Composite symmetry protected topological order and effective models

# Composite symmetry protected topological order and effective models

## Abstract

Strongly correlated quantum many-body systems at low dimension exhibit a wealth of phenomena, ranging from features of geometric frustration to signatures of symmetry-protected topological order. In suitable descriptions of such systems, it can be helpful to resort to effective models which focus on the essential degrees of freedom of the given model. In this work, we analyze how to determine the validity of an effective model by demanding it to be in the same phase as the original model. We focus our study on one-dimensional spin- systems and explain how non-trivial symmetry protected topologically ordered (SPT) phases of an effective spin model can arise depending on the couplings in the original Hamiltonian. In this analysis, tensor network methods feature in two ways: On the one hand, we make use of recent techniques for the classification of SPT phases using matrix product states in order to identify the phases in the effective model with those in the underlying physical system, employing Künneth’s theorem for cohomology. As an intuitive paradigmatic model we exemplify the developed methodology by investigating the bi-layered -chain. For strong ferromagnetic inter-layer couplings, we find the system to transit into exactly the same phase as an effective spin model. However, for weak but finite coupling strength, we identify a symmetry broken phase differing from this effective spin- description. On the other hand, we underpin our argument with a numerical analysis making use of matrix product states.

## I Introduction

Strongly correlated quantum many-body systems exhibit a wide range of intriguing properties, such as long range magnetic ordering de la Cruz et al. (2008); Klyushina et al. (2016) or notions of intrinsic or symmetry-protected topological order Zeng et al. (2015); Chen et al. (2011a); Pollmann et al. (2010); Schuch et al. (2010). In systems exhibiting geometric frustration the phenomenology is enriched by effects such as, spin-ice behavior, the emergence of magnetic monopoles, and fractionalisation coming into play Castelnovo et al. (2012); Gardner et al. (2010). Such strongly correlated and geometrically frustrated systems enjoy a significant experimental interest, and are probed with techniques such as neutron diffraction allowing for a window into their physics Anderson et al. (2009); Lake et al. (2005) At the same time, the theoretical and numerical description of these systems constitute significant challenges, at least in more than one spatial dimension. Even powerful numerical methods have difficulties to identify the essential ground state features in such systems accurately, as can be seen on the example of the race to identify the ground state properties of the anti-ferromagnetic spin-1/2 Heisenberg model on the Kagomé lattice Yan et al. (2011); Kolley et al. (2015); Mei et al. (). In layered materials, properties of such models become even more complex as the system can favor the creation of compounds. Here, behavior such as the formation of a spin liquid based on a novel frustration mechanism can appear Balz et al. ().

In general, the description of such interacting quantum many-body systems is often simplified by the use of effective models in which the description is reduced to the essential degrees of freedoms of the system needed in order to capture for instance the ground state properties. In order to be valid, the effective model should of course resemble the essential physics of the original one. In the most rough reading of this requirement, both systems should be in the same phase.

In this work, we investigate the problem of how to validate an effective model in the tractable setting of one-dimensional spin- systems in order to highlight the essential features of the method. In such systems, effective models are for instance derived by grouping several spins into a single site of definite higher spin. In the easiest instance we think of a ladder of spin- spins which have a horizontal ferromagnetic coupling of pairs and an arbitrary vertical coupling. Intuitively, for strong ferromagnetic couplings the pair of spins form a compound of spin for low energies such that an effective spin theory, i.e., neglecting the spin sector, should allow to capture the ground state properties of this system accurately. In such a setting, we require the blocked model and the effective model to be in the same symmetry protected topologically ordered (SPT) phase — ensuring that the most essential features of the system are accounted for. While we put emphasis on a paradigmatic situation to be specific, the overall approach pursued here can be applied to more elaborate settings.

