Z n solitons in intertwined topological phases
Daniel González-Cuadra, Alexandre Dauphin, Przemysław R. Grzybowski, Maciej Lewenstein, Alejandro Bermudez
ZZ n solitons in intertwined topological phases D. Gonz´alez-Cuadra, ∗ A. Dauphin, P. R. Grzybowski, M. Lewenstein,
1, 3 and A. Bermudez ICFO - Institut de Ci`encies Fot`oniques, The Barcelona Institute of Science and Technology,Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Pozna´n, Poland ICREA, Lluis Companys 23, 08010 Barcelona, Spain Departamento de F´ısica Te´orica, Universidad Complutense, 28040 Madrid, Spain
Topological phases of matter can support fractionalized quasi-particles localized at topological defects. Thecurrent understanding of these exotic excitations, based on the celebrated bulk-defect correspondence, typicallyrelies on crude approximations where such defects are replaced by a static classical background coupled tothe matter sector. In this work, we explore the strongly-correlated nature of symmetry-protected topologicaldefects by focusing on situations where such defects arise spontaneously as dynamical solitons in intertwinedtopological phases, where symmetry breaking coexists with topological symmetry protection. In particular, wefocus on the Z Bose-Hubbard model, a one-dimensional chain of interacting bosons coupled to Z fields, andshow how solitons with Z n topological charges appear for particle/hole dopings about certain commensuratefillings, extending the results of [1] beyond half filling. We show that these defects host fractionalized bosonicquasi-particles, forming bound states that travel through the system unless externally pinned, and repel eachother giving rise to a fractional soliton lattice for sufficiently high densities. Moreover, we uncover the topolog-ical origin of these fractional bound excitations through a pumping mechanism, where the quantization of theinter-soliton transport allows us to establish a generalized bulk-defect correspondence. This in-depth analysisof dynamical topological defects bound to fractionalized quasi-particles, together with the possibility of im-plementing our model in cold-atomic experiments, paves the way for further exploration of exotic topologicalphenomena in strongly-correlated systems. CONTENTS I. Introduction
Topological solitons and boson fractionalization Z n solitons: doping and pinning 41. Solitons with Z -valued topological charges 42. Solitons with Z -valued topological charges 6B. Boson fractionalization 7C. Polaron excitations and fractional soliton lattices 9III. Bulk-defect correspondence Z Bose Hubbardmodel 14IV.
Conclusions and Outlook
I. INTRODUCTION
Solitons are waves that, despite arising in dispersive me-dia, propagate with a well-defined velocity and an unperturbed ∗ [email protected] shape due to non-linear effects [2]. These solitary waves, firstobserved in hydrodynamics [3], have fascinated scientists eversince, with premieres in areas as diverse as biology [4], op-tics [5], high-energy physics [6] and condensed matter [7]. Itis in these two later disciplines where, despite the vast dif-ference in energy scales, the origin of solitons has a commonground: the interplay of symmetry and topology [8, 9].The importance of symmetry in modern physics canbe hardly exaggerated [10]. Invariance with respect toglobal/local symmetries has been the key principle to find thefundamental laws of nature. Additionally, the spontaneousbreaking of those symmetries is paramount to understand theemergence of a wide variety of phenomena from the same mi-croscopic laws [11]. For instance, different phases of mattercan be characterized by the values of a local order parame-ter, consistent with the spontaneous symmetry breaking (SSB)mechanism [12]. In certain situations, such as during phasetransitions [13, 14], the order parameter may adopt inhomo-geneous configurations interpolating between those differentvalues. This gives rise to a freely-propagating localized exci-tation that cannot be removed by smooth local deformations,nor understood using perturbation theory: a topological soli-ton .Solitons, also known as defects in this context, are relicsof the original disordered phase that distort the symmetry-broken groundstate around a certain core/center where the or-der parameter vanishes. Moreover, the winding of the orderparameter around such defect cores yields a topological in-variant, underlying the topological characterization of suchdefects. These winding numbers can only change via non-local deformations, guaranteeing the robustness of the solitonto physical, primarily local, perturbations [15]. Let us remarkthat topological solitons are finite-energy non-perturbative so- a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a r lutions of classical field equations. Despite the fact that theycan be quantized formally [16], the interesting interplay ofsymmetry and topology is, in this case, a classical feature.Genuine quantum effects can become manifest when thesesolitons interact with quantum matter. As first predicted inrelativistic quantum field theories [17, 18] and, independently,in linear conjugated polymers [19] and p -wave superconduc-tors [20], topological solitons can bind quasi-particles with afractional numbers and exotic quantum statistics. As such,these bound fractionalized quasi-particles cannot be adiabat-ically connected to the original particle content of the theory,as typically occurs in more standard situations such as Fermiliquids [21] or, more generally, perturbatively-renormalisedquantum field theories [22]. Let us note that, in contrast tothe topological soliton, these quasi-particles are not necessar-ily protected by any quantized topological invariant.With the advent of topological insulators [23–26], however,a quantum-mechanical protection mechanism has been un-veiled [27], giving rise to the concept of symmetry-protectedtopological defects (SPT-d). Under certain symmetry con-straints, the characterization of the bulk gapped matter com-prised in between two of these solitons requires yet anothertopological invariant. In this case, the relevant topological in-variant is no longer the winding number of a classical field,such as the order parameter, but is instead determined bythe Berry connection of the matter sector via the quantum-mechanical wavefunction. We note that these invariantsare quantized and cannot change under external symmetry-preserving perturbations, unless these perturbations suffice toclose the bulk energy gap, inducing a quantum phase transi-tion where the protecting symmetry is spontaneously broken.Moreover, the so-called bulk-defect correspondence connectsthis bulk topological invariant to the fractional quasi-particlesbound to the defect, and justifies the protection of these quasi-particles with respect to symmetry-preserving perturbations.The current theory of SPT-d typically assumes a back-ground solitonic profile that is static and externally fixed, fo-cussing on the properties of the matter sector and the robust-ness of the bound quasi-particles [28]. This simplifies thedescription, as one deals with non-interacting matter in aninhomogeneous classical background, allowing one to iden-tify the mechanism responsible for their protection, and evento classify all possible SPT-d according to the underlyingglobal/crystalline symmetries [29]. Let us emphasize that,in this limit, there is no intrinsic soliton dynamics, nor anySSB order parameter that would justify the topological protec-tion of the soliton. Therefore, by assuming/engineering suchexternally-adjusted solitonic profiles, one is missing half ofthe topological robustness of the SPT-d. Moreover, from afundamental point of view, this approach misses a key prop-erty: SPT-d can only arise in intertwined topological phases ,which simultaneously display SSB and topological symmetryprotection. Although one typically finds claims in the litera-ture about the absence of any local order parameter charac-terizing the SPT-d, the reality is that the matter sector canalso display long-range order as a consequence of the SSB.It is the possibility of simultaneously encompassing both SSBlong-range order and topological symmetry protection, which makes these intertwined SPT-d so exotic and interesting.Thisambivalent role of symmetry is a consequence of interactionsand, by exploring the full non-perturbative nature of SPT-d,as well as the back action of the matter on the semi-classicaltopological soliton, we expect that exotic many-body effectswill be unveiled, and new avenues of research will be open.To address this challenge, one can start by exploring sim-plified models that are still complex enough to capture the in-tricate nature of the problem. In particular, reducing dimen-sionality allows for efficient numerical techniques that can beused to test the validity of analytical predictions, and confirmthe existence of SPT-d and the interplay of SSB and topo-logical symmetry protection. In this work, we explore theexistence of SPT-d in the so-called Z Bose-Hubbard model( Z BHM) [30–32] (see Fig. 1). In addition to the standardBose-Hubbard dynamics of a one-dimensional chain of inter-acting bosons [33], the Z BHM includes Z quantum fields(i.e. Ising spins) that sit on the lattice bonds and dress thetunnelling of bosons according to the following Hamiltonian H = − ∑ i (cid:104) b † i ( t + ασ zi , i + ) b i + + H.c. (cid:105) + U ∑ i n i ( n i − )+ ∆ ∑ i σ zi , i + + β ∑ i σ xi , i + . (1)Here, b i ( b † i ) is the bosonic annihilation (creation) operator atsite i ∈ { , · · · , N } , and n i = b † i b i counts the number of bosons,such that the interaction strength U > σ xi , i + , σ zi , i + are Pauli operators describing the Z field at the bond ( i , i + ) , the fluctuations of which are con-trolled by the transverse field β . This model is reminiscent ofa lattice gauge theory [34], where the tunneling of bosons getsdressed with a relative strength α by the Z fields. However,we emphasize that the appearance of the bare boson tunnelling t , and the bare spin energy ∆ , explicitly breaks the local Z gauge symmetry. We stress that in the rest of the work, we fix α = . t , and measure the energies in units of t .As discussed in [30–32], the Z BHM (1) gives rise to abosonic version of the so-called Peierls dielectrics with in-teresting topological features. Due to the Peierls’ mecha-nism [35], one-dimensional (1D) metals are known to be un-stable towards a dielectric insulator, where a periodic latticedistortion develops in combination with a charge/bond densitywave. As shown in [30, 31], despite lacking a Fermi surface,the Z BHM hosts similar Peierls’ phenomena with interestingpeculiarities: (i) the bosonic 1D system is not always unsta-ble, but allows for a rich phase diagram. In particular, onefinds a bosonic quasi-superfluid in a polarized Z -field back-ground [see Fig. 1 (a) ], which is transformed at strong interac-tions into a bond-ordered wave in two possible N´eel-orderedbackgrounds (A- or B-type, as displayed in Fig. 1 (b) ). (ii) Athalf-filling, the bond-ordered wave in the B-type SSB sectoris a neat example of an intertwined topological phase, whichdisplays both long-range order and a non-zero topological in-variant signalling the existence of edge states. This topologi-cal bond-ordered wave (TBOW / ) is a bosonic counterpartof certain Peierls insulators, which are protected by inver- AA A A C B A B C A a bc de BB FIG. 1.
