Matrix Product States: Symmetries and Two-Body Hamiltonians

0.82 MB·English

by  M. Sanz

Matrix Product States: Symmetries and two-body Hamiltonians
M. Sanz1, M. M. Wolf2, D. P´erez-Garc´ıa3, and J. I. Cirac1
1Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany
2Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
3Facultad de Matematicas UCM, Plaza de Ciencias 3, 28040 Madrid, Spain
Wecharacterizetheconditionsunderwhichatranslationallyinvariantmatrixproductstate(MPS)
is invariant under local transformations. This allows us to relate the symmetry group of a given
statetothesymmetrygroupofasimpletensor. Weexploitthisresultinordertoproveandextend
a version of the Lieb-Schultz-Mattis theorem, one of the basic results in many-body physics, in the
context of MPS. We illustrate the results with an exhaustive search of SU(2)–invariant two-body
Hamiltonians which have such MPS as exact ground states or excitations.
9
0
I. INTRODUCTION try. They have served as toy models to understand cer-
0
tain physical behavior in real physical systems, like the
2
Matrix Product States (MPS) [1, 2] encapsulate many existenceofaHaldanegap[6]inspinchainswithinteger
n
a ofthephysicalpropertiesofquantumspinchains. Ofpar- spin,orthephenomenonofdimerization[5],respectively.
J ticular interest in various physical contexts is the subset Despitetheirkeyroleintheunderstandingofspinchains,
5 of translationally invariant (TI) MPS, originally intro- thereareveryfewotherexamplesknownofTIMPSwith
1 duced as finitely correlated states [1]. Their importance SU(2) symmetry and with a two-body parent Hamilto-
stemsfromthefactthatwithasimpletensor,A,onecan nian [1, 8, 9].

]
fully describe relevant states of N spins, which, at least InthisworkwefirstgeneralizetheresultsofRef. [3]to
l
e in principle, should require to deal with an exponential arbitraryTIMPS.Thisenablesustoderivesomegeneric
-
r numberofparameterswhenwritteninabasisinthecor- properties about those states, as well as to obtain a sim-
st responding Hilbert space H⊗N. Thus, all the physical pleproofforaversionoftheLieb-Schultz-Mattistheorem
. properties of such states are contained in A. It is there- [10]. This celebrated theorem states that all Hamiltoni-
t
a foreimportanttoobtainmethodstoextractthephysical ans with SU(2) symmetry are gapless for semi-integer
m
propertiesdirectlyfromsuchatensor,withouthavingto spin (dim(H) = n+1/2, n = 0,1,...). In our case, we
- resort to H⊗N. canprovethatallTIMPScorrespondingtosystemswith
d
AnimportantphysicalpropertyofaTIstate,Ψ,isthe semi-integerspinscannotbetheuniquegroundstateofa
n
o symmetrygroupunderwhichitisinvariant. Thatis,the local frustration-free Hamiltonian. Furthermore, we can
c group G such that extend the proof to other groups, like U(1) for spin 1/2
[ systems,andfindcounterexamplesforthislastcasewhen
1 u⊗gN|Ψ(cid:105)=eiθg|Ψ(cid:105), (1) the spin is 5/2 or larger.
v InthesecondpartofourworkweconcentrateonMPS
where g ∈G and u is a unitary representation on H. In
3 g that are eigenstates (not necessarily grounds states) of a
arecentpaper[3]weshowedthatforcertainkindofMPS
2
(so–called ’parent’) Hamiltonian which has SU(2) sym-
(those fulfilling theso–called injectivitycondition [1, 2]),
2
metry and contains two–body interactions only. We find
2 thissymmetrygroupisuniquelydeterminedbythesym-
other families of Hamiltonians beyond the well–known
. metrygroupofA(withatensorproductrepresentation).
1
AKLT and Majumdar-Gosh with those features. Fur-
Roughly speaking this means that by studying the sym-
0
thermore, we find the first examples of MPS that cor-
9 metries of A we can obtain those for the whole state Ψ.
respond to excited states of SU(2)-invariant Hamiltoni-
0 This result allows us, for example, to shed a new per-
: spective into string order [3], a key concept in strongly ans. There is a new example of state with spin 1, which
v
is never the ground state of any frustration free SU(2)-
correlated states in many–body quantum systems.
i
X Another relevant property of MPS is that they are all invariant two-body hamiltonian. In order to make a sys-
tematicsearchofallthoseMPSwedevelopasimpletech-
r exact ground states of short–range interacting (frustra-
a nique that allows for a numerical systematic search.
tion free) Hamiltonians [1, 2]. In particular, for every
TIMPSwecanalwaysbuilda(so–called)’parent’Hamil- Thispaperisorganizedasfollows. InSectionIIwere-
tonian for which it is the ground state. Of particular in- viewsomeofthebasicpropertiesofTIMPSandestablish
terests are TIMPS with two–body parent Hamiltonians; thenotationthatwillbeneededinthefollowing. InSec-
that is, whose parent Hamiltonian consist of two–body tion III we establish the relation between the symmetry
interactions only. And among those, the ones which group of a TIMPS and that of the tensor A defining the
have a large symmetry group, like SU(2). The reason MPS.Forcontinuoussymmetries,suchasSU(2),wewill
is that those are the ones that naturally appear in con- seethatthesetofsymmetricTIMPSisintimatelyrelated
densed matter problems. Two prominent examples are tothesetofClebsch-Gordancoefficients. SectionIVthen
the AKLT [4] and the Majumdar-Gosh [5] states, who provides an MPS version of the Lieb-Schultz-Mattis the-
have two-body parent Hamiltonians with SU(2) symme- oremandinSectionVwegiveadetailedinvestigationof
2
SU(2) symmetric TIMPS which are eigenstates of two- Property 1 (Span property). The set of products P =
body Hamiltonians. {A ···A }, with n the collected spins, spans the vec-
i1 in
tor space of all matrices with the same block diagonal
structure.
