Intertwined topological phases induced by emergent symmetry protection

The dual role played by symmetry in many-body physics manifests itself through two fundamental mechanisms: spontaneous symmetry breaking and topological symmetry protection. These two concepts, ubiquitous in both condensed matter and high energy physics, have been applied successfully in the last decades to unravel a plethora of complex phenomena. Their interplay, however, remains largely unexplored. Here we report how, in the presence of strong correlations, symmetry protection emerges from a set of configurations enforced by another broken symmetry. This mechanism spawns different intertwined topological phases, where topological properties coexist with long-range order. Such a singular interplay gives rise to interesting static and dynamical effects, including interaction-induced topological phase transitions constrained by symmetry breaking, as well as a self-adjusted fractional pumping. This work paves the way for further exploration of exotic topological features in strongly-correlated quantum systems.

T he notion of symmetry is paramount to unveil the fundamental laws of Nature, while spontaneous symmetry breaking (SSB) is essential to understand Nature's different guises 1 . In particular, at long length scales, various phases of matter can be understood by the pattern of SSB and the corresponding local order parameters 2 . Although different SSB patterns tend to compete with one another, a genuine cooperation can also arise in strongly correlated systems with intertwined orders 3 . More recently, topology has been recognized as an exotic driving force shaping the texture of Nature, and leading to phases characterized by topological invariants rather than by local order parameters 4 . It is no longer the breaking of certain symmetries, but actually, their conservation 5 , which gives rise to novel states of matter, the so-called symmetry-protected topological (SPT) phases 6 . In the noninteracting limit, topological insulators and superconductors provide well-understood examples of this paradigm 7 , while current research aims at understanding strongcorrelation effects, such as the competition of SPT and SSB phases, due to interactions 8 .
Alternatively, a cooperation between SPT and SSB may allow for intertwined topological phases that simultaneously display a local order parameter and a topological invariant. For integer and fractional Chern insulators, such intertwined orders have been already identified in the literature [9][10][11] . Nonetheless, in these cases, the topological phases exist in the absence of any protecting symmetry. In more generic situations, the existence of intertwined topological phases will depend on how the symmetry responsible for the SPT phase can be embedded into the broader symmetry-breaking phenomenon. Arguably, the first instance of this situation is the Peierls instability 12 in polyacetylene, neatly accounted for via the Su-Schrieffer-Heeger (SSH) model at halffilling 13 . Here, the instability leads to a dimerized lattice distortion and a bond-order wave (BOW), where electrons are distributed in an alternating sequence of bonding and antibonding orbitals. A closer inspection shows that inversion symmetry is never broken in such a SSB pattern, which leads to a topological quantization of the electronic polarization 14 , and is ultimately responsible for the protection of the SPT phase.
In this work, we study a hitherto unknown possibility: the occurrence of an intertwined topological phase when the SSB pattern does not generally imply the existence of a protecting inversion symmetry (Fig. 1). Instead, this protecting symmetry emerges from a larger set of configurations allowed by the SSB, such that its interplay with topology and strong correlations endows the system with very interesting, yet mostly unexplored, static and dynamical behavior. We demonstrate this topological mechanism in the Z 2 -Bose-Hubbard model 15,16 , a microscopic lattice model that displays strongly correlated intertwined topological phases at various fractional fillings. At one-third and twothird filling, and for sufficiently strong interactions, we find a period-3 BOW with a threefold degenerate ground state that displays a nonzero topological invariant: the total Berry phase. We show that inversion symmetry emerges from the larger SSB landscape of a bosonic Peierls' mechanism, protecting the intertwined topological BOW, and making it fundamentally different from other non-topological BOWs. We unveil a rich phase diagram with firstand second-order quantum-phase transitions caused by the interplay of this emergent symmetry, topology, and strong correlations. We also identify a dynamical manifestation of the underlying topology that is genuinely rooted in strong correlations and the interplay of the emergent and symmetry-broken symmetries: a self-adjusted fractional pump. As discussed by Thouless et al. 17,18 , the quantization of adiabatic charge transport in weakly interacting insulators uncovers a profound connection to higher-dimensional topological phases, as recently exploited in cold-atom experiments 19,20 . Strong interactions can lead to fractional pumped charges 21,22 , showing a clear reminiscence to the fractional quantum Hall effect (FQHE) [23][24][25][26] . We show that, following a dynamical modulation of the interactions in the Z 2 -Bose-Hubbard model, the system self-adjusts within the landscape of SSB sectors, allowing for a cyclic path that displays a fractional pumped charge 1/3, such that the correlated intertwined topological phase has no free-particle counterpart.
