A topological classification of molecules and chemical reactions with a perplectic structure
AA topological classification of molecules and chemical reactions with a perplecticstructure
Lukas Muechler Center for Computational Quantum Physics,The Flatiron Institute, New York, New York, 10010, USA (Dated: March 4, 2019)In this paper, a topological classification of molecules and their chemical reactions is proposed ona single particle level . We consider zero-dimensional electronic Hamiltonians in a real-space tight-binding basis with spinless time-reversal symmetry and an additional spatial reflection symmetry.The symmetry gives rise to a perplectic structure and suggests a Z invariant in form of a pfaffian,which can be captured by an entanglement cut. We apply our findings to a class of chemicalreactions studied by Woodward and Hoffmann, where a reflection symmetry is preserved along aone-dimensional reaction path and argue that the topological classification should contribute to therate constants of these reactions. More concretely, we find that a reaction takes place experimentallywhenever the reactants and products can be adiabatically deformed into each other, while reactionsthat require a change of topological invariants have not been observed experimentally. I. INTRODUCTION
The advancements of the topological descriptionof non-interacting crystalline matter within the lastdecade have revolutionized the field of condensed matterphysics.
Topological considerations first predicted theexistence of unremoveable exotic surface or edge statesand have lead to the discovery of many new phases ofmatter in crystals , e.g. exotic fermionic states withoutanalogues in high-energy physics. A state of matter istopologically non-trivial, if the ground state of the systemcannot continuously be deformed into the atomic limitwithout gap-closing. Spatial and non-spatial symmetriesdetermine the way this atomic limit is approached andgive rise to a large variety of different non-interactingand interacting topological phases.
These methodshave recently found to be relevant for other areas ofphysics, e.g. mechanical systems, electric circuits andeven weather phenomena
In like manner, it is known that topological effects playan important role in molecular systems and their chemi-cal reactions. For example, the geometric phase acquiredby moving around a conical intersection has been shownto strongly influence the reaction rate of simple chem-ical reactions due to interference of different reactionpaths.
Further, it has been proposed that the sur-face states of Weyl semimetals or topological insulatorscould influence the outcome of chemical reactions as cat-alysts.
Chemical reactions are rare events of the quantum dy-namics on the Born-Oppenheimer (BO) surface and gen-erally are complicated dynamical problems. Despite em-pirical rules that provide strong guidelines, it is not fullyunderstood why certain chemical reactions work the waythey do, e.g. the Woodward-Hoffman rules (WHR). In this paper we approach this problem from a topolog-ical point of view and propose a topological classifica-tion for molecules and their reactions that are describedby these rules. The way this classification manifests it-self in molecules is necessarily different from the solid state, since molecules are finite sized objects: When twocrystals with different topological invariants are broughtin contact, the spectral gap has to close at the inter-face, since a topological invariant can only change ata gap closing point . At the interface, one thereforefinds topologically required gapless states [Fig. 1 (b)]. Inmolecules, such states will not appear due to their zero-dimensional (0D) nature; the interface states can gener-ically be gapped out or are not well defined.Instead, we here propose that a topological classifica-tion of molecules can manifest itself in their chemicalreactions. We study chemical reactions of a set of re-actant molecules R that transform into a set of product molecules P ( R → P ) and describe the transformation ofthe ground state (GS) of the reactant Hamiltonian H R