Intertwined Topological Phases induced by Emergent Symmetry Protection

TL;DR

This paper uncovers a new mechanism where emergent symmetry protection in strongly correlated systems leads to intertwined topological phases, revealing novel static and dynamic phenomena.

Contribution

It introduces a novel mechanism for symmetry protection emerging from broken symmetries, leading to intertwined topological phases in strongly correlated systems.

Findings

Emergent symmetry protection arises dynamically from broken symmetries.

Intertwined topological phases coexist with long-range order.

Interaction-induced topological phase transitions and fractional pumping are observed.

Abstract

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 dynamically from a set of configurations enforced by another broken symmetry. This novel 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.

Equations

$$H=$$

$$+\frac{\Delta }{2}\sum _i{\sigma }_{i,i+1}^z+\beta \sum _i{\sigma }_{i,i+1}^x,$$

Full text

Introduction.–

The 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 Gross (1996). In particular, at long length-scales, various phases of matter can be understood by the pattern of SSB and corresponding local order parameters Landau (1937). Although different SSB patterns tend to compete with one another, a genuine cooperation can also arise in strongly-correlated systems with intertwined orders Fradkin et al. (2015). More recently, topology has been recognised as an exotic driving force shaping the texture of Nature, and leading to phases characterised by topological invariants rather than by local order parameters Wen (2017). It is no longer the breaking of certain symmetries but, actually, their conservation Chiu et al. (2016), which gives rise to novel states of matter, the so-called symmetry-protected topological (SPT) phases Senthil (2015). In the non-interacting limit, topological insulators and superconductors provide well-understood examples of this paradigm Qi and Zhang (2011), while current research aims at understanding strong-correlation effects, such as the competition of SPT and SSB phases due to interactions Rachel (2018).

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 Raghu et al. (2008); Sun et al. (2009); Kourtis and Daghofer (2014). 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 Peierls (1955) in polyacetylene, neatly accounted for via the Su-Schrieffer-Heeger (SSH) model at half-filling Heeger et al. (1988). 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 anti-bonding orbitals. A closer inspection shows that inversion symmetry is automatically fixed by such a SSB pattern, which leads to a topological quantization of the electronic polarization Hughes et al. (2011), and is ultimately responsible for the protection of the SPT phase.

In this work, we study a hitherto unknown possibility: the emergence of an intertwined topological phase when the SSB pattern does not automatically impose the protecting 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 behaviour. We demonstrate this novel topological mechanism in the $\mathbb{Z}_{2}$-Bose-Hubbard model González-Cuadra et al. (2018, 2019), a microscopic lattice model that displays strongly-correlated intertwined topological phases at various fractional fillings. At third-filling and for sufficiently-strong interactions, we find a period-3 BOW with a three-fold degenerate ground state that displays a non-zero 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 first- and 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. Thouless (1983a); Niu and Thouless (1984), 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 Nakajima et al. (2016); Lohse et al. (2016). Strong interactions can lead to fractional pumped charges Marra et al. (2015); Taddia et al. (2017a), showing a clear reminiscence to the fractional quantum Hall effect (FQHE) Tao and Thouless (1983); Bergholtz and Karlhede (2008); Guo et al. (2012); Budich and Ardonne (2013). We show that, following a dynamical modulation of the interactions in the $\mathbb{Z}_{2}$-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.

$\mathbb{Z}_{2}$-Bose-Hubbard model.–

We consider a 1D system of interacting bosons coupled to a dynamical $\mathbb{Z}_{2}$ field and described by the lattice Hamiltonian

[TABLE]

where $b^{\dagger}_{i}$ is the bosonic creation operator at site $i$, $n_{i}=b^{\dagger}_{i}b_{i}$ is the number operator, and $\sigma^{x}_{i,i+1},\sigma^{z}_{i,i+1}$ are the Pauli matrices associated to the $\mathbb{Z}_{2}$ 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 $\mathbb{Z}_{2}$ fields have an energy difference between the local configurations $\Delta$, and a transverse field of strength $\beta$ that is responsible for their quantum fluctuations. The $\mathbb{Z}_{2}$ fields renormalize the bosonic hopping via $\alpha$, and we set $\alpha=0.5t$ and $\Delta=0.85t$.

