Introduction

Exploring and controlling novel phases of matter has been a prominent subject in physics. Apart from variation of an external parameter such as temperature, pressure, or magnetic field, a synthetic gauge field has been proven to be a powerful tool in constructing and manipulating novel phases of matter, especially for neutral particle1,2,3. In particular, the non-Abelian gauge field4,5,6,7,8,9, manifest in a wide range of distinct fields in physics such as high-energy physics, condensed-matter physics and electromagnetism, offering an effective way to synthesize exotic topological phases10,11,12,13,14,15,16,17,18,19,20. Such a kind of gauge field is associated with the concept of the non-Abelian group in mathematics, as they hold non-commutative (non-Abelian) matrix-form components, which provides a framework for describing the statistics of spinful particles.

It has been growing interest in synthesizing non-Abelian gauge fields in diverse platforms. Compared to continuum systems21,22,23,24,25,26, a non-Abelian gauge field in a lattice model has only been demonstrated in a few platforms, such as ultracold atoms27,28,29,30, electrical circuits31,32 and photonic systems33. However, these works focus on the realization of an SU(2) gauge field, while a non-Abelian gauge field does not necessarily belong to the SU(2) group. For example, the complex generalization of the SU(2) group [i.e., the SL\((2,{\mathbb{C}})\) group] can also fulfill the requirement of non-commutativity. This leads to a question of whether an SL\((2,{\mathbb{C}})\) gauge field can result in more fruitful topological phase transition phenomena and how such a gauge field can be constructed in experiments. However, for the existing scheme, the spin-flipped hopping terms for different spins are usually not independent, namely they are generated by the same light fields/elements27,28,29,30,31,32,33, hence lacking flexibility. This poses challenges in realizing the SL\((2,{\mathbb{C}})\) gauge field, which requires a flexible tailoring to achieve independent spin-flipped hopping for different spins. Nevertheless, recent advance in constructing non-Abelian lattice models in photonic synthetic frequency dimensions33, shows great potential in tackling such a task and exploring the underlying physics of an SL\((2,{\mathbb{C}})\) gauge field.

Here, we generalize the SU(2) non-Abelian gauge field (linear superposition of Pauli matrices with real coefficients) to an SL\((2,{\mathbb{C}})\) one featuring non-Hermitian matrix-form components (complex coefficients), which can be realized by introducing imbalanced spin-flipped hoppings to the Hamiltonian model in synthetic frequency dimension34,35,36,37,38. Such kind of gauge field results in various kinds of transitions of Dirac semimetals39, including semimetal-insulator transition, different Dirac-type semimetal transition, annihilation and generation of Dirac points. Furthermore, the imbalanced spin-flipped hopping leads to the intrinsic competition between two hopping mechanisms that results in the transition of the orientation of the edge state projection on the pseudospins, changing from nearly constant to a sinusoidal variation during this process. Moreover, we find that the nontrivial \({{\mathbb{Z}}}_{2}\) topological phase can be protected by a more generalized symmetry under the SL(2, \({\mathbb{C}}\)) gauge field. Our work offers the first experimentally feasible proposal to study the underlying physics of SL\((2,{\mathbb{C}})\) gauge fields, which enriches the context of topological physics.

Results

Model

We consider the model composed of a one-dimensional ring resonator array including resonant site rings (green) and off-resonant link rings (yellow) in Fig. 1a. Each site ring supports resonant modes at frequencies ωm = ω0 + mΩ, where ω0 is the reference frequency and Ω is the frequency spacing. A pair of two coupled site rings (labeled by A and B) can form a photonic molecule40,41, where each resonant mode at ωm hybridizes into symmetric and antisymmetric supermodes at ωm,± = ωm ± γ. Here, γ is the coupling strength between the two rings. Electro-optic modulators (EOMs) are placed in ring A (or B) in the n-th photonic molecule at modulation form \({J}^{{{{\rm{A/B}}}}}(t)=\pm 2{\kappa }_{1,n}\cos ({\mathit{\varOmega}} -2\gamma )t+2{\kappa }_{2,n}\cos {\mathit{\varOmega}} t\pm 2{\kappa }_{3,n}\cos ({\mathit{\varOmega}}+2\gamma )t\) where κ1,n, κ2,n, and κ3,n are spatially-dependent modulation strengths. The dynamical modulation on the light field by the EOMs produces sidebands from the original frequency, determined by the modulation frequencies of the modulation signal JA/B(t), and hence connects different frequency modes in the same photonic molecules with frequency spacing Ω − 2γ, Ω, and Ω + 2γ, respectively (details are provided in the Supplemental Note 1).

Fig. 1: The proposed scheme for constructing a two-dimensional non-Abelian lattice gauge field by a photonic molecule array.
Fig. 1: The proposed scheme for constructing a two-dimensional non-Abelian lattice gauge field by a photonic molecule array.
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a Schematic view of the system composed of coupled ring resonators being dynamically modulated by EOMs. The hybridization of the modes in two individual rings causes a frequency splitting regarding to the original resonant frequency ωm. b Such a system can be mapped to a lattice supporting the two-dimensional Hamiltonian.