Topological solitons in the Z Bose-Hubbard model:
Bosonic particles (blue spheres) can tunnel between neighboring sites of achain, and their tunneling depends on the state of the Z fields (red spheres) at the bonds. (a) For weak Hubbard interactions, the groundstateof a half-filled system ( ρ = /
2) is described by a bosonic quasi-superfluid delocalized over the chain, and a polarized background where all Z point in the same direction. (b) At stronger interactions, the Z fields order anti-ferromagnetically according to two degenerate patterns(A and B), spontaneously breaking translational invariance. Simultaneously, the bosons display a long-range order with an alternating bonddensity. Note that, in addition to the bond order wave, one of the SSB sectors (B) hosts a symmetry-protected topological phase. (c) As aconsequence of the SSB, the Z fields may adopt an inhomogeneous configuration with topological solitons, such as the ABA backgroundhereby displayed. As a consequence, bosonic quasi-particles are bound to the solitons. (d)-(e) Analogue situation for the trimer bond-orderedwaves phases (valid for ρ = / ρ = / Z fields order ferri-magnetically according to three possible patterns (A, Band C), and a richer variety of topological solitons with bound quasi-particles can form. sion symmetry, and described by the so-called Su-Schrieffer-Hegger (SSH) model [19]. (iii) In contrast to the fermionicSSH model, which supports a TBOW / that can be adiabat-ically connected to a free-fermion SPT phase, the bosonicTBOW / of the Z BHM requires sufficiently-strong inter-particle interactions, and cannot be adiabatically connectedto a free-boson SPT phase even in the limit of a static lat-tice. It is thus a clear example of an interaction-induced SPTphase. (iv)
When considering other fractional fillings [seeFig. 1 (d) ], further distinctions with respect to the fermioniccase arise. For one-third (two-third) filling [32], the ground-state develops a TBOW / (TBOW / ) with no counterpartin the SSH model [36, 37], which only hosts non-topologicalbond-ordered waves with the analogue of the A-, B- or C-typeorderings of Fig. 1 (d) . As shown in [32], the Z BHM has adifferent region in parameter space where the three-fold de-generate groundstate displays ¯A-, ¯B- or ¯C-type ferrimagneticordering of the Z fields, obtained by a global Z inversion ofthe corresponding A, B or C ferrimagnets of Fig. 1 (d) . Forsuch backgrounds, the bosonic sector is described by a topo-logical bond-ordered wave. In contrast to the half-filled case,where the SSB of translation invariance leads to a 2-site unitcell, and directly imposes inversion symmetry as the protec-tion mechanism of the intertwined topological phase; the 3-site unit cell of one-third or two-third fillings does not neces-sarily impose the inversion symmetry. Instead, this symmetryprotecting the intertwined TBOW / (TBOW / ) does emergeat low energies and becomes responsible for various exotic ef-fects, such as a fractional topological pumping.The phenomena discussed above are driven by the interplayof SSB and topological symmetry protection, both of which occur simultaneously in the groundstate of the Z BHM (1)at commensurate fillings. However, as mentioned in the in-troduction, SSB also allows for the existence of topologicaldefects/solitons and SPT-d. Given the pioneering results onsolitons in the fermionic SSH model [38], and the numberof similarities discussed in the previous paragraphs, it is afair question to assess if topological solitons and fractional-ization of bosons could be observed in systems described bythe Z BHM. This question is even more compelling giventhe possibility of implementing this model using ultracoldbosonic atoms in periodically-modulated optical lattices, fol-lowing the lines of [39–42]. As substantiated in the followingsections, this implementation would allow for a real break-through in the field: the first direct experimental observationof fractionalization of matter by a non-static soliton.The soliton model of SSH has been argued to play a keyrole in the physics of a linear conjugate polymer, namely poly-acetylene [38]. Despite indirect evidence, there has alwaysbeen a certain degree of controversy about the role of solitonsin the properties of polyacetylene [43]: long-range dimerizedorder, let alone a solitonic configuration, has never been ob-served directly. Moreover, the spin-full character of electronsin polyacetylene, which is half-filled, masks the fractionaliza-tion. Despite leading to reversed spin-charge relations, whichyield an indirect evidence [38], this spin doubling of poly-acetylene forbids a direct experimental confirmation of thebound fractionalized nature of quasi-particles. Accordingly,the existence of topological solitons and bosonic fractional-ization shown below, together with the possible implementa-tion of the Z BHM in experiments of ultra-cold atoms, opensa new promising route in the study of SPT-d.In this work we summarize and extend the results of [1],where the Z Bose-Hubbard model was studied for incom-mensurate densities close to half filling, and characterizedthe different symmetry-protected topological defects that ap-pear for various other densities. The paper is organized asfollows. In Sec. II, we show how topological solitons ap-pear spontaneously in the groundstate of the Z BHM whenit is doped above/below certain commensurate fillings, andwe characterize them in terms of topological charges associ-ated to the underlying SSB sectors. Moreover, we demon-strate how these defects bind fractionalized bosonic quasi-particles. These composite objects can propagate through thechain, repelling each other at short distances. For a finitedensity of defects, this interaction gives rise to a fractionalsoliton lattice. In Sec. III, we explore how the topologicalproperties of the matter sector can bring extra protection tothese bound quasi-particles. In particular, we characterize thedifferent SSB sectors as intertwined topological phases usingsymmetry-protected topological invariants. This allows oneto track the origin of fractionalization in the system througha bulk-defect correspondence. Remarkably, the later can begeneralized to situations where the regions separated by thedefects are in the same topological sector. This requires ex-tending the system to two dimensions using a pumping mech-anism, where the 2D topological invariant, the so-called Chernnumber, can be recovered by measuring the quantized inter-soliton transport. The conclusions and outlook are presentedin Sec. IV.
II. TOPOLOGICAL SOLITONS AND BOSONFRACTIONALIZATIONA. Z n solitons: doping and pinning As advanced previously, topological solitons are stablefinite-energy excitations that may arise when different valuesof the order parameter are allowed by SSB [44]. These excita-tions can be dynamically generated by crossing a symmetry-breaking critical point in a finite time. In this way, the orderedphase gets distorted by such solitons, the density of whichscales with the crossing rate according to the so-called Kibble-Zurek scaling [13, 14]. Let us remark that, once the phasetransition has been crossed, an extensive number of excita-tions are present in the system. These solitons evolve in time,scattering off each other, or escaping through the edges ofthe system, which gives rise to a complex out-of-equilibriumproblem. In this subsection, we argue that the Z BHM (1) canhost solitons in the groundstate, and that they can be pinnedexternally, leading to a simpler equilibrium situation.The crucial condition to find solitons directly in the ground-state is to allow for their coupling with matter, which turnsthe situation into a very interesting quantum many-body prob-lem. This is predicted to occur in the SSH model of poly-acetylene by doping above/below half-filling [45–47] and, re-cently, also in a system of fermionic atoms inside an opticalwaveguide [48]. As shown below, solitons also appear in the Z BHM (1), with characteristic differences. For instance, a single boson above half-filling can fractionalize, giving riseto two quasi-particles bound to a soliton-antisoliton pair inthe groundstate (see the qualitative scheme in Fig. 1 (c) ). Forone-third and two-third fillings, a single boson can give riseto a richer profile of topological solitons and bound frac-tional quasi-particles [see the qualitative scheme in Fig. 1 (e) ].One of the key differences with respect to the soliton modelof polyacetylene is that, during the doping process in thepolymer, all sorts of additional disorder and randomness areinebitably introduced [49]. This disorder is likely the underly-ing source of difficulties in providing an unambiguous proofof the dimerised long-range order and the associated solitonicprofiles in polyacetylene [43]. In this context, an advantageof the Z BHM (1), and its potential cold-atom relisation, isthat the atomic filling does not introduce impurities. In thisway, one gets access to the groundstate of the doped sys-tem in a pristine environment, where the existence of soli-tons/fractionalization is not masked by uncontrolled disorder.Let us now present a more systematic study of topologicalsolitons in the Z BHM, and provide quantitative evidence ofthe correctness of Figs. 1 (c) and (e) . In the introduction, wepresented solitons as localized finite-energy solutions movingat constant speed and, yet, Figs. 1 (c) and (e) represent staticsolitonic configurations. In a continuum model of polyacety-lene [50], the solitons are indeed free to move. However, in arealistic situation, charged impurities appear upon doping, andplay an additional important role: they pin the charged quasi-particles bound to the solitons at random positions [38]. In thiswork, we introduce a simple and deterministic pinning mecha-nism which, as shown below, allows us to precisely control theposition of the solitons and fractional bosons in the Z BHM.Such a deterministic pinning is achieved by introducing alocal perturbation of the Z fields, H → H + H p , where H p = ∑ i β i σ xi , i + , β i = ∑ j p ∈ P β ε ( δ i − , j p + δ i , j p ) . (2)Here, ε stands for the relative strength of the pinning poten-tial, and we sum over all pinning centers labelled by j p ∈ P.Essentially, this perturbation modifies the transverse field attwo consecutive bonds that surround each of the pinning cen-ters, β → β = β ( + ε ) , selecting the corresponding solitonicprofile that will depend on the particular filling.