Results Z 2 -Bose-Hubbard model. We consider a 1D system of interacting bosons coupled to a dynamical Z 2 field and described by the lattice Hamiltonian where b y i is the bosonic creation operator at site i, n i ¼ b y i b i is the number operator, and σ x i;iþ1 ; σ z i;iþ1 are the Pauli matrices associated with the Z 2 fields on the bond (i, i + 1). The bare bosonic Hamiltonian depends on the hopping strength t, and the on-site Hubbard repulsion U > 0. Likewise, the Z 2 fields have an energy difference between the local configurations Δ, and a transverse field of strength β that is responsible for their quantum fluctuations. The Z 2 fields renormalize the bosonic hopping via α.  . The spontaneous breaking of translation symmetry results in a phase with a three-site unit cell, represented in (b) with different arrows accounting for the three possible magnetizations, which may not respect the inversion symmetry, lacking a nonzero topological invariant. Remarkably, such inversion symmetry can emerge from all the possible configurations constrained by the SSB pattern, leading to the low-energy sectors depicted in (c, d). Note that these two phases are not only distinguished by the SSB pattern, but also, and more importantly, by topology. Accordingly, whereas (c) is topologically trivial, (d) presents both a local order parameter and a nonzero topological invariant, and thus corresponds to an intertwined topological phase where the protecting symmetry emerges realizing the Z 2 -Bose-Hubbard model experimentally using ultracold atoms in optical lattices.
This model (1) hosts Peierls-type phenomena analogous to the fermionic SSH model 13 , but remarkably, in the absence of a Fermi surface 15 . There are, however, important differences: at halffilling and in the slow-lattice limit relevant for polyacetylene 30 , a Peierls' instability inevitably occurs for arbitrarily small fermionlattice couplings 12 . In this limit, the fermionic ground state in one of SSB sectors is adiabatically connected to a free-fermion SPT phase 5 . In contrast, our bosonic Peierls phases allow for a genuinely correlated topological bond-ordered wave (TBOW 1/2 ) protected by bond-inversion symmetry, which cannot be adiabatically connected to a free-boson SPT phase 16 . We remark that the symmetry protecting the TBOW 1/2 is completely fixed from the SSB pattern of the Z 2 fields at any energy scale, and is thus not an emergent symmetry.
We now describe a richer situation at filling n = 2/3 (similarly n = 1/3). For Δ ( t and Δ ) t the spins are uniformly polarized in the z direction, hσ z i;iþ1 i ¼ σ 0 , with σ 0 > 0 and σ 0 < 0, respectively. For intermediate values, a Peierls-type SSB leads to a trimerization of the Z 2 fields, namely a periodic repetition of a three-site unit cell, with bonds characterized by arbitrary expectation values hσ z 1;2 i; hσ z 2;3 i; hσ z 3;4 i. The resulting phase is an insulator, with a gap that increases with the value of the coupling α (see Supplementary Note 1 for details). Note that this trimerization still leaves freedom for various bond configurations that do not necessarily imply a protecting symmetry for the bosons (Fig. 1b). One of the main results of our work is to show how, for certain parameter regimes, such a protecting symmetry becomes effective at low energies, whereas higher-energy excitations of the Z 2 fields do not necessarily lead to it. Therefore, the inversion symmetry can be understood as an emergent symmetry that is crucial to protect the intertwined TBOW 2/3 (Fig. 1d). In the following, we set α = 0.5t and Δ = 0.85t.