We assume the spatial spacing between nearby photonic molecules as d, where off-resonant link rings are added to induce spatial coupling g42,43. Note that we introduce a small vertical shift on the center of link rings with respect to the center of site rings (see Fig. 1a), and therefore the light propagating from left to right accumulates a different phase from that of light propagating from right to left (see Supplemental Note 1). The additional propagation phases are expressed by αm,± ∝ ϕm ± φ for symmetric and antisymmetric supermodes, with the effective  +(−)αm,± for the connection from left (right) to right (left) photonic molecules with a multiple of 2π44. Here, (ϕm ± φ) ∝ (ωm ± γ) originates from the propagation phase accumulation for all supermodes, where we choose ϕm to be multiples of 2π and hence α ≡ αm,± = ±φ for each pair of supermodes.

By further assuming all coupling strengths are relatively small and applying the rotating wave approximation, we obtain the Hamiltonian of the lattice (see Fig. 1b) with n (m) being the space (frequency) index in the synthetic space (detailed deviation of the Hamiltonian is provided in Supplemental Note 1):

$$H= {\sum}_{n,m}\left(\begin{array}{c}{c}_{m+1,n,+}^{{{\dagger}} }{c}_{m+1,n,-}^{{{\dagger}} }\end{array}\right)\left(\begin{array}{cc}{\kappa }_{2,n} & {\kappa }_{3,n}\\ {\kappa }_{1,n} & {\kappa }_{2,n}\end{array}\right)\left(\begin{array}{c}{c}_{m,n,+}\\ {c}_{m,n,-}\end{array}\right)\\ +g\left(\begin{array}{c}{c}_{m,n+1,+}^{{{\dagger}} }{c}_{m,n+1,-}^{{{\dagger}} }\end{array}\right)\left(\begin{array}{cc}{e}^{i\varphi } & 0\\ 0 & {e}^{-i\varphi }\end{array}\right)\left(\begin{array}{c}{c}_{m,n,+}\\ {c}_{m,n,-}\end{array}\right)+H.c.,$$
(1)

where \({c}_{m,n, \pm }^{{{\dagger}} }\) (cm,n,±) is the creation (annihilation) operator of the symmetric or antisymmetric supermode at ωm,± for the n-th photonic molecule. Each pair of the symmetric and antisymmetric supermodes can be treated as a pair of pseudospins. The Hamiltonian can be rewritten by

$$H\equiv {H}_{0}+{H}_{1},$$
(2)
$${H}_{0}= {\sum}_{n,m}\left(\begin{array}{cc}{c}_{m+1,n,+}^{{{\dagger}} }&{c}_{m+1,n,-}^{{{\dagger}} }\end{array}\right)(\cos n\theta {\sigma }_{0}+i\sin n\theta {\sigma }_{2})\left(\begin{array}{c}{c}_{m,n,+}\\ {c}_{m,n,-}\end{array}\right)\\ +\left(\begin{array}{cc}{c}_{m,n+1,+}^{{{\dagger}} }&{c}_{m,n+1,-}^{{{\dagger}} }\end{array}\right)(\cos \varphi {\sigma }_{0}+i\sin \varphi {\sigma }_{3})\left(\begin{array}{c}{c}_{m,n,+}\\ {c}_{m,n,-}\end{array}\right)+H.c.,$$
(3)
$${H}_{1}=\left(\begin{array}{cc}{c}_{m+1,n,+}^{{{\dagger}} }&{c}_{m+1,n,-}^{{{\dagger}} }\end{array}\right)a\sin n\theta {\sigma }_{1}\left(\begin{array}{c}{c}_{m,n,+}\\ {c}_{m,n,-}\end{array}\right)+H.c.,$$
(4)

where σ0, σ1, σ2, and σ3 are Pauli matrices, \({\kappa }_{1,n}\equiv (a-1)\sin n\theta\), \({\kappa }_{2,n}\equiv \cos n\theta\), \({\kappa }_{3,n}\equiv (a+1)\sin n\theta\), and g = 1 (i.e., the Hamiltonian is dimensionless and all coupling strengths can be scaled versus g, and a ∈ R, θ ∈ [0, 2π]). Importantly, the parameter a represents the imbalance between the pseudospin-dependent spin-flipped hopping term, namely, \(| {\kappa }_{1,n}| -| {\kappa }_{3,n}|=2a\sin n\theta\) (for 0 ⩽ a ⩽ 1), which modifies the hopping along the frequency dimension from an SU(2) type of matrix to an SL\((2,{\mathbb{C}})\) one, and correspondingly marks the appearance of an SL\((2,{\mathbb{C}})\) gauge potential. The corresponding non-Abelian gauge field is written by

