Topological nature of spinons and holons: Elementary excitations from matrix product states with conserved symmetries
V. Zauner-Stauber, L. Vanderstraeten, J. Haegeman, I.P. McCulloch, F. Verstraete
TTopological nature of spinons and holons: Elementary excitations from matrixproduct states with conserved symmetries
V. Zauner-Stauber, L. Vanderstraeten, J. Haegeman, I.P. McCulloch, and F. Verstraete
1, 2 Vienna Center for Quantum Technology, University of Vienna, Boltzmanngasse 5, 1090 Wien, Austria Ghent University, Faculty of Physics, Krijgslaan 281, 9000 Gent, Belgium ARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics,The University of Queensland, St Lucia, QLD 4072, Australia
We develop variational matrix product state (MPS) methods with symmetries to determine disper-sion relations of one dimensional quantum lattices as a function of momentum and preset quantumnumber. We test our methods on the XXZ spin chain, the Hubbard model and a non-integrableextended Hubbard model, and determine the excitation spectra with a precision similar to the oneof the ground state. The formulation in terms of quantum numbers makes the topological nature ofspinons and holons very explicit. In addition, the method also enables an easy and efficient directcalculation of the necessary magnetic field or chemical potential required for a certain ground statemagnetization or particle density.
I. INTRODUCTION
Matrix product state (MPS) based methods suchas DMRG,
TEBD and VUMPS have proven tobe invaluable tools for simulating ground states of onedimensional quantum lattice models. By formulatingthose MPS methods in terms of manifolds and tangentspaces, it has recently been shown that excitation spec-tra or dispersion relations as a function of momenta canreadily be determined once the ground state is writ-ten in terms of a uniform (translation invariant) groundstate. Those tangent space methods extend the worksof ¨Ostlund and Rommer, in which a slightly more lim-ited ansatz was used. In this paper, we extend thosetangent space methods to accommodate for U (1) symme-tries, which are necessary to simulate quantum systemsexhibiting a large amount of entanglement to good pre-cision, such as e.g. the Hubbard model. The symmetricformulation further allows for targeting excitations withcertain quantum numbers only, which greatly helps indisentangling rich excitation spectra of models which e.g.host several different types of elementary excitations.In Sec. II we develop the theory of symmetric uniformMPS, while in Sec. III we introduce the necessary toolsfor formulating the excitation ansatz in the presence ofsymmetries, where we also generalize to multi site unitcells. This is done both for topologically trivial and non-trivial excitations which are domain wall like, such asspinons and holons. In Sec. IV, we demonstrate the use-fulness of the methods by simulating excitation spectra ofthe integrable XXZ and Fermi Hubbard model, as well asthe non-integrable extended Fermi Hubbard model. Theuse of symmetries makes the topologically nontrivial na-ture of spinons and holons very clear and intuitive. Fi-nally, we conclude with a summary and outlook in Sec. V. II. SYMMETRIC UNIFORM MPS
We begin by defining properties for symmetric uni-form MPS (suMPS), where we restrict the discussion tothe case of abelian symmetries (e.g. Z n parity, or U (1)like particle number or magnetization). While symmet-ric tensor networks and the use of conserved quantitiesin tensor network algorithms have been addressed in nu-merous previous works , we reiterate here in detailtheir consistent use in the context of MPS in the ther-modynamic limit.In the following we closely use and follow notation andnomenclature of Ref. 9 and restate here only the mostimportant concepts. For details we refer the reader toRef. 9 (in particular Sec. II.A and II.E.). We consider atranslation invariant uniform MPS in the thermodynamiclimit in the mixed canonical representation | Ψ( A ) (cid:105) = (cid:88) σ ( . . . A σ n − L A σ n C A σ n +1 R . . . ) | σ (cid:105) (1a)= (cid:88) σ ( . . . A σ n − L A σ n L CA σ n +1 R A σ n +2 R . . . ) | σ (cid:105) . (1b)Here A L , A R ∈ C D × d × D , with d the local Hilbert spacedimension and D the MPS bond dimension, both de-scribe the same state in different gauge representationsand are related by the gauge transformation matrix C via A σL C = CA σR = A σC , (2)where we have also defined the center site matrix A C . A L and A R then fulfill the left and right gauge constraints (cid:88) σ A σL † A σL = (cid:88) σ A σL CC † A σL † = CC † (3a) (cid:88) σ A σR A σR † = (cid:88) σ A σR † C † C A σR = C † C. (3b)and the singular values of C correspond to the Schmidtvalues of a bipartition of the state | Ψ( A ) (cid:105) . a r X i v : . [ c ond - m a t . s t r- e l ] A p r An N site unit cell uMPS, which is invariant under atranslation over N sites, is described by N independentMPS matrices A ( k ) , k = 1 , . . . , N , which define the unitcell tensor A Σ n = A (1) σ nN +1 . . . A ( N ) σ nN + N , (4)where Σ n = ( σ nN +1 , . . . , σ nN + N ) is a combined index.For an N site unit cell uMPS we then write | Ψ( A ) (cid:105) = (cid:88) σ ( . . . A Σ n − A Σ n A Σ n +1 . . . ) | σ (cid:105) , (5)where the integer n labels unit cells, not sites. Here wehave not explicitly specified the gauge representations ofthe individual MPS matrices within the unit cell, butwe will henceforth assume all states to be in the mixedcanonical representation (1).Global symmetries of an MPS that is invariant un-der certain symmetry operations can be easily encodedin symmetry properties of the local MPS matrices A .For MPS with abelian symmetries this simply amountsto attaching quantum numbers to all indices appearingin tensor contractions and constraining the MPS matri-ces A to transform as irreducible representations of theglobal symmetry group. In practice, states on the phys-ical and virtual level are therefore grouped into distinctquantum number sectors, and the matrices are of sparseblock form. The action of the symmetry group then de-termines which combinations of quantum numbers areallowed, i.e. which blocks are non zero. We denote suchsymmetric MPS matrices as A ( s,σ )( a,α )( b,β ) b = a ⊕ s (6)Here, a and b are quantum numbers labeling symme-try sectors on the virtual level and α , β label stateswithin these sectors (degeneracy indices), while s denotesthe quantum numbers associated with the local physicalstates σ , determined by a choice of numerical represen-tation s ( σ ) (see below for examples), and ⊕ denotes thegroup action. In the presence of several symmetries,the quantum numbers are multi valued. In the followingwe will often omit quantum number labels or degeneracyindices for better readability. To avoid ambiguity, wedenote quantum numbers with Latin letters and physi-cal/degeneracy indices with Greek letters. At times wealso write – in a slight abuse of notation – A σab , where itis understood that b = a ⊕ s ( σ ).In most cases the group action reduces to a simple(perhaps modular) addition/subtraction of properly de-fined quantum numbers, which we represent as (tuplesof) rational numbers. In the following we will thereforedenote a ⊕ b just as a + b , and the action with the inverse a ⊕ ¯ b as a − b , where b ⊕ ¯ b = 0. Furthermore, we regard s ( σ ) as a numerical representation of the physical statefor quantum number arithmetic only, not necessarily asa strict group representation.Without loss of generality, in (6) we have implicitlydefined s and a as ingoing , and b as outgoing quantum numbers. Upon concatenating symmetric MPS matri-ces, outgoing indices are then connected to ingoing in-dices only, and only sector blocks with matching quantumnumbers are contracted. For example, a concatenation oftwo such matrices then yields C σ σ ac = (cid:88) b A σ ab B σ bc c = a + s ( σ ) + s ( σ ) . (7)Quantum states on finite systems of size L and with acertain quantum number Q can then be constructed