1. Solitons with Z -valued topological charges Let us start by discussing the simplest situation, and addressthe appearance of topological solitons in the groundstate of the Z BHM doped above/below half-filling. As described above,there is a bosonic Peierls’ transition where the Ising spins de-velop an antiferromagnetic N´eel-type order [see Fig. 1 (b) ]. Inthis case, the order parameter can be defined as ϕ = N u . c . N u . c . ∑ j = ϕ j , ϕ j = ∑ i ∈ u.c. sin (cid:16) π ( i − ) (cid:17) (cid:68) σ zi , i + (cid:69) , (3)where N u . c . is the number of unit cells (u.c.) labelled by j ,and ϕ j is averaged over the elements of such unit cell i ∈ u . c . ϕ is the total average staggered magnetization, and has twopossible values ϕ = ± ϕ → ± (c) ].The existence of such solitons can be understood startingfrom the β = U → ∞ , the Peierls transition is soleley controlled by ∆ . In this case, one can find analytical solutions [30, 31] show-ing that N´eel order coexists with a bosonic bond-ordered waveif ∆ ∈ [ ∆ − c , ∆ + c ] , where ∆ ± c = t / π [ δ ± ( E ( − δ ) − )] andwe have introduced δ = α / t and the complete elliptic integralof the second kind E ( x ) . Conversely, for ∆ / ∈ [ ∆ − c , ∆ + c ] , thespins polarise along the same direction and the bosons form aquasi-superfluid state.In this section, we explore a new situation that goes be-yond the scope of these previous studies [30–32]: we set ∆ ∈ [ ∆ − c , ∆ + c ] , but explore lattices with an odd number of sites,such that perfect half-filling is never possible. In this case,one can analytically show that the groundstate is not a sin-gle N´eel antiferromagnet in the Ising sector and a dimerisedbond-ordered wave for the bosons, but that it is composed ofneighboring domains displaying the possible SSB orders. Inparticular, we find that the groundstate can accommodate fora single domain wall connecting the two N´eel states, and thatthis domain wall may come in two flavours: ther are describedby consecutive ”up-up” or ”down-down” spins and, conse-cuently, ”strong-strong” (SS) or ”weak-weak”(WW) tunnel-ing bonds for the bosons pattern respectively. The SS andWW domain walls couple differently to the itinerant bosons,and which one is stabilised in the groundstate shall depend onthe pinning and the particular number of sites.As β is increased, we show below that these domain wallswiden into a soliton profile that coincides with the kink solu-tions of the (1+1) relatisvistic ϕ theory [51], namely ϕ j = tanh (cid:18) j − j p ξ (cid:19) , (4)where j P is the soliton center and ξ is the width in latticeunits. By analogy with the relativistic scalar quantum fieldtheory, one can define a topological charge [44] as follows Q = (cid:0) ϕ j p + r − ϕ j p − r (cid:1) , (5)where the order parameter is evaluated at points that are wellseparated from the soliton center, namely r / ξ → ∞ . In thiscase, the topological charge of the soliton is Q = +
1, whereasanti-solitonic solutions carrying Q = − ϕ j → − ϕ j . These Z topo-logical charges, which cannot be modified by local perturba-tions, guarantee the robustness of the soliton.To make contact with our previous analytical results, wenote that the β = Z fields should yieldan order parameter ϕ j ∼ θ ( j − j p ) , where we have introducedthe Heaviside step function. In this case, the topological soli-ton would have a vanishing width, and be localized within i − h σ z i , i + i ( a ) β = 00 5 10 15 j − ϕ j ( b ) A B Q = 1 0 10 20 30 i − h σ z i , i + i ( c ) β = 0 . t j − ϕ j ( d ) A B ∼ ξQ = 1 FIG. 2. Z topological defects: (a) Spin magnetization (cid:104) σ zi , i + (cid:105) with a single topological defect modifying the dimerized pattern ofthe BOW / phase, where different colours are used for the two sub-lattices. The defect corresponds to a domain wall ( β =
0) connect-ing the two SSB sectors. (b)
The defect interpolates between twovalues of the order parameter ϕ j , which converges to − + j = r − j p and j = r + j p , respectively, with r (cid:29) ξ , and has atopological charge Q = (c)-(d) Analogous topological defect for β = . t , where we observe how quantum fluctuations broaden thedefect, leading to a soliton of finite width ξ . The order parame-ter, which was discontinuous for a domain wall, is smoothened for β >
0, and can be accurately fitted to Eq. (4). The parameters of themodel are fixed to U = t and ∆ = . t , and we use a chain with L =
31 sites and N =
16 particles. a single lattice site. To verify this prediction, we study the Z BHM on a chain of L =
31 sites, filled with N =
16 bosons.The ground-state has been obtained numerically using a den-sity matrix renormalization group (DMRG) algorithm basedon matrix product states (MPS) [52]. For the rest of the ar-ticle, we use open boundary conditions and bond dimension D = n =
2, which is sufficient for strong interactions andlow densities [30]. Figure 2 (a) shows the magnetization of the Z fields in the groundstate. As can be clearly observed, anSS domain wall is formed by interpolating between the twopossible anti-ferromagnetic N´eel patterns. In this case, thedefect is generated in the ground state since the odd num-ber of sites does not allow for perfect half-filling, but rather ρ = N / L > /
2. The existence of the domain wall becomesmore transparent in Fig. 2 (b) , where the corresponding orderparameter (3) displays the aforementioned step-like behavior.Let us note that, in this classical limit, the solitons can becentered around any lattice site j p within the bulk of the sys-tem (i.e. the groundstate has an extensive degeneracy). More-over, in contrast to continuum field theories where solitons arefree to move, lattice solitons are static as they find finite en-ergy barriers that inhibit their transport. These energy penal-ties, known as Peierls-Nabarro barriers, arise due to the lackof translational invariance on the lattice, and are typically ar-ranged forming a periodic Peierls-Nabarro potential [53, 54].As the quantum fluctuations of the Z fields are switchedon, β >
0, the topological solitons start tunnelling throughthese barriers, and delocalize along the lattice. In order tostudy static soliton properties in this quantum-mechanical
FIG. 3.
Effect of quantum fluctuations on topological defects:(a)
Ground state energy E g ( j p ) with one topological defect as wevary the pinning location j p on a chain of L =
61 sites, measuredwith respect to some reference E , for β = . t . Odd and even sitesare represented in different colors, showing the Peierls-Nabarro po-tential between them. Note that, far from the boundaries, the state isdegenerate due to translational invariance. (b) Peierls-Nabarro bar-rier ∆ E g as a function of β / t , which goes to zero as the quantumfluctuations increase, for L =
91. For β = ∆ E g / t (red square). In this limit, the defects are immo-bile. For β (cid:54) = (c) The pinningperturbation (Eq. (2)) localizes the defect at a certain position. Herewe show how the soliton width ξ varies as we increase the pinningstrength β / β , converging to a finite value for an infinite pinning.This finite-size scaling was performed by fitting the points to a line(black). In every panel, ξ should be understood as the convergedvalue. (d) Soliton width ξ as a function of β for different values ofthe Hubbard interaction U , and we now work with a chain of L = / phase, with U = t and ∆ = . t . regime, it is important to switch on the pinning terms ofEq. (2), which will effectively localize the soliton to the de-sired pinning center j p . However, as a result of quantum fluc-tuations, the soliton is no longer strictly localized to a singlesite, but will spread over a width ξ > j p , we can confirm the existence of such Peierls-Nabarro bar-riers by numerically computing the ground-state energy as afunction of the position of the soliton E g ( j p ) (see Fig. 3 (a )).Due to the Peierls’ dimerization, we find a periodic arrange-ment of energy barriers ∆ E g separating neighboring unit cells.As shown in Fig. 3 (b) , the height of these barriers convergesto a non-zero value when β →
0, leading to the aforemen-tioned Peierls-Nabarro barrier. This numerical result justifiesour previous statement about the static nature of the solitonsin the classical regime: the finite barriers inhibit the move-ment of the topological solitons. The small magnitude of thePeierls-Nabarro barrier also justifies our previous claim: assoon as quantum fluctuations are switched on, the solitons tun-nel and delocalize, justifying the requirement of pinning (2).We note that the strength of the Peierls-Nabarro barriers maybe larger for other choice of parameters, such that the tun-nelling of solitons only gets activated at second-order in thetransverse field β . In this particular regime, the very na- ture of the solitons is different, as they split into two distinctbranches (i.e. weak-weak and strong-strong bonds).In Fig. 2 (c) , we represent such a pinned solitonic profile inpresence of quantum fluctuations. The pinning center is po-sitioned at the middle j p = ( L − ) / L = (d) , the order parameter in-terpolates between the two SSB groundstates according to thesmooth profile of Eq. (4), and an accurate fit can be used toextract the soliton width ξ . In Figure 3 (c) , we show how thesoliton width gets modified as one increases the strength of thepinning potential, eventually converging to a fixed value as thepinning strength is sufficiently strong β / β = / ( + ε ) → (d) , we represent the widths ξ of the topological soliton as a function of the transversefield strength β , for different values of the Hubbard repulsion.We note that in all numerical simulations, we use a pinningstrength such that convergence of the soliton width has beenreached. As expected, the width increases (i.e. more delo-calized defects) as the quantum fluctuations are raised. Onthe other hand, we observe that the soliton width, at fixed β ,decreases as the Hubbard repulsion U is increased. This is aclear demonstration of the back-action of the matter sector onthe topological soliton that was briefly mentioned in the in-troduction: as the bosons become more repulsive, eventuallyreaching the hard-core constraint that forbids double occupan-cies, they can be accommodated more comfortably within the Z soliton, which then becomes more localized.