$${{{\bf{A}}}}\equiv \left\{\begin{array}{l}({A}_{{{{\rm{f}}}},\uparrow },{A}_{{{{\rm{x}}}}})=\left(\eta (\theta ){\sigma }_{2}-ia\eta (\theta ){\sigma }_{1},\varphi {\sigma }_{3}\right)\\ ({A}_{{{{\rm{f}}}},\downarrow },{A}_{{{{\rm{x}}}}})=\left(\eta (\theta ){\sigma }_{2}+ia\eta (\theta ){\sigma }_{1},\varphi {\sigma }_{3}\right)\end{array}\right.,$$
(5)

where Af,↑ (Af,↓) is the gauge field component along the positive (negative) direction along the frequency axis, \(\eta (\theta )=\frac{1}{2\sqrt{{a}^{2}-1}}\log \frac{\cos n\theta+\beta }{\cos n\theta -\beta }\), and \(\beta=\sqrt{({a}^{2}-1){\sin }^{2}n\theta }\). One can see the component Af,↑ (Af,↓) gives the hopping that mixes the pseudospins along frequency dimension, while the component Ax provides an opposite hopping phase  ± φ along spatial dimension. In particular, A does not belong to an SU(2) gauge potential when a ≠ 0 due to the appearance of the non-Hermitian matrix  − iaη(θ)σ1 in Af,↑ (Af,↓), instead, it refers to an SL\((2,{\mathbb{C}})\) gauge potential, which is a complex generalization of the SU(2) one (see Supplemental Note 2). Note that the asymmetric gauge field component Af,↑ (Af,↓) preserves the Hermiticity of the Hamiltonian.

Annihilation of Dirac points and rotation of pseudospin

We investigate how the bandstructure changes when a ≠ 0, for the Hamiltonian under the non-Abelian magnetic flux (θ, φ) = (2π/3, π/2) (so the lattice has the period of 2π/θ = 3 along the spatial axis). In Fig. 2a, b, we show the band structures of H with a = 0 and a = 1 (see “Methods” for the Bloch Hamiltonian in the reciprocal space), where kf (kx) is the quasi-momentum reciprocal to the frequency (space) axis and ε is the quasienergy. The six bands are separated, labeled from 1–6, which reflects the non-Abelian nature of the gauge field (see Supplemental Note 3). When a = 0, there are four band-crossing points between band 4 and band 5 in Fig. 2a, which refer to type-I Dirac points39. However, when a = 1, there is only a trivial gap between band 4 and band 5 in Fig. 2b, indicating a semimetal-insulator transition. Different Dirac-type semimetal transition and topological-normal insulator transition can also be presented if one chooses different flux (θ, φ) (see Supplemental Note 4).

Fig. 2: The annihilation of the Dirac points with the changes of the localization features at the edge and the variation of ESPP.
Fig. 2: The annihilation of the Dirac points with the changes of the localization features at the edge and the variation of ESPP.
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Bandstrucutre of the model H in Eq. (2) by varying a from (a) a = 0 to (b) a = 1. Red numbers in a denote the band labels. c Projected bandstructure with different strengths of the spin-flipped hopping a. Red dashed lines denote the position of the Dirac points, which correspond to kfΩ = 0.8π, 1.2π and kfΩ = π, respectively. Blue dashed lines denote the corresponding momenta in Fig. 3. d The quantity Λ for the localization of the edge states. The colors of the lines are in accordance with the colored bands in (c). e, f ESPP \(\left\langle {\sigma }_{i}\right\rangle\) of the blue band and green band in (c), respectively.

Properties of edge states with a spatial open boundary condition for N = 31 sites are investigated. We define a quantity \({\mathit{\varLambda}}=| {v}_{{n}_{{{{\rm{b}}}}}}{| }^{2}/{\sum }_{n}| {v}_{n}{| }^{2}\) to characterize the localization amplitude of an eigenstate v on the edge, where nb = 1 or 31 labels one edge. Furthermore, we define the edge-state pseudospin projection (ESPP) as \(\left\langle {\sigma }_{i}\right\rangle=\left\langle {v}_{{n}_{{{{\rm{b}}}}}}\left\vert {\sigma }_{i}\right\vert {v}_{{n}_{{{{\rm{b}}}}}}\right\rangle\), which partially reflects the changes in the spin-orbit coupling of the Hamiltonian H.