Specifically, in this work we establish a connecting between the SPT phases of the different models in order to validate the use of an effective model in general. In particular, we show how to obtain from a symmetric spin- system (which is always in a trivial phase) a system in a non-trivial SPT phases by composing sites. This transition roots on the blocking of two spin- spins into an indivisible unit cell. The symmetries of the resulting blocked system are then identified with the ones of the effective spin model which enables us to compare the corresponding SPT phase of the system. For this, we rely on well established tools for the classification of SPT phases using matrix product states (MPS)Chen et al. (2011a); Schuch et al. (2011); Pollmann et al. (2010) and the cohomology of the respective symmetry groups. Using the identification of the symmetry groups of the blocked and effective model, we then derive an order parameter for the detection of this SPT phase of the effective model on the level of the underlying spin- system giving a rigorous tool to compare the effective spin physics with the physics of the original model.

Further, we illustrate the approach on a geometrically frustrated one-dimensional system — the bi-layered chain (BDC) (cf. Fig. 2) with an inter-layer ferromagnetic coupling and an anti-ferromagnetic coupling inside each layer. This system serves as a one-dimensional archetype of geometrically frustrated systems and appears in the effective description of more complex higher dimensional layered systems Balz et al. (); Kikuchi et al. (2011). For the BDC, we investigate the validity of an effective spin- description, for which the ground state is numerically found to be in the Haldane phase, depending on the strength of the inter-layer coupling using perturbation theory and numerical tensor network (TN) methodsVerstraete et al. (2008); Orus (2014); Eisert (2013); Eisert et al. (2010); Schollwöck (2011). By this argument, we analytically find that starting at a finite coupling strength the blocked system can be well described using the effective spin- theory where the exact critical coupling strength is determined numerically to be with and denoting the corresponding ferromagnetic and anti-ferromagnetic coupling strength. Finally, applying our order parameter we align our results with studies on the spin- Heisenberg ladderWhite (1996); Pollmann et al. (2012); Dagotto and Rice (1996); Totsuka et al. (1995); Matsumoto et al. (2004).

This work is structured into two distinct parts. We first review the classification of SPT phases in one-dimensional systems using matrix product states and explain in mathematical terms how non-trivial SPT phases emerge from blocking sites on the example of spin- systems and establish a connection between the order parameters of the blocked and the effective model. In the second part, we discuss the concrete situation of the BDC as an example and apply the developed methods.

## Ii Symmetry protected topological order

We begin this section by reviewing the classification of SPT phases using MPS. We then argue that the symmetries of a blocked spin- system agree with the ones of an effective spin- theory and establish an order parameter that can be used to check the validity of the effective model.

### ii.1 Group cohomology and SPT in 1D

Much of what follows will build on the connection between symmetry protected topological order in one-dimensional systems and the second cohomology class of the respective symmetry. In this subsection, we will briefly review some notions made use of later in a language of MPS White (1992); Fannes et al. (1992); Perez-Garcia et al. (2007). As is well-known, for such one-dimensional systems, matrix product states approximate ground states arbitrarily well, at the expense of a bond-dimension that scales moderately with the system size Schuch et al. (2008). This is essentially rooted in the observation of ground states satisfying an area law for suitable entanglement entropies for gapped models Hastings (2007); Eisert et al. (2010). In such a language of TN states, notions of topological order can be particularly concise and at the same time rigorously captured Zeng et al. (2015); Schuch et al. (2013); Chen et al. (2011a); Schuch et al. (2011); Pollmann et al. (2010); Schuch et al. (2012, 2010, 2013). In this mindset and in this formalism, an emphasis is put on ground states, while local Hamiltonians reenter stage by means of the concept of a parent Hamiltonian.

To be specific, consider a uniform matrix product state (uMPS) vectorHaegeman et al. (2011) in canonical form parametrized by the tensor , where reflect the virtual indices with bond dimension and runs from to the local physical dimension. When the state vector is symmetric under a local transformation for some symmetry group , in that , together with a transformation of the matrix with indicating either complex conjugation or transposition, then it is shown in Ref. Perez-Garcia et al. (2008) that is reflected on the virtual level as , where is a projective representation of , so that are unitary and uniquely defined up to a phase. Let now be symmetric with respect to the group . Then, for any we obtain through this formula a which form a representation of on the virtual level. Fixing the gauge freedom in the virtual representatives and iterating this formula we find for

 VgVh=ω(g,h)Vgh. (1)

