number of boxes on the same
l=1...m row at the right plus the number of boxes in the same
4
column below plus the box itself (see Fig.1 d). Applying sites with m = 2 particles per site, in the case where
those rules to the fully antisymmetric (resp. symmetric) N ≥4, the two SYTs
shapes with m boxes in one column (resp. row) allows
one to obtain the following formulas (with (cid:15) = +1 for 1 2 5 1 2 6
symmetric and (cid:15)=−1 for antisymmetric): 3 4 7 3 4 7
6 9 10 5 9 10 (15)
N(N +(cid:15))...(N +(cid:15)(m−1))
d(cid:15),m = . (12) 8 11 12 8 11 12
N m!
belong to the same class for the symmetric case because
This number can be very high even for small numbers
each pair of numbers 2(k−1)+1 and 2k belong to the
of particles and could prohibit the diagonalisation of the
same pair of locations in the two SYTs.
Hamiltonian even on small clusters since the full Hilbert
ThesecondimportantobservationisthatalltheSYTs
space has dimension (d(cid:15)N,m)Ns, where Ns is the number belonging to the same equivalence class lead, after pro-
of sites. For instance, for SU(10) and m = 2 in the
jection, to the same (non vanishing) linear combination
symmetric irrep, d+10,2 = 55. Yet, we have been able, of permutations. This can be shown using some prop-
thanks to the method we present below, to find the ex- erties of the orthogonal units and a counting argument
actgroundstateenergyfortheantiferromagneticHeisen- based on the Itzykson-Nauenberg rules (cf Ref 42 and
berg model on the 20 sites chain for this system, while Appendix VD).
(d+10,2)20 (cid:39)6.42×1034. Then, we need to keep only one state per class, that
wewillcallarepresentative. Toselectonerepresentative
ineachclassofSYTs,onecanforinstanceusetheclassi-
1. SelectionofrelevantsymmetricandantisymmetricSYTs fication of the last letter sequence and pick the smallest
state. An algorithm that allows to construct this family
of SYTs is presented in Appendix VE. To proceed fur-
The first important observation (which will be proven
ther, we need to specify the form of the projector, hence
in Appendix VB) is that a SYT will give 0 after pro-
to work separately for the antisymmetric and the sym-
jection onto a symmetric or antisymmetric irrep unless
metric cases.
the particles at any given site satisfy a simple symme-
try condition: for the antisymmetric case, the numbers
corresponding to a given site should be in different rows,
3. The equivalence classes in the antisymmetric case
whileforthesymmetriccase,thenumberscorresponding
toagivensiteshouldbeindifferentcolumns. SuchSYTs
are said to be relevant. Generally, for a given shape α with Nsm boxes, there
Forinstance,form=2andN =4,thefollowingSYT are f˜α ≤fα equivalence classes and representatives. We
is relevant for the antisymmetricscase: denote the representatives as S˜r for 1 ≤ r ≤ f˜α, clas-
sified according to the last letter sequence. Due to the
selectionrulesestablishedinthepreviousparagraph,the
1 3
f˜α representatives are SYTs of shape α with additional
2 5
internal constraints: if we call y(q) the row (between 1
4 7 (13)
and N) where the number 1 ≤ q ≤ mN is located in
6 8 s
the considered SYT, one must have y(m(k−1)+1) <
y(m(k −1)+2) < ... < y(mk) (where 1 ≤ k ≤ N is
while the following one is not: s
the index of the site). Unfortunately, we are not aware
of the equivalent of the hook length formula to calculate
1 4 directly the number f˜α from its shape α43. However,
2 5
and indeed certainly more importantly in view of doing
3 6 (14) computational calculations, we have an efficient way to
7 8 generateallthef˜αSYTswithproperinternalconstraints
for a shape α given as an imput (see Appendix VE).
since the row of ’8’ which is the number of the second Moreover, one can perform the following decomposition
particleofthefourthsiteisinthesamerowas’7’,which ofthefullHilbertspace: (cid:78)Ns (HS )=⊕ Vα. WhenVα
is the number of the first particle of the fourth site. i=1 i α
stands for the SU(N) singlets collective irrep (α being a
rectangle of dimension N × Nsm), the dimension of Vα
N
(number of independent SU(N) singlets) is directly f˜α.