When a = 0, the Dirac points in Fig. 2a are located at kfΩ = 0.8π and 1.2π. There are four bands that support large localization Λ for a range of kf, indicating the corresponding edge localization (see Fig. 2d), where two of them (labeled in blue and red) are merged into the upper bulk bands for kf out of two Dirac points and the other two (labeled in green and yellow) belong to edge states. Two dips in the curves of Λ appear in the vicinity of the Dirac points as bulk bands close into the degeneracy at the Dirac points, so edge modes do not hold the localization feature at the two dips (see Supplemental Note 5). The distribution of the pseudospin for the upper blue band and lower green band is given in Fig. 2e, f, respectively. The upper red band (bottom yellow band) has the opposite distribution of the ESPP \(\left\langle {\sigma }_{i}\right\rangle\) over the entire kf as the one for the upper blue band (bottom green band). The ESPP \(\left\langle {\sigma }_{i}\right\rangle\) for these two bands is nearly a constant with respect to kf, so the ESPP is pinned at the y-axis in the Bloch sphere with kf varying, except for a sudden jump to the z-axis when kfΩ = 0 and π due to the vanishing of the spin-orbit coupling term σ2kf (see Supplemental Note 6). As a increases, the Dirac points move closer to each other, then merge at about a = 0.55, and finally vanish due to their opposite Berry curvature (see Supplemental Note 7), resulting in gap opening, as illustrated in Fig. 2c. The additional spin-flipped hopping (i.e., the term with a ≠ 0) also causes the hybridization between green and yellow bands and the hybridization between blue and red bands (see Supplemental Note 6), where the former hybridization results in the monomorphic edge band and the latter one pushes one of the resulting band into the upper bulk (so the red band is not shown for a ≠ 0). One can see, at a = 0.55 (Fig. 2c), the Dirac points get close to each other near kfΩ = π, and Λ of the edge state is nearly 0 in the vicinity of kfΩ = π. When a ≠ 0 and such hybridization occurs, the ESPP \(\left\langle {\sigma }_{i}\right\rangle\) at the vicinity of kfΩ = 0 and π is modified, and so the ESPP is rotated in the xy-plane of the Bloch sphere with a non-constant velocity by varying kf. As a further increase to 1, which marks the largest imbalance between the spin-flipped hopping terms κ1,n = 0, and \({\kappa }_{3,n}=2\sin n\theta\) the Dirac points merge and the gap opens with an edge state showing the localization feature throughout the entire kf. Moreover, the ESPP \(\left\langle {\sigma }_{i}\right\rangle\) of the two bands becomes a sinusoidal form in Fig. 2d–f, corresponding to a rotation of the ESPP with a constant velocity. We find a similar change of the ESPP under different choices of (θ, φ) when a varies from 0 to 1 (see Supplemental Note 8).

The additional spin-flipped hopping term with a ≠ 0 hybridizes the bands and consequently modifies the pseudospin of the eigenstates regardless of the non-Abelian flux (i.e., the values of θ and φ). To further investigate such a pseudospin rotation phenomenon, we perform the numerical simulation of the evolution with a being adiabatically decreased from 1 to 0 in Fig. 3, with the choice of N = 19 (details of the simulation is given in Supplemental Note 9).

Fig. 3: Adiabatic evolution of ESPP and the intensity distribution for different states.
Fig. 3: Adiabatic evolution of ESPP and the intensity distribution for different states.
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a–d Temporal evolutions with a tuned from 1 to 0 adiabatically of the pseudospin \(\left\langle {\sigma }_{i}\right\rangle\) of the states with different kf. Violet regions show the time interval with a being kept as 1 to excite the desired eigenstate. e–h Temporal evolution of the total intensity over all frequency modes versus each spatial site n with different choices of kf.

We choose four different states on the green bands with kfΩ = π/3, π/2, 2π/3, and 20π/21, tuned from a = 1 to a = 0 with the bandstructures given in Fig. 2c. When 0 ⩽ t ⩽ 50g−1, we keep a = 1 to excite the desired eigenstate. After t = 50g−1, a is linearly changed from 1 to 0. One can find that even though the four states start with different pseudospins, they all converge to the one with σ2 = 1 at t = 200g−1 (see Fig. 3a–d), which indicates how the additional spin-flipped hopping term gradually rotates the pseudospin of the eigenstate. We also show the evolution of the spatial intensity of all frequency modes on each spatial site in Fig. 3e–h. Despite the rotations in pseudospin, the spatial intensities are almost unchanged and localized at the edge in Fig. 3e–g. However, for the case of kfΩ = 20π/21 in Fig. 3h, the state exhibits a transition starting with an edge state, then to a bulk state, and finally becoming an edge state located at the other side. Such different dynamics occur due to the fact of different occupation of choices of kfΩ = π/3, π/2, 2π/3 and the choice of kfΩ = 20π/21 appearing on different sides of the Dirac point at kfΩ = 0.8π while one tunes a from 1 to 0 (see Fig. 2c). Therefore, when kf crosses the Dirac point in Fig. 2c, the edge state crosses through the bulk towards the other edge adiabatically. Note that if the evolution time is too short, the adiabatic approximation cannot be maintained, resulting in the inefficient transition to the target eigenstates (see Supplemental Note 9).

Phase transition in the (a, b) plane

One sees that there are two kinds of spin-flipped hopping terms \(i\sin n\theta {\sigma }_{2}\), and \(a\sin n\theta {\sigma }_{1}\) in H, while we only tune the lateral one in the above cases. The independent coupling strengths κ1,n and κ3,n allow us to make the other one also tunable by manipulating \({\kappa }_{1,n}\equiv (a-b)\sin n\theta\), \({\kappa }_{3,n}\equiv (a+b)\sin n\theta\) where b is a real number, and hence the Hamiltonian H is extended to a more general one \(H^{\prime}\) with a modified SL\((2,{\mathbb{C}})\) gauge field \({{{\bf{A}}}}^{\prime}\) (see Supplemental Note 2).