II. MATRIX PRODUCT STATES
It is an open conjecture stated in [2] and verified in
manyparticularcases,thatanupperboundforthenum-
Let us consider a system with periodic boundary con-
berofsiteswhichhavetobegatheredtoachieveproperty
ditions of N (large but finite) sites, each of them with
1 depends only on the dimension D of the Kraus opera-
an associated d-dimensional Hilbert space. A transla-
tors. Whenthereisonlyoneblockintheabovecanonical
tionally invariant MPS on this system can be defined
decomposition the MPS is usually called injective, since
with a valence bond construction in the following way:
the linear operator mapping boundary conditions to the
Let us consider another couple of D dimensional ancil-
resultingstatesisindeedinjective[1, 2]whentakingsuf-
lary/virtual Hilbert spaces associated to each site and
ficiently many particles. The definition reads:
connected to the real/physical d dimensional space by a
mapA=(cid:80) A |i(cid:105)(cid:104)αβ|. Then,byintroducingmax- Property 2 (Injectivity). There exists n such that the
iαβ i,αβ (cid:80)
imally entangled states connecting every pair of neigh- mapΓn(X)= i1,...,intr(XAi1···Ain)|i1···in(cid:105)isinjec-
boring virtual Hilbert spaces (usually called entangled tive.
bonds), it is not difficult to prove that the state can be
For each MPS |ψ(cid:105) one can construct a Hamiltonian,
written as
called parent Hamiltonian, for which |ψ(cid:105) is an eigenstate
(cid:88) with eigenvalue 0 .
|Φ(cid:105)= tr[A ···A ]|i ···i (cid:105)
i1 iN 1 N
i1,...,iN Definition 3 (Parent Hamiltonian). Let ρ(k) be the re-
duced density matrix of |ψ(cid:105) for k particles (k will be
wherewecallthematricesK={A ∈M , i=1,...,d}
i D called the interaction length of the parent Hamiltonian).
Kraus operators. A way to work simultaneously with all Let us suppose that {|v (cid:105)}r , with r ≥ 1, is an or-
of them is to define the map thonormal basis for ker(cid:2)iρ(ki)=(cid:3)1. Taking any linear com-
(cid:88) bination of projectors h((cid:126)a) = (cid:80)r a |v (cid:105)(cid:104)v |, we define
V = Ai⊗|i(cid:105). (2) H =(cid:80) τ (h)⊗1 ,whereτ isthi=e1trainsilatiionoperator.
i i i rest i
If a ≥ 0, then the Hamiltonian is positive semidefi-
i
nite and |ψ(cid:105) is indeed a ground state. Moreover H is
frustration free, since |ψ(cid:105) minimizes the energy locally.
Injectivity has now a deep physical significance. If it is
reached for n particles and every a > 0, it ensures that
i
theMPSistheonlygroundstateofits(n+1)-localparent
Hamiltonian, that it is an exponentially clustering state
and that there is a gap above the ground state energy
[1, 2].
In this work we will focus on symmetries of states in-
stead of Hamiltonians. There is however a close connec-
tion between the two approaches. On the one hand, it is
clearthattheuniquegroundstateofasymmetricHamil-
tonianhastokeepthesymmetry. Ontheotherhand,we
have the following
Proposition 4. If an MPS |ψ(cid:105) is invariant under a rep-
resentation of a group, one can choose its parent Hamil-
tonian H invariant under the same representation.
Toseethatitisenoughtonoticethatthesymmetryin
FIG. 1: This figure represents the MPS construction. A pair of thestate(1)impliestheinvarianceofker(cid:2)ρ(k)(cid:3)underthe
virtual spins which are connected to their neighbors via a maxi-
mally entangled state |Ω >= Pd |α α > are mapped into the same symmetry. Symmetrizing ker(cid:2)ρ(k)(cid:3) (i.e., averaging
α=1
physical spins (below). All properties of the state originate from it) w.r.t. the considered group will then yield a symmet-
the mapping between physical and virtual system.
ric h which still constitutes a parent Hamiltonian.
For each MPS there exists a canonical form [2, The-
orem III.7, Lemma IV.4] which assures that one may III. LOCALLY SYMMETRIC MPS
choose all matrices A with a block diagonal structure
i
[22], in such a way that after gathering enough spins to- In this section we analyze the implications of a given
gether, the Kraus operators fulfil: symmetry for a MPS. First, we show that the symmetry
3
transfers to the Kraus operators—generalizing the find- Since X (cid:54)= 0, there exists one block, let us say X ,
1
ingsof[1,3]. Inasecondstepweshowthatthesymmetry different from 0. Then, summing with appropriate coef-
in the Kraus operators imposes that they are essentially ficients again we get that there exists a matrix Y (cid:54)= 0
uniquelydefinedintermsofClebsch-Gordancoefficients. such that
Finally, for the special case of SU(2) one can simplify
even further and analyze the qualitative differences be- tr(cid:2)YA1 A1 A1 (cid:3)=tr(cid:2)X B1B1B1(cid:3), ∀i ,i ,i
i3 i4 i5 1 i3 i4 i5 3 4 5
tween integer and semi-integer spin.
We can now argue as in [2, Lemma IV.4] to conclude the
proof.
A. Characterization of symmetries
If we have now a symmetry given by a compact con-
It was demonstrated in [3] that the Kraus operators nected Lie group G, that is, (1) holds for any g ∈G and
which describe any injective state symmetric under a a representation g (cid:55)→ug, we obtain the following.
group G fulfil the condition (cid:80) ug A = U A U†, where
i ij i g j g Theorem7(Continuoussymmetries). Themapg (cid:55)→P
uandU arerepresentationsofG. Weprovideinthissec- g
isarepresentationofGandthereforethetrivialone. The
tion a generalization in which injectivity is not required.
The N appearing in the proof must be sufficiently large maps g (cid:55)→ eiθgb and g (cid:55)→ Vgb are also representations of
G.
to obtain property 1 after collecting N/5 spins.