into the GS of the Hamiltonian H P describing the prod-ucts. A common way to model this process is to define areaction path via a reaction coordinate τ . This allows usto describe a chemical reaction as a continuous deforma-tion of a reaction Hamiltonian H ( τ ) = f ( τ ) H R + g ( τ ) H p ,with f (0) = g (1) = 1 and f (1) = g (0) = 0 , which onecan classify topologically.This approach allows us to distinguish between two dif-ferent cases as displayed in Fig. 1(c)&(d). In the firstcase, reactants and products posses the same topologicalinvariant. By definition, the GS of H R can be smoothly,i.e. adiabatically, deformed into the GS of H P ; concor-dantly the GS of H ( τ ) is separated from the excited statesby a gap for all τ . In the second case, the reactants andproducts differ in their topological invariants; the gap of H ( τ ) has to close along the reaction path and the GSof reactants and products cannot be adiabatically trans-formed into each other, i.e. the reaction has to proceedin a non-adiabatic fashion.The quantum mechanical observable associated withchemical reactions is the reaction rate. For adiabaticreactions, i.e. the first case, the Born-Oppenheimer ap-proximation is usually valid and the reaction rate can becalculated by solely focusing on the GS of H ( τ ) . Therate is determined by the energy barrier of the reaction, a r X i v : . [ phy s i c s . c h e m - ph ] M a r adiabatic transition E n e r gy reaction coordinate non-adiabatic transition ν A = ν B reaction coordinate E n e r gy Material A Material BVBMCBM ν R = ν P ν A ≠ ν B VBMCBMMaterial A Material B E n e r gy E n e r gy reactants products reactants products (a) (b)(c) (d) ν R ≠ ν P Δ R k ∼ e −Δ R k ∼ p LZ = 0 FIG. 1. (a) An interface of two materials with the same topological invariant. While the valence band maxima (VBM) andconduction band maxima (CBM) change due to interface effects, the gap does not close at the interface (b) An interface of twomaterials with different topological invariants. The spectral gap has to close at the interface and topologically protected surfacestates emerge. (c) A chemical reaction in which the reactants and products have the same topological invariant. The spectralgap does not close along the reaction path and the reaction proceeds adiabatically and the reaction rate k is determined by theactivation barrier ∆ R . (d) A chemical reaction in which the reactants and products have a different topological invariant. Here,the spectral gap closes along the reaction path. The reaction rate is proportional to the probability p RP to transition betweenthe GS of the reactants | R (cid:105) to the ground state of the products | P (cid:105) , which can be estimated from Landau-Zener theory. which is determined by the energy of a transition state[Fig. 1(c)]. In the second case, transition state theory isnot valid, as the higher energy states of H ( τ ) cannot beignored due to the gap closing point, as e.g. in the non-adiabatic regime of Marcus Theory. In a simplifiedpicture, the reaction rate can be approximated from Lan-dau Zener theory close to the gap closing point. For ex-ample, the rate can be computed as k = (cid:82) dE p ( E ) e − βH ,where β is the inverse temperature, H is the Hamiltonianin the micro-canonical ensemble and p ( E ) is the proba-bility to jump from the left surface to the right surface fora given energy E [Fig. 1(d)]. Close to the crossing point,one can approximate this probability with the Landau-Zener probability p ( E ) (cid:39) p LZ = 1 − e − ξ ∆ , where ∆ isthe gap between the two potential energy surfaces and ξ is a constant that depends on the details of the Hamilto-nian. One therefore finds a vanishing reaction rate k incase of a crossing ( ∆ = 0 ), while it is exponentially smallin the presence of a small gap. The physical picture isthe following: instead of ending up in the GS of H P , thefinal state of the reaction will be a linear combinationof excited states and therefore the rate for the reaction,vanishes in this case. II. OUTLINE