Different Dirac point transitions between the fourth band and fifth band through changing both parameters a and b are illustrated by the phase diagram in Fig. 4. One see that there are always four type-I Dirac points between the fourth and fifth band under an SU(2) gauge field (line a = 0). While for the SL(2, \({\mathbb{C}}\)) gauge field [the (a, b) plane], various transitions between six different phases are involved, which indicates the intriguing opportunities in topological physics with SL\((2,{\mathbb{C}})\) gauge field.

Fig. 4: Various phase transitions in the (a, b) plane.
Fig. 4: Various phase transitions in the (a, b) plane.
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Phase diagram under different choices of the parameters a and b. Bandstructures of the model \(H^{\prime}\) for different (a, b) in spin-flipped hopping terms with (θ, φ) = (2π/3, π/2).

As an example, we show the phase transition from type-I to type-II Dirac points with a = b fixed, where only the spin-flipped hopping (κ3,n) between cm,n,− and cm+1,n,+ is allowed, while the other one between cm,n,+ and cm+1,n,− is prohibited (κ1,n = 0). For a = 0, the model degrades to the Abelian gauge-field case without the spin-flipped hopping terms, and two degenerated edge states with opposite ESPP σ3 = ±1 link the Dirac points (see Fig. 5a). As a slightly increases to 0.1, the corresponding gauge field becomes non-Abelian, and the degeneracy of the two bands is lifted. In contrast to the cases we study in Fig. 2, the ESPP of the bands changes abruptly to a sinusoidal form, as shown in Fig. 5c, d. If a further increase to 0.5, one can find that both the position and the tilt of Dirac cones are changed, which gives the type-II Dirac points (see Fig. 5a and Supplemental Note 10 for the bandstructures). The ESPP for a = b = 0.5 remains the same as the ones for a = b = 0.1, which implies the sinusoidal varying ESPPs are the result of the balance between the two kinds of hopping mechanisms associated with a and b in the spin-flipped hopping terms (see “Methods”). To get a better insight, we consider the reciprocal space. The term \(a\sin n\theta {\sigma }_{1}\propto \cos {k}_{{{{\rm{f}}}}}\) (or the other term \(ib\sin n\theta {\sigma }_{2}\propto \sin {k}_{{{{\rm{f}}}}}\)) with corresponding eigenvectors same as σ1 (or σ2), so the effect of the spin-flipped hopping becomes strongest at kfΩ = 0, π (or kfΩ = 0.5π, 1.5π) and vanishes at kfΩ = 0.5π, 1.5π (or kfΩ = 0, π). As a consequence of the competition between these two mechanisms from spin-flipped hopping terms, the orientation of the ESPP rotates in the xy-plane as kf varies, which implies one can possibly have a kf-dependent pseudospin beyond sinusoidal variation by a delicate design of the spin-flipped hopping terms.

Fig. 5: Transition from type-I Dirac points to type-II Dirac points with the changes with the localization features at the edge and the variation of ESPP.
Fig. 5: Transition from type-I Dirac points to type-II Dirac points with the changes with the localization features at the edge and the variation of ESPP.
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a Projected bandstructure with different strengths of the spin-flipped hopping a and b. Red dashed lines denote the position of the Dirac points. Inserted figures show the bandstructure at the vicinity of the Dirac points with the periodic boundary condition taken in both space and frequency dimensions. b The quantity Λ for the localization of the edge states. The colors of lines are in accordance with the colored bands in (a). c, d ESPP \(\left\langle {\sigma }_{i}\right\rangle\) of the blue band and green band in (a), respectively.

Difference between SL(2, \({\mathbb{C}}\)) and SU(2) gauge fields

To further illustrate the difference between the SL(2, \({\mathbb{C}}\)) gauge field and the SU(2) gauge field, we consider an Abelian example with (θ, φ) = (2π/3, π), where the effect of the non-Abelian gauge field is excluded, and the phase transition process is fully attributed to an SL(2, \({\mathbb{C}}\)) gauge field. The energy spectrum of the Hamiltonian H is plotted in Fig. 6a with a varying from 0 to 1. One sees that there is no gap closing during this process, which usually denotes the absence of a topological phase transition. However, the \({{\mathbb{Z}}}_{2}\) topological invariant ν is changed from 1 to 0 (see Supplemental Note 8).

Fig. 6: Topological phase transition under SL(2, \({\mathbb{C}}\)) gauge fields.
Fig. 6: Topological phase transition under SL(2, $${\mathbb{C}}$$ C ) gauge fields.
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a Energy spectrum of the Hamiltonian Hk. b, c Bandstructures of the Hamiltonian Hk with (a, b) = (0, 1) and (a, b) = (1, 1). d–f Phase diagrams in the (ρa, ρb) plane for the model with different choices of ∣a∣ and ∣b∣, where the yellow lines denote the non-trivial phase with a \({{\mathbb{Z}}}_{2}\) topological invariant ν = 1, while the blue regions correspond to a topological trivial phase with ν = 0. Red dots denote the values of ρa and ρb when four-fold band-crossing points appear. g–i The corresponding band structures with ∣a∣ = ∣b∣ = 1 and a different choice of ρa and ρb.