2. Equivalence classes and representatives AndwhenαstandsforanSU(N)irrepwithstrictlypos-
itive quadratic Casimir, exactly as in the m = 1 case13,
If two relevant SYTs only differ by permutations Vα can itself be decomposed into dα equivalent subsec-
N
among the particles of given sites, we say that they be- tor on which the Hamiltonian is invariant. Each of them
long to the same equivalence class. For instance, for 6 has the dimension f˜α. Thus, the decompostion of the
5
full Hilbert space leads to the following equality for the per site. First of all, due to the rules reviewed in section
dimensions: IIA, Proj×oα is a linear superposition of oα , where
r1 p(r)1
the indices p(r) designate SYTs belonging to the same
dim((cid:79)Ns(HSi))=dim(HSi)Ns =(d−N,m)Ns =(cid:88)f˜αdαN, equItiviamlepnlcieescltahsastasfoSr˜rt.wo representatives SYTs S˜ ,S˜
i=1 α r r(cid:48)
(16) (1 ≤ r < r(cid:48) ≤ f˜α), (cid:104)Ψαr(cid:48)|Ψαr(cid:105) = δr,r(cid:48)since the two classes
of S˜ and S˜ are disjoint. It also implies that each state
r r(cid:48)
where α stands for all the shapes of Nsm boxes and no |Ψα(cid:105)belongstothesectorofglobalSU(N)symmetryα.
r
morethanN rows. Importantly, suchanequalitycanbe Finally, since ∀j =1...N :
s
straightforwardly(andindependently)obtainedfromthe
Itzykson-Nauenberg rules that we review in Appendix Pm(j−1)+l,m(j−1)+gProj(j)=−Proj(j) ∀1≤l<g ≤m,
VC. (22)
the local antisymmetry is satisfied.
From a conceptual point of view, we could stop here,
4. The basis states in the antisymmetric case
but we want to give additional details to make the ac-
tual implementation easier. If one identifies each state
Now, to build basis states for a given shape α, we just oα |Φα(cid:105)
p1 1 withthecorrespondingtableauS ,onecanex-
need to apply a projection operator Proj to the orthog- ||oα|Φα(cid:105)|| p
11 1
onal units oα of the representatives: press the normalizing projection operator N−1Proj as
r1
an operator on the SYTs , for 1≤r ≤f˜α:
(cid:89)Ns
Proj = Proj(k), (17) N−1Proj oαr1|Φα1(cid:105) ≡N−1ProjS˜ =(cid:0)(cid:89)Ns ς (cid:1)S˜ , (23)
k=1 ||oα |Φα(cid:105)|| r j r
11 1 j=1
where Proj(k) imposes the local antisymmetry at site k:
where ς is a superposition of m! operators that in-
j
1 (cid:88) terexchange between each other the numbers m(j−1)+
Proj(k)= m! (cid:15)(σ)σ. (18) 1, m(j−1)+2..., mj intheSYTtableauS˜r. According
σ∈Sm(k) to the rules controling the effect of successive transpo-
sition reviewed in IIA and to the definition of the nor-
In the last equation, the sum runs over a group that can
malizedprojectoroperator(cfEq. (18)), itiseasytosee
be named S (k), which gathers all the permutations σ
m that for m=2, ς must be:
that interexchange between each other the m particles j
of the site k, whose numbers are m(k −1)+1,....,mk. (cid:114) (cid:114)
1+ρ(j) 1−ρ(j)
The function (cid:15)(σ) is the signature of the permutation σ. ς = I − T(j), (24)
j 2 d 2
It is equal to +1 (resp. −1) for an even (resp. odd)
permutation σ. Thus, for instance, for the site k = 1, if whereρ(j)istheinverseoftheaxialdistancebetweenthe
m=2, we have: numbers 2j−1 and 2j (which is necessarly non negative
due to the internal constraints on the tableau S˜ ), I
1 r d
Proj(1)= (I −(1,2)), (19) is defined as the identity operator on the SYT, while
2 d T(j) switches 2j−1 and 2j in the tableau on which it is
while if m=3: applied. For example, one has:
1(cid:8) (cid:9) 1 3 1 4
Proj(1)= I −(1,2)−(1,3)−(2,3)+(1,2,3)+(1,3,2) ,
6 d 2 5 2 5
T(2) = . (25)
(20)
4 7 3 7
where I is the identity, (1,2)=P is the permutation
d 1,2 6 8 6 8
1↔2, and so on.