bydefining the quantum number of the left virtual boundarystate to be zero, and the quantum number of the rightvirtual boundary state to be the desired quantum num-ber Q . Using symmetric MPS matrices (6), only basisstates | σ . . . σ L (cid:105) fulfilling the constraint (cid:80) j s ( σ j ) = Q will then contribute to the overall state. This procedureis good practice and widely used in implementations ofsymmetry exploiting MPS algorithms. However, quantum numbers of ground states with U (1)symmetry in particular usually scale with the systemsize L (e.g. particle number N = L/ and to subtract the value of the desired quan-tum number density ˜ q = Q/L as part of a modified groupaction for each individual MPS matrix b = a + s − ˜ q = a + ˜ s (8)Here we have defined a shifted quantum number˜ s = s − ˜ q (9)for the physical index, which is the original quantumnumber s , offset by the desired overall density ˜ q .This scheme is now easily scalable to the thermody-namic limit and we endow uniform MPS matrices withthis modified group action (8) to obtain symmetric uni-form MPS (suMPS) with well defined quantum number densities . The generalization to multi site unit cells isstraightforward by endowing every matrix within the unitcell with the modified group action (8) and a consistentdefinition of the quantum number density. Consequently,the unit cell size N has to be chosen in accordance withthe desired quantum number density ˜ q . For example, aspin S = 1 / N even in order to host an equal number of up and downspins. Choosing a single site unit cell in this case resultsin a superposition of possible zero magnetization suMPSand hence a non injective MPS (i.e. the transfer matrixhas multiple dominant eigenvalues with magnitude one).We show concrete examples for shifted quantum num-bers ˜ s and required unit cell sizes N for spin S = 1 / m in Table I,and states of spinful electrons with fixed particle andmagnetization densities ( n, m ) in Table II. For an il-lustration of the conventional finite size scheme and themodified scheme for infinite systems, see Fig. 1. m − / − / |↑(cid:105) / / / / |↓(cid:105) − / − / − / − / N − s ( σ ) and required unit-cell sizes N for various magnetization densities m for a S = 1 / | σ (cid:105) = {|↑(cid:105) , |↓(cid:105)} ). The first column denotes the chosen(unmodified) quantum number representations s ( σ ).( n, m ) − (1 ,
0) (2 / ,
0) (5 / , − / | (cid:105) (0 ,
0) ( − ,
0) ( − / ,
0) ( − / , / |↓(cid:105) (1 , − /
2) (0 , − /
2) (1 / , − /
2) ( − / , − / |↑(cid:105) (1 , /
2) (0 , /
2) (1 / , /
2) ( − / , / |↓↑(cid:105) (2 ,
0) (1 ,
0) (4 / ,
0) (3 / , / N − s ( σ ) and required unit cellsize N for various particle number and magnetization den-sities ( n, m ) for a chain of spinful electrons (with | σ (cid:105) = {| (cid:105) , |↓(cid:105) , |↑(cid:105) , |↓↑(cid:105)} ). The first column denotes the chosen (un-modified) quantum number representations s ( σ ). It is worth noting that due to the quantum numberdensity offset the shifted quantum numbers in general donot directly correspond to true quantum numbers of thesymmetry group. This is because we have distributedthe quantum number shift (to achieve a certain quantumnumber density) homogeneously over the unit cell, in-stead of applying a single offset at the unit cell boundary.Consequently, the shifted quantum numbers can be inter-preted as a combination of the true quantum numbers ofthe symmetry group combined with quantum numbersof fractional applications of the unit cell translation op-erator (i.e. translations over n < N sites). The shiftedquantum numbers therefore also bear information aboutthe location within the unit cell. For example, in the3 site unit cell suMPS with particle density n = 2 / /
3, and fi-nally on the third bond they are integers shifted by +2 / III. SYMMETRIC VARIATIONAL ANSATZFOR ELEMENTARY EXCITATIONS
We now generalize the variational ansatz for low energyexcitations presented in Ref. 11 to multi site unit cellsand abelian symmetries. As a foundation we start froma variationally optimized suMPS ground state approxi-mation | Ψ( A ) (cid:105) of bond dimension D in mixed canonicalrepresentation, where for now we focus on single site unitcells and generalize to multi site unit cells later. A suit-able suMPS ground state approximation can be obtainedfrom e.g. a symmetric implementation of the algorithmpresented in Ref. 9 using suMPS as described in Sec. II.In a slight reformulation of the ansatz in Ref. 11 we write for a low-energy excitation with momentum p inthe mixed canonical representation | Φ p ( B ) (cid:105) = (cid:88) n, σ e ipn (cid:16) . . . A σ n − L B σ n ˜ A σ n +1 R . . . (cid:17) | σ (cid:105) . (10)Here, the matrices left and right of B are in left and rightcanonical representation respectively. Furthermore, for atopologically trivial excitation A L and ˜ A R represent thesame state, i.e. | (cid:104) Ψ( A ) | Ψ( ˜ A ) (cid:105) | = 1. For a topologi-cally nontrivial excitation A L and ˜ A R represent differentground states (e.g. within the degenerate ground spacein a symmetry broken phase), and the excitation is of do-main wall type. The above ansatz only captures elemen-tary excitations and their bound states well, (or moreprecisely, isolated excitations branches ) while an accu-rate representation of scattering states in a continuumrequires a more complicated ansatz, whose symmet-ric formulation for multi site unit cells we leave for futurework.In the above mixed canonical representation, the pa-rameterization of the perturbation matrix B in the leftand right tangent space gauge then reduces to B σ = V σL x L , (11a) B σ = x R V σR . (11b)Here V σL and V σR are the left and right null spaces of A σL and ˜ A σR respectively, i.e. (cid:80) σ V σL † A σL = (cid:80) σ ˜ A σR V σR † =0, and the matrices x L , x R ∈ C D × ( d − D contain the( d − D variational degrees of freedom for the ansatz.Without loss of generality we will henceforth use the lefttangent space gauge (11a) and drop the subscripts L/R for V and x .Variational approximations of excited states with acertain fixed momentum p can then be obtained fromsolving an effective eigenvalue problem of a (momentum-dependent) effective Hamiltonian defined in the space ofthese variational parameters H eff p (cid:126)x [ j ] = e [ j ] p (cid:126)x [ j ] , (12)where j = 1 , . . . , ( d − D and (cid:126)x denotes the vector-ization of x . Notice that in contrast to the original for-mulation, in the above mixed canonical representationall necessary operations can now be performed withouttaking any (possibly) ill-conditioned inverses.For an N site unit cell ansatz, we again start from avariationally optimized (but here N site unit cell) groundstate approximation | Ψ( A ) (cid:105) in mixed canonical represen-tation and introduce local perturbations in the form oflocal perturbation matrices B ( k ) on each site. We collectall of these single site contributions into one single unit a) b) ... ... +0,+1 +1/3,-2/3 +1/3,-2/3 +1/3,-2/3 n = 1/2 n = 2/3 +0,+1+0,+1+0,+1+0,+1+0,+1
102 102
Figure 1. Examples for possible quantum number sectors of MPS on a system of spinless fermions with fixed particle number.The lines represent possible paths, corresponding to valid sequences of available quantum number sectors. (a) Possible quantumnumber sectors in the conventional finite size representation of a 6 site system at half filling (i.e. particle density n = 1 / s = 0 ,
1. (b) Possible quantum number sectors for an infinite system at 2 / s = 1 / , − /