Typical bandstructures are given in Fig. 6a, b with (a, b) = (0, 1) (topological nontrivial) and (a, b) = (1, 1) (topological trivial). Moreover, we find that such a phase transition occurs immediately once a ≠ 0, indicating that the symmetry which protects the nontrivial topological phase is broken by the introduction of the SL(2,\({\mathbb{C}}\)) gauge field. Note that for an SU(2) gauge field, the hopping term U is an SU(2) matrix, and the spin-flipped hopping terms satisfy an anti-conjugate relation \({U}_{12}=-{U}_{21}^{*}\)45, while the variation of the parameter a breaks this condition. To further investigate such a phase transition process, we now define a and b as two complex numbers \(a=| a| {e}^{i{\rho }_{a}}\) \(b=| b| {e}^{i{\rho }_{b}}\). The phase diagram for ∣a∣ = ∣b∣ = 1 is plotted in Fig. 6d. The blue region (topological trivial phase with ν = 0) covers most of the area in the phase diagram, and the nontrivial phase with ν = 1 occurs at ρa = ρb ± 0.5π and ρa = ρb ± 1.5π, denoted by the yellow lines. Similar phase diagrams are also found for ∣a∣ ≠ ∣b∣ as shown in Fig. 6e, f. Moreover, in the (ρa, ρb) plane, the corresponding bandstructure always shows an insulator phase except for (ρa, ρb) = (0, ± 0.5π) and (ρa, ρb) = (π, ± 0.5π) denoted by the red dots, where the gaps close and four-fold band-crossing points appear (e.g., Fig. 6g). Typical bandstructures for the topological nontrivial phase and the topological trivial phase are given in Fig. 6h, i with (ρa, ρb) = (0.1π, 0.6π) and (ρa, ρb) = (0, 0.6π). One can rewrite the condition for the topological nontrivial phase as ∣κ1,n∣ = ∣κ3,n∣ or equivalently ∣U12∣ = ∣U21∣. In other words, the symmetry that protects the nontrivial topological phase under an SU(2) gauge field is broken by the introduction of an SL(2, \({\mathbb{C}}\)) gauge field, but it can be retrieved in a more generalized form under an SL(2, \({\mathbb{C}}\)) gauge field.

Discussion

Our proposal can be feasible for experimental demonstrations. The fiber optic system could be a potential candidate as the combination of photonic molecules and synthetic frequency dimension46,47,48 have already been demonstrated in experiments, which provide the building block of our theoretical proposal. Specifically, for a fiber-based experimental setup with a cavity free spectral range  ~10MHz and cavity length  ~10m, the required amplitude of the voltage signal sent to the EOMs is  ~1V46,48. The quality factor of the cavities ideally desires an order of magnitude  ~109 to avoid the over-broadening of bands, which is challenging, but can be potentially doable with the development of state-of-art photonic technologies48,49 (see Supplemental Note 11). A precise control on the modulation depth and frequency of the EOMs is required to minimize the distortion of the measured bandstructure (see Supplemental Note 12). The disorder in the coupling between the rings can lead to the inefficient excitation of the edge state in experimental observation (see Supplemental Note 13). On the other hand, in the field of integrated photonics, the photonic molecule41,50 and synthetic frequency dimension51,52,53 have been individually experimentally demonstrated for on-chip ring resonators. Moreover, a recent experiment54 also shows the possibility to introduce synthetic frequency dimension in a one-dimensional ring resonator array. All these state-of-art nanophotonic technologies provide the potential to realize our proposal in the near future.

In summary, we extend SU(2) non-Abelian gauge fields to the SL\((2,{\mathbb{C}})\) regime, and give a theoretical proposal for simulating topological materials under such a gauge field. The underlying physics is unveiled in two aspects: various Dirac phase transitions beyond the conventional SU(2) gauge field and the transition of the orientation of the ESPP. Moreover, the extension to SL(2, \({\mathbb{C}}\)) gauge field unveils a more generalized symmetry that protects the nontrivial \({{\mathbb{Z}}}_{2}\) topological phase. We expect the proposed system to be a unique and versatile platform for studying topological physics with non-Abelian gauge fields. Future studies may include the employment of the non-Abelian topological charge55,56, which describes the topology of multiband systems, offering a useful tool to elucidate the nature of the phase transition discussed. The proposed system may also find potential applications on topological insulator lasers in synthetic space-frequency dimensions57,58 from pseudospin manipulations via introducing an SL(\(2,{\mathbb{C}}\)) gauge field, which provides an extra degree of freedom to modify the spectrotemporal shape of the output pulses.

