Interacting topological frequency converter

We show that an interacting two-spin model subjected to two circularly polarized drives enables a feasible realization of a correlated topological phase in synthetic dimensions. The topological observable is given by a quantized frequency conversion between the dynamical drives, which is why we coin it interacting topological frequency converter (ITFC). The ITFC is characterized by the interplay of interaction and synthetic dimension. This gives rise to striking topological phenomena that have no counterpart in the non-interacting regime. By calculating the topological phase diagrams as a function of interaction strength, we predict an enhancement of frequency conversion as a direct manifestation of the correlated topological response of the ITFC.

Introduction.-Recent advances in the realization and manipulation of quantum systems far from thermal equilibrium have triggered the quest for identifying and observing topological phenomena in such settings [1][2][3][4]. In particular, it has become clear that periodic driving in time can induce intriguing topological features , numerous of which have been shown to be unique to Floquet systems [28][29][30][31][32][33][34][35][36][37][38]. Many of these novel non-equilibrium phases rely on the fact that a time-periodic driving may formally be viewed as a dimensional extension of the system [39,40]. The archetypal example in this context is provided by the Thouless pump in one spatial dimension, which maps onto an integer quantum Hall scenario upon interpreting time as an extra dimension [41][42][43]. This line of reasoning can be generalized to multi-frequency driving [39,44,45]. There, the harmonics of different drives are viewed as lattice sites along different synthetic spatial dimensions, which allows for the realization of a wide range of topological effects [46][47][48][49][50][51][52]. Along these lines, it has been shown in Refs. [46,50] that driving a qubit with two fields of incommensurate temporal periodicity generates a dynamical analog of a Chern insulator [53,54], where the Hall response translates into a quantized frequency conversion between the external fields.
In this work, we extend the notion of topologically quantized frequency conversion to interacting spin systems. We focus on a minimal model of two interacting spins exposed to two circularly polarized incommensurate drives (see Fig. 1 for an illustration). Despite its simplicity, this setup already offers a striking example of how the interplay of interaction and the aforementioned dynamical dimensional extension can have a profound impact on the resulting topological properties. Most prominently, while in a non-interacting system of two identical spins the topological charge determining the frequency conversion is constrained to be even, in the interacting case also odd integers are allowed. This feature may, in turn, result in an enhancement by interactions of the topological response. For these reasons, we call our setup an interacting topological frequency converter (ITFC). We provide a The spins A/B coupled by spin-spin interaction are subjected to a static magnetic field with amplitude B0 and two circularly polarized drives with frequencies ω1 and ω2. The interaction is controlled by the coupling parameters Ji {i = x, y, z}.
simple interpretation of our results in terms of two-body spin configurations and corroborate the observed topological phase diagram with an explicit calculation of the system's dynamics and the related frequency conversion.
Model.-We analyze the ITFC schematically shown in Fig. 1, in which two spins coupled by spin-spin interaction are exposed to a dynamical magnetic field B( ϕ t ). The corresponding Hamiltonian is given bŷ where we have introduced the vector of Pauli matrices σ α = (σ αx , σ αy , σ αz ) acting on the individual spins α = A/B, and assumed anisotropic Heisenberg interaction with coupling parameters J i {i = x, y, z}. We look at the case of spin 1/2. For simplicity, we have set h = µ B = 1 and g = 2. In addition to a static magnetic field with amplitude B 0 in z−direction, the external field is composed of two circularly polarized drives with timedependent phases ϕ t = (ϕ 1t , ϕ 2t ) = ω t + φ and amplitudes λ 1/2 > 0, where the frequencies and offset-phases are parametrized by ω = (ω 1 , ω 2 ) and φ = (φ 1 , φ 2 ). In the following, we set λ 1/2 = λ for simplicity.