3. In typical low energy states,quantum number sectors corresponding to high fluctuations around the desired particle number density will in general besuppressed by small Schmidt values, which will be truncated away in a finite bond dimension MPS approximation, leaving afinite number of remaining sectors, even in infinite systems. Notice also the appearance of negative labels, as quantum numbershere are only defined up to an arbitrary global offset. In both cases we have marked in- and outgoing quantum numbers witharrows. cell perturbation matrix B Σ n = B (1) σ nN +1 ˜ A (2) σ nN +2 R . . . ˜ A ( N ) σ nN + N R + A (1) σ nN +1 L B (2) σ nN +2 . . . ˜ A ( N ) σ nN + N R + . . . + A (1) σ nN +1 L . . . B ( N − σ nN + N − ˜ A ( N ) σ nN + N R + A (1) σ nN +1 L . . . A ( N − σ nN + N − L B ( N ) σ nN + N (13)and write the full multi site unit cell ansatz with momen-tum 0 ≤ p < πN as | Φ p ( B ) (cid:105) = (cid:88) n, σ e ipNn (cid:16) . . . A Σ n − L B Σ n ˜ A Σ n +1 R . . . (cid:17) | σ (cid:105) . (14)Here again the integer n enumerates unit cellsand we parameterize B σ ( k ) = V σ ( k ) x ( k ), where (cid:80) σ V ( k ) σ † A ( k ) σL = 0 for k = 1 , . . . , N . The varia-tional energy is then a quadratic function of the concate-nation of all N parameter vectorizations (cid:126)x = (cid:76) k (cid:126)x ( k ),and multi site unit cell excited state approximations canbe obtained from solving an effective eigenvalue problemof the same type as (12), but with a larger and morecomplex effective Hamiltonian H eff p (for an explicit con-struction of H eff p see Appendix A). Note that, contrary toblocking sites in e.g. a regular DMRG calculation, herethe number of variational parameters scales linearly with N , enabling an efficient treatment of large unit cell sizeswithout sweeping.By definition and construction of the variational ansatz and H eff p , the excitation energies e [ j ] p obtained from (12)are (positive) energy differences to the extensive groundstate energy E . Hence, while E is of O ( L ), the excita-tion energies e [ j ] p are of O (1). Likewise, while U (1) quan-tum numbers – like particle number or magnetization –are extensive for ground states, low lying excitations arecharacterized by quantum number differences of O (1) tothe ground state. Popular examples for this are spin flipor few particle excitations.In the context of the variational ansatz (10), such rela-tive differences of O (1) can be perfectly well understoodas being caused by the single perturbation matrix B ontop of the homogeneous, extensive ground state back-ground generated by the MPS matrices A L and ˜ A R . Wecan therefore control and fix quantum numbers of excitedstates through the quantum numbers of B .More specifically, in the parameterization (11), V L and V R necessarily have the same symmetry sectors and quan-tum number labels as A L and ˜ A R . We can however at-tach a non-trivial quantum number q to the matrix x that contains the variational parameters, i.e. x is block(off-)diagonal and we write x [ q ] ab , b = a − q. (15)Note, that q is an (outgoing) quantum number associ-ated with x itself, rather than a physical index. Thisis very intuitive, as V takes care of the homogeneousground state density contribution on that site, while thequantum number q of x directly controls the quantumnumber difference of the excitation with respect to theground state. For B we then have B [ q ] σab = (cid:88) c V σac x [ q ] cb , b = a + ˜ s ( σ ) − q. (16)and we denote the generated symmetric excitation ansatzas | Φ [ q ] p ( B ) (cid:105) .From the structure of (10) it is clear that the values of q have to be such that the outgoing quantum numbers of B match the ingoing quantum numbers of ˜ A R . Dependingon A L and ˜ A R , only certain values of q are thereforeallowed.For example, in a system of spinless fermions with s ( σ ) = 0 ,
1, a single particle excitation is character-ized by q = 1, while a hole excitation corresponds to q = −
1. Likewise, on a S = 1 / s ( σ ) = ± /
2, single magnons (spin flips) are character-ized by q = ±
1. This holds regardless of the ground stateparticle/magnetization density, which is encoded in V .The above are typical examples for topologically trivialexcitations, where ˜ A R = A R , i.e. A L and ˜ A R have thesame quantum number sectors, and the quantum numberof the excitation is well defined. The quantum numbersof fractional excitations such as spinons and holons how-ever necessarily require them to be of topologically nontrivial nature (see below), where ˜ A R (cid:54) = A R and the quan-tum numbers of A L and ˜ A R can in principle differ by anarbitrary offset. Just as the momentum p (cf. Ref. 11),the quantum number q of a topologically nontrivial exci-tation therefore seems to be completely arbitrary. Again,this is an artifact of open boundary conditions and fixingthis ambiguity depends on the nature of the excitation(see Sec. IV A and IV C). IV. RESULTSA. Spinons and Magnons in the S=1/2 XXZAntiferromagnet
As a first prototypical example we study low energyexcitations of the one dimensional S = 1 / H XXZ = (cid:88) j X j X j +1 + Y j Y j +1 + ∆ Z j Z j +1 − hZ j . (17)Here X , Y and Z are S = 1 / h is an external magnetic field.The energies for the ground state and low energy excita-tions in the thermodynamic limit are known exactly. We consider the antiferromagnetic case ∆ >
0. Therethe ground state in zero field has zero magnetization, andthe elementary excitations are given by spinons . Incontrast to simple spin flip excitations (magnons) – whichare integer spin excitations – spinons have fractional spin S = 1 / In the following we will demonstrate that in the context of the symmetric ansatz (14), spinon excita-tions must necessarily be of topologically nontrivial na-ture, i.e. they are domain wall like and cannot be createdby a single (or few) local spin flips.For a zero magnetization ( m = 0) suMPS groundstate approximation, we require a unit cell size N = 2.We use a shifted quantum number ˜ s ( σ ) = s ( σ ) = ± / q of the excitation directlycorresponds to the magnetization m of the excitation. Without loss of generality we assume integer quantumnumbers on even bonds, and half-integer quantum num-bers on odd bonds. Consider the contribution A (1) σ L B (2) [ q ] σ = A (1) σ L V (2) σ x [ q ] . (18)For a topologically trivial excitation, the next unit cellstarts again with ˜ A (1) R = A (1) R . The outgoing quan-tum numbers of V (2) and A (2) however are the same,which are in turn also equal to the ingoing quantum num-bers of A (1), all of which are integers. This means thatonly integer values q are possible, such that both in- andoutgoing quantum numbers of x [ q ] are integers. The lo-cal (single-mode approximation like) nature of the ansatzthus leads to the well known fact that excitations gen-erated by localized spin flips can only generate integerspin excitations, i.e. magnons (where e.g. q = ± q to be half-integer in order togenerate a spinon , is for the unit cell ˜ A to the right of B to start with half-integer instead of integer quantumnumbers. This can be achieved by using a translated unitcell for ˜ A , i.e. ˜ A (1) = A (2), ˜ A (2) = A (1), or | Ψ( ˜ A ) (cid:105) = T | Ψ( A ) (cid:105) with T the (single site) translation operator.Due to translation invariance of (17), | Ψ( ˜ A ) (cid:105) is also avalid ground state approximation with the same energyas | Ψ( A ) (cid:105) .Indeed, in the gapped antiferromagnetic phase ∆ > staggered magnetization density m s = L (cid:80) j ( − j (cid:104) Z j (cid:105) (cid:54) = 0, and the above | Ψ( ˜ A ) (cid:105) (cid:54) = | Ψ( A ) (cid:105) are good ground state approximations. For − < ∆ ≤ D the above suMPS ground stateapproximations however artificially break translation in-variance, such that still | Ψ( ˜ A ) (cid:105) (cid:54) = | Ψ( A ) (cid:105) are groundstate approximations with the same variational energyand m s (cid:54) = 0 and we can use them to build spinon exci-tations. This symmetry is restored in the limit D → ∞ ,where m s → | (cid:104) Ψ( ˜ A ) | Ψ( A ) (cid:105) | → B carrying twohalf integer spins (one for the local physical S = 1 / V , and one for the spinon excitation carriedby x ) and the resulting ansatz is topologically nontrivial.Due to the 2 site unit cell, the momentum of the spinonis also restricted to 0 ≤ p ≤ π , i.e. to half of the first ... ... Magnon
102 023/21/25/2 1 3/21/25/2 ... ...