The paper is structured as follows: We begin withthe topological classification by introducing a simple toy-model in Sec. III, where we summarize and discuss themain results without derivation. In the following section,Sec. IV, we review the theory of bisymmetric matricesand derive the most general form of a spinless, time-reversal symmetric Hamiltonian of even matrix dimen-sion with a reflection symmetry. In addition, we derivea general expression for the Z topological invariant thathas been introduced in the first section.In Sec. V, we discuss the implications of the topologicalclassification on the theory of chemical reactions, in par-ticular the case of the Woodward-Hoffmann rules thatare described by our models. III. A SIMPLE EXAMPLE
Before discussing the most general case, we start witha simple example of a × Hamiltonian H ( t ) at half fill-ing, which depends on a single tuning parameter t thattakes on the role of the reaction coordinate while theother parameters a, h, g remain constant. The Hamilto-nian models the reaction of two ethylene molecules thatapproach each other along a reflection symmetric reac-tion pathway, which is modeled by tuning t (see Sec. Vfor a more detailed discussion). The Hamiltonian in thesite basis is given as H ( t ) = g − µ t h at − g − µ a hh a − g − µ ta h t g − µ , (1)where µ is the chemical potential, which will be usedto define a suitable reference energy. The Hamiltonian istime-reversal symmetric with T = K , where K is the anti-unitary complex conjugation operator. H ( t ) possesses areflection symmetry J , J = (2)such that [ H ( t ) , J ] = 0 for all t . Note that the tuning-parameter t is not affected by any symmetry. Since [ H ( t ) , J ] = 0 , there exists a orthogonal matrix K whichblock-diagonalizes both J and H ( t ) , K = 1 √ − − . (3)We arrive at H B ( t ) = K H ( t ) K T = (cid:18) H − ( t ) 00 H + ( t ) (cid:19) , (4)where the blocks corresponding to the ± eigenspaces of J are obtained as H ± = (cid:18) ± a + g − µ t ± ht ± h ± a − g − µ (cid:19) . A. Spectrum of H B ( t ) The two eigenvalues of H + are given as (cid:15) e ± ( t ) = a ± (cid:112) g + ( h + t ) − µ (5)and the eigenvalues of H − are given as (cid:15) o ± ( t ) = − a ± (cid:112) g + ( h − t ) − µ (6)For the sake of simplicity, we assume a > g > h > aswell as t > . Now, while tuning t , there can be a levelcrossing between two states of the different blocks, (cid:15) e − ( t ) = (cid:15) o + ( t ) ⇔ t = t c := a (cid:112) a − g − h (cid:112) ( a − h )( a + h ) . (7)This crossing point is a gap closing point between thetwo eigenspaces of J , and therefore the Hamiltonians H B ( t < t c ) and H B ( t > t c ) should be topologically dif-ferent. We expect that this can be characterized by atopological invariant ν ( t ) that completely characterizesthe 0-D Hamiltonian for each t . The invariant shouldnot change if trivial bands are added and should be ro-bust to deformations that do not close the gap betweenthe occupied states. Due to the lack of a chiral or par-ticle hole-symmetry, there is no natural zero of the en-ergy, which we need to define the topological invariant.We therefore define the zero of the energy to be at thecrossing point t c and set µ = a − (cid:113) ( t c + h ) + g , whichenforces half-filling. B. Topological invariant and Z structure We here propose, with the derivation given in sec IV,that this invariant derives not directly from the Hamil-tonian, but from the matrix S ( t ) = Σ H B ( t )= − a − g + µ − h − t − h − t − a + g + µa − g + µ h − t h − t a + g + µ , (8)where the σ i are the Pauli-matrices and Σ = − σ ⊗ σ .The invariant is given as ν ( t ) = Pf [ S ( t )] = sign (cid:2) − ( h + t c ) + ( h + t ) (cid:3) , (9)which means that ν ( t ) = − for t < t c and ν ( t ) = 1 for t > t c , while it jumps abruptly at the crossing point t = t c . The invariant is a Z invariant since it can onlytake on the values ± . For t < t c and t > t c there existsa gap between the lowest two energy-eigenstates and thehighest two energy eigenstates [Fig. 2(a)].To prove that the topological classification is indeed a Z -classification, we now double the size of the Hamilto-nian H D ( t ) = H B ( t ) ⊗ σ . (10)Similarly J D = J B ⊗ σ . There exist several symme-try preserving mass terms M , such that { M, H D ( t ) } =[ M, J d ] = [ M, T ] = 0 , which indicates that the sys-tem becomes topologically trivial upon doubling the sys-tem. For example, M can be chosen to be the matrix M = σ ⊗ σ ⊗ σ , which gaps out every crossing, whilepreserving the spatial symmetry J and TRS [Fig. 2(b)]. C. Entanglement spectrum