Note that in case where 2j −1 and 2j are in the same
Then, the desired set of states can be defined as: column (necessarly one above the other), then ρ(j) = 1
and ς =I : the constraint P directly gives −1 on
j d 2j−1,2j
(cid:110)|Ψα(cid:105)=N−1Proj× oαr1|Φα1(cid:105) (cid:111) , (21) such a SYT.
r ||oα11|Φα1(cid:105)|| r=1...f˜α Thus,forfoursiteswithm=2particlespersite,inthe
case where N ≥4, one state of the irrep α=[2,2,2,2] is
where N is some normalization constant, and where
for instance:
the index r runs over the representatives SYTs S˜ for
r
1 ≤ r ≤ f˜α. Note that ||ooαrα1||ΦΦαα1(cid:105)(cid:105)|| is a normalized Nsm- 1 3
11 1 2 5
particles state of SU(N) symmetry α , that has no de- N−1Proj (26)
4 7
fined property of local symmetry, i.e it appears in the
Hilbert space of a system of N m sites with one particle 6 8
s
6
which is equal to the superposition:
1 4 7 1 4 7 1 5 7
1 3 1 4 1 3 1 4
√ √
2 5 10 2 6 10 2 6 10
2 5 √ 2 5 √ 2 6 2 6 5 5 2 5
2 2 2 1 3 8 11 − 3 8 11 + √ 3 8 11
4 7 − 3 7 − 4 7 + 3 7 , (27) 9 18 3 6
3 3 3 3 6 9 12 5 9 12 4 9 12
6 8 6 8 5 8 5 8
1 4 8 1 5 8 1 4 8
√ √
which is obviously normalized. For m=3, the operators 5 2 2 5 10 5 2 6 10 5 2 6 10
ς are a bit more complicated: − 3 7 11 − √ 3 7 11 + 3 7 11
j 18 6 3 18
6 9 12 4 9 12 5 9 12
5 1 4 9 1 4 9 1 5 9
(cid:88)
ςj = ηq(j)Tq(j). (28) √ 2 5 10 √ 2 6 10 2 6 10
5 5 1
q=0 + √ 3 7 11 − √ 3 7 11 + 3 7 11 .
3 6 6 3 6
6 8 12 5 8 12 4 8 12
The vector of coefficients (η(j))q=0...5 reads: (32)
 (cid:112)1+ρx(j)(cid:112)1+ρy(j)(cid:112)1+ρz(j)  For m ≥ 4, the construction of ςj is still based on the
(cid:112) (cid:112) (cid:112) definitionoftheprojectorshowninEq. (18),butitwould
− 1−ρx(j) 1+ρy(j) 1+ρz(j)
 (cid:112) (cid:112) (cid:112)  require 4!=24 coefficients.
1 − 1+ρx(j) 1+ρy(j) 1−ρz(j)
η(j)= √  (cid:112) (cid:112) (cid:112) ,
6 1+ρx(j) 1−ρy(j) 1−ρz(j) 
 (cid:112) (cid:112) (cid:112) 
 1−ρx(j) 1−ρy(j) 1+ρz(j) 
5. The equivalence classes in the symmetric case
(cid:112) (cid:112) (cid:112)
− 1−ρx(j) 1−ρy(j) 1−ρz(j)
(29) ForagivenshapeαwithN mboxes,therearef¯α ≤fα
s
equivalences classes, or reprentatives. We denote them
where ρx(j) is the inverse of the axial distance from the withabarS¯ (for1≤r ≤f¯α),inthesamespiritaswhat
r
number 3j−2 to 3j−1, ρy(j) is the inverse of the axial isdoneinsectionIIC4fortheantisymmetriccase. They
distance from the number 3j−2 to 3j, and ρz(j) is the are also classified according to the last letter sequence.