Spinon
102 02 3/21/25/21
A(2)
A(1) A(2)
V(1) x(1)A(2) A(2) A(1)V(1) x(1) q=1 q=1/2 a) b)
Figure 2. Possible constructions and quantum number sectors for (a) a topologically trivial (magnon) and (b) a topologicallynontrivial (spinon) excitation in the antiferromagnetic S = 1 / x in (b) to enable half integer excitation quantum numbers. p= : e mp m = ' = p= : e mp m = ' p= : e mp m = ' = p= : e mp m = ' m = 0Bethe Figure 3. Variational low energy dispersion for the antiferromagnetic XXZ model (17) at ∆ = 3 , h = 0 and bond dimension D = 70. We show results for half integer magnetization (topologically nontrivial) excitations on the left, while results for integermagnetization (topologically trivial) excitations are shown on the right, together with exact results from Bethe ansatz, where the black solid line denotes the exact elementary branch, and the blue, green and purple areas the exact 2, 3 and 4particle scattering continua respectively. We show the first 20 lowest energies for each magnetization. While excitations inthe multi particle continua get partially reproduced, the elementary spinon with m = ± / Brillouin zone, which is also consistent with Ref. 26 and28.Here, generating ˜ A from translating A also removesthe above mentioned ambiguity of the excitation quan-tum number q , arising from the fact that A and ˜ A can inprinciple have arbitrary differences in the global quantumnumber offset. For a graphical representation of the con- struction of topologically trivial (magnon) and nontrivial(spinon) excitations, see Fig. 2.For the case N = 2 and m = 0 we calculate thevariational excitation energy dispersion with integer andhalf integer magnetizations for the XXZ Antiferromagnet(17) at ∆ = 3 , h = 0 for bond dimension D = 70. Thenumerical results are shown in Fig. 3, together with ex- p= : -2-1.5-1-0.500.511.522.5 e m ;p m = +1 = m = ! = m = +2 = m = ! = e mp e mp + hm Figure 4. Dispersion relations e m ,p of elementary excitationswith fractional magnetizations m = ± / m = ± / m = 1 / | Ψ (cid:105) with respect to H . The energies are shifted by∆ e = hm with respect to the excitation energy dispersion e mp of H h , for which | Ψ (cid:105) is the overall ground state. Thesolid lines show the exact dispersion e mp obtained from Betheansatz for reference. act results from Bethe ansatz. Excitations with halfinteger magnetizations are of topologically nontrivial na-ture, while topologically trivial excitations carry integermagnetization. The elementary excitations are given byspinons with magnetization m = ± /
2, and the entirespectrum of excitations can be generated from composi-tions of multiple spinon states into scattering or boundstates.As mentioned below (10), the ansatz only captures ele-mentary excitations and their bound states well. Conse-quently, the elementary spinon branch is accurate to ma-chine precision, while excitations in multi particle con-tinua are only partially reproduced. Nevertheless, thelow energy boundaries of these continua are still surpris-ingly well reproduced, where accuracy however decreasesquickly with higher particle number. An interesting ad-vantage of the suMPS ansatz is that excitations very highup in the spectrum with high magnetizations (e.g. boundstates) can now easily be targeted, whereas in the non-symmetric approach such states would be buried high upin a multitude of other states and targeting them wouldrequire more involved numerical procedures. This alsoenables a systematic estimation of unknown lower boundsof high up multi particle continua by targeting high mag-netization excitations.
B. Magnetic Field for Ground State fromExcitations in the XXZ Model
Apart from directly targeting and identifying excita-tions with certain quantum numbers, we can also usesuMPS excitations to calculate the magnetic field h required for the ground state of (17) to have a cer-tain magnetization density m (or total magnetization m ∆ = 0 . e m ,p with respect to H forelementary excitations with fractional magnetizations m = ± / , ± / D = 200 for momentum p =0 on top of a m = 1 / H . m ∆ = 0 . h necessary for theground state of (17) to have magnetization density m = 1 / m = 1 / , /
3, together with exact val-ues from Bethe Ansatz. M = Lm ). This is possible, as the magnetic field term H M = (cid:80) j Z j commutes with the rest of the Hamiltonian H , and changing h just changes the eigenenergies, butnot the eigenstates of (17).More specifically, we write H XXZ = H h = H − hH M and assume | Ψ M (cid:105) to be the ground state of H h withground state energy E (0) h and (total) magnetization M for a suitable (unknown) value of h . For a low lying exci-tation | Φ mp (cid:105) of H h with magnetization m and momentum p we have( H h − E (0) h ) | Φ mp (cid:105) = e mp | Φ mp (cid:105) (19)( H − E ) | Φ mp (cid:105) = ( e mp + hm ) (cid:124) (cid:123)(cid:122) (cid:125) e m ,p | Φ mp (cid:105) (20)with E = E (0) h + hM the corresponding eigenenergy of H . Note that while e mp ≥ e m ,p with respect to H need not be positive. In fact,as a function of p we obtain the true excitation energydispersion e mp of H h shifted by a constant energy offset hm .Let us now obtain the overall ground state | Ψ M (cid:105) of H h as the lowest energy state with magnetization M andenergy E from H . In order to infer h we then constructa variational excitation | Φ mp (cid:105) on top of | Ψ M (cid:105) . Its vari-ational energy e m ,p with respect to H is then given by(20), from which we can calculate h if we know e mp .From symmetry arguments we however know that e mp = e − mp , such that we can additionally construct | Φ − mp (cid:105) and use both variational excitation energies to ob-tain e mp = ( e m ,p + e − m ,p ) / . (21)We demonstrate this method to calculate the mag-netic field h required for a ground state with magneti-zation density m = 1 / . H with m = 1 /
6, which requires a N = 3 site unit cell suMPS representation. Due to a fi-nite bond dimension representation the obtained groundstate approximation for ∆ = 0 . B inan excitation ansatz with (fractional) quantum numbers q = ± / , ± /
3. There are 3 possibilities for each quan-tum number, totaling in 12 possible states. See Ap-pendix B for more details on obtaining excitations ener-gies for well defined (fractional) magnetizations.Without loss of generality we choose momentum p = 0and calculate variational excitation energies e m ,p with re-spect to H with D = 200 and fractional magnetizations m = ± / , ± / . , The numerical re-sults are given in Table III. From these values we furtherobtain the true excitation energies e mp with respect to H h from (21). These energies are known to be exactlyzero and we obtain values of the order O (10 − ), dueto finite bond dimension. Finally, we calculate h from h = ( e m ,p − e mp ) /m (22)The numerically obtained values for h are given in Ta-ble IV. In comparison to the exact values from BetheAnsatz the errors are of order O (10 − ) and thusquite low a for moderate bond dimension of D = 200.The (shifted) variational energy dispersions are shown inFig. 4.This method is not restricted to determining necessarymagnetic fields h for a certain ground state magnetization(density), but is generally applicable to all Hamiltonianswhich contain one (or more) generators of their globalsymmetries as a parameterized term. It is especially use-ful for models that are not exactly solvable, where other-wise one would have to perform a large number of groundstate calculations in a grid search with small variations of h , while here h is calculated directly from a single groundstate calculation followed by two (or few) excited statecalculations. C. Spinons, Holons and Electrons in the FermiHubbard Model
In the following two sections we consider as a secondexample the low energy spectrum of the (extended) Fermi Hubbard model H HUB = − t (cid:88) σ,j c σ,j c † σ,j +1 − c † σ,j c σ,j +1 + U (cid:88) j (cid:18) n ↑ ,j − (cid:19) (cid:18) n ↓ ,j − (cid:19) + V (cid:88) j ( n j −