2. Solitons with Z -valued topological charges Let us note that, around half-filling, it is only possible toobtain two types of topological defects, solitons with charge Q = + Q = −
1. By doping, thegroundstate configuration can only correspond to a successionof neighboring soliton and anti-solitons, such that the over-all charge is either Q tot =
0, or Q tot = ± (d) ]. The situation isanalogous for filling one-third. The order parameter that candifferentiate between these SSB groundstates is˜ ϕ = N u . c . N u . c . ∑ j = ˜ ϕ i , ˜ ϕ j = √ ∑ i ∈ u.c. sin (cid:18) π ( i − ) (cid:19) (cid:68) σ zi , i + (cid:69) , (6) i − h σ z i , i + i ( a ) ρ = 29 / . . . . . j ˜ ϕ j ( b ) A B ∼ ξQ = − i − h σ z i , i + i ( c ) ρ = 29 / . . . . . j − ˜ ϕ j ( d ) A C ∼ ξQ = − FIG. 4. Z topological defects: (a) Spin magnetization (cid:104) σ zi , i + (cid:105) with a topological defect distorting the trimerized pattern, and corre-sponding to an anti-soliton on the BOW / phase connecting two ofthe three SSB sectors, A and B, for a chain of L =
43 sites and N = (b) We represent the order parameter ˜ ϕ j , where j is the unitcell index. The black line is obtained by fitting ˜ ϕ j using eq. (7) andeq. (9). Far from the defect, the order parameter distinguishes thedifferent SSB sectors, and leads to an associated topological chargeof Q = −
1. The parameters of the model are fixed to β = . t , ∆ = . t and U = t . (c)-(d) Same as (a)-(b) , but for an anti-soliton connecting the sectors A and C, for L =
44 and N =
29. Thisdefect has an associated topological charge of Q = − which attains the following values for the the three ground-states ˜ ϕ ∈ {− , , } . As there are more SSB vacua, therewill be more types of solitons that interpolate between them.For instance, an anti-soliton interpolating between the A andB groundstates (see Fig. 1 (e) ) can be described as˜ ϕ AB j = (cid:18) − tanh (cid:18) j − j AB ξ (cid:19)(cid:19) , (7)which has topological charge Q AB = − ϕ BC j = (cid:18) + tanh (cid:18) j − j BC ξ (cid:19)(cid:19) , (8)which also has topological charge Q BC = −
1. Finally, thesoliton interpolating between C and A can be written as˜ ϕ CA j = (cid:18) j − j CA ξ (cid:19) , (9)which has a larger topological charge Q CA = +
2. We note thatsolitons with the inverse orderings BA, CB and AC, are alsopossible, and would lead to reversed topological charges, suchthat solitons are restricted to a Z -valued topological charge.In Figs. 4 (a) and (c) we provide numerical confirmation ofthis phenomenon, displaying two examples of defects con-necting different SSB sectors of the BOW / phase. In bothcases, the order parameter (6) adjusts very accurately to theexpected shapes in Eqs. (7) and (8) [see Figs. 4 (b) and (d) ].As it can be appreciated in the figure, different types of de-fects are generated depending on the bosonic density. In the next section we will analyze this mechanism in detail. How-ever, before turning into this discussion, let us emphasize thatthe wider variety of solitons hereby discussed leads to fur-ther possibilities regarding their scattering. In the ABCA se-quence of Fig. 1 (e) , we see that this fractional filling allowsfor a richer multi-solitonic profile with respect to the half-filling. In particular, one is no longer restricted to neighbor-ing soliton-atisoliton pairs, but it becomes possible to findtwo neighboring defects with the same topological charges Q AB = Q BC = −
1. In this case, by externally adjusting thepinning potentials, these two defects can collide and lead to alarger conserved charge Q AB + Q BC = −
2, which correspondsto a new type of anti- soliton AC with charge Q AC = − B. Boson fractionalization
So far, we have only focused on the Ising-spin sector, andproved the existence of various topological solitons in thegroundstate of the Z BHM. Although the existence of thesesolitons is triggered by the coupling of the Z fields to thebosonic quantum matter, we recall that these solitons can befully understood classically, as they are characterized by thetopological charge of a classical field, i.e. the order param-eter of Eq. (5). In this section, we will explore the bosonicsector, and show that there are bound quasi-particles local-ized within the soliton, which clearly display the bosonic ver-sion of the quantum-mechanical phenomenon of fractionaliza-tion. Charge fractionalization was first predicted for relativis-tic quantum field theories of fermions coupled to a solitonicbackground [17, 56], and find a remarkable analogue in thephysics of conjugated polymers [47]. As outlined previously,the direct experimental observation of this effect would be areal breakthrough in the field, overcoming past limitations inlightly-doped and disordered polyacetylene [43].Arguably, the clearest manifestation of fractionalizationarises by doping a Peierls-type system with a single particleabove/below a given commensurate filling, such as half fill-ing. As it turns out, in order to accommodate for this ad-ditional particle, the groundstate of the Z BHM develops asoliton/anti-soliton pair of the Z fields, each of which hostsa bound quasi-particle/quasi-hole with a fractionalized num-ber of bosons, i.e. the boson splits into two halves. We can,once more, build our understanding starting from the β = L and periodicboundary conditions: (i) the homogeneous N´eel order, (ii) apair of SS domain walls , and (iii) a pair of WW domain wallsand, finally, (iv) one SS and one WW domain wall, all of themseparated by a large distance.In the U → ∞ limit, the diagonalization of the model for anyIsing spin configuration can be carried out semi-analytically,and gets reduced to a simple matrix diagonalization. This al-lows us, in particular, to compare the exact groundstate ener-gies (i) - (iv) to understand the nature of the groundstate. Set- j − ϕ j ρ = 46 / a ) A B A j − ˜ ϕ j ρ = 61 / d ) A B C A j − ˜ ϕ j ρ = 59 / g ) A C B A i . . . h n i i ( b ) 0 30 60 90 i . . . h n i i ( e ) 0 30 60 90 i . . . h n i i ( h )0 30 60 90 i . . . N i ( c ) 0 30 60 90 i . . . . N i ( f ) 0 30 60 90 i − . − . − . . N i ( i ) FIG. 5.
Fractionalization of bound quasi-particles: (a)
We represent the order parameter ϕ j (3) at each unit cell j for the ground stateconfiguration when we add one extra particle on top of half-filling. In this case a soliton-antisoliton pair is created, dividing the chain into ABAconsecutive sectors. In (b) , we show the bosonic occupation (cid:104) n i (cid:105) in real space. We observe how, inside the bulk of each sector, the occupationcorresponds to 0 .
5. Around the defects, however, we find peaks where the occupation increases, with a different profile for each sublattice.The black line corresponds to a fit to Eq. (10). (c)
The integrated particle number N i = ∑ j < i (cid:104) n j (cid:105) shows how each peak is associated with half aboson. Each bound bosonic quasi-particle is, thus, fractionalized. The situation is similar around two-third filling when adding or substractinga single particle above (d) , or below (g) the corresponding filling. We represent the order parameter ˜ ϕ j (6), showing how three defects appear,creating the SSB patterns ABCA, or ACBA, respectively. The bosonic occupation shows (e) peaks and (h) drops with respect to 2 / (f) / (i) − /
3. The calculations were performed using a chain of L =
90 sites and taking U = t . In the firstcolumn we took N =
46 particles, ∆ = . t and β = . t . In the center and right columns we used ∆ = . t and β = . t , with N = N =
59 particles, respectively. The solitons were pinned as discussed in the previous section. ting α = . t , we obtain ∆ − c = . t and ∆ + c = . t , andset ∆ within this range. We now dope the system with one ex-tra boson/hole, which can stabilise the configurations (ii)-(iv) in detriment of (i) that controls the exact half-filling regime.Comparing the analytical gorundstate energies, we find thatthey are degenerate at ∆ = . t . On the otehr hand, for ∆ closer to ∆ ± c , the energy differences in energies are very large,and only one of the configurations will be stablished in thegroundstate. Specifically, for ∆ = . t , we find that the con-figuration (iii) of two WW domain walls has a considerablylower energy, while for ∆ = . t it is configuration (ii) of twoSS domain walls which arises in the groundstate.Interestingly, these results provide some further confir-mation of the existence of the Peierls-Nabarro barriers dis-cussed in the preceding section (see Figs. 3 (a)-(b) ) [53, 54].By analyzing the semi-analytical energies, we see that for ∆ ≈ . ∆ ≈ .