The interaction favors ferromagnetic (J i < 0) or antiferromagnetic (J i > 0) alignment of the spins along the respective quantization axis in the ground state. The interplay of interaction and magnetic field B( ϕ t ) can be conveniently investigated by introducing the total spin S = 1 2 (σ A + σ B ) and the associated eigenstates |ψ s,mz determined by the quantum numbers (s, m z ): where {s = 0, 1} and {m z = −s, . . . , s}. The Hamiltonian of Eq. (1) commutes with the total spin [Ĥ,Ŝ 2 ] = 0, so that the total spin quantum number s is conserved. Thus, the singlet state |ψ 0,0 is an eigenstate of the interacting system with trivial dynamics. For this reason, we restrict ourselves to study the Hilbert subspace with s = 1. In this case, apart from a global constant, the Hamiltonian (1) can be written as: in the basis of triplet states {|ψ 1,1 , |ψ 1,0 , |ψ 1,−1 }. We have introduced the transverse components B ± = B x ± i B y and the effective interaction strengths J x±y = (J x ± J y )/λ and J z = J z /λ − J x+y /2 for ease of notation. The interaction entersĤ T with the parameters J z and J x−y , while the third parameter J x+y only affects the energy E 0,0 = −λ (2 J x+y + J z ) of the decoupled singlet state |ψ 0,0 . Below we focus on the influence of J z setting J x−y = 0 for simplicity. Additional results are presented in the Appendix showing that our main predictions are not qualitatively affected for non-vanishing J x−y . Topological phase diagrams.-For each qubit, the noninteracting part of Eq. (1) is equivalent to a Chern insulator [53,54,56] with mass parameter M = B 0 /λ. Hence, the dynamics of the magnetic field B( ϕ t ) of Eq. (2) can induce non-trivial topology in the individual subspaces. As the phases ϕ 1t and ϕ 2t vary between 0 and 2π, they define a two-dimensional torus, analogously to a 2D Brillouin zone (BZ). Therefore, each of the two eigenstates of the single-spin Hamiltonian can be characterized by a Chern number: ν = ±1 (non-trivial) for |M | ≤ 2 or ν = 0 (trivial) for |M | ≥ 2 [57,58]. In the interacting case, we can diagonalize the projected Hamiltonian in Eq. (3) and determine the Chern number [59][60][61] of the respective eigenstates |Ψ n ( ϕ) {n = 0, 1, 2} according to The resulting topological phase diagrams as a function of mass parameter M and effective interaction strength J z (J x−y = 0) are displayed in Fig. 2. Interactions have two striking effects on the topology of the eigenstates. First, phases with odd Chern numbers C n = ±1, ±3 emerge. This observation is a genuine interaction effect since for J z = 0 the two qubits are independently exposed to the Topological phase transitions caused by Jz lead to inversions between separable and maximally entangled states, which is why it is no longer possible to specify a winding number for the individual constituents. To describe the topology, we consider each separable state that contributes to the linear combination in |ψ1,0 individually according to its topological features. Depending on the number {k = 0, ..., 3} of HSPs [55] with a maximally entangled state |ψ1,0 as ground state, this leads to 2 k combinations for the spin configurations formed by the equally weighted quantum states |↑↓ and |↓↑ at different HSPs. same magnetic field B( ϕ t ), so that the global topological invariant can only change by an even number ∆C n = ±2, ±4. Secondly, a finite interaction strength J z = 0 can induce a non-trivial topology for states that were trivial in the non-interacting regime.
The topological phase transitions are caused by band inversions at high-symmetry points (HSPs) of the analogue of the BZ, accompanied by Dirac gap closings for mass parameters |M | = 2 or M = 0, and effective interaction strengths J z shown in Tab. I. Notably, gap closings and reopenings at (0, π) and (π, 0) occur simultaneously, so that the change in topology is twice as large as in (0, 0) or (π, π). At HSPs, the Hamiltonian (1) commutes withŜ z , makingĤ T diagonal in the |ψ s,mz basis: Gap closings caused by J z (M = const.) lead to inversions between the ferromagnetic triplet states |ψ 1,1 , |ψ 1,−1 and the anti-ferromagnetic triplet state |ψ 1,0 . The corresponding topological phases are bounded by straight lines in Fig. 2. Topological phase transitions for mass parameters |M | = 2 or M = 0, however, only involve the ferromagnetic states |ψ 1,1 and |ψ 1,−1 , which is why the polarization of both spins reverses upon band inversion at the respective HSP. The global topological invariant then changes by an even number ∆C n = ±2, ±4. Conversely, a band inversion containing the anti-ferromagnetic state |ψ 1,0 is the reason for the generation of the odd topological phases with C n = ±1, ±3.