1) ( n j +1 − − µ (cid:88) j n j , (23)where c σ,j , c † σ,j are creation and annihilation operatorsof electrons of spin σ on site j , n σ,j = c † σ,j c σ,j and n j = n ↑ ,j + n ↓ ,j are the particle number operators. Here, t is the hopping amplitude, U and V are the on site andnearest neighbor interactions respectively and µ is thechemical potential (we do not consider an external mag-netic field).Due to the phenomenon of spin charge separation, the elementary excitations are fractionalized quasi-particles of either spin or charge alone, which cannot beconstructed from the bare electrons, as those carry both spin and charge. Rather, electrons can in turn be in-terpreted as bound states of these elementary spin andcharge excitations, known as spinons and holons respec-tively. Consequently, we use quantum number represen-tation ( n, m ) of the local physical electronic states, with n the particle number and m the magnetization. Spinonsand holons carry quantum numbers q s = (0 , ± /
2) and q c = ( ± ,
0) and excitations with electronic quantumnumbers q e = ( ± , ± /
2) are therefore combinations ofholons and spinons.We first focus on the integrable case V = 0, wherethe ground state energy and elementary excitations areknown exactly from Bethe ansatz, at half filling( n , m ) = (1 , U >
0, the finite bond di-mension suMPS ground state approximation again arti-ficially breaks translation invariance, and the translatedground state | Ψ( ˜ A ) (cid:105) = T | Ψ( A ) (cid:105) (cid:54) = | Ψ( A ) (cid:105) is an equallygood ground state approximation. From these we cannow construct topologically nontrivial excitations whichallow for quantum numbers q c and q s and we can gener-ate elementary spinons and holons that way. Conversely,topologically trivial excitations carry quantum numbersthat are even combinations of q s and q c , for examplespin-spin, charge-charge, or spin-charge excitations.For the case N = 2 and half filling we calculate vari-ational excitation energy dispersions for U = 5 and p= : e ( n;m ) p n = 0, m z = ' = p= : e ( n;m ) p n = ' m z = ' = n = ' m z = ' p= : e ( n;m ) p n = 0, m z = ' n = 0, m z = 0elem. spin (Bethe) p= : e ( n;m ) p n = ' m z = 0 n = ' m z = 0elem. charge (Bethe) Figure 5. Variational low energy dispersion for the Fermi Hubbard model (23) in the integrable case V = 0 at U = 5 and halffilling ( n = 1, m = 0), for bond dimension D = 600. We show results for pure spin excitations on the left, while results forcharge and spin-charge excitations are shown on the right. We show the first 8 lowest obtained variational energies for eachquantum number. We also show the exact elementary branches and multi-particle continua from Bethe ansatz, where on theleft the purple area is the continuum of spin-spin excitations, while on the right the green, red and orange areas correspond tocharge-charge, spin-charge and spin-charge-charge excitation continua respectively. bond dimension D = 600 for several different quan-tum numbers. There, the elementary charge excita-tions are gapped, while elementary spin excitations aregapless. The numerical results are shown in Fig. 5,together with exact results from Bethe ansatz, where weshow pure spin excitations in the left plot and excitationswith nonzero charge quantum numbers in the right plot.We find that the elementary spinon is reproduced up toan excellent accuracy of O (10 − ) by the lowest varia-tional excitation branch with m = ± /
2. The higherup branches for integer and half integer magnetizationslie within the continuum of spin-spin scattering stateswhich are not well captured by our ansatz. Neverthelessthe variational energies reproduce the low end of thiscontinuum surprisingly well.Due to the elementary spinon being gapless, the exactelementary holon branch lies completely within the con-tinua of scattering states of one charge and arbitrarilymany spin excitations. Out of these, e.g. the charge-spin-spin continuum (red area in Fig. 5) also containsexcitations with quantum number q = ( ± ,
0) (charge +spin singlet or triplet with m z = 0) equal to q c . The vari-ational excitation ansatz for this q tries to reproduce ex-actly these excitations, as due to the smaller bandwidthof the spinon branch they are at lower energies than theelementary charge excitations (shown as black solid linesin Fig. 5 on the right), except at p = 0 , π , where thelower bound of the continuum is exactly the elementary charge branch. Around these momenta, the variationalansatz with q = ( ± ,
0) indeed yields the lowest energiesand reproduces the elementary holon up to an accuracyof O (10 − ). Away from p = 0 , π the same ansatz tries toreproduce a three particle scattering state, and energiesfor q = ( ± , ± /
2) – which try to reproduce two particlespin-charge scattering states – yield in fact slightly lowerenergies. A similar effect can be observed for q = ( ± , p ≈ .
15. There the ansatz tries to repro-duce a spin-spin-charge-charge instead of a charge-chargescattering state. Also, the exact elementary charge branch spans the en-tire Brillouin zone p ∈ [0 , π ), while the momentum of atwo-site ansatz is necessarily restricted to half of the Bril-louin zone. However, it turns out that the two-site unitcell is required only by the spin quantum numbers, whilehalf filling for charge quantum numbers alone could beachieved with a single site unit cell ansatz. Consequently,charge excitations are reproduced twice by this ansatz,with a relative shift of p = π over the entire Brillouinzone. For that reason we also draw the exact elementarycharge branch from Bethe ansatz twice with the corre-sponding momentum shift in Fig. 5.A possible way to remedy this fact is to realize that thevariational ground state – even though not translation in-variant under pure translations T – is invariant under atranslation followed by a spin flip, i.e. under application0of T F S , where T is the translation and F S is the spin flipoperator. The entire Brillouin zone for charge excitationscould therefore be recovered by using a restricted ansatz B (2) = e i p F S B (1), which indeed yields an eigenstate of T F S with eigenvalue e − i p where p ∈ [0 , π ) can be in-terpreted as quasi-momentum covering the full Brillouinzone.Additional ways to only target the elementary chargebranch within the charge-spin scattering continuumwould be to either minimize the energy variance of thevariational ansatz (see 25, especially Appendix 4), in-stead of the energy itself, or to distinguish excited statesby higher conservation laws, which could be intro-duced as artificial penalty terms into the Hamiltonian.The latter option however requires analytical knowledgeabout these higher conserved quantities and is unsuitedfor non-integrable models. We leave all these avenues tobe explored in future studies.Overall it is demonstrated that the new symmetricsuMPS ansatz allows for an efficient separation of excita-tion sectors with different quantum numbers, which waspossible in the original proposal in Ref. 11 only a posteri-ori and with great effort. For example, charge excitationsonly start appearing above the U dependent charge gapabove a continuum of pure spin excitations. In the non-symmetric original ansatz these excitations are next toimpossible to single out or target, especially if the valueof the charge gap is unknown.In fact, even despite the elementary charge branch ly-ing completely within multi-particle continua, the chargegap can still be calculated with the new symmetric ansatzto excellent precision of the same order as the groundstate. D. Spin to Charge Density Wave Phase Transitionin the Half Filled Extended Fermi Hubbard Model
As a final example, we show the qualitative changeof the low energy spectrum of the extended Fermi Hub-bard model at half filling at the spin to charge densitywave (SDW to CDW) transition for
U, V > Inparticular we consider the case of U = 10 fixed and vary V around the critical point V c ≈ .