inverse of the axial distance from the number 3j −1 to Interestingly, there is an other way to define f¯α: it is
3j. Notethatbydefinition 1 + 1 = 1 . (seealso the number of semi-standard Young tableaux of shape α
ρx(j) ρz(j) ρy(j)
Fig.2). TheoperatorsTq(j)permutethenumberscorre- andofcontent1,1,..1,2,2,..2,...,Ns,Ns,..Ns (eachnum-
pondingtositej onaSYT.Thecorrespondancebetween ber j between 1 and Ns appearing m times). A semi-
the (T (j)) and the permutation of the symmetric standard Young tableau is a tableau filled up with num-
q q=0...5
group S is the following: bersinnon-descendingorderfromlefttorightinanyrow
3
andinascendingorderfromtoptobottominanycolumn.
Such a number is by definition called a Kostka number
T (j)−→I (30)
0 d (SeeChapter7ofRef.44). Byrealizingthatonecanpass
T (j)−→(3j−2,3j−1)
1 from the antisymmetric case to the symmetric case by
T (j)−→(3j−1,3j) performingbasicallysomeconjugationoftableaux(which
2
T (j)−→(3j−2,3j,3j−1) consists in transforming rows into columns and columns
3
into rows), one can prove that for the same number of
T (j)−→(3j−2,3j−1,3j)
4 particle per site m :
T (j)−→(3j−2,3j)
5
f¯α =f˜ , (33)
αT
where αT is the transposition of the shape α. The de-
SeealsoFig. 2. Thus,forfoursiteswithm=3particles
compostion of the full Hilbert space can be done exactly
per site, in the case where N ≥ 4, one state of the irrep
likeintheantisymetriccase,anditleadstothefollowing
α=[3,3,3,3] (and indeed the only one) is :
equality for the dimensions:
1 4 7 n
dim((cid:79)(HS ))=(d+,m)n =(cid:88)f¯αdα. (34)
2 5 10 i N N
N−1Proj 3 8 11 (31) i=1 α
6 9 12
Again, such an equality could be obtained as well from
the Itzykson-Nauenberg rules that we review in Ap-
which is equal to: pendix VC.
7
6. The basis states in the symmetric case which is equal to the normalized superposition:
We assign to each representative SYT S¯ a specific 1 2 5 1 2 6 1 2 5 1 2 6
r
superposition of orthogonal units oα that allows it to 3 4 7 √ 3 4 7 √ 3 4 8 3 4 8
r1 3 15 15 5
satisfythelocalconstraintsbyusingtheprojectionoper- 6 9 10 + 5 9 10 + 6 9 10 + 5 9 10.
N(cid:81)s 8 8 11 12 8 8 11 12 8 7 11 12 8 7 11 12
atorProj = Proj(k)whereProj(k)isthesymmetric
k=1 (41)
version of the projector defined in Eq. (18):
For m=3, the operators ς are given by:
1 (cid:88) j
Proj(k)= σ. (35)
m!
σ∈Sm(k) (cid:88)5
ς = η (j)T (j), (42)
j q q
Thus, for m=2, and for the first site,
q=0
1
Proj(1)= (I +(1,2)), (36) where the vector of coefficients (η(j)) becomes:
2 d q=0...5
while if m=3: (cid:112)1−ρx(j)(cid:112)1−ρy(j)(cid:112)1−ρz(j)
(cid:112) (cid:112) (cid:112)
Proj(1)= 61(cid:8)Id+(1,2)+(1,3)+(2,3)+(1,2,3)+(1,3,2)(cid:9). η(j)= √1 (cid:112)(cid:112)11+−ρρxx((jj))(cid:112)(cid:112)11−−ρρyy((jj))(cid:112)(cid:112)11+−ρρzz((jj)),
(37) 6 1−ρx(j) 1+ρy(j) 1+ρz(j)
(cid:112) (cid:112) (cid:112) 
Then, the set of states:  1+ρx(j) 1+ρy(j) 1−ρz(j)
(cid:112) (cid:112) (cid:112)
1+ρx(j) 1+ρy(j) 1+ρz(j)
(cid:110) oα |Φα(cid:105) (cid:111)
|Ψα(cid:105)=N−1Proj× r1 1 , (43)
r ||oα11|Φα1(cid:105)|| r=1...f¯α
with the ρa(j) (a = x,y,z) and the operators T (j) are
can be proved to have the appropriate properties with q
defined as before.