Using Eq. (1) iteratively on for any one finds the relation

 1=ω(h,g)ω(hg,k)ω(h,gk)ω(g,k). (2)

Eq. (2) is the -cocycle equation. Any such defines a projective representation of with elements . Given a group , the set of possible cocycles over the -module is not arbitrary but corresponds to , namely the second cohomology group of over . With this in mind, consider a family of MPS parametrized by in a specific symmetry sector of the symmetry . In Refs. Chen et al. (2011a); Schuch et al. (2011); Pollmann et al. (2010) it is shown that the cocycle corresponding to can only change from to if the gap of the parent Hamiltonian corresponding to these MPS closes for an . Moreover, they show that any two MPS with the same corresponding cocycle can be connected by a smooth path along which the respective parent Hamiltonian remains gapped. This, however, means that the possible SPT phases with respect to , which are characterized by symmetry protected long range entanglementChen et al. (2010), correspond to the non-trivial elements of its second cohomology group,

 SPT phases of G↔H2(G,U(1)). (3)

### ii.2 Effective symmetry group

The specific question addressed in this work is how the SPT phases change when changing from the original model described by the Hamiltonian to a blocked model described by the Hamiltonian , and how they relate to an effective model. Therefore, let us first introduce the notions of blocked and effective models and identify the collection of symmetries with respect to which and its ground state are invariant.

As explained in the introduction, we would like to focus our attention to one-dimensional spin- lattice systems. Depending on the interactions present in the Hamiltonian, such a system might be well described by an effective model that captures only the relevant degrees of freedom which we want to assume to originate from a blocked set of original sites. To be precise we block two spin- sites of the system into a new site with local Hilbert space and consider them to be the natural unit of the system. The representation of the Hamiltonian with respect to this unit cell will be denoted by . The effective model in our case is then the corresponding spin- system arising from the decomposition of the local Hilbert space into and then neglecting the spin degrees of freedom.

We assume the systems under investigation to be invariant under transformations. Further those systems frequently favor additional discrete symmetries such as a time reversal (TR) or lattice inversion (I) symmetry. In layered models such as considered in the second part of this work also an additional layer exchange (LE) symmetry can be present.

In order to tell what happens due to the blocking it is important to observe how these symmetries transform under the blocking procedure. Assume a bi-layered model with the symmetries introduced above and block two opposing vertices in each layer. The TR symmetry remains and just changes its representation. The LE symmetry becomes a local unitary transformation and the I symmetry becomes an I symmetry with respect to a vertex in the blocked model. The most important changes come from the local unitary transformations. Blocking a pair of vertices maps

 SU(2)→G={g⊗g:g∈SU(2)} (4)

where we use the fundamental SU(2) representation on (i.e., the spin- representation). In the following we will denote by the full symmetry group under which is invariant.

It is known that the second cohomology class of the is trivial, implying the fact that there are no corresponding non-trivial SPT phases for spin- systems. However, the new insight is that the respective symmetry group due to the blocking is isomorphic to having non-trivial second cohomology giving rise to two SPT phases — a trivial phase and an ordered phase which we refer to as Haldane phase here. This essentially roots in the fact that by blocking two spin- sites one restricts the possible representations to the integer spin sector, locally corresponding to a faithful representation, as the total number of sites is thereby restricted to be even. In what follows, we make this isomorphism explicit and show using Künneth’s theorem that the topological phases with respect to manifest themselves in the topological phases of .

### ii.3 Isomorphism between G and So(3)

In this section, we establish the isomorphism between and . It essentially uses the fact that as the tensor product factors out . Then we show and use the fact that is a projective orthogonal representation of . We lay out the details here as understanding this isomorphism allows for a clearer interpretation of the order parameter derived in the subsequent section.

###### Lemma 1 (Group isomorphism).

There exists a group isomorphism .