We start by proving the case of discrete symmetries,
Proof. Let us start with the map g (cid:55)→ P . From eq. (3)
extendingthedemonstrationtocontinuousgroupsbelow. g
we get
Theorem 5 (Discrete symmetries). Let {A }d be the
i i=1 (cid:88)
Kraus operators which describe a locally invariant MPS W U A U† = ug2g1A =
g2g1 g2g1 h g2g1 jh j
|ψ(cid:105) with respect to a single unitary u, i.e. u⊗N|ψ(cid:105) = j
(4)
eiθ|ψ(cid:105). Then, the symmetry in the physical level can (cid:88)ug2ug1A =W W U U A U† U†
be replaced by a local transformation in the virtual level. jk kh j g2 g1,Pg2 g2 g1 h g1 g2
jk
This means that there exists a unitary U – which can be
taken block diagonal with the same block structure as the
where W is the same unitary as W but with the
A’s in the MPS and composed with a permutation matrix g1,Pg2 g1
blockspermutedaccordingtothepermutationP . Since
among blocks, i.e. U =P(⊕ V ) – such that g2
b b P W = W P and W commutes with all other
g(cid:48) g g,Pg(cid:48) g(cid:48) g
(cid:88) terms appearing in eq. (4), we can multiply successively
u A =WUA U† (3)
ij j i and use property 1 (with L the required block size), to
j get, for all n≥L and all X block-diagonal,
with W =⊕beiθb1b. Wn U XU† =(W W )nU U XU† U† .
g2g1 g2g1 g2g1 g2 g1,Pg2 g2 g1 g1 g2
(5)
Proof. We follow here a reasoning as in the proof of [2,
By taking X = 1 for each block b, we get that P P
LemmaIV.4]. Wecollectthespinsinfivedifferentblocks, b g2 g1
each one of them with property 1. Applying u⊗N gives must be Pg2g1. But since we are assuming the group G
connected, this in turn implies that P = 1 for all g.
usthesameMPS(weincorporatetheglobalphaseinthe g
Withthiswecansplitequation(5)intoblockstoget,for
new matrices) with different matrices B’s, but with the
each b, each n≥L and each matrix X,
sameblockdiagonalformandalso(aftergathering)with
property 1. We now require the following lemma, which
is demonstrated below.
Lemma 6. For each block in the A’s, for instance the einθgb2g1Vgb2g1XVgb2g†1 =ein(θgb1+θgb2)Vgb2Vgb1XVgb1†Vgb2† (6)
one given by matrices A1, there is a block in the B’s, Taking X =1 we obtain
i
given by matrices B1, which expands the same MPS.
i
Since both are now canonical forms of the same in- ein(θgb2g1) =ein(θgb1+θgb2)
jective MPS, by [2, Theorem 3.11], [23], they must be
In particular, when n = L, we get that
related by a unitary and a phase: V1A1iV1† = eiθ1Bi1, L(θb )=L(θb +θb )+2k π and when n=L+1 that
which finishes the proof of the theorem. g2g1 g1 g2 0
(L+1)(θb ) = (L+1)(θb +θb )+2k π. Gathering
Letusprovenowthelemma. Byusingproperty1and g2g1 g1 g2 1
both results, the L can be removed and we obtain
summing with appropriate coefficients, it is possible to
θb =θb +θb +2(k −k )π.
show that there exists a block diagonal D ×D matrix g2g1 g1 g2 1 0
X (cid:54)=0 such that
Finally, to show that g (cid:55)→ Vb is a representation, it is
g
enough to notice that eq. (6) implies that Vb†Vb†Vb
tr(cid:2)A1 ···A1 (cid:3)=tr[XB ···B ], ∀i ,...,i commutes with every matrix. g1 g2 g2g1
i2 i5 i2 i5 2 5
4
does the job if we sum over i’s corresponding to equiva-
lent representations ci = w . The next lemma guaran-
g g
tees that this is all.
Lemma 8. All possible solutions of Equation (7) are
given by (8).
Proof. Any Ω verifying eq. (7) gives
Ω†Ω=w Ω†Ωw†
g g
FIG. 2: The unitary ug applied on the physical level is reflected which means by Schur’s lemma that Ω†Ω = αI and we
in the virtual level as a pair of unitaries Ug.
may assume that, if there is a non-zero solution, it can
be taken an isometry. Moreover, introducing V = C†Ω,
which verifies V†V =I, one has
Atrivialconsequenceofthesetheoremsisthefactthat
having an irreducible representation Ug in the virtual Vwg =(cid:0)⊕icig(cid:1)V (9)
level implies that the MPS has to be injective. We give
an alternative proof of this fact in the appendix without From there one gets that P = VV† is a rank d pro-
having to rely on the MPS canonical form. There we jector (d the dimension of the representation w ) that
g
analyze also when the reverse implication holds. commutes with (cid:0)⊕ ci(cid:1) for all g. By Schur’s lemma, it
i g
is supported on ⊕ H with i’s such that ci = w and in
i i g g
this subspace it is of the form
B. Uniqueness of the construction method
 |β |21 β¯β 1 ···
1 d 1 2 d
Once the theorem which provides the condition that β1β¯21d |β2|21d ···=|β(cid:105)(cid:104)β|⊗1d .
the Kraus operators must fulfil in order to generate in- ··· ··· ···
variant MPS has been established, the next step is to
prove that they can always be constructed by means of This implies that V =|β(cid:105)⊗W for a given d×d unitary
Clebsch-Gordan coefficients. To do that, it is more con- W. Butifwesubstitutethisin(9),sinceweareassuming
venient to work with the map V defined in (2). From a unique fixed representative for each class of equivalent
the definition it is clear that the condition (cid:80)iugijAi = representations, we get W =1d and Ω=(cid:80)iβiφi.