Methods

Bloch Hamiltonian in the reciprocal space

We assume θ satisfies θ = 2π/P, where P is an integer, and therefore, the system can still maintain periodicity in both the spatial and frequency dimensions. For the choice of θ = 2π/3, the lattice has a period of 2π/θ = 3 along the spatial axis, as the hopping along the frequency dimension (associated with θ) depends on the spatial index n. The Bloch Hamiltonian in the reciprocal space is then given by (detailed derivation for other choices of θ is given in the Supplemental Note 1)

$${H}_{{{{\bf{k}}}}}=\left({c}_{{{{\bf{k}}}},1,+}^{{{\dagger}} }{c}_{{{{\bf{k}}}},1,-}^{{{\dagger}} }{c}_{{{{\bf{k}}}},2,+}^{{{\dagger}} }{c}_{{{{\bf{k}}}},2,-}^{{{\dagger}} }{c}_{{{{\bf{k}}}},3,+}^{{{\dagger}} }{c}_{{{{\bf{k}}}},3,-}^{{{\dagger}} }\right) \\ \tilde{{H}_{{{{\bf{k}}}}}}{\left({c}_{{{{\bf{k}}}},1,+}{c}_{{{{\bf{k}}}},1,-}{c}_{{{{\bf{k}}}},2,+}{c}_{{{{\bf{k}}}},2,-}{c}_{{{{\bf{k}}}},3,+}{c}_{{{{\bf{k}}}},3,-}\right)}^{T},$$
(6)
$$\tilde{{H}_{{{{\bf{k}}}}}}=\left(\begin{array}{cccccc}{B}_{1}({k}_{{{{\rm{f}}}}},\theta ) & {C}_{1}({k}_{{{{\rm{f}}}}},\theta ) & {e}^{i\varphi }{e}^{-i{k}_{{{{\rm{x}}}}}d} & 0 & {e}^{-i\varphi }{e}^{i{k}_{{{{\rm{x}}}}}d} & 0\\ {C}_{1}^{*}({k}_{{{{\rm{f}}}}},\theta ) & {B}_{1}({k}_{{{{\rm{f}}}}},\theta ) & 0 & {e}^{-i\varphi }{e}^{-i{k}_{{{{\rm{x}}}}}d} & 0 & {e}^{i\varphi }{e}^{i{k}_{{{{\rm{x}}}}}d}\\ {e}^{-i\varphi }{e}^{i{k}_{{{{\rm{x}}}}}d} & 0 & {B}_{2}({k}_{{{{\rm{f}}}}},\theta ) & {C}_{2}({k}_{{{{\rm{f}}}}},\theta )&{e}^{i\varphi }{e}^{-i{k}_{{{{\rm{x}}}}}d} & 0\\ 0 & {e}^{i\varphi }{e}^{i{k}_{{{{\rm{x}}}}}d} & {C}_{2}^{*}({k}_{{{{\rm{f}}}}},\theta ) & {B}_{2}({k}_{{{{\rm{f}}}}},\theta ) & 0 & {e}^{-i\varphi }{e}^{-i{k}_{{{{\rm{x}}}}}d}\\ {e}^{i\varphi }{e}^{-i{k}_{{{{\rm{x}}}}}d} & 0 &{e}^{-i\varphi }{e}^{i{k}_{{{{\rm{x}}}}}d} &0 &{B}_{3}({k}_{{{{\rm{f}}}}},\theta ) &{C}_{3}({k}_{{{{\rm{f}}}}},\theta )\\ 0 &{e}^{-i\varphi }{e}^{-i{k}_{{{{\rm{x}}}}}d} & 0 &{e}^{i\varphi }{e}^{i{k}_{{{{\rm{x}}}}}d} & {C}_{3}^{*}({k}_{{{{\rm{f}}}}},\theta ) &{B}_{3}({k}_{{{{\rm{f}}}}},\theta )\end{array}\right),$$
(7)

where \({B}_{j}({k}_{{{{\rm{f}}}}},\theta )=2\cos j\theta \cos {k}_{{{{\rm{f}}}}}{\mathit{\varOmega}}\), \({C}_{j}({k}_{{{{\rm{f}}}}},\theta )=2a\sin j\theta \cos {k}_{{{{\rm{f}}}}}{\mathit{\varOmega}} -2i\sin j\theta \sin {k}_{{{{\rm{f}}}}}{\mathit{\varOmega}}\), and j = 1, 2, 3 being the index of the sites in the reciprocal space. By diagonalizing Hk, one can obtain the bandstructure. One can also rewrite \({\tilde{H}}_{{\mathbf{k}}}\) in spin-orbit coupling form