8) there is large energy penalty for finding SS(WW) domain walls in the groundstate, which consists of apair of WW(SS) domain walls. This is indeed a manifesta-tion of some underlying Peierls-Nabarro barrier, since movingthe soliton by a single lattice site requires changing its naturefrom WW(SS) to SS(WW), and this is energetically penalizeddue to the aforementioned energy differences. We find thatfor ∆ ≈ . t , the Peierls-Nabarro barriers vanish, and soli- ton hopping occurs through first-order perturbation processproportional to β , leading to widening of the domain wallsforming a soliton/anti-soliton configuration. Conversely, or ∆ ≈ . ∆ ≈ .
8, the barriers restrict soliton hopping, whichcan only occur virtually as a second-order process propor-tional to β .Let us now contrast these theoretical predictions with nu-merical results. Figure 5 (left column) contains all the numeri-cal evidence that supports this mechanism in a chain of L = N =
46 bosons. Note that this yields preciselyan extra particle above half filling ρ = N / L = / + / L = ρ (cid:63) + / L .As shown in Fig. 5 (a) , the Ising sector of the Z BHMgroundstate clearly displays the predicted soliton/anti-solitonprofile ABA, and its finite width is a result of the finite quan-tum fluctuations of the Ising spins. In Fig. 5 (b) , we representthe bosonic particle number for the different sites of the chain.As can be observed, away from the soliton/anti-soliton, the av-erage particle number is consistent with the half-filling condi-tion. Remarkably, as one approaches the topological defects, abuild-up in density becomes apparent, which signals the pres-ence of a quasi-particle bound to the topological soliton/anti-soliton. Moreover, the density profile of this fractional quasi-particle can be accurately fitted to (cid:104) : n j : (cid:105) = (cid:104) n j (cid:105) − ρ (cid:63) = ξ sech (cid:18) j − j p ξ (cid:19) , (10)where j = i or j = i + ρ (cid:63) isthe closest commensurate filling (i.e. ρ (cid:63) = / Z /scalar field, the profiles of the solitons and the fraction-alized quasi-particles are completely equivalent.Finally, in order to display clearly the fractionalization phe-nomenon, let us compute the integrated number of bosonsabove the half-filled vacuum N i = ∑ j ≤ i (cid:104) : n j : (cid:105) , (11)As shown in Fig. 5 (c) , this integrated boson number displaystwo clear plateaux connected by jumps of 1 / (b) indeedcarry a fractional number of bosons, namely 1 / (d) and (g) ), sep-arating different SSB sectors with a density of 2 / (e) and (h) ). Moreover, the integrated den-sity of bosons (11) displayed in Figs. 5 (f) and (i) is fully con-sistent with the fractional ± / ± / ± /
3. In addition to the situation discussed in the previ-ous paragraph, we note that one extra particle could also befractionalized into two + / − / ± / Q AB + Q BC = −
2, whichcorresponds to a new type of anti-soliton AC with charge Q AC = −
2. This picture is also consistent with the boundnumber of bosons, as the AC defect of Fig. 6 is associatedto a fractional value of 2 / = / + / FIG. 6.
Topological defects and fractionalized states:
The tablesummarizes the particle numbers associated to the localized statesbound to the topological solitons, when the state is occupied (parti-cle) or empty (hole). We write defect A-B-C-A, for example, refer-ring to the defects corresponding to each pair of consecutive sectors:AB, BC and CA, which have the same particle numbers. We high-light the configurations that are generated spontaneously by dopingthe commensurate densities ρ (cid:63) with one particle or hole. In shadedareas, we highlight the instances realised in the groundstate. C. Polaron excitations and fractional soliton lattices
Let us now comment on the possibility that the extrabosons/holes about the commensurate fillings, instead of frac-tionalizing into quasi-particles bound to the solitons/anti-solitons, lead to a simpler excitation: a topologically-trivialpolaron. In fact, in the fermionic SSH model, a single fermionabove the half-filled groundstate does not lead to the frac-tionalized pair of quasi-particles. Instead, a lower-energyquasi-particle is formed, corresponding to the electron beingsurrounded by a cloud of phonons, the so-called electronicpolaron [45, 47]. We note that, in the context of the SSHmodel, this polaron solution can be understood as a confinedsoliton-antisoliton pair. The separation between both defects d is smaller than their corresponding widths ξ , such that thefermion is not fractionalized into a pair of 1/2 charges boundto each defect, but instead distributed across the entire po-laron with the same total charge [38, 47]. If one studies theenergy of the soliton-antisoliton pair as a function of the dis-tance E g ( d ) , a global energy minimum is found for the sepa-ration of the polaron mentioned above [55].In the Z BHM, we can calculate numerically the ground-state energy as a function of the soliton-antisoliton distance E g ( d ) , which is controlled though the pinning centers of theperturbation (2). In Fig.7 (a) , we show how this energy doesnot present any global minimum corresponding to a polaronicsolution, which in this case would stand for the bosonic parti-cle surrounded by spin-wave fluctuations.According to this result, we can rule out the existence ofany confining mechanism, and ensure that the groundstate cor-responds to the distant soliton-antisoliton pair with fraction-alized bound quasi-particles. This results are also useful todiscuss the following type of groundstate, the so-called soli-ton lattice. Since the energy decreases exponentially with thesoliton distance (see the fit in Fig.7 (a) ), solitons tend to re-0 x E g / t − E / t × − ∼ e − γx ( a ) 0 10 20 30 40 50 j − ϕ j ( b ) 0 20 40 60 80 100 i . . . h n i i ( c ) FIG. 7.
Polarons and the soliton lattice : (a) Groundstate energy E g ( x ) for a chain of L =
120 sites filled with N =
62 bosons, asa function of the distance x between the pinned soliton-antisolitonpair. The absence of a miniminum for a given distnace x showsthat topologically-trivial polarons are absent in the groundstate ofthe Z BHM. (b)
The order parameter ϕ j (3) for a chain of L = N =
55 bosons self-assembled in a periodic configu-ration of soliton-antisolitons, maximizing the inter-soliton distance,and leading to the so-called soliton lattice. The dotted line representsthe corresponding staggered polarization of the half-filled ground-state without any topological defect. (c)
The local boson density dis-plays the localized nature of the bound quasi-particles, which againdisplay a fractionalized density upon a closer inspection. pel each other seeking for groundstate configurations with themaximal inter-soliton distance. If we keep on adding addi-tional particles above/below half-filling, but do not fix theirrelative positions by the external pinning (2), the solitons-antisolitons pairs will self-assemble in a crystalline configura-tion that maximises the inter-soliton distance: a soliton lattice.Such type of solutions were originally predicted for the SSHmodel of polyacetylene [57, 58], and are also known as kink-antikink crystals in relativistic field theories of self-interactingfermions at finite densities [59, 60].In Fig. 7 (b) , we represent the Ising sector of the Z BHMgroundstate for a chain with L =
100 sites and N =
55 bosons.As can be readily appreciated, in the absence of pinning (2),the Z fields self-assemble in a periodic configuration ofsoliton-antisoliton pairs that delocalize over the chain whilemaximising the inter-soliton distance. In this way, the back-ground spins form a soliton lattice, and lead to a periodicconfiguration of fractionalized bosonic quasi-particles: a frac-tional soliton lattice [see Fig. 7 (c) ]. As can be observed inFig. 7 (b) , the value of order parameter in between a con-secutive soliton-antisoliton pair does not reach the value ofthe defect-free configuration at precisely half-filling. As oc-curs for the SSH model [57, 58], this can be considered asevidence that the energy gap of the soliton lattice is smallerthan that of the defect-free configuration or the pinned-solitongroundstate, signalling that the topological defects in the soli- ton lattice are not independent excitations above the perfectly-dimerised groundstate. Instead, the topological defects in thesoliton lattice are coupled and form an energy band, leadingto groundstates that are fundamentally different from a col-lection of uncoupled topological solitons. We would like toemphasize that, although this type of solutions has been an-alytically predicted before [57–60], our DMRG results forthe Z BHM provide, to the best of our knowledge, the firstnumerically-exact confirmation of their existence without theapproximations underlying previous analytical works (e.g.large- N methods) . III. BULK-DEFECT CORRESPONDENCE
So far, we have solely focused on the topological propertiesof the Z -field sector, and discussed the fractionalized natureof the bound bosonic quasi-particles. In this section, we delveinto the full topological characterization of the defects, andshow how the fractionalized matter can bring an extra topo-logical protection to the groundstate, making connections tothe notion of symmetry-protected topological defects (SPT-d). We remark, once more, that previous studies typicallyconsider the properties of SPT-d in the presence of externally-adjusted static solitons [27, 28]. Here, on the other hand, wefocus on the full many-body problem where the solitons havetheir own dynamics, and the back action of the matter sectoron the solitons becomes relevant. Moreover, we will unveilthe particular topological origin of the associated fractionalbosons through a generalized bulk-defect correspondence. A. Symmetry protection of bound quasi-particles
Paralleling our discussion of the previous sections, let usstart with the simplest situation: the half-filled configura-tion, where the Peierls’ mechanism gives rise to a two-folddegenerate groundstate with a dimerization of the Z field(see Fig. 8). In the hardcore limit U / t → ∞ , the bosonscan be mapped onto fermions by means of a Jordan-Wignertransformation [61], which shows that the resulting ground-state has both inversion symmetry and a sub-lattice, so-called chiral, symmetry. According to the general classifica-tion of symmtery-protected topological phases [29], the bulkTBOW / is a clear instance of a BDI topological insula-tor [31]. The two degenerate ground states A and B can becharacterized as trivial or topological, respectively, with thehelp of a topological invariant, namely the Berry phase γ [62].The latter takes the value γ = γ = π in the topological B phase. Furthermore, the chiralsymmetry ensures the so-called bulk-edge correspondence: agroundstate that displays a Berry phase γ = π will have oneedge state at each boundary of the chain.Let us now revisit the case of one extra particle abovehalf filling. Figure 9 (a) shows the real-space occupation ofbosonic in the ground state of a finite system of L =
90 sites.The system displays the pattern A-B-A with two solitons sep-arating the degenerate SSB configurations. To understand the1 a b A B C A B ¯A = B ¯B = A P Z P Z P Z P Z P Z ¯C = A ¯B = C ¯A = B FIG. 8.