###### Proof.

To start with, it can be easily seen that the map which is defined by is a faithful representation (where denotes the equivalence class and the map is independent of the representative of that class). Now, we use the fact that the map from the tensor product of matrices into the set of completely positive maps

 ι:G→G+={Πg:=g(⋅)g†|g∈SU(2)},g⊗g↦g(⋅)g† (5)

is an isomorphism if and are interpreted as groups. Next we construct a isomorphism from the latter to . Consider therefore the real space of Hermitian traceless complex matrices, where the are the Pauli matrices. It is natural to define a scalar product in as

 ⟨ab|ba⟩H3=αTr[ab] (6)

for any pre-factor and any . Using the commutation relations of Pauli matrices

 σiσj=δi,j+iϵi,j,kσk (7)

it turns out that choosing renders the map

 S:R3→H3:v↦v⋅σ, (8)

which is surjective, an isometry (preserving the scalar product) such that there exists an inverse . Let us now make full use of in order to connect and . For any we define the map which can by definition be expressed as for some . Obviously is linear and we find (using the fact that is an isometry)

 Missing or unrecognized delimiter for \right =12Tr[v⋅σw⋅σ]=⟨vw|wv⟩R3. (9)

Hence, , the set of isometries on . The other way around, for any we can define a map such that for any it holds

 Π(R)(v⋅σ)=(Rv)⋅σ. (10)

Clearly, as is the set of basis transformations in we can translate the basis transformation in induced by to a basis transformation in with such that . Hence, the isometry induces a group isomorphism which we also denote by , where the homomorphic structure follows from the properties of the scalar product. Finally, we can define which defines the claimed group isomorphism. ∎

### ii.4 SPT phases of GTot

We now argue that the SPT phases with respect to are embedded in the SPT phases of , i.e., we explain how the additional discrete symmetries of the system affect the cohomology. To do so, we employ Künneth’s theorem for cohomology which states that for two groups and , and corresponding -module Gerig ()

 Hn(G1×G2,M)≃[⨁i+j=nHi(G1,M)⊗Hj(G2,M)]⊕ [⨁p+q=n+1TorZ1(Hp(G1,M),Hq(G2,M))]. (11)

Eq. (11) holds true even for non-trivial actions of the on given that one of the -modules is -free. In our case we use with , being the remaining discrete and finite symmetries and . The action on is trivial for all symmetries but TR.

Note that the Tor functor is trivial given is a free module and . Moreover, is symmetric given is abelian. Hence, as and , the Tor functor in Eq. (11) is trivial in our caseChen et al. (2011b); Gerig (). Similarly, we find that as well as is trivial, such that the possible SPT phases in our system reduce to

 H2(Gtot,U(1))=H2(SO(3),U(1))⊕ H2(Z2×Z2×ZT2,U(1)). (12)

In particular, as is a subgroup of this, the group element corresponding to the Haldane phase is contained in which corresponds to the possible non-trivial SPT phase in our system.

### ii.5 Detection of SPT phases

In this section we are going to derive an order parameter for the symmetry characterizing the Haldane phase that emerges due to the blocking of opposing lattice sites. We therefore follow the construction of the order parameter for the standard Haldane phase in the spin 1 chain as described in Pollmann et al. (2010); Pollmann and Turner (2012). As shown in Ref. Perez-Garcia et al. (2008), given a uMPS parametrized by the 3-tensor and a local (anti-) unitary symmetry operation parametrized by the matrix on the physical level (and complex conjugation of in case of anti-unitaries), then acts on the uMPS as explained in Fig. 1.

Let us now consider the action of looking at the two elements

 Rj=exp(iπσj/2)⊗exp(iπσj/2)∈G (13)

for . By the isomorphism of and explained in Sec. II.3, we can interpret those operators as rotations around the axes in . Using the relation displayed in Fig. 1 a) we obtain for each the respective virtual representation . Iterating the relation in Fig. 1 b) twice and using as well as the defining formulas for the left canonical gauge

 ∑jM†jIMj=I;∑jMjΛM†j=Λ (14)

we obtain . Hence, using the gauge freedom of the ’s we can substitute such that we assume in the following. Consecutively applying and and iterating the relation in Fig. 1 b) twice and using Eq. (14), however, we obtain . As the gauge of the is already fixed the phase can not be absorbed in the . Moreover, using the unitarity of the ’s we can rewrite this in a gauge invariant form

 eiθx,zI =VxVzVxVz =VxVzeiθx/2eiθz/2e−iθx/2e−iθz/2VxVz =~Vx~Vz~V†x~V†z

where the ’s are the virtual representation in an arbitrary gauge, such that we obtain