13 of the first orderSDW-CDW transition. While the charge excitations arealways gapped in this parameter regime, the spin excita-tions are gapless in the SDW phase (
V < V c ) and becomegapped in the CDW phase ( V > V c ).We show results for variational excitation energy dis-persions for V = { . , . } and bond dimensions D = { , } in Fig. 6 as well as V = { . , . } and bonddimensions D = { , } in Fig. 7. We plot the low-est 10 variational energies for various excitation quan-tum numbers. It can be seen that for V < V c in theSDW phase (top two panels in Fig. 5) the spin excita-tions are gapless and the dispersion looks very similar tothe integrable case V = 0 in Sec. IV C. For V > V c in theCDW phase however, the spin excitations become gapped and the nature of the low energy spectrum changes com-pletely: rather than one single elementary spin and onesingle elementary charge branch, there is now a multi-tude of isolated elementary (or bound state) excitationbranches which lie below the multi particle scatteringcontinua.In particular, there are now two lowest excitationbranches with spinon quantum numbers ( n = 0 , m = ± /
2) which have a level crossing at p = π/
2. Likewise,there are also several isolated charge excitation brancheswith holon quantum numbers ( n = ± , m = 0). Here,the elementary holon is now also restricted to half ofthe first Brillouin zone, as the particle density n showsa strong dimerization in the ground state. While thereare several level crossings between these charge branchesfor V slightly above V c , there remain two lowest crossingbranches which become more and more separated fromthe rest with increasing V .In addition, there are further isolated branches withspinon and holon quantum numbers, and also withelectronic, or multi-particle spin or charge quantumnumbers, which correspond to bound states. Thesebranches lie completely or partially below the spin-spin,charge-charge and spin-charge multi-particle continuaconstructed from the lowest possible elementary spin orcharge excitations (see colored areas in Fig. 6 and Fig. 7).For example, for V (cid:38) n = ± , m = ± / V. CONCLUSION
We present a formulation of the variational MPSansatz for elementary excitations first proposed in Ref. 11with conserved symmetries for multi site unit cells, wherethe computational cost and the number of variational pa-rameters scales linearly in the number of sites N withinthe unit cell. The resulting ansatz allows for an effi-cient separation of the low energy excitation spectruminto certain desired quantum number sectors, which canbe targeted individually. This is a great advantage overthe original proposal, where an identification of differ-ent quantum numbers is only possible a posteriori, andthere is no mechanism to target excitations with certainquantum numbers.We show through the structure of the symmetricansatz, that elementary excitations in the antiferromag-netic XXZ model (spinons) and in the Fermi HubbardModel (spinons and holons) are necessarily of topologi-cally nontrivial domain wall nature. Even though such1 p= : e ( n;m ) p V = 5 : n = 0, m z = ' = n = 0, m z = ' p= : e ( n;m ) p V = 5 : n = 0, m z = ' = n = 0, m z = ' p= : e ( n;m ) p V = 5 : n = ' m z = 0 n = ' m z = 0 p= : e ( n;m ) p V = 5 : n = ' m z = 0 n = ' m z = 0 p= : e ( n;m ) p V = 5 : n = ' m z = ' = n = 0, m z = 0 p= : e ( n;m ) p V = 5 : n = ' m z = ' = n = 0, m z = 0 Figure 6. Variational low energy dispersions for various quantum numbers of the extended Fermi Hubbard model (23) in the nonintegrable case V = 5 . V = 5 . U = 10 on top of a half filled ground state ( n = 1, m = 0). We show thefirst 10 lowest energies for each quantum number, represented by colored symbols. In addition we show spin-spin, charge-chargeand spin-charge scattering continua constructed from the variational elementary spin and charge excitations as purple, greenand red areas respectively. The spectrum on the left is in the SDW phase V < V c ≈ .
13, while the spectrum on the right isin the CDW phase
V > V c . The SDW phase looks very similar to the integrable case in Fig. 5, while in the CDW case bothspin and charge excitations are gapped and there is a multitude of additional isolated elementary (or bound state) branches.In the SDW phase – like in the integrable case – the lowest variational charge energies are only suboptimal approximations ofmulti-spin-charge scattering states; the charge-charge and spin-charge continua constructed from the variational energies aretherefore not exact, but are kept for reference and marked with dashed boundaries. In the CDW phase the lowest excitationbranches are isolated and the accuracy of variational energies – and consequently also of the multi particle continua – is expectedto be excellent. p= : e ( n;m ) p V = 5 : n = 0, m z = ' = n = 0, m z = ' p= : e ( n;m ) p V = 6 : n = 0, m z = ' = n = 0, m z = ' p= : e ( n;m ) p V = 5 : n = ' m z = 0 n = ' m z = 0 p= : e ( n;m ) p V = 6 : n = ' m z = 0 n = ' m z = 0 p= : e ( n;m ) p V = 5 : n = ' m z = ' = n = 0, m z = 0 p= : e ( n;m ) p V = 6 : n = ' m z = ' = n = 0, m z = 0 Figure 7. Variational low energy dispersions for various quantum numbers of the extended Fermi Hubbard model (23) in thenon integrable case V = 5 . V = 6 . U = 10 on top of a half filled ground state ( n = 1, m = 0). Weshow the first 10 lowest energies for each quantum number, represented by colored symbols. In addition we show spin-spin,charge-charge and spin-charge scattering continua constructed from the variational elementary spin and charge excitations aspurple, green and red areas respectively. Here, both spectra are in the CDW phase V > V c (see also caption of Fig. 6). The performance of the proposed ansatz is demon-strated by calculating variational low energy dispersionswith different quantum numbers for the antiferromag-netic XXZ model and the (extended) Fermi Hubbardmodel. In cases where exact Bethe ansatz solutions ex-ist for comparison, the elementary spinon excitations arereproduced by the variational ansatz to excellent preci-sion. In the gapped CDW phase of the (non-integrable)extended Fermi Hubbard model, we observe a large num-ber of new bound states below the multi particle con-tinua, which are not present in the gapless SDW phase. Itwould be interesting to explore the physical consequencesof their appearance.As the gapped elementary holon excitations in the (in-tegrable) Fermi Hubbard model completely lie within amulti particle continuum with the same quantum num-bers, the ansatz tries to reproduce lower lying statesin this continuum instead, except around momentum p = 0 , π , where the elementary excitation has the sameenergy as the lower boundary of the continuum. Possi-ble ways to remedy this fact are discussed in Sec. IV Cand are left to be explored in future