Interpretation of the results.-From the previous considerations, it seems useful to investigate the relationship between the spin configuration of the eigenstates and the respective topological invariants. In our case, it is sufficient to focus on HSPs, which completely determine the topology of the system [62]. In Fig. 3, the spin configurations at HSPs are schematically shown for an increasing interaction strength J z ≥ 0 and fixed mass parameter M = 1.2 [compare also to Fig. 2]. For J z = 0, the ground state |Ψ 0 ( ϕ) is always a separable state, and the global topological features can be interpreted by examining the qubits separately: each one has a well defined single-particle Chern number corresponding to the winding number in the single-qubit Bloch sphere as ϕ varies. The global Chern number C n is then simply the sum of the two single-particle ones. Since the two qubits at (0, 0) are polarized in opposite directions with respect to the other HSPs, each spin winds once around its Bloch sphere resulting in the combined Chern number C 0 = 2.
By increasing the interaction strength to 0.8 < J z < 1.2, the separable ground state at (0, 0) is substituted by the maximally entangled ground state |ψ 1,0 . Thus, since it is no longer possible to specify a winding number for the individual constituents, we exploit the following idea: We examine the topological features of each separable state that contributes to the linear combination in |ψ 1,0 individually. For instance, for 0.8 < J z < 1.2 the topology for the equally weighted quantum states (a) |↑↓ = 1. In both cases (a) and (b), the single-particle Chern numbers add up to a global Chern number C 0 = 1. This picture provides an intuitive explanation for the odd topological phase: each band inversion between an anti-ferromagnetic (maximally entangled) state and a ferromagnetic (separable) state causes a change in the global Chern number by ∆C n = ±1.
For 1.2 < J z < 3.2, the separable ground states at (0, π), (π, 0) are additionally replaced by the maximally entangled ground states |ψ 1,0 . By applying the previous picture, we have to consider four [55] combinations for the spin configurations formed by the equally weighted quantum states |↑↓ and |↓↑ at different HSPs:
Observable consequences.-The dynamics of the system can also be described within a two-dimensional frequency lattice with built-in "electric" field [46,50,64,65], where the Hall response in the non-trivial regime is given by a transverse current in this frequency domain. For a system initially prepared in an eigenstate ofĤ T , the topological features of Fig. 2 then translate into a quantized frequency conversion between the circularly polarized drives [46,50]. As first realized in Ref. [46], energy is pumped between the two fields at a time-averaged rate P 12 n = −P 21 n = C n 2 π ω 1 ω 2 (5) when the system is confined to an energy level of Chern number C n . A necessary condition for the observation of a quantized rate, besides quasi-adiabaticity [63], is that the two frequencies are incommensurate in such a way that the dynamics effectively samples the whole BZ. Notably, the enhancement of the frequency conversion for C n = ±3 is a direct consequence of the correlated topological response. At the same time, the topological origin makes the pumping of energy highly robust against variations in the intrinsic parameters of the ITFC (not explicitly shown here). We investigate the topological energy pumping by numerically solving the Schrödinger equation associated toĤ [66]. We then compute the expectation value of the "current operator"Ĵ = ∇ ϕĤ , which is associated to the total energy transfer rate as it can be seen from the Heisenberg equation: d dt Ĥ = ∂ tĤ = ω Ĵ . If the initial state is the eigenstate |Ψ n ( ϕ(t = 0)) ,  Fig. 2. This applies as long as the system shows quasi-adiabaticity [63], while otherwise the topological response breaks down (gray regimes).