The topological transition at t = t c is accompanied bya change in the single-particle entanglement spectrum (a)(b) (c) ��� ��� ��� ��� ��� - � - � - ����� � � (d) ��� ��� ��� ��� - �� - ������� � � � � ( � � ) Eig(H e ) ��� ��� ��� ��� ��� - � - � - ����� � �
12 34
A B ⌫ ( t ) E FIG. 2. (a) Eigenvalues of H ( t ) and topological invariant ν ( t ) as function of t for a = 1 , g = 0 . and h = 0 . . (b) Eigenvaluesof the doubled Hamiltonian H D ( t ) with M = 0 . (c) Real space picture of entanglement cut and definition of subsystem A and B . (d) Entanglement spectrum for a = 1 , g = h = 0 as a function of t . The symmetry breaking mass has been set to m = 0 . . between two spatial blocks A and B [Fig. 2(e)] of theHamiltonian H ( t ) that are connected by varying t . Theentanglement spectrum is the spectrum of the entangle-ment Hamiltonian H Be which is defined through the re-duced density matrix of subsystem B via ρ B = 1 T r B (cid:2) e − H Be (cid:3) e − H Be . (11)For non-interacting systems, it can conveniently be ob-tained from the eigenvalues λ m of the flattened Hamilto-nian Q projected on subsystem B as Q B ( t ) = − P B (cid:32) (cid:88) n ∈ occ | n, t (cid:105) (cid:104) n, t | (cid:33) P B (12)where P B is the projector on subsystem B and | n, t (cid:105) isan eigenstate of the Hamiltonian H ( t ) . The eigenvalues p i of the entanglement Hamiltonian H Be can be obtainedfrom inverting the relation λ n = 12 tanh (cid:16) p n (cid:17) . (13)In this part, we solve the problem for the analyticallytractable case of g = h = 0 . In order to define the oc-cupied bands for all t , we have to introduce an infinites-imally small symmetry breaking term M = mσ σ . Inthe limit of m → the projected flat band Hamiltonian is then given as Q B ( t ) = m (cid:16) | t − | + t +1 (cid:17) (cid:16) sgn (1 − t ) (cid:17) (cid:16) sgn (1 − t ) (cid:17) m (cid:16) − | t − | − t +1 (cid:17) (14)Expanding around m → again, the eigenvalues λ n tolowest order in m are , , (cid:40) ± tm t − + O (cid:0) m (cid:1) t > ± (cid:104) − − m t − + O (cid:0) m (cid:1)(cid:105) t < . (15)Keeping only the nonsingular entanglement eigenvalues,we arrive at p ± = (cid:40) ± (cid:104) m ) + log (cid:16) t − (cid:17) − (cid:105) t < ± tmt − t > (16)which is plotted in Fig. 2(d). The entanglement spectrumshows a discontinuous jump at t = t c = 1 . For t < ,the spectrum is nonzero and depends on the values of m and t , whereas it becomes quantized to zero for t > as m → , which is a general indication of a topologicalphase transition. IV. GENERAL THEORY OF BISYMMETRICHAMILTONIANS
In this section, we introduce the general theory of real n × n Hamiltonians H which commute with a reflectionsymmetry J . For this, we choose a special basis in whichthe reflection symmetry takes on the form of the so-calledexchange matrix J = (cid:18) JJ (cid:19) , (17)where J is the n × n matrix with ’s along the anti-diagonal and ’s everywhere else, such that J i,j = δ i,n − j +1 . J is an involution and therefore the eigenvaluesare ± .We begin with a few preliminary definitions. A matrix X , for which X = JXJ , is called perplectic or centrosym-metric . Matrices Y obeying Y = JY T J , where Y T isthe transpose of Y , are called persymmetric . Symmetriccentrosymmetric or equivalently symmetric persymmet-ric matrices H = H T , H = JHJ are called bisymmetric .Similarly, skew-persymmetric matrices V are defined via V = − JV T J , skew-centrosymmetric S via S = − JSJ and finally we call a matrix doubly-skew if it is skew-symmetric and skew-centrosymmetric.The above-mentioned types of matrices have been stud-ied extensively and therefore we restrict ourself to onlya brief review of the most relevant properties.