 ~Vx~Vz=eiθx,z~Vz~Vx. (15)

Obviously, the phase is connected to the cocycle evaluated at Pollmann et al. (2010). Hence, any corresponds to a non-trivial cocycle and it turns out that for it holds that where corresponds to the topological non-trivial Haldane phase. Keeping this in mind it is straightforward to define the order parameter as

 OSO(3)=1DTr[~Vx~Vz~V†x~V†z] (16)

for symmetric MPS and else, where we use the gauge free representation .

In order to make this more precise, from Künneth’s theorem we find that

 H2(Gtot,U(1))≃ H2(SO(3),U(1))⊕H2(ZL2×ZT2×ZF2,U(1)), (17)

as explained in II.4. Hence, given we find a non trivial phase for some of the form and , we can deduce from this that the corresponding factor is corresponding to a non trivial element from that is from the sector in (II.5). In other words we deduce from this the fact that we are in a topologically non trivial SPT phase with respect to , the Haldane phase. Now, this order parameter allows us to compare the SPT phase of the blocked model with the phase of an effective symmetric spin- model in order to explore the validity of the effective model as we will illustrate in more details in the next part on the example of the bi-layered delta chain.

## Iii Case study: a bi-layered Δ-chain

In the focus of attention in this work, as a proxy for similar microscopic models allowing for an effective description, are bi-layered -chains (BDC) with anti-ferromagnetic Heisenberg interaction within the layer and ferromagnetic Heisenberg interaction between the layers (see Fig. 2). The corresponding model Hamiltonian is given by

 H=JAFHAF+JFHF (18)

where the ferromagnetic and anti-ferromagnetic Heisenberg terms and are given by

 HAF=∑⟨i,j⟩∈EAFsi⋅sj, (19) HF=−∑⟨i,j⟩∈EFsi⋅sj. (20)

Here denotes the anti-ferromagnetic edge set (blue bonds in Fig. 2) and denotes the ferromagnetic edge set (red bonds in Fig. 2). Moreover, throughout the paper we use the convention .

The ratio determines the physics of the model. Therefore, in view of performing a numerical investigation of the system we chose to parametrize the couplings using a compactly supported parameter by defining

 H(θ)=cos(θ)HAF+sin(θ)HF. (21)

Here, interpolates between weak ferromagntic inter-layer couplings () and very strong ferromagnetic interactions (). As explained in the introduction, we are going to draw the connection between the original double layered spin- model and the effective single-layered spin- model on the level of phases. To be precise, the effective model here is defined as a single-layered delta chain with spin- particles on each site and anti-ferromagnetic coupling just as in Eq. (19). In order to establish the comparison of the effective and the original model, it is crucial to introduce the blocking in the original spin- model of the ferromagnetically coupled edges into a single unit cell. This leads to a new Hamiltonian describing a single-layered -chain of generalized quantum particles supported on a local Hilbert space . For the sake of distinction we will refer to the blocked Hamiltonian as . The main reason for the emergence of the Haldane phase in a system basically described by a spin- ladder can be traced to the annihilation of the phases in the tensor product .

In order to obtain an intuition for the BDC we are first going to analytically investigate its strong and weak coupling limits in the following three sections. Subsequently we employ numerical MPS based methods in order to investigate the order parameter of the blocked model probing the validity of the effective spin- model explained above.

### iii.1 single-layered spin-1/2 and uncoupled Δ-chains

To start with, we consider an isolated spin- -chain with anti-ferromagnetic Heisenberg interaction. We can write the Hamiltonian of the system as , where we sum over each triangle in the chain, and refers to the local Hamiltonian on the respective triangle When coupling two spin- particles the respective Hilbert space decomposes as . Doing the same with three spins we end up with . We can easily construct a (non-orthogonal) basis for the ground state space of a single triangle as follows. Any ground state has a total spin of . So we take two of the three spins and couple them in a singlet. The third spin can be chosen arbitrarily, such that the three spins form a state. In particular, we find four independent states by putting the singlet either on the edge or and putting the third spin either in the state vector or . In particular, in what follows we will denote these states on a single triangle as