U A U† reads then U ⊗u V = VU . Notice that we
g j g g g g
have removed the dependence on the phase. By Theo- From this we can now conclude:
rem7thiscanbedoneforgroupswithacomplexenough
structure,asSU(2),forwhichthereisnonon-trivialone- Theorem 9. Let us consider a group G and two repre-
dimensional representation. sentationsug (irrep)andUg =(cid:76)iUgDi. Then,thestruc-
Given a compact group G, the tensor product of two ture of all possible maps V fulfilling Ug⊗ugV =VUg is
irreps –we are choosing a single representative for each
class of equivalent irreps– can always be decomposed as α11VDD11 α12VDD12 ··· α1nVDD1n
a direct sum of irrueps⊗v C =C(cid:77)ci V =α21V...DD21 α22V...DD22 ·.·.·. α2nV...DD2n (10)
g g g α VD1 α VD2 ··· α VDn
i n1 Dn n2 Dn nn Dn
where C is a unitary whose elements are called Clebsch- whereVDj isasolution,accordingtoLemma8,toUDi⊗
Gordan coefficients. In what follows we will denote by Di g
φi : Cdi → Cd ⊗ Cd(cid:48) the matrix associated to the re- ugVDDij =VDDijUgDi.
striction of C to the d -dimensional invariant subspace
i
H associatedto theirrep ci, withd,d(cid:48) beingthedimen-
i g
sions of the representations u and v respectively. C. The case of SU(2)
g g
We are interested in possible solutions of
Let us apply the results of the previous section to the
u ⊗v Ω=Ωw ∀g . (7)
g g g case in which G=SU(2). Our construction is a natural
generalization of the one used in [1, 16].
where u ,v ,w are irreps of a given compact group G.
g g g
We consider from now on irreducible representations
It is clear that taking
u of the symmetry on the physical spin. Nevertheless,
g
Ω=(cid:88)β φ (8) a substantial part of the results can be straightforwardly
i i
extended to the reducible case. Hence, we are interested
i
5
in analyzing the restrictions that SU(2) impose in the IV. LIEB-SCHULTZ-MATTIS THEOREM
general solution given by Theorem 9 to the equation
TheLieb-Schultz-Mattistheoremstatesthat,forsemi-
(U ⊗J)V =VU (11)
integer spin, a SU(2)-invariant 1D Hamiltonian cannot
where, with some abuse of notation, J is the SU(2) ir- have a uniform (independent of the size of the system)
rep corresponding to spin J and U = (i ⊕...⊕i ⊕ energygapaboveauniquegroundstate. Thatis,symme-
1 n
s ⊕...⊕s ) is the virtual representation composed of try imposes strong restrictions on the possible behaviors
1 m
n integer irreps and m semi-integer irreps. Note that in of a system. In this section we want to go a step further
theClebsch-GordandecompositionofSU(2)allrepresen- andanalyzewhichimplicationsonecanobtainfromhav-
tations appear with multiplicity one. Therefore there is ing a single symmetric state in a semi-integer spin chain.
onlyoneterminthesumin(8). Atthispointoneshould By restricting our attention to the class of MPS we will
distinguish the cases of J integer or semi-integer. If J is show
integer, zero is the only solution to (i ⊗J)Ω=Ωs and
j k
Theorem 10. Any MPS with an SU(2) symmetry in
(s ⊗J)Ω = Ωi for all j,k, and we get in (10) a block
k j
the sense of (1) with u irrep and even physical dimen-
diagonal structure: g
sion d cannot be injective. By Theorem 11 of [2] this
α1Vi1 ··· αnVin 0 ··· 0  implies that it cannot be the unique ground state of any
 1...i1 ... 1...i1 ... ... ...  frustration free Hamiltonian.
α1Vi1 ··· αnVin 0 ··· 0  Proof. LetusassumethattheMPSisinjectiveandprove
V = n0in ··· n0in αn+1Vs1 ··· αn+mVsm the theorem by contradiction. Theorems 5 and 7 guar-
 ... ... ... n+1... s1 ... n+1... s1  antee that
(cid:88)
0 ··· 0 αnn++m1 Vssm1 ··· αnn++mmVssmm ugjkAj =UgAkUg† . (12)
j
The paradigmatic example in this case is the AKLT
state[4],whichcorrespondstothecaseofJ =1,U =1/2 We consider u=eiJz with (Jz)j,k =δj,k(cid:0)k−(d+1)/2(cid:1),
in (11). In [1], the authors generalized the AKLT model k =1,...,d. Then, eq. (12) gives
to arbitrary integer J and U irreducible. We will call
the resulting MPS FNW states. It is shown in [1] how eiϕkAk =UAkU† (13)
for U = J FNW states are unique ground states of frus-
2
trationfreenearest-neighborinteractions. Analternative foraunitaryU andϕk half-integer. Wefinishbyproving
construction focused on the restrictions imposed by the thatifN isodd, tr(Ak1···AkN)=0andhencetheMPS
SU(2) symmetry on the density matrix instead of the cannotbeinjective. From(13)wegettr(Ak1···AkN)=0
Kraus operators can be found in [18]. unless (cid:80)N ϕ = N(d+1)/2. The latter is, however,
i=1 ki
If J is semi-integer, zero is the only solution to (s ⊗ impossibleforN oddasthenthel.h.s. isintegerwhereas
j
J)Ω = Ωs and (i ⊗J)Ω = Ωi for all j,k, and we get the r.h.s. is half-integer.
k k j
in (10) an off-diagonal structure:
From the proof one may get the impression that
V = only U(1) symmetry is required, and this is indeed the
case if the generator of such symmetry has eigenvalues
 0 ··· 0 αn+1Vs1 ··· αn+mVsm
1 i1 1 i1 −m/2,...,m/2 as above. The next example shows that
 ... ... ... ... ... ...  this is, however, not true for any U(1) symmetry, which
 0 ··· 0 αn+1Vs1 ··· αn+mVsm in turn shows that a larger symmetry like SU(2) is re-
α1 Vi1 ··· αn Vin n 0 in ··· n 0 in  quired for the Lieb-Schultz-Mattis theorem.
