Topological pumping of quantum correlations

T. Haug1,2,∗ L. Amico2,3,4,5,6, L.-C. Kwek2,5,7,8, W. J. Munro1, and V. M. Bastidas1† 1NTT Basic Research Laboratories & Research Center for Theoretical Quantum Physics, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa, 243-0198, Japan 2Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore 3CNR-MATIS-IMM and Dipartimento di Fisica e Astronomia, Universitá Cnia, Via S. Soa 64, 95127 Catania, Italy 4INFN Laboratori Nazionali del Sud, Via Santa Sofia 62, I-95123 Catania, Italy 5MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore 6 LANEF ’Chaire d’excellence’, Universitè Grenoble-Alpes & CNRS, F-38000 Grenoble, France 7Institute of Advanced Studies, Nanyang Technological University, 60 Nanyang View, Singapore 639673, Singapore and 8National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616, Singapore (Dated: January 20, 2020)

Topological pumping and duality transformations are paradigmatic concepts in condensed matter and statistical mechanics. In this letter, we integrate them within a single scheme: we demonstrate how dualities enable us to perform topological pumping of highly-entangled collective excitations in spin chains. We explore the topological pumping of spin-flips, kinks and cluster-like excitations under the effect of disorder and interactions. Highly entangled states like kinks and cluster states have their entanglement reduced during the pumping process, while lowly entangled spin-flips become highly entangled. This change in entanglement is related to the transport mechanism and may have potential applications in quantum computation and quantum information processing. We find that interaction enhances the robustness of topological pumping to disorder in a wide class of interacting spin systems related by dualities.
In 1983 it was demonstrated by D. Thouless in a seminal paper that the topological properties of the wave function of a driven extended system can be exploited to realize quantum pumps [1][2][3][4][5][6]. Such a mechanism can be used to implement topologically-protected quantum transport, which is robust against disorder and weak interactions [2]. The remarkable progress of quantum technology has allowed the implementation of the Thouless idea to unprecedented degrees. Recently, quantum pumping of particles has been realized in diverse platforms ranging from ultracold optical superlattices [7,8] to waveguide arrays [9,10]. In the context of quantum simulation, quantum pumping in low-dimensional systems can be used to simulate higher dimensional quantum systems [11]. Experiments have demonstrated that topological pumping in two-dimensional systems can be used to explore the exotic physics of Quantum Hall effect in four dimensions using cold atoms [12] and photonic systems [13].
Dualities are important in diverse fields, because they can relate different physical regimes of a given theory [14][15][16][17][18][19][20]. The duality transformations are constructed by identifying the effective degrees of freedom of the system that, in the different physical regime, can be of very different nature. This idea traces back to classical electrodynamics, in which electric and magnetic fields can be suitably swapped together with electrical charges and currents. In the context of manybody physics, the Kramers-Wannier duality establishes a powerful relation between low-and high-temperature limits of two-dimensional classical Ising model [21][22][23], defining one of the most well known and important paradigms in statistical physics [24,25]. Through the classical-quantum correspondence the aforementioned duality is applied to quantum many-body Hamiltonians [14]. Since then, dualities have been demonstrated to be insightful in different contexts [26,27].
In this letter, we show that by using dualities it is possible to extend to idea of topological pumping in order to transport highly entangled spin excitations in the one-dimensional quantum Ising model. Due to its topological nature, the proposed quantum pumping is protected against disorder and interactions. In our scheme, we carry out topological pumping in quantum systems that are connected by a duality transformation. While the duality can change the entanglement of the collective excitations [20,28], the topological properties of the energy bands remain unaltered. To illustrate our approach, we show how to perform topological pumping of spin flips, kinks and cluster excitations, which are related by duality as depicted in Fig. 1. We find that the dynamics of bipartite entanglement can be dramatically affected by duality and pumping: spin flips become highly entangled at the anticrossings of the spectrum. Contrary to this, the entanglement present in kinks and cluster states is reduced or stays constant for specific bipartite divisions. One of the most appealing features of topological pumping is its robustness against disorder. In most cases, introducing interactions between excitations destroys topological transport [29]. However, we find for a particular type of interaction topological pumping is still possible, and furthermore robustness against certain types of disorder is strongly enhanced.