then the average pumping rate P 12 n can be extrapolated through linear regression of the energy transfer E i (t) = ω i t 0 dt Ĵ i (t ) [46]. We have chosen the frequencies ω 1 = 0.1 λ and ω 2 = γ ω 1 , where γ = 1 2 (1 + √ 5) is the golden ratio. The offset-phases are φ 1 = π/10, φ 2 = 0. In Fig. 4, the extrapolated pumping rate of the ground state is shown as a function of J z for parameters M = 1.2 and M = 2.2. A quantized frequency conversion occurs in an excellent agreement (white regimes) with the topological phase diagrams of Fig. 2. This applies as long as the ITFC shows quasi-adiabaticity. Otherwise, the topological response is suppressed (gray regimes), and perfect quantization breaks down.
Conclusion.-We have demonstrated that combining two-body interaction with the notion of dynamically induced synthetic dimensions gives rise to remarkable topological phenomena that go beyond a non-interacting implementation. In particular, we predict that an interacting topological frequency converter (ITFC) realizes correlated topological phases with odd Chern numbers C n = ±1, ±3. The topological response of the ITFC is given by a quantized frequency conversion between the dynamical drives, and can be enhanced for C n = ±3, as compared to its non-interacting counterpart. This amplification is found to be more pronounced as the number of interacting spins increases, as we explicitly confirmed for the example of three spins where Chern numbers up to C n = ±5 are observed. An experimental realization of the ITFC proposed in our present work might be implemented in a singlet-triplet qubit exposed to circularly polarized driving fields [67].
This work was supported by the DFG (Grants No. SPP1666 and No. SFB1170 "ToCoTronics"), the Würzburg-Dresden Cluster of Excellence ct.qmat (EXC2147, Project No. 39085490), and the Elitenetzwerk Bayern Graduate School on "Topological Insulators". We acknowledge Caterina De Franco for technical advices. In the following, we investigate the influence of a non-vanishing interaction strength J x−y on the topological phase diagrams of Fig. 2 of the main text. For a finite J x−y = 0, the Hamiltonian (1) does not commute withŜ z at HSPs of the BZ, leading to a coupling of the ferromagnetic triplet states |ψ 1,1 , |ψ 1,−1 . A band inversion at HSPs then occurs between the anti-ferromagnetic triplet state |ψ 1,0 and a superposition of ferromagnetic triplet states |ψ 1,1 and |ψ 1,−1 . Accordingly, the topology of the system can no longer be determined by considering the spin configurations at HSPs only, but has to be calculated by the full integral of Eq. (4) of the main text. Thus, we have to diagonalize the projected Hamiltonian (3) for a finite J x−y = 0, and calculate the Chern number of the respective eigenstates |Ψ n ( ϕ) {n = 0, 1, 2} by Eq. (4) of the main text.
The resulting topological phase diagrams as a function of mass parameter M and interaction strength J z are displayed in Fig. 5 for different interaction strengths J x−y = 0. For a finite J x−y , the topological phases are no longer bounded by straight lines as in Fig. 2 of the main text. Especially, phase transitions that corresponded to the horizontal lines at |M | = 2 and M = 0 now show dispersive character. Although the critical values of J z and M are strongly modified by the coupling of the ferromagnetic triplet states, the corresponding topological invariants of the phases do not vary from those of the main text. In particular, the striking topological phenomena with odd Chern numbers C n = ±1, ±3 are still present and are influenced at most only by a shift of the delimited lines of the phase transitions. Accordingly, the phenomenological results of the main text are not qualitatively affected by a non-vanishing interaction strength J x−y .  Fig. 2 of the main text, which is why the topological phase transitions are quantitatively modified by a finite Jx−y = 0. The topological invariants of the phases, however, do not differ from those of the main text. In particular, the striking topological phenomena with odd Chern numbers Cn = ±1, ±3 are still present, and only influenced by a shift of the boundaries. The phenomenological results of the main text are therefore not qualitatively influenced by a finite interaction strength Jx−y.