the same arguments as before. In particular, the local
Thus, for four sites with m = 3 particles per site, in
required symmetry is a consequence of the equality:
the case where N ≥ 3, the only state of the irrep α =
P Proj(j)=Proj(j) ∀1≤l<g ≤m. [3,3,3,3] is :
m(j−1)+l,m(j−1)+g
(38)
1 2 3 4
Finally, by identifying each state oαp1|Φα1(cid:105) with the cor- N−1Proj 5 6 7 8 (44)
||oα|Φα(cid:105)||
responding tableau S , one can a1ls1o 1express the nor- 9 10 11 12
p
malizing projection operator N−1Proj as an operator
(for 1 ≤ r ≤ f¯α) on the SYTs N−1Proj oαr1|Φα1(cid:105) ≡ which is equal to:
||oα|Φα(cid:105)||
11 1
N−1ProjS¯r =(cid:0) N(cid:81)s ςj(cid:1)S¯r, where ςj is now for m=2: 1 2 3 4 √ 1 2 3 5
1 5
j=1 5 6 7 8 + √ 4 6 7 8
6 6 3
(cid:114) (cid:114) 9 10 11 12 9 10 11 12
1−ρ(j) 1+ρ(j)
ςj = 2 Id+ 2 T(j), (39) √ 1 2 3 6 √ 1 2 3 4
10 5
+ √ 4 5 7 8 + √ 5 6 7 9
with the same notation as in Eq.(24). 6 3 6 3
9 10 11 12 8 10 11 12
Note that if 2j−1 and 2j are in the same line (neces-
√
sarly one before the other), then ρ(j)=−1 and ςj =Id: 5 1 2 3 5 5 2 1 2 3 6
the permutation P directly gives +1 on such a + 4 6 7 9 + 4 5 7 9
2j−1,2j 18 18
SYT. 8 10 11 12 8 10 11 12
Thus, for 6 sites with m = 2 particles per site, in the √ √
1 2 3 4 1 2 3 5
case where N ≥ 4, the first (out of the five) state of the 10 5 2
+ √ 5 6 8 9 + 4 6 8 9
irrep α=[3,3,3,3] is for instance: 6 3 18
7 10 11 12 7 10 11 12
1 2 5 1 2 3 6
5
3 4 7 + 4 5 8 9 . (45)
9
N−1Proj 6 9 10 (40) 7 10 11 12
8 11 12
8
FIG. 2. Action of the operators T (j) on a SYT.
q
III. SU(N) ANTIFERROMAGNETIC
HEISENBERG CHAIN IN THE FULLY
SYMMETRIC REPRESENTATION
In this section, we apply this method to perform ED
oftheHeisenbergSU(N)modelonanantiferromagnetic FIG. 3. Gap of the SU(N) antiferromagnetic Heisenberg
chaininthefullysymmetricirrepswithm=2and3par- chain with two particles per site in the symmetric irrep with
ticlespersite. Theapplicationtotheantisymmetriccase periodic boundary conditions. The gap has been determined
can be found in the recent paper [34], where analytical as the energy difference between the first excited state (al-
predictionsaboutthenatureofthegroundstate(gapped ways located in the irrep of smaller non vanishing quadratic
orcritical)duetoAffleck45,46 havebeennumericallyveri- Casimir) and the ground state, which is a SU(N) singlet in
thesystemswehaveconsidered,wherethenumberofsitesisa
fiedbyacombinationofEDcalculationsperformedalong
multipleofN. ForSU(2)(thespin-1chain),onecaninferthe
the lines of the present paper and of variational Monte-
presenceoftheHaldanegapfromtheresultsobtainedforrel-
Carlo simulations.
ativelysmallchains(uptoN =16),whereas,forN ≥3,the
s
The basic results are the energies for the SU(N) sym-
resultsforsimilarsizesareconsistentwithagaplessspectrum
metric chain for m=2,3 particles per site in the singlet in the thermodynamic limit.
subspace,and,wheneverpossible,insomeirrepsofsmall
quadratic Casimir. We have employed the Lanczos algo-
rithm whose key part is the product of the Hamiltonian
in the ordered list of constrained SYTs (through for in-
(restricted to a given invariant sector) times a vector.
stance a binary search or a more sophisticated indexing
We have achieved this task by using a 4-step procedure.
function that goes beyond the scope of this paper).