Topological Bond Order Wave phases: (a)
The Bond Order Wave phase at half filling (BOW / ) is doubly degenerate. The twoSSB sectors, A and B, are related by a one-site translation, and can be distinguished by their topological properties: while A is topologicallytrivial, B is a symmetry-protected topological phase [31]. Note that, by applying a P Z inversion σ z → − σ z , we can transform states in the Asector to states in the B sector and viceversa. (b) This property is not satisfied, however, at one-third and two-third fillings. For each of thesedensities, we find again a BOW phase with a three-fold degeneracy. A, B and C denote the three SSB sectors, connected again by one-sitetranslations. Note that now the P Z transformation takes states from these sectors to a different phase. We denote the later as TBOW / (respectively TBOW / ), which is again three-fold degenerate ¯A, ¯B and ¯C, and appears at stronger Hubbard interactions [32]. The latter is asymmetry-protected topological phase, as opposed to the BOW / (respectively BOW / ), which possesses a zero topological invariant [32].The fact that, as opposed to the half-filled case, P Z do not coincide with the one-site translation has an important consequence: while in theformer case topological defects separate regions with different topological properties in the bulk, this is not true for the latter, and localizedstates associated to the defects are not expected a priori based on topological arguments. We will see, however, that a topological origin forsuch states can be recovered by extending the system to two dimensions through a pumping mechanism. topological origin of these states, we show in Fig. 9 (b) our nu-merical calculation of the local Berry phase in real space [63],computed on the intercell bond (i.e. the bond joining twoneighboring unit cells). As can be observed in this figure,the local Berry phase in the configuration A is equal to γ = γ = π . In sucha situation, the theory of topological defects [27, 28] predictsthat localized and topologically-protected boundary states willappear at the interface of the two topologically-distinct re-gions. Furthermore, as a consequence of chiral symmetry,these states have support in just one of the two sub-lattices.This makes them robust against perturbations that respect thechiral symmetry, but are not sufficiently strong to close the gapof the system. Therefore, apart from the inherent robustness ofthe classical topological solitons, the total defects formed bya soliton and a fractionalized bosonic quasi-particle are alsoprotected against chiral-preserving perturbations, leading tothe aforementioned SPT-d [27, 28].From a pragmatic point of view, these SPT-d constitute analternative to observe topological edge states in a cold atomexperiment. We note that the presence of topological edgestates at the boundaries of a cold-atom system can be some-times hampered by the presence of an additional trapping po-tential [64]. Previous attempts to overcome these difficul-ties rely on externally adjusting inhomogeneous configura-tions [65], imposing background solitonic profiles on a su-perlattice structure [66], or by shaking the optical lattice [67].Here, on the contrary, the topological solitons are dynamicallygenerated by doping the system, and self-adjust to certain po-sitions of the chain depending on the doping. In particular,there are certain configuration where the SPT-d can be found at the middle of the system (see Fig. 2), where the deleteriouseffect of the trapping potential would be absent.Having clarified the topological protection of these SPT-d in the simpler half-filled and hardcore limits, let us nowturn into more complex situations, and assess the nature ofthe topological protection of these defects (i) away from thehardcore constraint, or (ii) around other fractional fillings.For finite values of the Hubbard repulsion U , the chiralsymmetry is explicitly broken even in the half-filled case.Nevertheless, the system still possesses inversion symmetry,which can lead to a quantized non-zero Berry phase and anintertwined topological phase. We emphasize that the bulk-boundary correspondence is no longer guaranteed for thesephases: the edge states break the inversion symmetry, andare therefore not protected by the topology of the bulk [31].Let us now discuss the situation for fillings above/below half-filling. Figure 9 (c) shows the real-space bosonic occupationfor U = t . The soliton-antisoliton pair is still present forsuch finite interactions but, as a direct consequence of the lossof chiral symmetry, the support of the fractionalized modes isno longer restricted to a single sub-lattice. Nonetheless, Fig-ure 9 (d) shows that the local Berry phase is still quantizedto γ = γ = π in the bulk of the trivial and topologicalconfigurations, respectively. This quantization is preservedeven in the absence of chiral symmetry, as there still existsthe discrete inversion symmetry. We observe how, in the re-gion where the solitons interpolate between the two inversion-symmetric groundstates, and inversion symmetry is thus notmaintained, the Berry phase attains intermediate values con-necting the two quantized values that appear far away from thesoliton cores. In summary, the bulk can be characterized by a2 i . . . h n i i ( a ) U = ∞ i . . . h n i i ( c ) U = 10 t j . . . | γ | / π ( b ) A B A j . . . | γ | / π ( d ) A B A
FIG. 9.
Topological invariant and fractionalization: (a)
Occupa-tion number (cid:104) n i (cid:105) for a chain with L =
90 sites in the hardcore limit, U / t → ∞ , when we dope a state in the BOW phase with one extraparticle above half filling. At the location of the defects, two peaksin the occupation can be observed, each one localized only in onesublattice (represented in different colors) as a consequence of chi-ral symmetry. (b) Local Berry phase γ calculated on the bonds thatseparate different unit cells j . This quantity is quantized to 0 and π in the different SSB sectors, and interpolates between these twovalues in the region where the defects are located. (c) For a finitevalue of U we can still observe peaks in the occupation. These areno longer localized in specific sublattices, since chiral symmetry isbroken. However, the topological Berry phase is still quantized farfrom the defects (d) since inversion symmetry is still preserved. Inthe hardcore limit we used the parameters ∆ = . t and β = . t ,and for the softcore case ∆ = . t and β = . t . topological invariant, but there is no direct bulk-defect corre-spondence responsible for the protection of the defects. In thefollowing section, we shall revisit this scenario in search for ageneralised bulk-defect correspondence.Let us now turn to two-third filling, where we recall thatthe Peierls’ mechanism gives rise to a threefold degenerategroundstate with a trimerization of the Z fields (see Fig. 8).The resulting phases are insulators that can either be topolog-ically trivial or non-trivial, depending on the strength of theHubbard interactions [32]. Figure 8 (b) depicts the differentground state configurations for U = t (trivial) and U = t (topological): the trivial phase has a BOW pattern with twostrong bonds and one weak bond (A), whereas the topologicalphase has a BOW pattern with two weak bonds and one strongbond ( ¯A). The topology of these ground states can be charac-terized with the help of the inter-cell local Berry phase, whichis equal to γ = γ = π in the topological phases ¯A, ¯B and ¯C. We note that thethree degenerate ground states A, B and C (resp. ¯A, ¯B and ¯A)are related to each other by a translation of one site, and arethus equivalent in the thermodynamic limit. We emphasizehere that the three ground states have the same topology, un-like the half-filled case, where B = ¯A has a different topologywith respect to A (see Fig. 8).We now move away from the commensurate fillings, andaddress the topological characterization of the bound quasi-particles that appear when doping the system with one parti-cle above the two-third filled groundstate, which would cor-respond to the trivial BOW phase for U = t . As dis-cussed in previous sections, three topological solitons arise leading to the A-B-C-A configuration [See Fig. 8 (b) ], eachof which hosts a localized bosonic quasi-particle with a frac-tional charge of 1 /
3. Notice that, in contrast to half-filling, thedefects are no longer separating regions with different topo-logical bulk properties. According to the theory of SPT-d,since the solitons do not interpolate between topologically-distinct regions, there is no reason to expect that a quasi-particle will be bound to the soliton. Nonetheless, such boundquasi-particles do appear, carry fractionalized charges, and wewould like to understand if they have some generalised topo-logical origin. In the next section, we show that these quasi-particles can be understood as remnants of topologically pro-tected defect modes of an extended 2D system, even if thiscannot be inferred a priori in the 1D system.