Let us begin by considering the one-dimensional quantum Ising model in a transverse field [30].
which is a paradigmatic model in condensed matter and statistical physics. Here, σ α j with α = {x, y, z} represents the usual Pauli-matrices acting on the j-th site and J  [30], which are generated by nearest-neighbor spin-spin interaction and are highly entangled. Further dual transformation relates them to cluster excitations that are generated by three-spin-interaction term. The spectrum is preserved by duality transformations, enabling topological transport of all these different kinds of excitations.
is strength of spin-spin interaction. The transverse field where ω is the frequency of the drive with period is T = 2π/ω. The parameters φ 0 and 1/b determine the initial phase shift and the spatial period of the modulation, respectively. In our work, we consider σ α j = σ α j+N and for simplicity, we explore the specific case b = 1/3. However, our results remain valid for any rational value b = p/q, where p and q are coprimes.
Due to the spatial modulation of the transverse field, the Ising Hamiltonian of Eq. (1) exhibits topological features that we can exploit to perform topological pumping. To gain some intuition about this, let us consider the weak-coupling regime g 0 J, where the collective excitations are spin-flips. Since the interaction between neighboring spins is small, the total number of excitationsN = 1/2 N j=1 (1 + σ x j ) is approximately conserved. Thus, the spectrum is divided into different bands associated with a fixed number of spin flips. Within a given band, the Hamiltonian (1) exhibits the same dynamics as the Aubry-Andre (AA) model [9,31], where the total number of spin-flips corresponds to the number of fermions. Crucially, by adiabatically modulating the parameters of the AA model [9], one can perform robust transport of particles, which is referred to as topological pumping [1,2]. The current of transported particles is intimately related to the Chern numbers of the Harper-Hofstadter model [32,33].
As the Hamiltonian (1) is related to the AA model in the regime g 0 J, let us analyze in detail its topological properties. Fig. 2 depicts the instantaneous spectrum of Hamiltonian (1) over one period of the modulation. For J = 0, any two levels cross periodically at times t n = nT/3, where n is an integer. A non-zero J opens an energy gap ∆E ≈ 2J, realizing separate bands with non-trivial topology characterized by the Chern number C [32,33]. For b = 1/3 there is one band with C = 2 and two with C = −1, [34,35]. By driving the system adiabatically along the bands, one can pump particles with topological protection against disorder and weak interactions [1,2]. In particular, at the anti-crossings, states hybridize and the excitations move by one site [34,35]. Due to the topological properties of Hamiltonian (1), we can pump spin-flip excitations and its dual quasiparticles, as we show below. Now let us consider the effect of Krammers-Wannier dual- on topological pumping. Here, τ α j are the Pauli matrices after the dual transformation [21,22]. We restrict ourselves to the positive parity sector of the parity operatorΠ = exp[iπ/2 N j=1 (σ x j +1)]. While the duality does not change the spectrum, its non-local character changes dramatically the nature of the eigenstates. The dual version of Hamiltonian (1) readŝ For g 0 > J, the fundamental excitations of this model are highly entangled states, known as kinks [30]. The dual transformation maps the spin-flip excitations to entangled kinks (see Table I). As the duality preserves the topological properties of the spectrum, by adiabatically modulating G j (t) in For spin-flip X j (t) = σ x j , for kinks X j (t) = τ z j τ z j+1 and for cluster-Ising models X j (t) = µ z j µ x j+1 µ z j+2 . b) Expectation value of the non-driven operator of the model Hamiltonians. For spin flips Y j (t) = σ z j σ z j+1 , for kink model Y j (t) = τ x j and for cluster-Ising model Y j (t) = µ z j µ z j+1 . The initial state is the eigenstate with one excitation per trimer in the lowest band with Chern number C = −1. Parameters for all graphs are N = 9, g 0 = 10J, g 1 = 3J, initial phase φ 0 = 0, and frequency ω = 0.02J.