We first study a system of L = 30 with sites and openboundary conditions using DMRG 31 , for β = 0.01t and different Hubbard interactions U. For weak interactions (U ⪅ 9t), the Z 2 field is polarized along the same axis (Fig. 1a), and the bosons display a quasi-superfluid behaviour with algebraically decaying off-diagonal correlations. Increasing the interactions leads to a bosonic Peierls transition, whereby translational symmetry is spontaneously broken, leading to a threefold degenerate ground state with ferrimagnetic-type ordering hσ z 1;2 i ¼ hσ z 2;3 i > hσ z 3;4 i, together with a bosonic period-3 BOW that displays inversion symmetry with respect to the central intercell bond (see Fig. 2a, b, left panel). The nature of a similar qSF-BOW phase transition is analyzed in ref. 16 for the Z 2 -Bose-Hubbard model at half-filling. The BOW phase described here exhibits similar properties to the charge density waves in extended Hubbard models 25,26 , albeit without the need of longer-range interactions. We note that a fermionic counterpart of this phase has been predicted in chargetransfer salts 32,33 . To characterize its topology, we use the local Berry phase is the μth ground state of Hamiltonian (1) with a single bond twisted according to t → te iθ 34 . The left panel of Fig. 2c depicts the local Berry phase for one of the ground states, which clearly vanishes on the intercell bonds relevant for the inversion symmetry of Fig. 1. We note that the three possible ground states become degenerate in the thermodynamic limit, which can be characterized by the total Berry phase γ ¼ P μ γ μ . In this limit, the value of γ μ for the three degenerate states on a fixed bond coincides, up to permutations, with the value of this quantity on the three bonds of the unit cell for each one of the states. Therefore, the sum gives the same value in both cases. For the present BOW 2/3 , we find γ = 0, indicating that this phase is topologically trivial.
By further increasing the interactions, a phase with a different SSB pattern hσ z 1;2 i ¼ hσ z 2;3 i < hσ z 3;4 i arises (right panels Fig. 2a, b). Although the ferrimagnetic and BOW patterns look rather similar to the previous case, the local Berry phase at the intercell bonds is now quantized to γ μ = π (right panel Fig. 2c). Note again that this phase presents three degenerate ground states in the thermodynamic limit, and we find a total Berry phase γ = π, indicating a nontrivial TBOW 2/3 phase. This exemplifies the scenario of Fig. 1: from all the trimerized configurations possible a priori, the system chooses one with additional bond-centered inversion symmetry, allowing for a topological crystalline insulator 5 . In combination with the local order parameters (right panel Fig. 2a, b), this shows that the TBOW 2/3 is an interaction-induced intertwined topological phase, in which, contrary to half-filling 15 , the protecting symmetry is emergent and not fixed a priori by the SSB pattern. The ocurrence of this mechanism is a hallmark of our Z 2 -Bose-Hubbard model and does not have an analog in the standard SSH model 32,33 .
Interaction-induced topological phase transitions. Topological phase transitions delimiting free-fermion SPT phases, and those found due to their competition with SSB phases, are typically continuous second-order phase transitions. In the presence of strong correlations, however, first-order topological phase transitions may arise [35][36][37][38] . We now discuss how critical lines of different orders delimit the intertwined TBOW 2/3 in a strongly interacting region of parameter space, showing that the TBOW 2/3 cannot be adiabatically connected to a free-boson SPT phase.