This model (1) hosts Peierls-type phenomena analogous to the fermionic SSH model Heeger et al. (1988) but, remarkably, in absence of a Fermi surface González-Cuadra et al. (2018). There are, however, important differences: at half filling and in the slow-lattice limit relevant for polyacetylene Pouget (2016), a Peierls’ instability inevitably occurs for arbitrarily-small fermion-lattice couplings Peierls (1955). In this limit, the fermionic groundstate in one of SSB sectors is adiabatically connected to a free-fermion SPT phase Chiu et al. (2016). In contrast, our bosonic Peierls phases allow for a genuinely-correlated topological bond-ordered wave (TBOW1/2) protected by bond-inversion symmetry, which cannot be adiabatically connected to a free-boson SPT phase González-Cuadra et al. (2019). In this case, the symmetry protecting the TBOW1/2 is completely fixed from the dimerization SSB pattern of the $\mathbb{Z}_{2}$ fields.

We now describe a richer situation at filling $n=2/3$ (similarly $n=1/3$). Here, the Peierls-type SSB leads to a trimerization of the $\mathbb{Z}_{2}$ field, namely a periodic repetition of a 3-site unit cell with bonds characterized by arbitrary expectation values $\langle\sigma^{z}_{1,2}\rangle,\langle\sigma^{z}_{2,3}\rangle,\langle\sigma^{z}_{3,4}\rangle$. Note that this trimerization still leaves freedom for various bond configurations that do not necessarily imply a protecting symmetry for the bosons (Fig. 1(b)). One of the main results of our work is to show how such a protecting symmetry actually emerges dynamically at low energies, giving rise to an intertwined TBOW2/3 (Fig. 1(d)).

We first study a system of $L=30$ sites with DMRG Hauschild and Pollmann (2018), for $\beta=0.01\,t$ and different Hubbard interactions $U$. For weak interactions, the $\mathbb{Z}_{2}$ field is polarized along the same axis (Fig. 1(a)), and the bosons display a quasi-superfluid behaviour. Increasing the interactions leads to a bosonic Peierls transition, whereby translational symmetry is spontaneously broken, leading to a three-fold degenerate ground-state with ferrimagnetic-type ordering $\langle\sigma^{z}_{1,2}\rangle=\langle\sigma^{z}_{2,3}\rangle>\langle\sigma^{z}_{3,4}\rangle$, together with a bosonic period-3 BOW that displays inversion symmetry with respect to the central inter-cell bond (see Fig. 2(a,b), left panel). This phase exhibits similar properties to the charge density waves in extended Hubbard models Guo et al. (2012); Budich and Ardonne (2013), albeit without the need of longer-range interactions. We note that a fermionic counterpart of this phase has been predicted in charge-transfer salts Su and Schrieffer (1981); Su (1983). To characterize its topology, we use the local Berry phase $\gamma^{\mu}={\rm i}\int_{0}^{2\pi}{\rm d}\theta\langle\Psi^{\mu}(\theta)|\partial_{\theta}\psi^{\mu}(\theta)\rangle$, where $\ket{\psi^{\mu}_{\theta}}$ is the $\mu$-th groundstate of the Hamiltonian (1) with a single bond twisted according to $t\to t{\rm e}^{{\rm i}\theta}$ Hatsugai (2006). The left panel of Fig. 2(c) depicts the local Berry phase for one of the groundstates, which clearly vanishes on the inter-cell bonds relevant for the inversion symmetry of Fig. 1. We note that the three possible groundstates become degenerate in the thermodynamic limit, which can be characterized by the total Berry phase $\gamma=\sum_{\mu}\gamma^{\mu}$. For the present BOW2/3, we also find $\gamma=0$, indicating that this phase is topologically trivial.

By further increasing the interactions, a phase with a different SSB pattern $\langle\sigma^{z}_{1,2}\rangle=\langle\sigma^{z}_{2,3}\rangle<\langle\sigma^{z}_{3,4}\rangle$ arises (right panels Fig. 2(a,b)). Although the ferrimagnetic and BOW patterns look rather similar to the previous case, the local Berry phase at the inter-cell bonds is now quantized to $\gamma_{\mu}=\pi$ (right panel Fig. 2(c)). Note again that this phase presents three degenerate groundstates in the thermodynamic limit, and we find a total Berry phase $\gamma=\pi$, indicating a non-trivial TBOW2/3 phase. This exemplifies the scenario of Fig. 1: from all the trimerized configurations possible a priori, the system chooses the one with additional bond-centered inversion symmetry, allowing for a topological crystalline insulator Chiu et al. (2016). In combination with the local order parameters (right panel Fig. 2(a,b)), this shows that the TBOW2/3 is an interaction-induced intertwined topological phase in which, contrary to half-filling González-Cuadra et al. (2018), the protecting symmetry is emergent and not fixed a priori by the SSB pattern. The ocurrence of this mechanism is a hallmark of our $\mathbb{Z}_{2}$-Bose-Hubbard model and does not have an analogue in the standard SSH model Su and Schrieffer (1981); Su (1983).