 |↗⟩⊗|s⟩=1√2(|0,1⟩−|1,0⟩)⊗|s⟩, (22) |s⟩⊗|↖⟩=1√2|s⟩⊗(|0,1⟩−|1,0⟩) (23)

where the arrow indicates a singlet, which we will also refer to as dimer, on the up, respectively downwards facing edge and . It is easy to compute

 |(⟨s|⊗⟨↖|)(|↗⟩⊗|~s⟩)|≤1√2. (24)

Using this observation we can construct the ground states of the full (periodic) chain of sites as follows. Either we place a singlet on any of the up edges, or we put it on any of the down edges. That these states correspond to ground states of the system is clear from the observation that they are locally the ground states of each of the and therefore the minimal possible energy expectation value and it is also clear that those are the only states corresponding to this energy. Hence, periodic single-layered -chain has a two fold degenerate ground state space spanned by the dimer states formed by singlets sitting on all up or down edges. We denote these states by

 |↗⟩N=|↗⟩⊗N/2 (25) |↖⟩N=|↖⟩⊗N/2 (26)

where we will drop the index if it is clear from the context that we talk about the many particle wave function. It is worth mentioning that the state in the definitions (25) and (26) is implicitly assumed to be placed on edges facing upwards whilst the state is assumed to be placed on edges facing downwards. This implies that the tensor products in the definitions (25) and (26) act between different local Hilbert spaces. Moreover, it is easy to see that the ground state degeneracy for an open -chain would grow proportional in system size as for any site the state is a ground state.

Having identified the ground state of a single chain we can easily define the ground state space of the uncoupled bi-layered -chain. As the Hamiltonian of the uncoupled system is simply the sum of the individual Hamiltonians of each layer , the ground state space is spanned by the tensor products of the ground states of the individual layers. The two ground states of a single layer give therefore rise to four states spanning the ground state space of the bi-layered system.

Using the notation introduced above we can write a basis spanning the four dimensional ground state space of the uncoupled BDC as

 |1⟩:=|↗⟩⊗|↗⟩, |2⟩:=|↗⟩⊗|↖⟩, (27) |3⟩:=|↖⟩⊗|↖⟩, |4⟩:=|↖⟩⊗|↗⟩. (28)

It is worth mentioning that the scalar product between any of those states vanishes exponentially in the thermodynamic limit as

 |N⟨ij|ji⟩N|≤1√2N/4,i≠j (29)

for . Note that the state vectors and are of product form with respect to the unit cells introduced from the blocking of opposing spins and therefore correspond to a trivial SPT phase, where the other two states correspond to the non-trivial phase.

### iii.2 Strong coupling limit

Let us now characterize the ground states in the extremal coupling regimes. To start with we analyzed the strong coupling limit. Rewriting the Hamiltonian of the system as we find to be infinitely suppressed in the infinite coupling limit with Investigating for and large and using the local Hilbert space decomposition for two opposing vertices, we find that equals the effective model, the single-layered spin 1 -chain in the spin 1 subspace of the opposing vertices, plus a weak perturbation connecting the spin 0 and spin 1 spaces which is suppressed by . Hence, to zeroth order perturbation theory the ground state in the strong coupling limit is the ground state of the effective spin 1 model mapped onto the double layered -chain. Using the TDVP for uMPSHaegeman et al. (2011) we numerically computed the ground space for the spin 1 system. Unsurprisingly, as the spin 1 -chain is an one dimensional anti-ferromagnetic spin- Heisenberg model, its ground state is found to be in the same phase as the anti-ferromagnetic spin- Heisenberg model on a linear chain, the topologically non-trivial Haldane phase. From this consideration it is however unclear to what extend this result caries over to other coupling strengths as the ground state space of the uncoupled chain is four-fold degenerate and therefore in a different phase. In what follows we will investigate the weak coupling limit in order to show that a weak ferromagnetic interaction favours the space spanned by the topologically trivial states and leads to a symmetry broken phase. Hence, for some finite coupling strength a transition from symmetry broken to the SPT phase has to occur in the BDC.