 n+1 s1 n+1 s1 
 ... ... ... ... ... ...  Example 11. Let us consider a local symmetry gen-
α1 Vi1 ··· αn Vin 0 ··· 0 erated by G = eiβH for a hermitian matrix H. Let
n+m sm n+m sm us choose the physical dimension d = D2 − D, which
It is clear that the virtual representations must be re- is always even, and the set of Kraus operators K =
ducible now, which is very much related to the Lieb- (cid:8)A =|i(cid:105)(cid:104)j|, i(cid:54)=j(cid:9). Select α ,...,α ∈ R such
(i,j) 1 D
Schultz-Mattis theorem, as we will show in the following that α − α (cid:54)= 0 if i (cid:54)= j and H the diagonal ma-
i j
section. The paradigmatic example in this case is the trix H =(cid:80) (α −α )|(i,j)(cid:105)(cid:104)(i,j)| (which has in addi-
i(cid:54)=j i j
Majumdar-Ghosh model [5], which corresponds to J = 21 tion only non-zero eigenvalues). With Uβ = eiβΩ where
and U = 1 ⊕0. A generalization of this model for the Ω=diag[α ...α ] it is clear that
2 1 D
caseofarbitraryJ andU =F⊕0,wasrecentlyproposed
in [9]. eiβ(αi−αj)A =U A U†
(i,j) β (i,j) β
In general, it is possible to find a set of representa-
tions which fits into any model with SU(2) symmetry, sotheMPSgeneratedbymeansoftheKrausoperatorsK
for instance [8, 17, 19, 20]. hasthelocalsymmetryG. Moreover,theMPSistrivially
6
injective when D ≥ 3. We can prove this by choosing A. Completeness of the parent Hamiltonian
arbitrary k and k(cid:48). Since D ≥ 3, we can always find an method
l such that k(cid:48) (cid:54)= l (cid:54)= k and then |k(cid:105)(cid:104)k(cid:48)| = |k(cid:105)(cid:104)l|l(cid:105)(cid:104)k(cid:48)| =
A(k,l)A(l,k(cid:48)). Theorem 13. Given an MPS |ψ(cid:105), any translational in-
variant Hamiltonian having it as a local eigenstate is of
Let us remark that this counter-example is applicable
the form a1+H where H is a parent Hamiltonian for
to spin ≥ 5. Indeed, one can prove Theorem 2 for U(1)
2 |ψ(cid:105) in the sense of Definition 3.
and spin 1, which is the content of the following propo-
2
sition. The case of spin 3/2 remains an open question. Proof. Letuscallhthelocalhamiltonian. Byhypothesis
of local eigenstate,
Proposition 12. If |Φ(cid:105) is an MPS with physical dimen-
sion d=2 and invariant under U(1), then |Φ(cid:105) cannot be
hρ=λρ (14)
injective.
Proof. We will show it by contradiction. By choosing a for certain λ ∈ R. This implies [ρ,h] = 0 and hence one
(cid:80)
basis where the physical unitary u is diagonal, the con- canfindasetofprojectorsP ={Pi,i=1,...,r| iPi =
dition on the Kraus operators becomes 1} such that we can decompose both ρ and h by means
(cid:80) (cid:80)
of them, i.e. h = a P and ρ = b P , where C
i i i j∈C j j
eiλnφAn =eiHφAne−iHφ represents the set of projectors which describe the sup-
port of ρ. Using eq. (14) with this decomposition gives
whereH isthehermitiangeneratorofthesymmetry. Let
that a =λ for all i∈C and hence
i
us expand the expression for infinitesimal angles
(cid:88) (cid:88)
h= a P +λ P
[H,A ]=λ A i i i
n n n
i∈C⊥ i∈C
which is the equation of eigenvalues for the operator (cid:88)
= (a −λ)P +λ1.
L(•)=[H,•]. This can be transformed into an ordinary i i
eigenvalue equation for the matrix operator L = H ⊗ i∈C⊥
1−1⊗H¯. The diagonalization can be easily performed Then, the translational invariance hamiltonian is H =
by taking the spectral decomposition of H = (cid:80)iµiPi, (cid:80)jτj(h) ⊗ 1rest, where τ is the translation operator.
where Pi are orthogonal projectors. It straightforwardly The theorem follows from replacing the result for the
follows that the eigenvalues of L are λij = µi −µj and local hamiltonian and comparing this with Definition 3
the corresponding eigenoperators fulfil Aij =PiAijPj. of parent hamiltonian.
Let us focus now on the case d = 2. Then, we have
that A = P A P and A = P A P for some α,β,γ. This Theorem shows that, given an MPS |ψ(cid:105), looking
1 1 1 α 2 β 2 γ
If β =1 P X =X for all X ∈span{A ···A } and the forallpossibleparentHamiltoniansofinteractionlength
1 i1 in
MPS cannot be injective. The same happens if α = γ. k is equivalent to look for all possible solutions to the
So let us assume that β (cid:54)= 1 and γ (cid:54)= α. Now if α = 1, equation
wehaveA =P A P , A =(1−P )A (1−P )andthe
1 1 1 1 2 1 2 1
MPSisblockdiagonalandhencenon-injective. Thesame hρ(k) =λρ(k), (15)
happens if β = γ. So α (cid:54)= 1 and β (cid:54)= γ and this gives
A2 = 0 = A2 which implies that span{A ···A } = with λ = tr(cid:2)hρ(k)(cid:3). The next lemma gives yet another
1 2 i1 in equivalent formulation, which is the one we will use in
span{A A A A ··· ,A A A A ···}hasdimension≤2.
1 2 1 2 2 1 2 1
the sequel.
Lemma 14. Given a Hermitian matrix h and a density
matrix ρ, hρ=λρ if and only if
V. GENERAL CONSTRUCTION OF SU(2)
TWO-BODY HAMILTONIANS WITH MPS tr(cid:2)h2ρ(cid:3)−tr[hρ]2 =0. (16)
EIGENSTATES
Proof. One implication is clear. For the other, let us
We have seen in Definition 3 a way, called the par- write (cid:104)h(cid:105) for tr[hρ]1. By assumption
ent Hamiltonian method, to construct local SU(2)-
symmetric Hamiltonians with MPS as eigenstates. In tr(cid:2)(h−(cid:104)h(cid:105))2ρ(cid:3)=tr(cid:2)h2ρ(cid:3)−tr[hρ]2 =0.
this section we first prove that this method is the most
general one to find Hamiltonians having a given MPS as So ρ1/2(h−(cid:104)h(cid:105))2ρ1/2 =0, since it is a positive operator
local eigenstate, that is, being an eigenstate of each local
with trace 0. This implies that (h−(cid:104)h(cid:105))ρ=0 and hence
term in the Hamiltonian. Then, we show examples (in-
hρ=λρ.
cludingtheAKLTandMajumdar-Ghoshstates)ofMPS
that are excited eigenstates of local two-body transla- With this at hand we can systematically search for
tionally invariant SU(2)-symmetric Hamiltonians. More MPS that are excited local eigenstates of SU(2) invari-
examples are then provided in the appendix. ant Hamiltonians with two-body interactions. We will
7
proceed as follows. We start with a given SU(2) sym-
metric MPS |ψ(cid:105) and fix the interaction length n. Then
we look for possible solutions to Eq. (16) of the form
2J
h= (cid:88) (cid:88)a(α)(S(cid:126) ◦S(cid:126) )α+a 1, (17)
ij i j 0
i<j≤nα=1
to ensure SU(2) symmetry and two body interactions in
theHamiltonian. Finally,toguaranteethattheMPS|ψ(cid:105)
isanexcitedstate,wewillfindanotherSU(2)symmetric
MPS with less energy that will act as a witness. In the
nextsectionwewillillustratethisprocedurestartingwith
|ψ(cid:105) the AKLT, the Majumdar-Ghosh state, and general-
izations. Throughout we work in the thermodynamical
limit N →∞.