In order to explore which other types of excitations can be pumped, we perform a π rotation around the x-axis on the Pauli matrices, and apply the duality again. The result is a cluster-Ising Hamiltonian [36][37][38][39] where µ α j are the transformed Pauli matrices. The excitations of this model are cluster-like states (see Table I), which are highly entangled objects relevant for quantum computation [40]. The duality, once again, preserves the spectrum, which allows us to pump cluster states with topological protection agains perturbations. The x-rotation and duality transformation can be repeatedly applied to generate higher-order dual models. This creates a zoo of dual spin models with nontrivial interactions, which host excitations with a wide range of topological phases [41]. Moreover, the duality ensures that the excitations of these models can be pumped as well.
The duality allows us to infer the pumping dynamics of kinks, clusters and higher-order excitations by simply studying the spin-flip Ising model and its mapping to the AA model. As mentioned above, the topological Chern numbers and the spectrum remain invariant under the duality. Based on this, we can study the dynamics of all models within a single framework. After one period T of driving, the excitations of all models move by 3C sites, where C is the Chern number of the relevant band. As an example, the dynamics of expectation values X(t) of the state with one excitation per trimer is shown in Fig. 3a). The dynamics of topological pumping can be detected in the different models by measuring a local observable X j (t) = σ x j for spin flips, a two-point correlator X j (t) = τ z j τ z j+1 for kinks and a three-point correlator X j (t) = µ z j µ x j+1 µ z j+2 for cluster-like excitations. properties transform in a nontrivial way under the duality. We can calculate the states for a single excitation per trimer exactly in the limit g 0 J in Table I at and away from anticrossings. The complexity of the states increases by repeatedly applying the duality and so it does their entanglement, as we show below. For our chosen values (g 0 = 10J, g 1 = 3J), we find an overlap of the analytic states with the numeric results of more than 95%.
The pumping of kinks and clusters is equivalent to pump spin flips. However, kinks and clusters are highly entangled states, whereas spin flips are very close to product states. A natural question that arises is: what is the dynamics of entanglement during topological pumping and how this depends on the character of the excitations? To answer this question we divide the spin chain into two subsystems A and B, perform a partial trace over Aliceρ B = Tr A (ρ) and calculate the von-Neumann entropy S = −Tr B (ρ B logρ B ) [38]. Hereρ is the density matrix of of the total system. We consider two types of partitions: In partition 1 [ Fig. 4a)], Alice is the first trimer, and Bob the rest. In partition 2 [ Fig. 4b)], Alice is the first site of each trimer, and Bob the rest. We find that the entanglement exhibits dramatic changes during the pumping process. In particular, the entanglement of the spin flips increases at the anti-crossings. This result has implications for recent experiments [7,8], which have realized topological pumping of bosonic [7] and fermionic particles [8]. In contrast, the entangled kink and cluster states loose entanglement at the anti-crossings for the case 2 partition, and have nearly unchanged entanglement for case 1. The difference arises because partition 2 extends over all trimers, whereas partition 1 is measuring only a single trimer, highlighting the non-local structure of the quantum state. One important consequence of our results is that both the dynamics and the entanglement are topologically-protected against disorder, that may arise in experiment, as long as it is smaller than the energy gap.
Topological pumping is generally considered with noninteracting particles. A next step in our work is to study the effect of interactions and dualities. It is know that interactions can break topological pumping in some situations [29], while it is still possible in specific cases [34,35]. We find that the lowest band with one excitation per trimer (Chern away from anti-crossing at anti-crossing spin-flip x   TABLE I. Eigenstates with one excitation per trimer in the pumping process of the spin-flip, kink and cluster-Ising models. We show the analytic states at and away from the anti-crossing in the limit g 0 J. States are given in terms of the eigenstates of the z-basis |0 and |1 for kinks and in the x basis for spin-flip and cluster-Ising model. For the cluster-Ising model we define the ground state of cluster model as N i=1 cµ z i,i+1 |000 x , with cµ z i,i+1 being the control phase gate acting on site i, i + 1).