$${\tilde{H}}_{{{{\bf{k}}}}}=\left(\begin{array}{ccc}{D}_{1}({k}_{{{{\rm{f}}}}},\theta )&{e}^{i\varphi {\sigma }_{3}}{e}^{-i{k}_{{{{\rm{x}}}}}d}&{e}^{-i\varphi {\sigma }_{3}}{e}^{i{k}_{{{{\rm{x}}}}}d}\\ {e}^{-i\varphi {\sigma }_{3}}{e}^{i{k}_{{{{\rm{x}}}}}d}&{D}_{2}({k}_{{{{\rm{f}}}}},\theta )&{e}^{i\varphi {\sigma }_{3}}{e}^{-i{k}_{{{{\rm{x}}}}}d}\\ {e}^{i\varphi {\sigma }_{3}}{e}^{-i{k}_{{{{\rm{x}}}}}d}&{e}^{-i\varphi {\sigma }_{3}}{e}^{i{k}_{{{{\rm{x}}}}}d}&{D}_{3}({k}_{{{{\rm{f}}}}},\theta )\end{array}\right),$$
(8)

where \({D}_{j}({k}_{{{{\rm{f}}}}},\theta )=2\cos j\theta \cos {k}_{{{{\rm{f}}}}}{\mathit{\varOmega}} {\sigma }_{0}+2a\sin j\theta \cos {k}_{{{{\rm{f}}}}}{\mathit{\varOmega}} {\sigma }_{1}+2\sin j\theta \sin {k}_{{{{\rm{f}}}}}{\mathit{\varOmega}} {\sigma }_{2}\) following the form of typical spin-orbit terms59. The last two terms in Dj(kf, θ) correspond to the two different spin-flipped hopping mechanisms.

Quantitative analysis on the ESPP

To quantitatively analyze the change of pseudospin behaviors originating from the competition between the two spin-flipped hopping mechanisms. We consider a simplified model, which also exhibits the same pseudospin behaviors as H. The Hamiltonian is read as

$${H}_{{{{\rm{s}}}}}= {\sum}_{m}\left(\begin{array}{c}{c}_{m+1,+}^{{{\dagger}} }{c}_{m+1,-}^{{{\dagger}} }\end{array}\right)\left(\begin{array}{cc}\cos \theta &0\\ 0&\cos \theta \end{array}\right)\left(\begin{array}{c}{c}_{m,+}\\ {c}_{m,-}\end{array}\right)\\ +\left(\begin{array}{c}{c}_{m+1,+}^{{{\dagger}} }{c}_{m+1,-}^{{{\dagger}} }\end{array}\right)\left(\begin{array}{cc}0&a\sin \theta \\ a\sin \theta &0\end{array}\right)\left(\begin{array}{c}{c}_{m,+}\\ {c}_{m,-}\end{array}\right)\\ +\left(\begin{array}{c}{c}_{m+1,+}^{{{\dagger}} }{c}_{m+1,-}^{{{\dagger}} }\end{array}\right)\left(\begin{array}{cc}0&b\sin \theta \\ b\sin \theta &0\end{array}\right)\left(\begin{array}{c}{c}_{m,+}\\ {c}_{m,-}\end{array}\right)+H.c.,$$
(9)

where the second and third terms correspond to the two hopping mechanisms, respectively. One can obtain the eigenstate of the Hamiltonian by diagonalizing Hs

$$\left\vert \psi \right\rangle=\frac{1}{| \psi | }{\left(\pm \sqrt{\frac{a\cos k-ib\sin k}{a\cos k+ib\sin k}},1\right)}^{T},$$
(10)

where ∣ψ∣ is the normalization factor. One can further obtain the pseudospin projection \(\left\langle {\sigma }_{i}\right\rangle\). We take \(\left\langle {\sigma }_{1}\right\rangle\) as an example

$$\left\langle {\sigma }_{1}\right\rangle=\left\langle {\psi }_{+}| {\sigma }_{1}| {\psi }_{+}\right\rangle=\frac{1}{| \psi {| }^{2}}\left(\sqrt{\frac{a\cos k-ib\sin k}{a\cos k+ib\sin k}}+\sqrt{\frac{a\cos k+ib\sin k}{a\cos k-ib\sin k}}\right).$$
(11)

We first consider the two special cases. For a = 0 and b = 1, \(\left\langle {\sigma }_{1}\right\rangle=0\); for a = 1 and b = 1, \(\left\langle {\sigma }_{1}\right\rangle=\cos k\), which are in accordance with our model. To further illustrate the competition between the two hopping mechanisms, we consider a perturbative case, where b = 1 and a ≪ 1. In this case

$$\left\langle {\sigma }_{1}\right\rangle=\frac{1}{| \psi {| }^{2}}\left(\sqrt{1+\frac{-2ib\sin k}{a\cos k+ib\sin k}}+\sqrt{1+\frac{2ib\sin k}{a\cos k-ib\sin k}}\right),$$
(12)

One sees that \(\left\langle {\sigma }_{1}\right\rangle\) depends on the values of \(a\cos k\) and \(b\sin k\). For k = 0 and k = π, the nominator parts are vanished as \(b\sin k=0\), so \(\left\langle {\sigma }_{1}\right\rangle=1\), while for k ≫ 0, \(\left\langle {\sigma }_{1}\right\rangle \to 0\) as \(a\cos k\to 0\). Therefore, there are two peaks located at k = 0 and k = π for \(\left\langle {\sigma }_{1}\right\rangle\), which can also be found in the Supplemental Note 6.