B. Examples of SU(2) two-body Hamiltonians
1. Spin 1
FIG. 3: Space of parameters of the local Hamiltonian h for the
AKLT state and n=3. The orange volume represents the points
Let us consider the AKLT state as a first example. Its
√ where the state is the local (and hence the global) ground state.
Kraus operators are A = − 2σ−, A = σz, A =
√ −1 0 1 Thegreenvolumerepresentspointscorrespondingtoexcitedstates
2σ+. detected with the witness 3 ⊕ 1.
2 2
In the case n = 2 the only solution to Eq. (16) is the
AKLT Hamiltonian. In the case n=3, the solutions are
(cid:72)
given by (cid:72) j
(cid:72) 1 1 3 2 5 3 7
J (cid:72)(cid:72) 2 2 2 2
h=(−3v +v +3v )(S(cid:126) ◦S(cid:126) )+v (S(cid:126) ◦S(cid:126) )2+ 1 2 1 (cid:4) (cid:4) (cid:4) (cid:4) (cid:4)
1 2 3 1 2 3 1 2
1 1 2 — 5 3 (cid:4) (cid:4) (cid:4) (cid:4)
(−3v +v )(S(cid:126) ◦S(cid:126) )− (−3v +v )(S(cid:126) ◦S(cid:126) )2+
2 1 2 1 3 2 1 2 1 3 3 — — 4 2 2 1 (cid:4)
v (S(cid:126) ◦S(cid:126) )+v (S(cid:126) ◦S(cid:126) )2
2 2 3 1 2 3
TABLE I: Table of results for FNW states with physical spin J
wheretheeigenvaluecorrespondingtotheAKLTstateis
and virtual spin j. The numbers in the table are the number of
7v1−3v2−2v3. The total translational invariant Hamil- parameters the obtained families of Hamiltonians depend on. The
tonian is then (cid:4) represent the cases for which no solution was found.
H =(cid:88)(−3v +2v +3v )(S(cid:126) ◦S(cid:126) )+
1 2 3 i i+1
the local Hamiltonian h depends on cannot be reduced,
i
1 whichmeansthattherearenon-physicalparametersinit.
(v1+v3)(S(cid:126)i◦S(cid:126)i+1)2+ 2(−3v1+v2)(S(cid:126)i◦S(cid:126)i+2)− In Fig. 4 we have represented the problem above (n=3
and AKLT state) in terms of the physical parameters.
1
(−3v +v )(S(cid:126) ◦S(cid:126) )2 ThepositiveaxisbcorrespondstheretotheusualAKLT
2 1 2 i i+2
Hamiltonian.
which contains the usual AKLT model. It is not difficult Concerning FNW states, that is integer spin J and
to check that there is a region in the parameter space virtual irrep j, we have performed an exhaustive search
where the AKLT state is still the ground state of this andtableIgathersthemainresults. Thestudyhasbeen
Hamiltonian. Tofindregionswhereitisanexcitedeigen- carried out by increasing n and studying the number of
statewewilluseasawitnesstheSU(2)symmetricMPS parameters which the family of Hamiltonians depends
associatedtothevirtualrepresentation 3⊕1 (seeSection on (notice that the case of interaction length n contains
2 2
III). The result is plotted in fig. 3, where one sees the the case of interaction length n−1). We have increased
existence of points in this family of spin 1 Hamiltonians n until the number of parameters stops growing. In all
for which the AKLT state is an excited state. the cases considered in the table, a saturation occurs
Notethatitispossibletoperformachangeofvariables when n > 3, i.e. considering more than 3 particles does
in the total Hamiltonian, for instance a → 1(−3v + apparently not add new Hamiltonians.
2 1
v ) and b → v +v , such that it depends only on two
2 1 3
parameters. However, the number of parameters that Letusalsointroduceanewstateofspin1withvirtual
8
2. Spin 1
2
LetusconsidernowthetheMajumdar-Ghoshstateas
an example with semi-integer spin. The Kraus operators
are now
   
0 √1 0 0 0 0
2
A−1 =0 0 −1 , A1 =1 0 0
2 2
0 0 0 0 √1 0
2
As in the previous case, we do not find any solution for
n = 2 and only the Majumdar-Ghosh Hamiltonian for
the cases n = 3 and n = 4. For n = 5 the solutions to
Eq. (16) are given by
h=(v −v +v )(S(cid:126) ◦S(cid:126) )+(v −v +v )(S(cid:126) ◦S(cid:126) )+
1 2 4 1 2 1 2 4 1 3
v (S(cid:126) ◦S(cid:126) )+v (S(cid:126) ◦S(cid:126) )+v (S(cid:126) ◦S(cid:126) )
FIG.4: SpaceofphysicalparametersoftheglobalHamiltonianH 3 1 4 3 1 5 4 2 3
corresponding to n = 3 and the AKLT state. The orange points +(−v +v +v )(S(cid:126) ◦S(cid:126) )+v (S(cid:126) ◦S(cid:126) )+v (S(cid:126) ◦S(cid:126) )+
1 2 3 2 4 3 2 5 2 3 4
represent where the state is the local (and hence the global) GS.
Thegreensurfacerepresentspointscorrespondingtoexcitedstates v1(S(cid:126)3◦S(cid:126)5)+v1(S(cid:126)4◦S(cid:126)5) (18)
detected by means of the witness 3 ⊕ 1.