Themost general real perplectic n × n square block matrix X is given as X = (cid:18) U JV JV JU J (cid:19) , (18)with U, V ∈ R n × n .The most general real Hamiltonian which commutes with J therefore has to be bisymmetric, due to the additionalconstraint H = H T and is given as, H = (cid:18) A JBJB JAJ (cid:19) , (19)where A = A T is symmetric and B = JC T J is persym-metric. The Hamiltonian H posses n J -symmetric and n J -antisymmetric eigenvectors. A. Topological invariant
We wish to characterize the Hamiltonian in (19) topo-logically by defining N occ occupied states ordered by en-ergy. We define the zero of energy via a suitable shiftof the chemical potential, such that N occ states have anenergy ε ≤ . In the case of a degeneracy of the highestenergy state, we define the zero of energy at the point ofdegeneracy.Working in the basis in which J is diagonal, we arriveat H B = K H B K T (cid:18) A + JB A − JB (cid:19) (20) with K = 1 √ (cid:18) − J J (cid:19) (21)Thus the eigenstates of the different blocks of H B aregiven by the eigenstates of the symmetric matrices A ± JB .For each Hamiltonian of the form of (19) there ex-ists a one-to-one mapping to a non-symmetric skew-centrosymmetric matrix S via S = Σ H = (cid:18) A JBJ − B − JAJ (cid:19) , (22)with Σ = n × n ⊕ − n × n . The diagonalizable, non-symmetric matrix S , by definition, possesses a chiralsymmetry J SJ = −S . Therefore, the eigenvalues λ i ∈ C come in pairs: if ( x , λ ) is an eigenpair of S , ( J x , − λ ) is an eigenpair as well.Now, instead of characterizing the Hamiltonian H B , wechoose to characterize the S in the diagonal basis of JS B = K S K T = (cid:18) A + JBA − JB (cid:19) . (23)This is motivated by the observation that the null-spaceof H B is the null-space of S B , since S TB S B = H B . (24)It can be shown that only the null-space of S B can beexpressed in the basis of H B , while the non-zero eigen-vectors of S B are neither even nor odd under J B . Wenow assume that there is a degeneracy between an eigen-state | + (cid:105) of A + JB and an eigenstate |−(cid:105) of A − JB atzero energy. It follows, that the vector [ | + (cid:105) , |−(cid:105) ] T is azero-mode of S B , since S B (cid:20) |−(cid:105)| + (cid:105) (cid:21) = (cid:20) ( A + JB ) | + (cid:105) ( A − JB ) |−(cid:105) (cid:21) = (cid:20) (cid:21) (25)The zero modes of S B thus correspond to the doubledegeneracies between the different blocks of H B at zeroenergy. At this point, the real and imaginary parts ofthe eigenvalues coalesce at a so-called exceptional point[Fig 3]. Exceptional points have recently attracted inter-est as they are relevant for the topological classification ofnon-hermitian Hamiltonians in translationally invariantsystems. To measure this coalescence, we introduce ν = sgn Pf[ S B ] (26)as a topological invariant. The pfaffian Pf[ S B ] vanishesiff S B posses a zero mode, which corresponds two zeromodes of H B with opposite J -eigenvalues.The topological invariant proposed here can thereforemeasure if two eigenstates of H B cross at zero energy asone continuously varies a parameter of the system, e.g.a hopping as discussed in Sec. III, for a suitable definedchemical potential. t c ���� ���� ���� ���� ���� - ��� - ������������ � � (a)(b) FIG. 3. (a) Real and imaginary parts of the eigenvalues of S B ( t ) as function of t for a = 1 , g = 0 . and h = 0 . forthe model defined in Sec. III. The crossing point of H B ( t ) at t = t c corresponds to a zero mode of S B ( t ) highlighted bya red line. (b) Evolution of the real and imaginary parts ofthe eigenvalues of S B ( t ) as a function of t . The t = t c planeis highlighted and the zero-eigenvalues are highlighted by ablack ball. V. APPLICATION TO CHEMICAL REACTIONS
The model introduced in Sec. III describes the reac-tion of two ethylene ( C H ) molecules to cyclobutane( C H ) in the subspace of the carbon- p z orbitals. Re-actions of this type are called pericyclic reactions andtheir outcome can be predicted and rationalized via theWoodward-Hoffmann rules (WHR). The WHR are based on the number of π -electrons in-volved in a reaction. Reactions of n π -electrons, where n ∈ N , are ’forbidden’, while reactions involving n +2 π -electrons are ’allowed’. For example, in the cycloadditionof the two ethylenes there are four π -electrons involved,because each double bond contributes two π -electrons.Accordingly, the reaction does not take place under nor-mal conditions [Fig 4(a)]. In contrast, the cyclodaddi-tion of butadiene ( C H ) with ethylene to cyclohexene( C H ) is allowed according to the WHR, since thereare three double bonds involved, which corresponds tosix π -electrons [Fig 4(b)]. This reaction takes place read-ily in the lab and is frequently used in organic synthesis. A common rationalization of the WHR is based on ener-getic arguments: ’forbidden’ reactions have to overcomea large activation barrier ∆ R , because there is a crossingbetween the occupied and unoccupied states along the re-action path [Fig 4(c)]. Allowed reactions on the otherhand have a low reaction barrier ∆ R due to the absenceof any crossings [Fig 4(d)]. However, this