### iii.3 Weak coupling limit

Let us now investigate the weak coupling limit. We denote the ground state space spanned by the state in Eq. (27), (28) of the uncoupled periodic BDC with sites by and consider the perturbation to the uncoupled BDC. As we find for every ferromagnetic edge and all we need to find the minimizer of the second order with . Here is the projection onto the orthogonal complement of the ground state space of and is its ground state energy. Therefore we define the matrix in the thermodynamic limit as with

 Δi,j(N)=⟨iV(HAF−E0)−1P1Vj∣∣V(HAF−E0)−1P1V∣∣jiV(HAF−E0)−1P1V⟩, (30)

where refer to the ground states of the unperturbed Hamiltonian as defined in Eq. (27), (28). Our aim is now to give an estimate of the entries .

Writing for the projection onto the maximal eigenspace of and for the projection onto the orthogonal complement and using we can rewrite (30) as follows

 Δi,j(N)= −1E0⟨iV(I−HAFE0)−1P1Vj∣∣∣V(I−HAFE0)−1P1V∣∣∣jiV(I−HAFE0)−1P1V⟩ = −1E0⟨iV(I−HAFE0)−1PMVj∣∣∣V(I−HAFE0)−1PMV∣∣∣jiV(I−HAFE0)−1PMV⟩ −1E0⟨iV(I−HAFE0)−1P⊥MP1Vj∣∣∣V(I−HAFE0)−1P⊥MP1V∣∣∣jiV(I−HAFE0)−1P⊥MP1V⟩. (31)

Next, we observe that the energy of locally on each of the triangles is in with

 Elocalmax=−Elocalmin=34 (32)

and hence (and, as is easy to show, ). Therefore, we can expand the last term into a Neumann series as

 Missing or unrecognized delimiter for \right (33)

for all . We find

 Δi,j(N)= −1E0⟨iV(I−HAFE0)−1PMVj∣∣∣V(I−HAFE0)−1PMV∣∣∣jiV(I−HAFE0)−1PMV⟩ −1E0∑k⟨iV(HAFE0)kP⊥MP1Vj∣∣∣V(HAFE0)kP⊥MP1V∣∣∣jiV(HAFE0)kP⊥MP1V⟩. (34)

It is easy to see that

 Missing or unrecognized delimiter for \left ≤1|E0|∑k||V|i⟩||2∣∣ ∣∣∣∣ ∣∣HAFE0∣∣∣Img(P⊥MP1)∣∣ ∣∣∣∣ ∣∣k (35)

such that, due to the projectors, by construction for every the sum over in Eq. (III.3) is absolutely converging.

Next we estimate the first summand in Eq. (III.3) as well as each of the terms in the von Neumann series in the large limit. Therefore, we make use of the fact that the overlap of the excitations and the ground state space and the maximal energy eigenspace is small. In particular,

 |⟨iVj|V|jiV⟩|≤∑a|⟨iVaj|Va|jiVa⟩|≤N√2N/4−1(1−δi,j) (36)

as for for each of the summands the vector contains a triplet being orthogonal to the singlet in the dual vector, whilst for there are at least triangles being covered with different singlet configurations (cf. Eq. (24)). Similarly, for any it holds

 ⟨MVi|V|iMV⟩=∑a⟨MVai|Va|iMVa⟩=0 (37)

as each triangle in is covered by a spin configuration being orthogonal to the singlet configurations in . Therefore, using the continuity of the scalar product, we can drop the projections and in the thermodynamic limit and drop the first summand in Eq. (III.3). We obtain

 Δi,j =−∑klimN→∞1E0⟨iV(HAFE0)kVj∣∣∣V(HAFE0)kV∣∣∣jiV(HAFE0)kV⟩ Missing or unrecognized delimiter for \left (38)

Let us now estimate the off diagonal elements . Assume therefore that and , where the other cases work in the same fashion and only the exponent in the final estimation might change by a factor of two. Expanding the power of sums over triangles we can bound