2 2
and the energy associated to the state is −3(v +v ).
(cid:80) 4 1 4
The total Hamiltonian H = τ (h) is given by
i i
spin 1, given by the Kraus operators
H =(cid:88)2(v +v )(S(cid:126) ◦S(cid:126) )+(v +v +v )(S(cid:126) ◦S(cid:126) )
    1 4 i i+1 1 3 4 i i+2
0 1 0 1 0 0
1 1 i
A1 = √20 0 1 , A0 = √20 0 0  +2v3(S(cid:126)i◦S(cid:126)i+3)+v3(S(cid:126)i◦S(cid:126)i+4) (19)
0 0 0 0 0 −1
AsintheAKLTcase,bymeansofachangeofvariables
a→v and b→v +v , the number of physical param-
  3 1 4
0 0 0 eters in the total Hamiltonian is 2, compared with the
1
A = √ −1 0 0 four parameters the local Hamiltonian depends on. The
−1  
2
Majumdar-Ghosh state is an excited local eigenstate for
0 −1 0
a region in the space of parameters, which in this case is
The total translational invariant hamiltonian which has detectedbythewitness 1⊕1⊕0,asshowninfig. 5. The
2
this state as eigenstate is usual Majumdar-Ghosh Hamiltonian [24] corresponds to
the positive axis b.
H = (cid:88)(S(cid:126) ◦ S(cid:126) )2 − (S(cid:126) ◦ S(cid:126) ) − (S(cid:126) ◦ S(cid:126) )2
i i+1 i i+2 i i+2
i
3. Spin 3
This state is injective and a local excited state. The fact 2
that this state is an excited state of the global hamilto-
Let us consider as final example the SU(2) symmetric
nian can be checked as above by means of the witness
MPS corresponding to spin 3 and virtual representation
1⊕0. 2
3 ⊕0. For n=3, the solutions to Eq. (16) are given by
2
h=v (S(cid:126) ◦S(cid:126) )+v (S(cid:126) ◦S(cid:126) )2+v (S(cid:126) ◦S(cid:126) )3+
3 1 2 2 1 2 1 1 2
(2v −v +v )(S(cid:126) ◦S(cid:126) )+(4v −v )(S(cid:126) ◦S(cid:126) )2
1 2 3 1 3 1 2 1 3
+v (S(cid:126) ◦S(cid:126) )3+v (S(cid:126) ◦S(cid:126) )+v (S(cid:126) ◦S(cid:126) )2+
1 1 3 3 2 3 2 2 3
v (S(cid:126) ◦S(cid:126) )3 (20)
1 2 3
and the energy associated to the MPS is in this case
−15(165v −60v +16v ). TheglobalHamiltonianreads
64 1 2 3
9
as a ’laboratory’ where to search for some generic math-
ematical and physical properties of states that are rel-
evant in 1-dimensional spin chains. Later on, one may
use more powerful mathematical methods to try to ex-
trapolate those properties to general spin chains. Fur-
thermore, many of the techniques used in the present
FIG.5: SpaceofphysicalparametersofthetotalHamiltonianfor
n=5associatedtotheMajumdar-Ghoshstate. Theorangepoints
representwherethestateisthelocal(andhencetheglobal)ground
state. Thegreensurfacerepresentspointscorrespondingtoexcited
states detected by means of the witness 1 ⊕1⊕0.
2
now
FIG. 6: Space of parameters of the spin 3 model. The orange
2
H =(cid:88)2v (S(cid:126) ◦S(cid:126) )+2v (S(cid:126) ◦S(cid:126) )2 points are obtained numerically and they represent values of the
3 i i+1 2 i i+1 parameterswheretheMPSstateistheGS.Thegreenvolumerep-
i resentspointscorrespondingtoexcitedstatesdetectedwiththewit-
+2v1(S(cid:126)i◦S(cid:126)i+1)3+(2v1−v2+v3)(S(cid:126)i◦S(cid:126)i+2)+ ness 32 ⊕1⊕0.
(4v −v )(S(cid:126) ◦S(cid:126) )2+v (S(cid:126) ◦S(cid:126) )3 (21)
1 2 i i+2 1 i i+2
It is remarkable that in this case there are no spurious work are amenable of an extension to higher spatial di-
parameters in the local Hamiltonian h. Considering the mensions, where PEPS play the role of MPS. In Ref. [3]
family of states whose virtual representation is 3 ⊕1⊕0 some first results in this direction were derived, which
as a witness, it is possible to demonstrate that2there is will be generalized in a further publication.
a region in the space of parameters of the Hamiltonian
for which the MPS is an excited eigenstate, as shown in
Fig. 6.
VII. ACKNOWLEDGMENTS
VI. CONCLUSIONS M. Sanz would like to thank Dr. Miguel Aguado, Dr.
MarcoRoncagliaandDr. FrankVerstraetefortheirhelp-
Despite the fact that all our results are restricted to ful discussions and the QCCC Program of the EliteNet-
thefamilyofTIMPS,theirrelevanceismanifestedbythe zWerk Bayern as well as the DFG (FOR 635, MAP
factthatthosestatesapproximateallgroundstatesof1- and NIM) for the support. D. Perez-Garcia acknowl-
dimensional Hamiltonians with short range interactions. edges financial support from Spanish grants I-MATH,
Thus, one would expect that the properties derived for MTM2005-00082andCCG07-UCM/ESP-2797andM.M.
MPSwouldberelevantinamoregeneralcontext. More- WolfacknowledgessupportbyQUANTOPandtheDan-
over, due to their simplicity, MPS can be then thought ish Natural Science Research Council(FNU).
[1] M. Fannes, B. Nachtergaele and R. W. Werner, Comm. Cirac, Quantum Inf. Comput. 7, 401 (2007).
Math. Phys. 144, 443 (1992). [3] D.Perez-Garcia,M.M.Wolf,M.Sanz,F.Verstraeteand
[2] D. Perez-Garcia, F. Verstraete, M. M. Wolf and J. I. J. I. Cirac, Phys. Rev. Lett. 100, 167202 (2008)
10
[4] I.Affleck,T.Kennedy,E.H.LiebandH.Tasaki,Comm. uctstate. Therefore,thedesiredresultthatirrep implies
Math. Phys. 115, 477 (1988) injectivity is a consequence of the following theorem.