Each basis state is represented by a SYT with proper
In fact, since each permutation is decomposed into a
internal constraints (see previous paragraph). As a first
product of successive transpositions, this algorithm is
step, we develop such a basis state to express it as a
particularly suited for the study of chains with open
superposition of orthogonal units times a product state,
boundary conditions, since it is possible to index every
with coefficients given by expression Eq.(39) (for m=2)
pair of connected sites with consecutive numbers.47 In-
and Eq.(43) (for m = 3). Then, as a second step, we
cidentally, this also means that the computation of the
apply one interaction term (corresponding to one link
exactenergiesisfasterforopenboundaryconditionsthan
in the lattice) to such a superposition by employing the
for periodic boundary conditions.
rules reported in the paragraph IIA. We first write the
interaction term as a sum of permutations, like in Eq. For m=2, we list in Table I the ground state energies
(7). Then, each permutation is written as a product of per site for periodic boundary conditions (EP (N )), as
GS s
successivetranspositionswhoseeffectoneachorthogonal well as for open boundary conditions (EO (N )) since it
GS s
unit is known anddescribed in the paragraph IIA. After can be useful for benchmarking future DMRG studies.
step2,wehavealargersuperpositionoforthogonalunits WhenthenumberofsitesN isamultipleofthenumber
s
than one needs to express as a linear sum of the initial ofcolorsN,N =pN,thegroundstateisalwaysasinglet
s
symmetric basis states. Since the interaction term con- (for antiferromagnetic couplings), so that the minimal
servesthesymmetryofthewave-function,onejustneeds energy is obtained by diagonalizing the Hamiltonian in
to project the last superposition using the coefficients the SU(N) singlet sector, i.e the sector corresponding
givenbyexpressionEq.(39)(form=2)andEq.(43)(for to the shape α = [q,q,...,q], where q = N m/N = pm.
s
m = 3). One obtains a linear sum of symmetric states. We also provide in Table I the corresponding dimensions
Thefinalstepconsistsinfindingtheranksofthosestates f¯[q,...,q] thatgivethesizeofthematriceswediagonalized.
9
m=2 N f¯[q,...,q] EP (N ) EO (N ) DMRGvaluetakenfrom9(∆≈0.410),whichcomesfrom
s GS s GS s
SU(3) 15 6879236 -1.448589 -1.397889 the two different ways to write the interaction between
2 sites, either in terms of spin operators or in terms of
SU(3) 18 767746656 -1.402602
permutation operators. Indeed, if we use the notations
SU(4) 16 190720530 -1.687431 -1.619271
of the previous section, the spin 1 interaction between
SU(5) 15 25468729 -1.804955 -1.716275
sites 1 and 2 is related to the permutations through the
SU(6) 12 16071 -1.88243593 -1.752105 identity S(cid:126) .S(cid:126) = 1{P +P +P +P }−1.
SU(8) 16 3607890 -1.932087 -1.827798 1 2 2 1,3 1,4 2,3 2,4
InallcasesexceptSU(2), thedataareconsistentwith
SU(10) 20 1135871490 -1.869078 a vanishing gap in the thermodynamic limit, hence with
a gapless spectrum. So the difference between 1 and 2
TABLE I. Dimension of the singlet sector f¯[q,...,q], ground particles per site predicted by Haldane for SU(2) does
state energy per site with periodic boundary conditions
not seem to carry over to larger values of N.
EP (N ),andgroundstateenergypersitewithopenbound-
GS s Forthreeparticlespersite,thesizeswecanreachwith
ary conditions EO (N ) for SU(N) Heisenberg chains with
GS s our algorithm are smaller (N ≤ 12). The ground state
two particles per site in the symmetric irrep and N sites. s
s
energies are listed in Table III.