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 Amaricci et al. (2015); Barbarino et al. (2018). We now discuss how critical lines of different orders delimit the intertwined TBOW2/3 in a strongly-interacting region of parameter space, showing that the TBOW2/3 cannot be adiabatically connected to a free-boson SPT phase.

In the adiabatic regime $\beta\ll t$, we observe that the transition between trivial BOW2/3 and intertwined TBOW2/3 is of first order using an infinite DMRG algorithm Hauschild and Pollmann (2018). Figure 3(a) shows the Ising fields $\langle\sigma^{z}_{k,k+1}\rangle$ within the unit cell as the Hubbard interaction is increased, while keeping $\beta$ fixed. For $\beta=0.01t$ (left column) we observe an abrupt transition characterized by a discontinuity in the first derivative of the ground state energy ${\delta E_{\rm g}}/{\delta U}=({E_{\rm g}(U+\Delta U)-E_{\rm g}(U)})/{\Delta U}$ Barbarino et al. (2018), signaling a first-order phase transition (Fig. 3(b)). Introducing the bond observables, $\mathcal{O}_{k}=\bra{E_{\rm g}}\sigma^{z}_{k,k+1}-\sigma^{z}_{k+1,k+2}\ket{E_{\rm g}}$ with $k$ even or odd, we can characterise the corresponding bond-inversion symmetry within the unit cell. The inset of Fig. 3(b) shows how $\mathcal{O}_{1}$ displays a discontinuous jump. We note that the gap of the system remains open during this transition, while the total Berry phase, computed here with the help of the entanglement spectrum Zaletel et al. (2014), changes abruptly, as depicted in Fig. 3(c). 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 $\delta E_{\rm g}/\delta U$ and in $\mathcal{O}_{1}$ for $\beta=0.025t$. Remarkably, there is a finite region between the trivial and topological BOW phases where the $\mathbb{Z}_{2}$ fields have different expectation values, breaking the emergent inversion symmetry within the larger Peierls’ trimerization. These results are in accordance with the behaviour of the total Berry phase in Fig.  3(c), which shows a non-quantized 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 gap closing 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 3-site unit cell González-Cuadra et al. (2018). 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 $\gamma=\pi$ of Fig.  3(c). This non-trivial interplay between symmetry protection and symmetry breaking, driven solely by correlations, is another hallmark of our $\mathbb{Z}_{2}$-Bose-Hubbard model, absent at other fillings or in the fermionic SSH model Su and Schrieffer (1981); Su (1983).

Finally, we present the phase diagram as a function of $\beta$ and $U$ in Fig. 4(a) by depicting the product of $\mathcal{O}_{1}\mathcal{O}_{2}$: it can only attain a non-zero value if the bond inversion symmetry within the unit cell is broken (i.e. if the transition occurs continuously via an intermediate non-symmetric region). As can be neatly observed in this figure, for $\beta<\beta_{c}$, a first-order transition separates the trivial and topological BOWs. We find that $\beta_{\rm c}\approx 0.012t$, which is consistent with a triple point separating the first-order and continuous transitions that delimit the TBOW2/3 (Supplementary Material). Figure. 4(b) shows the phase diagram in terms of the total Berry phase, quantized to [math] and $\pi$ in the regions with inversion symmetry and with non quantized 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 Thouless (1983b). This topological pumping lies at the heart of our current understanding of free-fermion SPT phases Qi and Zhang (2011), and can also be generalized to weakly-interacting systems Niu and Thouless (1984). As advanced in the introduction, 1D and quasi-1D systems at sufficiently-strong interactions can exhibit a new type of fractional pumping Grusdt and Höning (2014); Grushin et al. (2015); Zeng et al. (2015, 2016); Li and Fleischhauer (2017); Taddia et al. (2017b) that cannot be accounted for using non-interacting topological pumping.

In this section, we show that adiabatic dynamics traversing through intertwined topological phases allows for a new effect: 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 $\mathbb{Z}_{2}$ fields, and raising/lowering the Hubbard interactions, the free $\mathbb{Z}_{2}$ fields self-adjust dynamically during the adiabatic cycle. As a consequence, the bosonic sector traverses a sequence of groundstates that are energetically favourable 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 Grusdt and Höning (2014); Grushin et al. (2015); Zeng et al. (2015, 2016); Li and Fleischhauer (2017); Taddia et al. (2017b). The details of this self-adjusted topological pumping are explained in Figure 5.