[5] C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10,
1399 (1969) Theorem 15. Let us take a completely positive map
[6] F.D.M. Haldane, Phys. Lett. A 93, 464 (1983); Phys. E : M −→ M that is covariant for an irrep of a
D D
Rev. Lett. 50, 1153 (1983). compact connected Lie group G. Then, either E is the
[7] O. Bratteli, P.E.T. Jorgensen, A. Kishimoto and R.F. ideal channel or it is trace preserving and the identity its
Werner, J. Operator Theory 43 97 (2000) unique fixed point.
[8] A. Kluemper, A. Schadschneider and J. Zittartz, Euro-
phys. Lett. 24, 293 (1993). Proof. Letusconsiderafixedpoint∆ofE. ThenU ∆U†
g g
[9] V. Karimipour and L. Memarzadeh, Phys. Rev. B 77, isalsoafixedpointbecauseofthecovariance. Therefore,
094416 (2008)
integrating under the Haar measure and using Schur’s
[10] E. Lieb, T. Schultz and D. Mattis, Ann. Phys. 16, 407 lemma,1isalsoafixedpoint. Asimilarargumentshows
(1961).
that E is also trace preserving.
[11] M. B. Hastings, Phys. Rev. B 69, 104431 (2004)
Now we can apply Lu¨ders’ theorem [7], which ensures
[12] N.Schuch,J.I.CiracandF.Verstraete,Phys.Rev.Lett.
that the set of fixed points P of E coincides with the
100, 250501 (2008)
[13] S. O¨stlund and S. Rommer, Phys. Rev. Lett. 75, 3537 commutantK(cid:48) ofthesetofKrausoperatorsofE. Thisis
(1995) trivially a C∗-subalgebra of MD. Moreover, we know by
[14] F. Verstraete and J. I. Cirac, Phys. Rev. B 73, 094423 theclassificationoftheC∗-subalgebrasinMD thatthere
(2006) exists a unitary V ∈ M such that VPV† = ⊕ (M ⊗
D i ni
[15] H. Nakazato, Y. Hida, K. Yuasa, B. Militello, A. Napoli 1 )=A.
n(cid:48)
and A. Messina, Phys. Rev. A 74, 062113 (2006) Ti he equivalent representation V = VU V† is also an
[16] S. Singh, H.-Q. Zhou and G. Vidal, arxiv:cond- g g
irrep and fulfils that V AV† = A. This means that the
mat/0701427v1 (2007) g g
block structure of A remains invariant under the action
[17] B. Kumar, Phys. Rev. B 66, 024406 (2002)
of V by conjugation. Now we use that
[18] J. Dukelsky, M. A. Martin-Delgado, T. Nishino and G. g
Sierra, Europhys. Lett. 43, 457 (1998).
[19] J.M. Roman, G. Sierra, J. Dukelsky and M.A. Martin- VgAVg† ⊂A⇔[J,A]⊂A for all generators J. (A1)
Delgado, J. Phys. A 31, 48 9729 (1998)
[20] S.RachelandM.Greiter,Phys.Rev.B78,134415(2008) This implies that J has the same block structure as A.
[21] D.SenandN.Surendran,Phys.Rev.B75,104411(2007) If there is more than one block, the representation is
[22] Eventuallyaftergatheringsomespinstogethertoneglect reducible. If A=M ⊗1 , then we use again eq. (A1):
n n(cid:48)
theperiodiccomponents—somethingthatwewillalways The Schmidt decomposition allows us to take J =
assume. (cid:80) A ⊗B wheretheB ’sformabasisofM ,withB =
[23] The condition of [2, Theorem 3.11] that the canonical 1.iThien,ieq. A1 givesithat (cid:80) [A ,M ]⊗nB(cid:48) = C ⊗11,
form in the OBC must be unique can be dropped by eq. which implies that A is proporitionial tno 1 foir all i ≥ 2.
(3) of [Lamata et al, PRL 101, 180506 (2008)] This gives J =1⊗Xi+Y ⊗1 and hence V =Vg⊗Vg,
[24] The family of hamiltonians constructed in [17] is quite g 1 2
which is reducible unless A = 1 or A = M (which im-
remarkable. In this paper the ground state of the hamil- N
plies that E is the ideal channel).
tonianwhichcorrespondstob=2aiscalculated,andfits
perfectly to our results.
Although the implication in the opposite direction
could also seem true, it is not, as shown by the following
example.
APPENDIX A: RELATIONS BETWEEN
IRREDUCIBILITY AND INJECTIVITY Example 16. Let us consider the family of SU(2) sym-
metric MPS of spin 1 with a reducible virtual represen-
tation 1 ⊕ 3 given by the following maps (see Section
Inthisappendixwegiveadirectproofofthefactthat 2 2
III).
anirreduciblerepresentationinthevirtuallevelofasym-
metric MPS implies that the MPS is injective. We also  1 3
see that the reverse inclusion is not true in general, but eiα11cosθ1V12 eiα12sinθ2V12
it holds under some conditions on the Kraus operators. V˜ = 2 2 
 
defiWneinhgaavne tMoPreScaKll=th{aAt, ,g.iv.e.n,Aa }se,twoefcKanraduesfionpeearantoarss- eiα21sinθ1V321 eiα22cosθ2V323
1 d 2 2
sociated completely positive map E(X) = (cid:80)d A XA†.
i=1 i i ItisnotdifficulttocheckthattheMPSisinjectiveexcept
The symmetry in the MPS transfers then to the covari-
in particular directions in space, such as those for which
aalnlcXe o.fIttheiscshhaonwnnel,inth[a1t, i2s],tEh(aUtgiXf UEg†i)s=traUcgeEp(Xre)sUerg†vifnogr the isometry breaks into blocks, i.e. θi =nπ2.
and has 1 as its unique fixed point, then the MPS is in- Althoughtheequivalenceisnottrueingeneral,wecan
jective. Moreover, it is trivial to see that if E is the ideal stillgiveasufficientconditionwhichapplies,forinstance,
channel (E(X) = X for all X), then the MPS is a prod- to the AKLT and other FNW states. Let us recall from