B. Quantized inter-soliton pumping
In this section, we show how a generalized bulk-defect cor-respondence can be stablished by extending the solitons totopological defects in a higher dimension through a Thoulesspumping argument [68].
1. Thouless pumping and the bulk-defect correspondence
In the beginning of the 80s, Thouless showed that gapped1D systems undergoing a slow cyclic modulation can displayquantised particle transport, and that the robustness of this ef-fect is rooted in an underlying non-zero topological invariantin higher dimensions [68]. We now briefly review this conceptfor the so-called Rice-Mele model [69], a generalisation of theSSH fermionic model in a static solitonic backround [70]. Weconsider the dimerized Hamiltonian with an extra chemicalpotential term following an adiabatic cycle, H ( ϕ ) = ∑ i − ( t + ( − ) i δ ϕ ) (cid:16) c † i c i + + h.c. (cid:17) + ( − ) i ∆ ϕ c † i c i (12)where δ ϕ = δ cos ( ϕ ) and ∆ ϕ = δ sin ( ϕ ) . Varying the param-eter ϕ in time, the groundstate starts in the topological con-figuration ¯A ( ϕ =
0) and adiabatically evolves into the trivialconfiguration A ( ϕ = π ). The onsite staggering potential ischosen such that the model remains gapped during the wholeadiabatic cycle. As a consequence of this cyclic evolution,the charge that is pumped across the chain, ∆ n , is quantizedto integer values. This can be shown by relating this quantityto the Chern number ν of an extended 2D system, where thetime-dependent parameter ϕ is taken as the momentum cor-responding to an additional synthetic dimension [70], namely ∆ n = − ν = − ∑ i ν i , where the ν i = π (cid:82) T (cid:82) BZ Ω i d τ d q are theChern numbers of the occupied bands, written in terms of theBerry curvature Ω i = i ( (cid:104) ∂ q u i | ∂ ϕ u i (cid:105) − (cid:104) ∂ ϕ u i | ∂ q u i (cid:105) ) of the cor-responding Bloch eigenstates | u n (cid:105) .Figure 10 (a) shows the full energy spectrum of the single-particle Hamiltonian (12) in terms of ϕ . At ϕ = . . . . . . ϕ/ π − − E / t . . . . . . ϕ/ π − − E / t . . . . . . ϕ/ π . . . . . . E / t FIG. 10.
Non-interacting pumping in a static lattice:
In the upper panels we show the 2D extension to a finite cylinder of a one-dimensionalfermionic chain in a static lattice through a pumping process (eq. (12)), where the pumping parameter ϕ corresponds to the momentum in theperiodic dimension. The associated Chern number in each region is denoted by ν . The lower panels show the corresponding spectral flowalong the pumping. Localized edge states inside the gap are represented in different colors that correspond to their position in real space,where the intensity is associated qualitatively to their localization length in 1D. (a) Dimerized lattice without defects. Two edge states, eachone localized at one boundary of the system, cross the gap connecting the two bands. These edge states are topologically protected in theextended 2D system since they can not be removed unless the gap closes. They have the same energy at the inversion-symmetric point ϕ = ϕ = π corresponds to the trivial case A. Alongthe pumping cycle, one particle is transported through the bulk from one edge to the opposite one, where the direction is different for eachstate and depends on the Chern number of the extended 2D cylinder. (b) Dimerized lattice with two domain walls. Apart from the edge statesdiscussed in the previous case, two extra topologically-protected localized states, each one associated to one domain wall, appear in the gapconnecting the bands. Here the two inversion-symmetric points correspond to the configurations ¯AA ¯A and A ¯AA, respectively. In this case,two particles are pumped from one defect to the other during the cycle, since the difference between the Chern numbers of the regions theyseparate is now ± (c) Trimerized lattice with three domain walls, where we represent the spectrum around the gap that opens at two-thirdfilling. In this case, apart from the two edge states, three extra localized states associated to the defects appear in each gap. However, onlytwo of them connect different bands. The third one (black line, localized at the center of the cylinder) starts and ends in the valence band, soit is not topologically protected as it can be removed without closing the gap. This can be understood using the bulk-defect correspondence,since that particular defect separates regions with the same Chern number. In this case, the two inversion-symmetric points correspond to theconfigurations ¯A ¯B ¯C and ABC, respectively. correspondence argument [71], the number of topologicallyprotected edge states at each boundary is equal to | ∆ n | and thesign of ∆ n determines the direction of the charge transport.These states connect the two bands as ϕ is modified, which iscommonly referred to as spectral flow, and are responsible fortransporting the charge during the pumping process. We finda Chern number of ν ¯A = − (a) can be interpreted as the spectrum of a 2Dsystem in a cylindrical geometry, where ϕ is the momentumin the synthetic dimension. The localized edge states become,in this picture, the 1D conducting edge states at the bound-aries of the synthetic cylinder. The phase of this synthetic2D system corresponds to a Chern insulator and, as opposedto the 1D case, the bulk-boundary correspondence guaran-tees the presence of topologically-protected edge states evenin the absence of chiral symmetry. Therefore, this 2D exten-sion through the Thouless pumping establishes a generalizedbulk-boundary correspondence, where the localized states in1D can be seen as remnants of protected edge states in 2D.This is true as long as the 1D bulk has a non-zero topolog-ical invariant, as it is the case in our bosonic model, wherethe topology is protected by inversion symmetry. Abusing the notation, we will associate Chern numbers to the 1D phases,keeping in mind that this requires the specification of not onlythe initial state for the pumping protocol, but also of its direc-tion. In this case, if we reverse the direction of the pumpingwe would get ν ¯A =
1. In the following, we will keep the samepumping protocol, specified by Eq. (12).Let us now use this bulk-boundary correspondence in thecontext of SPT-d and topological solitons. We first notice thatthe topological solitons in the bulk of the 1D chain can alsobe understood as extended interfaces between regions withdifferent Chern numbers in the adiabatic pumping. We nowconsider the configuration ¯AA ¯A for ϕ =
0, and its cyclic evo-lution to the configuration A ¯AA for ϕ = π . Far from thedefects, the bulk of the three regions can be characterized bydifferent Chern numbers: ν ¯A = − ν A = (b) ).The theory of topological defects predicts that the peri-odic particle pumping across one point of the chain will de-pend on the Chern numbers of the corresponding regions. Inparticular, the number of bound states that circulate aroundeach soliton is equal to the difference of Chern numbers be-tween the regions connected by the soliton. This can be ob-served in Figure 10 (b) , where we represent the spectral flow4 FIG. 11.
Inter-soliton pumping with Z solitons: (a) and (b) show the local Berry phase γ calculated at the middle of a finite chainwith L =
42 sites as a function of the pumping parameter ϕ . Bothcorrespond to half filling, but starting from the homogeneous config-urations ¯A and A, respectively. Using Eq. (14) we can calculate thecorresponding Chern numbers, obtaining ν ¯A = − ν A = +
1, re-spectively. The sign of the Chern number gives us the direction of thetransport, ∆ n = − ν . This can be observed in (c) , where we representthe real-space bosonic occupation (cid:104) n i (cid:105) through the pumping cycle ofa system with two domain walls, starting from the ¯AA ¯A configura-tion at ϕ =
0. Each region transports a quantized charge in the bulk,but the direction (dashed arrows) is different in each of them. of a finite dimerized chain with OBC and two domain walls.Apart from the two localized states associated to the edgesof the chain, two extra localized states cross the gap connect-ing both bands. These are associated with the domain walls,and cross at the two inversion-symmetric points, where theyare degenerate. The extended 2D cylinder consists of differ-ent regions with different Chern numbers, and the localizedstates associated to the solitons in 1D can be interpreted astopologically-protected conducting states that reside at the 1Dcircular boundaries between these regions. Note that thesestates circulate twice as fast compared to the states on theboundaries, since the Chern number difference is doubled.The explicit connection between the adiabatic pumping in1D and the Chern insulators in 2D (12) allows us to clarify thetopological origin of the localized states bound to the solitonsin the case of a non-interacting fermionic system on a staticdimerized chain, generalizing the bulk-edge correspondenceto a defect-edge correspondence. We will now apply this rea-soning to the full interacting system with dynamical solitons.
2. Inter-soliton pumping in the Z Bose Hubbard model
Let us now move away from the single-particle scenario,and explore the pumping of bosons between the topologi-cal solitons in the strongly-correlated Z BHM. We shall usethis pumping to discuss a generalised bound-defect correspon-dence that shines light on the topological origin of the bound
FIG. 12.