For finite systems, the pumped charge can be inferred from the center of mass (COM) $P_{L}(\tau)=\frac{1}{L}\sum_{j}(j-j_{0})\bra{\Psi(\tau)}\hat{n}_{j}\ket{\Psi(\tau)}$, where $j_{0}$ is the center of a chain of size $L$, and $\ket{\psi(\tau)}$ is the adiabatically-evolved state at time $\tau$. Figure 5(c) shows the DMRG results describing how the COM changes along the cycle connecting the BOW2/3 and TBOW2/3 possible groundstates for a finite chain of size $L=90$. After $\tau=T$, we observe a COM displacement of $\Delta n^{T}_{L=90}=P_{L=90}(T)-P_{L=90}(0)=0.316$, reflecting the fractional charge. To obtain precisely the charge, we perform a finite-size scaling analysis and find $\Delta n^{T}_{\infty}=\lim_{L\to\infty}\Delta n^{T}_{L}=1/3$ (inset). At $\tau=2T$, 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 groundstate degeneracy, and cannot be found for any non-interacting topological phase (Supplementary Material). In our present case, the adiabatic path in parameter space can be understood as a dynamical analogue of the spatial interpolation between the different groundstates, which leads to topological solitons and fractionally-quantized charges bound to them Su and Schrieffer (1981). During each period $T$, we interpolate between two such groundstates, 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 towards $-1/3$, as this is related to the presence of many-body edge states for a finite system Hatsugai and Fukui (2016), and can be used to define a bulk-boundary correspondence for our intertwined TBOW2/3. The transported charge across the bulk, $\Delta n^{3T}_{L}$, can be related to the discontinuous jumps during the cycle Hatsugai and Fukui (2016), namely $\Delta n^{3T}_{L}=-\sum_{i}\Delta P_{L}(\tau_{i}),$ where $\Delta P_{L}(\tau_{i})=P_{L}(\tau_{i}^{+})-P_{L}(\tau_{i}^{-})$ quantify the discontinuities occurring at instants $\tau_{i}$, and $\tau_{i}^{\pm}=\tau_{i}\pm\epsilon$ with $\epsilon\to 0$. In the thermodynamic limit, it converges to the quantized value of the pumped charge $\Delta n^{3T}_{\infty}=\lim_{L\to\infty}\Delta n^{3T}_{L}=1$ related to the integer Chern number in an extended 2D system Hatsugai and Fukui (2016). 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 Wang et al. (2013), and it has been used to reveal the topological properties of fermionic and bosonic SPT phases Nakajima et al. (2016); Lohse et al. (2016).

By estimating the discontinuity, we can extract the transported charge across the bulk during the whole adiabatic evolution that bring the BOW back to itself after $\tau=3T$, obtaining a nearly quantized value $\Delta n_{L=90}=0.92$. As it is 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 Hatsugai and Fukui (2016), even though this was not guaranteed a priori due to the lack of the global symmetries regarding the ten-fold classification of topological insulators. In particular, one may understand the edge states of the TBOW2/3 as remains of topologically-protected conducting edge states of an extended 2D system (Supplementary Material). 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 non-zero quantized charge can only be obtained when the parameter modulation encircles such a degeneracy point.

Conclusions and outlook.–

We have shown how symmetry protection can emerge dynamically in an interplay between symmetry breaking and strong correlations. In the $\mathbb{Z}_{2}$ Bose-Hubbard model, this novel mechanism gives rise to 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 Barbiero et al. (2018); Schweizer et al. (2019). The proposed self-adjusted pumping protocol, in particular, could be used to reveal the topological properties of the system and its fractional nature. Future research directions include the study of topological defects on top of the intertwined topological phases, where localized states with fractional particle number are expected to appear, signaling deeper connections to the physics of the FQHE.

Methods. –

The numerical calculations have been performed using a density matrix renormalization group algorithm (DMRG) Hauschild and Pollmann (2018). 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 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.

Supplementary Information.–

See Supplementary Material for further details on the topological phase transitions and the location of the tricritical point, as well as on the connection between the Thouless pumping and an extended 2D topological system, shedding light on the nature of fractionalization and the presence of edge states.

Acknowledgements.–

The authors thank L. Tagliacozzo for useful discussions. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 665884, the Spanish Ministry MINECO (National Plan 15 Grant: FISICATEAMO No. FIS2016-79508-P, SEVERO OCHOA No. SEV-2015-0522, FPI), European Social Fund, Fundació Cellex, Generalitat de Catalunya (AGAUR Grant No. 2017 SGR 1341 and CERCA/Program), ERC AdG OSYRIS, EU FETPRO QUIC, and the National Science Centre, PolandSymfonia Grant No. 2016/20/W/ST4/00314. A. D. is financed by a Juan de la Cierva fellowship (IJCI-2017-33180). A.B. acknowledges support from the Ramón y Cajal program RYC-2016-20066, MINECO project FIS2015-70856-P, and CAM/FEDER Project S2018/TCS-4342 (QUITEMAD-CM).