explanation interms of energetics has two main weaknesses:i) It does not take into account the strong non-adiabaticnature of the dynamics in case of a crossing, as discussedin the introduction. A transition state is not well definedin these cases and it is well known that non-adiabatic ef-fects such as surface hopping strongly influence chemicalreactions, often leading to a suppression of the reactionrate. ii) If the barrier height was the the only way to distin-guish an ’allowed’ reaction from a ’forbidden’ one, thereshould be a crossover between ’allowed’ and ’forbidden’reactions, e.g. by changing the temperature or the sol-vent. This, however, is not supported by experiment.Much rather, despite valiant efforts, no ’forbidden’ reac-tion has been reported in the literature starting from theGS of the reactants.We therefore want to suggest an alternative way of un-derstanding theses reaction rules based on topological ar-guments and non-adiabatic effects. The main idea hasbeen discussed in the intruduction and we briefly reviewit here: If reactants and products possess different topo-logical invariants, there has to be a crossing along thereaction path. This crossing induces very strong non-adiabatic effects, which prevent the reaction from pro-ceeding, e.g. by ending up in an excited state instead ofthe ground state. If the topological invariants of reac-tants and products do not differ, the reaction dynamicsare adiabatic, and therefore the reactions will proceedgiven the right experimental conditions. Experimentally,it has been verified that the reactions described by theWoodward-Hoffmann rules follow symmetry preservingreaction paths and therefore the topological classificationbased on the mirror symmetry J can be applied. Wegenerally find that the outcome of a pericyclic chemicalreaction described by the Woodward-Hoffmann rules cor-relates with the difference of the topological invariants ofreactants and products. If reactants and products sharethe same value of the topological invariant ν ( t ) defined inthe last section, they can be smoothly deformed into eachother along the reaction path without a crossing betweenoccupied and unoccupied states [Fig 4(d)], correspond-ing to an ’allowed’ reaction. If the topological invariantchanges during the reaction, no adiabatic symmetry pre-serving path exists and there has to be a crossing alongthe reaction path [Fig 4(c)]; in the language of the WHR,the reaction is forbidden. ⌫ ( t ) ⌫ ( t ) (a) (b)(c) (d) FIG. 4. (a) Reaction of two ethylene ( C H ) molecules to cyclobutane ( C H ). There are two double bonds involved in thereactions, which contribute π -electrons, which is forbidden by the WHR. (b) Reaction butadiene ( C H ) with ethylene tocyclohexene ( C H ).There are three double bonds involved in the reactions, which contribute π -electrons, which is allowedby the WHR. (c) Eigenvalues of H ( t ) and topological invariant ν ( t ) for the reaction depicted in (a). (d) Eigenvalues of H ( t ) and topological invariant ν ( t ) for the reaction depicted in (b). VI. SUMMARY AND CONCLUSIONS
In this paper, we have introduced a topological clas-sification of molecules with a reflection symmetry andtheir chemical reactions. In these reactions, the reflec-tion symmetry is preserved along the reaction path andresults in a Z invariant given in Eq. (26) in form ofa pfaffian, which is motivated by the theory of perplec-tic matrices. Our theory can be applied to chemical re-actions that are described by the Woodward-Hoffmannrules, i.e. pericyclic reactions. We find, that Woodward-Hoffmann allowed reactions are reactions in which thetopological invariant does not change along the reactionpath, while the invariant of Woodward-Hoffmann forbid-den reactions changes during the reaction. In light ofthese findings, we propose that certain chemical reac-tions can be described from a topological perspective,i.e. by computing a topological invariant for the reactantand product molecules. In the case where the invariantof the reactants and products is different, there has tobe a gap closing point along the reaction path, if thesymmetry defining the topological invariant is preservedalong the path. This gap closing point has strong effectson the dynamics and time-evolution of the system and should generically lead to a suppression of the reactionrate, since there is no possibility to adiabatically movefrom the reactants to the products.It remains to be shown that this approach is valid forother chemical reactions rules, e.g. the Wade-Mingosrules and its extensions and if more general statementsabout chemical reaction rules can be made via a topolog-ical approach. In addition it would be interesting to study the effect ofelectronic interactions in certain cases. Some molecularmany-body states along the reaction coordinate for ourtoy model can be viewed as molecular analogues of Mott-Insulators, so called fragile Mott-Insulators. It remainsan open questions if the many-body nature of this stateschanges the topological nature of the molecular states.
ACKNOWLEDGMENTS
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