Thecalculatationismoredifficultforperiodicboundarycon-
ditions because the matrix is less sparse, and it has not been
doneforSU(10)with20sitesandforSU(3)with18sitessince m=3 N f¯[q,...,q] EP (N ) EO (N )
s GS s GS s
itwouldrequireaparallelarchitecturewithseveralhundreds
SU(3) 12 3463075 -2.218913 -2.106611
of nodes.
SU(4) 12 10260228 -2.574628 -2.421300
SU(6) 12 1113860 -2.832493 -2.634385
A. Gap
TABLE III. Dimension of the singlet sector f¯[q,...,q], ground
state energy per site with periodic boundary conditions
We have applied this algorithm to determine the gap
EP (N ),andgroundstateenergypersitewithopenbound-
of the Heisenberg SU(N) symmetric chain. With both GS s
ary conditions EO (N ) for SU(N) Heisenberg chains with
periodic (EP(N )) and open (E0 (N )) boundary condi- GS s
ex s ex s three particles per site in the symmetric irrep and Ns sites
tions, as long as the number of sites N is a multiple
s
of N, the first excited state belongs to the adjoint irrep
Quitesurprisingly,theresultsforthegaparealsocon-
α = [q +1,q,..,q −1] with the smallest non vanishing
quadratic Casimir C . Its dimension f¯[q+1,q,..,q−1] is al- sistent with a vanishing gap in the thermodynamic limit
2 for SU(3), SU(4) and SU(6), as shown in Fig. 4. For
ways much larger than that of the singlet. For instance,
for SU(10) and 20 sites, f¯[5,4,4,4,....,4,3] ≈40×109, a size SU(3),theseresultsareincontradictionwiththeDMRG
resultsreportedinRef.35. Wewillcomebacktothisdif-
we could not handle with the computers at our disposal,
ference in the discussion section below.
whereasthedimensionofthesingletsectoris≈1.3×109.
WehavegatheredsomevaluesofEP(N )andE0 (N )in
ex s ex s
Table II.
B. Central charge
m=2 N f¯[q+1,q,..,q−1] EP(N ) EO(N )
s ex s ex s
When a 1D quantum system is critical, it is in gen-
SU(3) 15 44994040 -1.413231 -1.381094
eral possible to identify the universality class to which
SU(4) 16 2077175100 -1.6604222 -1.607230
thelowenergytheorybelongsintermsoftheunderlying
SU(5) 15 377182806 -1.778192 -1.705462
conformal field theory (CFT). In the case of Heisenberg
SU(6) 12 272712 -1.84503164 -1.738918 SU(N) models, the relevantCFTs arethe SU(N) Wess-
SU(8) 16 93683590 -1.915457 -1.822196 Zumino-Witten (WZW) models with topological integer
coupling coefficient k48. The corresponding algebra of
TABLE II. Dimension of the representation of smallest non-
such a theory is SU(N) , with a central charge c given
vanishing Casimir f¯[q+1,q,..,q−1] (corresponding to the triplet k
by:
sector for SU(2)), and first excited energies per site in that
sectorforSU(N)Heisenbergchainswithtwoparticlespersite
N2−1
inthesymmetricirrepandNssitesforperiodicboundarycon- c=k . (46)
ditions (EP(N ) ) and open boundary conditions (EO(N )). N +k
ex s ex s
Thecentralchargecanbeextractedfromtheexactdi-
For two particles per site, we have plotted the corre- agonalizationresultsintwosteps. Firstofall,onecanex-
sponding gaps in Fig. 3, in which the SU(2) case has tracttheproductofthecentralchargescwiththesound
beenaddedforcomparison. TheSU(2)casecorresponds velocity v from the dependence of the ground state en-
to the spin-1 chain known to exhibit the Haldane gap, ergy per site EP (N ) with the number of sites N 48–50:
GS s s
which can be clearly inferred from the results obtained
even for relatively small chains (N ≤16). Note the fac- 2πcv
s EP (N )=EP (∞)− +o(1/N2), (47)
tor2betweenourinterpolatedvalue(around0.8)andthe GS s GS 12N2 s
s
10
FIG. 5. Examples of finite size results that were used
to extract the central charge for SU(3) and m = 2. Left:
Excitation energy of the first excited state with momentum
k = 2π/N times the number of sites as a function of 1/N2.