In the completely adiabatic regime β = 0, we observe that the transition between trivial BOW 2/3 and intertwined TBOW 2/3 is of first order, using an infinite DMRG algorithm 31 . Figure 3a shows  Fig. 2 Simultaneous orders in intertwined topological phases: real-space configuration of (a) the Z 2 field hσ z i;iþ1 i and (b) the bosonic bond densities B i;iþ1 ¼ hb y i b iþ1 i þ c:c:, using different colors for each element of the unit cell at β = 0.01t. Different permutations within the unit cell lead to a threefold quasi-degenerate ground state, each obtained from one another by translating the modulation patterns of the ferrimagnetic and BOW orders. The quasi-degeneracy comes from the finite-size effects, but degeneracy is recovered in the thermodynamic limit. c) The local Berry phases γ μ display a quantized value of 0 (U = 10t) or π (U = 15t) on the bonds preserving the inversion symmetry of the unit cell, allowing us to distinguish between the trivial and topological BOW phases. We note that the topological BOW phase (right panels) does not have a fermionic analog 48 in the ground state of the SSH model 32 the Ising fields hσ z k;kþ1 i within the unit cell, as the Hubbard interaction is increased, while keeping β fixed. For β = 0 (left column), we observe an abrupt transition characterized by a discontinuity in the first derivative of the ground-state energy δE g /δU = (E g (U + ΔU) − E g (U))/ΔU 36 , signaling a first-order phase transition (Fig. 3b). Introducing the bond observables, O k ¼ hE g jσ z k;kþ1 À σ z kþ1;kþ2 jEs g i with k even or odd, we can characterize the corresponding bond-inversion symmetry within the unit cell. The inset of Fig. 3b shows how O 1 displays a discontinuous jump. The total Berry phase, computed here with the help of the entanglement spectrum 39 , also changes abruptly, as depicted in Fig. 3c. To the best of our knowledge, this is the first topological characterization of a first-order phase transition in an intertwined topological phase.
The situation changes as one departs from the adiabatic regime. Figure 3 (right panel) shows a continuous second-order transition both in δE g /δU and in O 1 for β = 0.025t. Remarkably, there is a finite region between the trivial and topological BOW phases, where the Z 2 fields have different expectation values, breaking the emergent inversion symmetry within the larger Peierls' trimerization. These results are in accordance with the behavior of the total Berry phase in Fig. 3c, which shows a nonquantized value in this intermediate asymmetrical region. In fact, the appearance of this region originates from a very interesting interplay between the emergent inversion symmetry and the Peierls SSB phenomenon: a direct continuous transition between the trivial and topological BOWs would require a gapclosing point in the bosonic sector, where every bond had the same expectation value and the BOW would disappear. However, this comes with an energy penalty, since the Peierls' mechanism favors the formation of a three-site unit cell 15 . Therefore, the system energetically prefers to keep the trimerized unit cell at the expense of breaking the bond-inversion symmetry within the unit cell, and continuously setting the emergent inversion symmetry responsible for the quantized Berry phase γ = π of Fig. 3c (see Supplementary Note 1 for details). This nontrivial interplay between symmetry protection and symmetry breaking, driven solely by correlations, is another hallmark of our Z 2 -Bose-Hubbard model, absent at other fillings or in the fermionic SSH model 32,33 . The intermediate phase could extend up to β = 0, although first-order transitions are also possible for low enough values of β. An extended numerical analysis would be required to distinguish between these two situations.
Finally, we present the phase diagram as a function of β and U in Fig. 4a by depicting the product of O 1 O 2 : it can only attain a nonzero value if the bond-inversion symmetry within the unit cell is broken (i.e., if the transition occurs continuously via an intermediate nonsymmetric region). Figure 4b shows the phase diagram in terms of the total Berry phase, quantized to 0 and π in the regions with inversion symmetry and with nonquantized values in the region where the symmetry is broken.