number C = −1) can be pumped even when adding the interaction termĤ flip int = K j σ x j σ x j+1 to the spin-flip Hamilto-nianĤ flip . However, interaction destroys pumping in the other bands. The pumping of that specific band persists as the single excitations of each trimer are highly localized and separated by a large distance of 3 sites. Thus, they couple only weakly to each other and the energy gap is nearly unchanged compared to the non-interacting case. From the duality argument, it immediately follows that pumping is also possible in the dual models, as long as we consider the dual interaction terms:Ĥ kink int = K j τ z j τ z j+2 for kink Hamiltonian and H cluster int = K j µ z j µ y j+1 µ y j+2 µ z j+3 for Cluster-Ising. Interaction has a profound impact on the robustness to disorder. We implement random spatial disorder ∆ j with strength δ in either variable G j (t) = G j (t) + ∆ j or J = J + ∆ j of our Hamiltonians, where ∆ j is randomly sampled between [−δ, δ]. In Fig. 5 we show the fidelity F = | Ψ(0)|Ψ(3T ) | 2 of the pumped state after 3 pump cycles. We observe that without interaction K = 0, the fidelity is reduced above a critical disorder, independent of whether disorder is applied to G j (t) or J. However, with interaction K = J, pumping is much more stable for disorder applied to G j (t). This effect cannot be attributed to a change in the energy gap, as it is nearly unchanged with interaction K. However, we observe that the system is more robust against disorder when the interaction operator commutes with the part of the Hamiltonian that is disordered, i.e., for spin-flip modelĤ flip int commutes with σ x i σ x i+1 , but not with σ z i σ z i+1 . We conjecture that the increased stability arises when the interaction Hamiltonian acts as a stabilizer on the pumped state, and leads to a renormalization of disorder due to interaction [42]. These findings on the stability against disorder can immediately transferred to all dual models via duality transformation.
In summary, we demonstrated topological pumping of a wide range of excitations that are related via duality transformations. An external drive allows us to transport spin-flips, kinks and cluster states via topological pumping. While the expectation values of states are related via a simple duality transformation, the type of states and entanglement changes non-trivially. In particular, we show how highly entangled and delocalized excitations like kinks and cluster states can be transported. Transport is robust against disorder due to topological protection and the interactions can even enhance ro- Without interaction K, fidelity starts to decay when disorder is on the order of the energy gap δ c ≈ 0.7J, independent of which variable is disordered. With interaction K, pumping is more robust for disorder in G j (t), however unchanged for J. Parameters are N = 9, g 1 = 3J, initial phase φ 0 = 0, and angular frequency ω = 0.02J.
bustness against specific types of disorder. Kinks and cluster states are highly entangled while spin-flips have low entanglement. However, for specific partitions the entanglement decreases for kinks and cluster states, while spin-flips become highly entangled. This change in entanglement is part of the transport mechanism and could be used to generate entanglement. The dualities are general for spin systems and could be applied to other fundamental models like the Haldane model [43]. It would be interesting to devise protocols to pump collective excitations with other types of interaction or beyond the duality mapping. Our proposed spin-flip and dual kink model are experimentally realizable in current quantum simulators [44][45][46] that are available in several platforms such as superconducting qubits [47][48][49], trapped ions [50], Rydberg [51] and cold atoms [52][53][54][55]. Cluster states have been realized in photonic systems [56] as well cluster Hamiltonians could be realized with cold atoms [57]. We thank D. G. Angelakis, M. Estarellas, M. Hanks and H. Price for valuable discussions. We thank National Research Foundation Singapore, the Ministry of Education Singapore Academic Research Fund Tier 2 (Grant No. MOE2015-T2-1-101), and the Japanese QLEAP program for support. The Grenoble LANEF framework (ANR-10-LABX-51-01) is acknowledged for its support with mutualized infrastructure. The computational work for this article was partially performed on resources of the National Supercomputing Centre, Singapore (https://www.nscc.sg).