s s
In a critical system, this is expected to tend to 2πv in the
FIG. 4. Gap of the SU(N) antiferromagnetic Heisenberg thermodynamic limit, where v is the sound velocity. Right:
chainwiththreeparticlespersiteinthesymmetricirrepwith GroundstateenergypersiteEGPS(Ns)asafunctionof1/Ns2.
periodic boundary conditions. The gap has been determined Inacriticalsystem,itisexpectedtoconvergetoEGPS(∞)lin-
as the energy difference between the first excited state (al- early with 1/Ns2, with a slope equal to 2πcv/12, where c is
ways located in the irrep of smaller non vanishing quadratic the central charge.
Casimir) and the ground state, which is a SU(N) singlet in
thesystemswehaveconsidered, wherethenumberofsitesis
a multiple of N. In the case N = 2 (spin-3/2 chain), which
Toavoiduncertaintiesduetoextrapolationsinextract-
is known to be gapless, the curve is slightly convex, consis-
ing the central charge c, we have used for the velocity v
tent with a vanishing gap in the thermodynamic limit. For
the value for the largest available size, and for the prod-
N =3,4,6, the concavity is opposite and the curves seem to
uct cv the slope deduced from the values of the ground
converge towards zero; however the maximal sizes reached in
the simulations are quite small (N ≤12). stateenergyforthetwolargestsizes. Thecentralcharges
s
extractedinthiswaycanbeexpectedtobeslightlyover-
estimated since the product cv decreases with the size
where EP (∞) is the ground state energy per site in the
GS while the velocity v increases with the size. The corre-
thermodynamic limit. The case SU(3) with m = 2 is
spondingestimatesforthecentralchargearelistedinTa-
shown as an example in the right panel of Fig. 5. The
bleIVform=2andinTableVform=3,togetherwith
scaling as 1/N2 is already quite accurate for the largest
s the theoretical values for SU(N)1 and SU(N)m. Quite
available sizes.
remarkably, in all cases, the results m particles per site
Secondly,onecanextractthesoundvelocityvfromthe
for are in good agreement with SU(N) (and slightly
m
energyofthefirstexcitedstateofmomentumk =2π/N
s above, as expected), and very far from SU(N) .
and non zero quadratic Casimir51: 1
This result is quite surprising since, according to field
EP (N )−EP (N )= 2πv +o(1/N ). (48) theory,theSU(N)k>1WZWmodelshaveatleastonerel-
2π/Ns s GS s N s evant operator allowed by symmetry46,52, implying that
s
one should adjust at least one parameter to sit at such a
where EP (N ) and EP (N ) are total energies. To
2π/Ns s GS s critical point, as in the case of integrable models53.
check the momentum of the excited state, which is not
available right away in our approach since we do not use
spatial symmetries, we had to extract the ground state SU(3) SU(4) SU(5) SU(6) SU(7) SU(8)
wave function, and to apply directly the translation op-
m=2 c 3.23 5.16 7.16 9.77 11.65 13.56
erator. This is tedious but straightforward because the
k=2 N2−1 3.2 5 6.86 8.75 10.67 12.60
translation operator can be written in terms of permu- N+2
k=1 N2−1 2 3 4 5 6 7
tations. It turns out that in all the gapless cases inves- N+1
tigated (m = 2,3 for N > 2), and as soon as N = pN
s
TABLE IV. Finite-size estimates of the central charge c (see
withp>1,thefirstexcitedstateintheadjointirrephas
main text and Fig. 5) for SU(N) Heisenberg chains with
a momentum k = 2π/N, and it is actually only the sec-
two particles per site in the symmetric irrp (m = 2), com-
ond excited state which has the momentum k = 2π/N .
s pared to the predictions for the SU(N) and SU(N) WZW
2 1
Examples of the resulting finite-size estimates of the ve-
universality classes.
locity v are given in the left panel of Fig. 5 for N = 3,
m=2.