Self-adjusted fractional pump. Topology can also become manifest through dynamical effects, such as the quantized transport of charge in electronic systems evolving under cyclic adiabatic modulations: Thouless pumping 17 . This topological pumping lies at the heart of our current understanding of freefermion SPT phases 7 , and can also be generalized to weakly interacting systems 18 . As advanced in the "Introduction" section, 1D and quasi-1D systems at sufficiently strong interactions can exhibit a fractional pumping 22,[40][41][42][43][44] that cannot be accounted for using noninteracting topological pumping.
In this section, we show that adiabatic dynamics traversing through intertwined topological phases allows for a self-adjusted fractional pumping, due to the interplay of the SSB mechanism and other gap-opening perturbations. By introducing guiding fields that only act on a subset of the Z 2 fields, and raising/ In the first case, there is an abrupt transition between the trivial and topological BOW. In the second case, the transition is continuous, and we find a finite region, where inversion symmetry is broken. b First derivative of the ground-state energy per unit cell E g through the transition. For β = 0, there is a discontinuous jump, signaling a first-order topological phase transition. The inset shows also a jump in the observable O 1 . For β = 0.025t, both quantities behave smoothly. c Total Berry phase, where the same behavior is observed. The results shown are obtained directly in the thermodynamic limit using iMPS (see the "Methods" section) We also present the total Berry phase. The latter has a nonquantized value in the region where the protecting inversion symmetry is broken. The phase diagram is calculated in the thermodynamic limit using iMPS (see the "Methods" section) lowering the Hubbard interactions, the free Z 2 fields self-adjust dynamically during the adiabatic cycle. As a consequence, the bosonic sector traverses a sequence of ground states that are energetically favorable due to the Peierls' mechanism. In this way, the system self-adjusts along this adiabatic sequence, allowing for an exotic fractional pumping induced by interactions 22,40-44 . The details of this self-adjusted topological pumping are explained in Fig. 5.
For finite systems, the pumped charge can be inferred from the center of mass (COM) P L ðτÞ ¼ 1 L P j ðj À j 0 ÞhΨðτÞjn j jΨðτÞi, where j 0 is the center of a chain of size L, and |ψ(τ〉) is the adiabatically evolved state at time τ. Figure 5c shows the DMRG results describing how the COM changes along the cycle connecting the BOW 2/3 and TBOW 2/3 possible ground states for a finite chain of size L = 90. After τ = T, we observe a COM displacement of Δn T L¼90 ¼ P L¼90 ðTÞ À P L¼90 ð0Þ ¼ 0:316, reflecting the fractional charge. To obtain precisely the charge, we perform a finite-size scaling analysis and find Δn T 1 ¼ lim L!1 Δn T L ¼ 1=3 (inset). At τ = 2T, the COM displacement reaches a value consistent with 2/3 in the thermodynamic limit. We note that these fractional values are characteristic of a strongly correlated SPT phase with ground-state degeneracy, and cannot be found for any noninteracting topological phase (see Supplementary Note 2 for details). In our present case, the adiabatic path in parameter space can be understood as a dynamical analog of the spatial interpolation between the different ground states, which leads to topological solitons and fractionally quantized charges bound to them 32 . During each period T, we interpolate between two such ground states, and a fractional charge is pumped without creating any spatial solitonic profile.
Let us now turn our attention to the discontinuous jump of the pumped charge toward −1/3, as this is related to the presence of many-body edge states for a finite system 45 , and can be used to define a bulk-boundary correspondence for our intertwined TBOW 2/3 . The transported charge across the bulk, Δn 3T L , can be related to the discontinuous jumps during the cycle 45 , namely Δn 3T L ¼ À where ΔP L ðτ i Þ ¼ P L ðτ þ i Þ À P L ðτ À i Þ quantify the discontinuities occurring at instants τ i , and τ ± i ¼ τ i ± ε with ε → 0. In the thermodynamic limit, it converges to the quantized value of the pumped charge Δn 3T 1 ¼ lim L!1 Δn 3T L ¼ 1 related to the integer Chern number in an extended 2D system 45 . Since these discontinuities depend on the presence of edge states in a finite system, the center-of-mass approach establishes a sort of bulk-boundary correspondence that can be explicitly proven via the adiabatic pumping. Moreover, the COM can be measured in cold-atomic experiments 46 , and it has been used to reveal the topological properties of fermionic and bosonic SPT phases 19,20 .