Inter-soliton pumping with Z solitons: (a) , (b) and (c) show the local Berry phase γ calculated at the middle of a finite chainwith L =
42 sites as a function of the pumping parameter ϕ . Theycorrespond to ρ = /
3, but starting from the homogeneous config-urations ¯A, ¯B and ¯C, respectively. Using eq. (14) we obtain Chernnumbers equal to (a) ν ¯A = − (b) ν ¯B = (c) ν ¯C = (d) Realspace bosonic occupation (cid:104) n i (cid:105) through the pumping cycle of a sys-tem with three domain walls, starting from the ¯A ¯B ¯C ¯A configuration.At the edges of the system, one particle is transported during the cy-cle, since the former act as boundaries between the region ¯A and thevacuum, with a the difference in Chern numbers of −
1. Two of thedefects separate regions where this difference is ±
2, and the chargepumped through these defects is two, while it is zero across the triv-ial soliton (center). The direction of the transport (dashed arrows)depends on the sign of the Chern number, with ∆ n = − ν . fractional bosons. In this case, we implement the adiabaticpumping by modifying the Z BHM (1) as follows H = − ∑ i (cid:104) b † i ( t + ασ zi , i + ) b i + + H.c. (cid:105) + U ∑ i n i ( n i − )+ ∑ i ∆ ϕ , i σ zi , i + + ∑ i β ϕ , i σ xi , i + , (13)where ∆ ϕ , i = ( − ) i δ cos ( ϕ ) and β ϕ , i = ( − ) i δ sin ( ϕ ) . Bychoosing δ (cid:29) t , we guarantee that the spins rotate period-ically, effectively generating a superlattice modulation simi-lar to the one we considered in the non-interacting Rice-Melemodel (12). In practice, it is enough to fix δ = t .In this case, one must consider the many-body ground-statedue to the presence of interactions. Therefore, the notion ofthe Chern number of the band must be generalized to thatof an interacting system. The latter can be done by defin-ing the Chern number in a 2D finite size system with the helpof twisted boundary conditions [72]. Alternatively, the Chernnumber can also be computed for a cylinder as the change ofthe Berry phase γ ( ϕ ) during the pumping cycle [73, 74] ν = π (cid:90) π d ϕ ∂ ϕ γ ( ϕ ) . (14)5We use the latter definition to infer the Chern number andcharacterize the topology of our pumping cycle.We first consider the case of solitons around half filling.Figure 11 (a) shows the change in the many-body Berry phaseduring the pumping, which starts from the configuration B = ¯Aat ϕ =
0. We note that the many-body Chern number can becalculated from the finite changes of this quantity, which yield ν ¯A ≈ −
1. The Chern number associated to the reversed pump-ing starting from the configuration A is ν A ≈
1, as shownin Fig. 11 (b) . These results allow us to draw an alternativepicture of the topological origin of the fractionalized boundquasi-particles that appear in the Z -BHM above/below halffilling. Similarly to the non-interacting case, the fractionalbosons bound to the defects can be understood as remnantsof the conducting states localized at the 1D cylindrical inter-faces that separate synthetic 2D regions with different Chernnumbers. The difference between the Chern numbers of theseregions predicts the number of bound modes, and its sign isrelated to the direction of the particle flow [75]. Figure 11 (c) shows how the real-space bosonic occupation evolves for a fi-nite chain with two domain walls. This figure clearly supportsthe above prediction: as a consequence of the different topo-logical invariants, quantised charge is transported between thetwo existing solitons. The latter separate regions with differ-ent Chern numbers in the extended 2D system, and the direc-tion of the transport is different for each one.As stated above, this pumping offers an alternative takeon the topological origin of the fractionalized bosons. How-ever, for the half-filled case, it is not essential as the topolog-ical solitons always separate regions with a different Berryphase, and one can develop a topological characterizationbased solely on the equilibrium properties of the1D model.Let us now move the case of two-third filling, where soli-tons interpolate between regions with the same Berry phase,and pumping becomes essential to unravel the topological as-pects of the fractionalized bosons. We first compute the Chernnumbers associated to the three degenerate configurations ¯A,¯B and ¯C. Figures 12 (a) , (b) and (c) show the Berry phasesfor the three configurations ¯A, ¯B and ¯C, the change of whichas a function of the pumping phase leads to Chern numbers ν ¯A = − ν ¯B = ν ¯C = (c) for the equivalent situationwith free fermions, the in-gap state of the corresponding de-fect does not connect the valence and conduction bands, andcan be thus removed without closing the gap.In the full many-body Z BHM (13), the aforementionedabsence of spectral flow means that the ¯B ¯C soliton will notaccumulate an integer pumped charge during the adiabatic cy-cle. In Figure 12 (d) , we represent the time evolution of thebosonic density through the cycle for a system with three soli-tons. It can be observed how the bosonic density is pumpedalong distinct directions in the the three different regions, andthat no particle number is being pumped in one of the de-fects, which is consistent with the above argument. In thecontrary, for the other ¯A ¯B and ¯C ¯A solitons, the extended in- terfaces have a different Chern number, and the quantizationof the adiabatic pumping can be thus used for the topologi-cal characterization of the SPTd. Let us highlight once more,that the Berry phase of all these composite defects are all thesame γ ¯A = γ ¯B = γ ¯C = π . Therefore, although the fractional-ized bound states do not arise in the interface of two topolog-ically distinct regions in the equilibrium situation, we see thatthe extended regions via the pumping do indeed interpolatebetween regions with a different Chern number, guaranteeingthus the topological robustness of the pumping between theseSPT-d.As a summary, the pumping argument has allowed us touncover the topological origin of the localized bosonic statesassociated to the Z solitons. As opposed to the case of Z solitons, in the later this origin could not be inferred from thetopological properties of the different SSB sectors in 1D, asthey all belong to the same topological phase. We note that, inthese simulations, we have pinned down the defects since ev-erything is computed in the ground state of the system, whichcoincides with the time-dependent calculation for an adiabaticevolution. In a cold-atom experiment, however, the defects areexpected to move during the cycle. The results for the trans-ported charge, however, should hold also in this situation. IV. CONCLUSIONS AND OUTLOOK
In this work, we have explored the existence of symmetry-protected topological defects in intertwined topologicalphases, where both spontaneous symmetry breaking and topo-logical symmetry protection cooperate, giving rise to exoticstates of matter. We have analyzed how the topological soli-tons, which interpolate between different symmetry-brokensectors, can host fractionalized matter quasi-particles, andhow the back action of the matter sector on the topologicalsolitons is crucial to encounter this phenomenon directly in thegroundstate. We have presented a thorough analysis of the Z Bose-Hubbard model, where such interesting effects appearvia a bosonic version of the Peielrs’ mechanism. This leadsto different types of Z n solitons, showing distinct fractional-ized bosonic densities bound to the solitons, and organizedaccording to a rich variety of layouts: from pinned configu-rations with few topological defects, to solitonic lattices withcorresponding crystalline densities of fractionalized bosons.The remarkable progress in cold-atom quantum simulators,together with the recent interest in bosonic lattice models with Z -dressed tunnelling, points to a very promising avenue forthe realization of the Z Bose-Hubbard model. As conveyedin this work, this experiment would allow for the first di-rect observation of dynamic topological solitons with boundfractionalized quasi-particles, overcoming past difficulties inconjugate-polymer science, and giving novel insights in gen-uinely quantum-mechanical topological defects.In addition to the aforementioned static phenomena, wehave also explored the quantization of particle transport inadiabatic pumping protocols. As a consequence of the topo-logical solitons and bound quasi-particles, we have observedthat an integer number of bosons can be pumped between the6topological solitons, and that this phenomenon can be usedto derive a generalised bulk-defect correspondence. Giventhe variety of intertwined topological phases in the Z Bose-Hubbard model, it turns out that one can find fractionalizedquasi-particles bound to solitons that separate regions withthe same topological Berry phase. It is only through the adi-abatic inter-soliton pumping of bosons, and the connectionto extended 2D system with the adiabatic parameter playingthe role of an extra synthetic dimension, that the topologi-cal origin of the bound quasi-particles can be neatly under-stood. We have shown that, despite appearing at the interfaceof SSB sectors with the same topological Berry phase, thesequasi-particles bound to the solitons can be understood, viathe pumping scheme, as remnants of protected edge states in2D that are bound to the boundaries separating synthetic re-gions with different Chern numbers. This generalised bulk-defect correspondence thus clarifies the topological origin ofthese quasi-particles via a Kaluza-Klein dimension reduction.So far, our discussion has focused on the topological fea-tures of the bosonic sector at zero temperature, which are al-lowed by their coupling to the solitons in the spin sector. Asan outlook, we would like to note that, at finite temperatures,some additional fractionalization can also take place in thespin solitons, which reminds of the exotic properties of thespinon branches in antiferromagnetic Heisenberg chains. Inthe regime of large Peierls-Nabarro barriers, the existence oftwo soliton branches dicussed in Sec. II A 1 has a direct con-sequence: the domain walls develop a fractional value of the statistical-interaction parameter g = /
2, which leads to frac-tional exclusion statistics with characteristic semionic thermo-dynamics [76]. At other fractional fillings, such as ρ = / g = / ACKNOWLEDGMENTS
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