studies. Despite thisfact, the charge gap can however still be calculated withexcellent precision of the same order as the ground state.We further show that the symmetric ansatz can be usedto calculate e.g. the magnetic field h required for a cer- tain ground state magnetization m of the antiferromag-netic XXZ model. As this strategy allows a direct cal-culation of h , involving a single variational ground stateand two (or few) variational excitation calculations only,it is particularly useful for non-integrable models, whereotherwise a large number of ground state calculations ina grid search with small variations of h are necessary.This procedure is applicable to all Hamiltonians, whichcontain generators of their global symmetries as param-eterized terms.The presented ansatz is also a perfect candidate fora more precise and efficient study of elementary excita-tions in two dimensional systems with topological orderon cylinders, such as e.g. in Refs. 43 and 44. More gen-erally, the presented ansatz may prove to be vital for anefficient study of elementary excitations in lattice gaugetheories, topological excitations on top of Projected En-tangled Pair State (PEPS) ground states in two di-mensions, and also for excitations of transfer matricesconstructed from topological PEPS. ACKNOWLEDGMENTS
We thank F. Essler, M. Ganahl, V. Korepin and G.Roose for inspiring and insightful discussions. This workis supported by an Odysseus grant from the FWO, ERCgrants QUTE (647905) and ERQUAF (715861), and theEU grant SIQS. V.Z.-S. and F.V. gratefully acknowledgesupport from the Austrian Science Fund (FWF): F4104SFB ViCoM and F4014 SFB FoQuS. I.P.M. acknowledgessupport from the Australian Research Council (ARC)Centre of Excellence for Engineered Quantum Systems,Grant No. CE110001013, and the ARC Future Fellow-ships Scheme No. FT140100625. The computationalresults presented have been achieved using the ViennaScientific Cluster (VSC). M. Fannes, B. Nachtergaele, and R. Werner, Comm. Math.Phys. , 443 (1992). D. Perez-Garcia, F. Verstraete, M. Wolf, and J. Cirac,Quant. Inf. Comput. , 401 (2007). F. Verstraete, V. Murg, and J. Cirac, Adv. Phys. , 143(2008). U. Schollw¨ock, Ann. Phys. , 96 (2011). S. White, Phys. Rev. Lett. , 2863 (1992). S. White, Phys. Rev. B , 10345 (1993). I. McCulloch, J. Stat. Mech. , P10014 (2007). G. Vidal, Phys. Rev. Lett. , 147902 (2003). V. Zauner-Stauber, L. Vanderstraeten, M. Fishman,F. Verstraete, and J. Haegeman, Phys. Rev. B , 045145(2018). J. Haegeman, T. Osborne, and F. Verstraete, Phys. Rev.B , 075133 (2013). J. Haegeman, B. Pirvu, D. J. Weir, J. I. Cirac, T. J. Os-borne, H. Verschelde, and F. Verstraete, Phys. Rev. B ,100408 (2012). J. Haegeman, S. Michalakis, B. Nachtergaele, T. Osborne,N. Schuch, and F. Verstraete, Phys. Rev. Lett. ,080401 (2013). S. Rommer and S. ¨Ostlund, Phys. Rev. B , 2164 (1997). S. ¨Ostlund and S. Rommer, Phys. Rev. Lett. , 3537(1995). D. P´erez-Garc´ıa, M. Wolf, M. Sanz, F. Verstraete, andJ. Cirac, Phys. Rev. Lett. , 167202 (2008). I. McCulloch, “Infinite size density matrix renormalizationgroup, revisited,” (2008), arXiv:0804.2509. M. Hastings, J. Math. Phys. , 095207 (2011). S. Singh, R. Pfeifer, and G. Vidal, Phys. Rev. B , 115125(2011). More precisely, it is the fusion structure of the irreduciblerepresentations, which in the case of abelian symmetries ishowever isomorphic to the group action. It is worth noting that the symmetric MPS matrices so de-fined behave like the transpose of the corresponding sym-metry operator: e.g. for the S = 1 / S + ab we would have a = b + 1. In that regard we could have also defined (6) that way, as the definition is just a mat-ter of choice, as long as this choice is consistently followedsubsequently. More specifically, we attach a single dummy ingoing quan-tum number a = 0 to the leftmost, and a single dummyoutgoing quantum number b = Q to the rightmost MPSmatrix. In fact, the left and right boundary quantum numbers neednot even be zero, they just have to be equal. Virtual U (1)quantum numbers are therefore always defined up to aglobal offset only. In general, the choice of the physical quantum numberlabels is not unique. For example, for spinful electrons,instead of particle and magnetization density one couldalso choose the densities of up and down spin electrons.In a practical implementation it is furthermore desirableto scale and shift all quantum numbers to e.g. integers toobtain an efficient representation. L. Vanderstraeten, J. Haegeman, T. J. Osborne, andF. Verstraete, Phys. Rev. Lett. , 257202 (2014). L. Vanderstraeten, F. Verstraete, and J. Haegeman, Phys.Rev. B , 125136 (2015). M. Takahashi,
Thermodynamics of one-dimensional solv-able models (Cambridge University Press, 1999). J. des Cloiseaux and M. Gaudin, J. Math. Phys. , 1384(1966). L. Faddeev and L. Takhtajan, Phys. Lett. , 375 (1981). Here and in the following we loosely speak of an excita-tion’s magnetization, which is to be understood as themagnetization difference to the ground state. In the general case of any translation invariant Hamilto-nian with N different N site unit cell ground state approx-imations with equal variational ground state energy, wecan in principle construct 2 N ( N −
1) different (but per-haps equivalent) topologically nontrivial excitations withfractional quantum numbers: There are N ( N −
1) possi-ble ordered ways to combine N different unit cells, and foreach combination there is one excitation with a positiveand one with a negative fractional quantum number. Seealso Ref. 46. Instead of domain wall excitations with fractional magneti-zation we could have also used topologically trivial magnonexcitations with integer magnetization. These excitationsare in general however scattering states of two or three el-ementary excitations, and the accuracy of their variationalenergy will be worse than the accuracy for elementary ex-citations. F. Essler, H. Frahm, F. G¨ohmann, A. Kl¨umper, andV. Korepin,
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In this appendix we describe the necessary terms forapplying the effective Hamiltonian H eff p onto a vector (cid:126)x of variational parameters, required for solving the effec-tive eigenvalue problem (12) using an efficient iterativeeigensolver. We restrict to the case of nearest neighborinteractions, i.e. the total Hamiltonian is a sum of near-est neighbor terms H = (cid:80) n h n,n +1 . The treatment oflong ranged Hamiltonians is straight forward to derive,but results in a dramatic increase of the amount andcomplexity of involved terms. A complete treatment forgeneral Hamiltonians given in terms of Matrix ProductOperators (MPOs) will be given elsewhere.
1. Single Site Unit Cells
This case has already been considered in the originalworks and we refer the reader to Refs. 10 and 11.