By estimating the discontinuity, we can extract the transported charge across the bulk during the whole adiabatic evolution that brings the BOW back to itself after τ = 3T, obtaining a nearly quantized value Δn L=90 = 0.92. As shown in the inset, a truly quantized charge is recovered in the thermodynamic limit, signaling the topological nature of the system. These results allow us to establish a bulk-boundary correspondence in the pumping process 45 , even though this was not guaranteed a priori, due to the lack of the global symmetries regarding the tenfold classification of topological insulators. In particular, one may understand the edge states of the TBOW 2/3 as remains of topologically protected conducting edge states of an extended 2D system (see Supplementary Note 2 for details). We note that, even if the topological degeneracy point does not appear in the phase diagram of the model, the quantized transported charge reveals its presence in an effective parameter space, as a nonzero quantized charge can only be obtained when the parameter modulation encircles such a degeneracy point 47 .

Discussion
We have shown how symmetry protection can emerge through an interplay between symmetry breaking and strong correlations. In the Z 2 -Bose-Hubbard model, this mechanism gives rise to an intertwined topological phase for certain fractional fillings. The unique properties of these phases are manifest in the special static and dynamical features discussed in this work. A realistic implementation of the model with cold atoms is suggested by recent experimental results 27,28 . The proposed self-adjusted pumping protocol, in particular, could be used to reveal the topological properties of the system and its fractional nature.  Fig. 5 Self-adjusted fractional pumping: a the trivial and topological BOW phases are threefold degenerate each. The six different states are represented here as different points on an effective parameter space characterized by the expectation values of the bond fields t k ¼ 1 þ α=thσ z k;kþ1 i, with k ∈ {1, 2, 3}. To define an adiabatic cycle through these different BOWs, the protecting inversion symmetry must be broken at intermediate states in order to enclose the degeneracy point at t 1 = t 2 = t 3 . b The Peierls mechanism forces the system to break this symmetry spontaneously when interactions are increased, connecting states in the trivial and topological BOW phases (Fig. 3b). In order to select which state from the degenerate manifold the system will transition to, we introduce an external inhomogeneous Z 2 field Δ k that is only applied to a subset of bonds within the unit cell. The fields partially break the degeneracy of the BOWs, and restrict the possible adiabatic evolution. A sequential combination of local fields and interaction-driven self-adjustments allows the system to cycle around the degeneracy point in the effective parameter space. Note that the protocol must be repeated three times for the ground state to reach the initial configuration. c COM P L (t) through the cycle for a finite chain of size L = 90 and for β = 0.025t. The discontinuous jump (red), related to the presence of edge states, allows us to obtain the total charge transported in the bulk during one cycle, Δn L=90 = 0.92. Inset: finite-size scaling yields a transported fractional charge at τ = T, and an integer charge at the end of the adiabatic path (τ = 3 T) states with fractional particle number are expected to appear, signaling deeper connections to the physics of the FQHE.

Methods
Numerical simulations. The numerical calculations have been performed using a density matrix renormalization group algorithm (DMRG) 31 . For the finite-size calculations, we used a matrix product state (MPS)-based algorithm with bond dimension D = 100. To directly access the thermodynamic limit, we used an infinite MPS (iMPS) with a repeating unit cell composed of three sites and D = 150. The Hilbert space of the bosons is truncated to a maximum number of bosons per site of n 0 = 2. This is justified for low densities and strong interactions.

Data availability
The data supporting the plots within this paper are available from the authors upon reasonable request. The figures have been produced with Python and adapted with Inkscape and Affinity Designer.