 ∣∣ ∣∣⟨3Va(∑ΔhΔE0)kVb1∣∣ ∣∣Va(∑ΔhΔE0)kVb∣∣ ∣∣13Va(∑ΔhΔE0)kVb⟩∣∣ ∣∣ ≤∑i∈[N2]k∣∣ ∣∣⟨3VahΔi1…hΔikEk0Vb1∣∣ ∣∣VahΔi1…hΔikEk0Vb∣∣ ∣∣13VahΔi1…hΔikEk0Vb⟩∣∣ ∣∣. (39)

Moreover, it is easy to compute

 Va|↗⟩N⊗|↗⟩N=|↗⟩N−1⊗|↗⟩N−1⊗ ⊗(1∑i,j=−1αi,j|i╱⟩⊗∣∣j╱⟩)a (40)

where denotes a triplet with sitting on the upwards facing edge of a triangle (similarly for the downwards facing edge), and . Also we implicitly assume the tensor products to be ordered in such a way that the last factor on the right hand side corresponds to the pair of triangles for which the singlets (before the action of ) are intersecting the ferromagnetic edge , indicated by the index . Then we can rewrite the summands in (39) as

 ⟨3VahΔi1⋯hΔikEk0Vb1∣∣ ∣∣VahΔi1⋯hΔikEk0Vb∣∣ ∣∣13VahΔi1⋯hΔikEk0Vb⟩= =N−1⟨↖|⊗N−1⟨↖|⊗(1∑i,j=−1αi,j⟨╲i|⊗⟨╲j∣∣)ahΔi1⋯hΔikEk0|↗⟩N−1⊗|↗⟩N−1⊗(1∑i,j=−1αi,j|i╱⟩⊗∣∣j╱⟩)b. (41)

We know that each independent of and similarly for the other singlet configuration. can act non-trivially only on triangles covered by a triplet and can propagate the excitation only to one of the neighbouring triangles (cf. Appendix V.1). Henceforth, can at most create a set of triangles of the triangles that may not be in the singlet configuration. Let now be the set containing the triangles excited by the local perturbations plus the at most triangles which are acted on non-trivially by the . Then we can bound Eq. (41) using , the Cauchy-Schwarz inequality and

 ∣∣ ∣∣⟨3VahΔi1⋯hΔikEk0Vb1∣∣ ∣∣VahΔi1⋯hΔikEk0Vb∣∣ ∣∣13VahΔi1⋯hΔikEk0Vb⟩∣∣ ∣∣≤∣∣ ∣∣N−k−4⟨31|13⟩N−k−4k1⟨↖|⊗k2⟨↖|⊗(1∑i,j=−1αi,j⟨╲i|⊗⟨╲j∣∣)a ×hΔi1⋯hΔikEk0|↗⟩k1⊗|↗⟩k2⊗(1∑i,j=−1αi,j|i╱⟩⊗∣∣j╱⟩)b∣∣ ∣∣≤2kNk√2N/2−k−4, (42)

where the states are meant to be supported on the complement of only, while the triangles in that are not excited by the excitation are labeled by and fulfilling . If we had chosen a different combination of states and in the beginning, the same argumentation would hold just that for a combinations such as and , where the states coincide in one of the layers, the exponent in the final estimate in Eq. (42) changes from to , as then only half the triangles are populated by a different covering. Finally, combining this bound with (38) and (39) we can conclude that

 |Δi,j|≤∑klimN→∞N2√2N/4−k−4=0 (43)

for .

Let us now investigate the diagonal elements . For those it is crucial to investigate the action of onto the states . In Appendix V.1 we investigate the action of the local terms on locally excited states of the form , the other configuration follows directly from inversion symmetry. We find that the excitation may be spread only to the neighbouring triangle to the right, while the triangle originally occupied by the excitation will always remain occupied by some triplet excitation. In particular, no combination of local terms can transform a triplet on the original triangle into a singlet state. This can be used straightforward in order to calculate the corresponding action of the local terms in the BDC on the states . It follows directly that can recombine only with itself, respectively that

 ⟨i|VaVb|i⟩=0⇒⟨i|VaHkAFVb|i⟩=0 (44)

Now it is easy to verify that for the parallel configurations