2. Multi Site Unit Cells
We consider the effective eigenvalue problem (12) for amulti site unit cell ansatz (14). For an efficient solutionfor low lying excited states we use an iterative Krylovsubspace eigensolver (such as e.g. Lanczos). For suchmethods, only the implementation of the action of H eff p onto some current vector (cid:126)x is necessary, which we de-scribe in the following.We divide the result (cid:126)x out = H eff p (cid:126)x in of one action of theeffective Hamiltonian into individual contributions persite (cid:126)x ( n ) out , which are computed separately and com-bined at the end. Furthermore, we describe all terms inthe space of B matrices, where B ( n ) σ in = V ( n ) σL x ( n ) in and x ( n ) out = (cid:80) σ V ( n ) σ † L B ( n ) σ out .The individual contributions to H eff p can be derivedby fixing the position of B ( n ) σ out and moving the posi-tions of B ( n ) σ in and the two site Hamiltonian h n,n +1 (cfRef. 11). Due to the gauge choice (11a) however, a good5part of these terms are zero, namely those where B ( n ) σ in is strictly left of h n,n +1 and B ( n ) σ out , and those where B ( n ) σ out is strictly left of h n,n +1 and B ( n ) σ in . Below wegive the remaining terms, collected and combined for effi-cient evaluation. Note that here the two site Hamiltonianhas been offset by the energy density of the ground state(i.e. h → h − e e = (cid:104) Ψ( A ) | h | Ψ( A ) (cid:105) ) in order toobtain positive energy differences to the ground state aseigenvalues of (12).We follow the notation of Refs. 9 and 11 and write( x | and | x ) for vectorizations of a D × D matrix x in the D dimensional “double layer” virtual space, onwhich (mixed) transfer matrices, such as T AB = (cid:80) σ ¯ A σ ⊗ B σ , or operator transfer matrices, such as O ABCD = (cid:80) σρµν O σρµν ¯ A σ ¯ B ρ ⊗ C µ D ν (with O σρµν = (cid:104) σρ | O | µν (cid:105) ), act.For better readability we also raise site indices to su-perscripts, e.g. A ( n ) σL → A n,σL and omit the tilde for A n,σR .Furthermore we write T nL = T A nL A nL for single site and T L = (cid:81) Nn =1 T nL for unit cell regular transfer matrices(and similarly for R ). For mixed single site and unitcell transfer matrices we consequently write T LR n = T A nL A nR and T LR = (cid:81) Nn =1 T LR n (and similarly for reversed L and R ). In all expressions it is understood that N + 1 ≡ ≡ N .We start by constructing quantities needed for terms,where B ( n ) in and B ( n ) out are on the same site.( h L | = ( | h A NL A L A NL A L ( h nL | = ( h n − L | T nL + ( | h A n − L A nL A n − L A nL , n > | h NR ) = h A NR A R A NR A R | | h nR ) = T nR | h n +1 R ) + h A nR A n +1 R A nR A n +1 R | , n < N, (A2)which collect Hamiltonian contributions within one unitcell. To collect contributions from all other unit cells wedefine further ( H NL | = ( h NL | [ − T L ] − ( H nL | = ( H n − L | T nL , n < N (A3)and | H R ) = [ − T R ] − | h R ) | H nR ) = T nR | H n +1 R ) , n > , (A4)where [ − T L ] − and [ − T R ] − are to be understood aspseudo-inverses (cf Ref. 11 and Appendix D in Ref. 9).Finally, we collect all left and right Hamiltonian contri-butions up to some site n into( H NL | = ( H NL | ( H nL | = ( H nL | + ( h nL | , n < N (A5) and | H R ) = | H R ) | H nR ) = | H nR ) + | h nR ) , n > B in and p and canbe precomputed as constants. They will also show up inother subsequent contributions.We now turn to quantities dependent on B in whichhave to be recalculated every time H eff p (cid:126)x in is invoked. Westart with terms where B n in is right of B n out from withinone unit cell | b NR ) = T A NR B N in | | b nR ) = T RL | b n +1 R ) + T A nR B n in | , n < N (A7)and from all other unit cells | B R ) = [ − e i pN T LR ] − | b R ) (A8a) | B nR ) = T RL n | B n +1 R ) , n > B n in right of B n out into | B R ) = e i pN | B R ) | B nR ) = e i pN | B nR ) + | b nR ) . (A9)These terms will be combined with ( H nL | in the final con-tributions to B n out .Next we consider quantities where both h n,n +1 and B in are left of B out ( hb L | =( H NL | T A L B + ( | h A NL A L A NL B +e − i pN ( | h A NL A L B N in A R ( hb nL | =( H n − L | T A nL B n in + ( | h A n − L A nL A n − L B n in +( | h A n − L A nL B n − A nR + ( hb n − L | T LR n , n > B n in within thesame unit cell and all h n,n +1 left of B n in . We proceed toinclude all contributions of B n in in all other unit cells( HB NL | = ( hb NL | [ − e − i pN T LR ] − (A11a)( HB nL | = ( HB n − L | T LR n , n < N (A11b)and finally combine( HB NL | = e − i pN ( HB NL | ( HB nL | = e − i pN ( HB nL | + ( hb nL | , n < N. (A12)The inverses in (A8a) and (A11a) are to be understood aspseudo-inverses in the case p = 0 and q = 0 only (i.e. forexcitations with zero momentum and the same quantumnumber(s) as the ground state) and can be fully invertedotherwise, as then the spectral radius of the transfer ma-trices is strictly smaller than 1.6We now have all the necessary quantities to compute B n out B n,σ out = H n − L B n,σ in + B n,σ in H n +1 R + (cid:88) ρµν h ρσµν ( A n − ,ρL ) † A n − ,µL B n,ν in + (cid:88) ρµν h σρµν B n,µ in A n +1 ,νR ( A n +1 ,ρR ) † + H n − L A n,σL B n +1 R + (cid:88) ρµν h ρσµν ( A n − ,ρL ) † A n − ,µL A n,νL B n +1 R + e i pNδ n,N (cid:88) ρµν h σρµν A n,µL ( B n +1 ,ν in + A n +1 ,νL B n +2 R )( A n +1 ,ρR ) † + e − i pNδ n, (cid:88) ρµν h ρσµν ( A n − ,ρL ) † B n − ,µ in A n,νR + HB n − L A n,σR , (A13)where the Kronecker symbols δ n,N and δ n, ensure that the corresponding momentum factors only contribute in thecases n = N and n = 1 respectively. Here, the first line corresponds to contributions where B n in and B n out are onthe same site, the second and third line where B n in is right of B n out , and the last line where B n in is left of B n out . For agraphical representation see Fig. 8. Appendix B: Magnetization of TopologicallyNontrivial Excitations from Translation SymmetryBroken Ground States in the XXZ Model
In this Appendix we elaborate on the topologicallynontrivial elementary excitations with fractional magne-tizations obtained from the threefold degenerate, trans-lation symmetry broken ground states with magnetiza-tion density m = 1 / m ,n (cid:54) = 1 / (cid:80) Nn ( m ,n − /
6) = 0 still holds within one unitcell. Denote the suMPS unit cells for these three de-generate ground states as A , A and A and assume | Ψ( A ) (cid:105) = T | Ψ( A ) (cid:105) = T | Ψ( A ) (cid:105) .The resulting magnetizations of domain wall excita-tions created from combining these three different groundstates are not exactly equal to the quantum numbers q .This is due to the fact, that the perturbation unit cell B consists of a superposition of different contributions from A i,L and A j,R ( i (cid:54) = j ), such that now (cid:80) n ( m ,n − m ) (cid:54) = 0within the perturbation unit cell. Consequently, B thenhas a magnetization different from N m and the excita-tion carries a magnetization slightly perturbed away from q . These perturbations are usually of the form m ,n − m .For example, if we take the m ,n to be the magnetizationsof the A i,L unit cell, the q = 1 / m = 1 / − ( m , − / q = − / m = − / m , − / q (which are related by single or two site overall transla-tions of the entire state) together have again a mean mag-netization of exactly m = q . For example, a q = 1 / A ,L with A ,R , A ,L with A ,R or A ,L with A ,R . To remedy the abovefact – which is once more an artifact of open boundaryconditions and finite bond dimension – we therefore com-pute and average over all three excitation energies foreach q . Exactly this has been done to obtain the valuesshown in Table III. Note that for topologically trivial excitations the mag-netization is always well defined and precisely corre-sponds to m = q . This is because the same MPS groundstate unit cell is used left and right of the perturbationmatrix B .7 ++ ++ + ++++=++ ++ + ++++=