Quantized fields induced topological features in Harper-Hofstadter model

Classical magnetic fields might change the properties of topological insulators such as the time reversal symmetry protected topological edge states. This poses a question that whether quantized fields would change differently the feature of topological materials with respect to the classical one. In this paper, we propose a model to describe topological insulators (ultracold atoms in square optical lattices with magnetic field) coupled to a tunable single-mode quantized field, and discuss the topological features of the system. We find that the quantized field can induce topological quantum phase transitions in a different way. To be specific, for fixed gauge magnetic flux ratio, we calculate the energy bands for different coupling constants between the systems and the fields in both open and periodic boundary conditions. We find that the Hofstadter butterfly graph is divided into a pair for continuous gauge magnetic flux ratio, which is different from the one without single-mode quantized field. In addition, we plot topological phase diagrams characterized by Chern number as a function of the momentum of the single-mode quantized field and obtain a quantized structure with non-zero filling factor.


Introduction
Since the observation of the quantized Hall effect [1], electronic topological quantum matter [2] became one of the most active subjects of condensed matter physics. The researchers have paid much attention to topological materials [3][4][5][6][7] including but not limited to Chern insulators (CIs) [8][9][10][11][12] in the past decades. Novel topological phases that correspond to different conducting edge or surface states are predicted and observed. Topological phase of matter exist not only in electronic systems but also in ultracold atomic gases in optical lattices [13][14][15][16]. The later system increases the modulation flexibility of topological materials and inspires a wide interest in topological insulators subject to external fields [17,18], for instance, quantum quench [19,20], thermalization [21,22] and decoherence [23][24][25].
Manipulating the topological features of matter by coupling the systems to electromagnetic fields becomes an active research area for many years. Various topological structures coupled to electromagnetic fields are studied for different issues, including topological phases induced photocurrent [26][27][28][29][30][31][32][33], topological order by dissipation [34][35][36], and optical Hall conductivity [37][38][39][40]. Interestingly, classical electromagnetic fields can change the energy band structure of the topological materials and induce nontrivial topological edge states in topological insulators such as HgTe/CdTe quantum well [41] or graphene [42]. In addition, the superradiant phase transition occurs in quantum spin Hall insulator for arbitrary weak coupling between the system and fields [43]. This provides us with a new perspective to study the topological features of topological matter coupled to a quantized field. Many problems remain open, including how topological features can take place in a system where the topological tight-binding system coupled to a single-mode quantized field with momentum, and what is *Correspondence: yixx@nenu.edu.cn 2 Center for Advanced Optoelectronic Functional Materials Research, and Key Laboratory for UV Light-Emitting Materials and Technology of Ministry of Education, Northeast Normal University, 130024 Changchun, People's Republic of China Full list of author information is available at the end of the article the behavior of the Chern number in such a situation? We will answer these questions in this paper.
In this paper, we first introduce our framework that consists of the Harper-Hofstadter model and a single-mode quantized field. The Harper-Hofstadter model contains the Harpers model [44] and the Hofstadter model [45] for optical lattices, which is realizable in experiments [46,47]. With the development of the Ultracold atoms, it has become an important platform for considering topological matter coupled to a quantized field, the experimental implementation of our scheme can be designed with several theoretical studies [48,49]. In order to connect 1D and 2D physics, we express the model in a mixed real-and momentumspace called mixed-space representation [50][51][52][53]. With these arrangements, we calculate the energy bands of the quantized light-matter interaction system for both open and periodic boundary conditions. And then we calculate the Chern number of the system and show the topological quantum phase transitions induced by a single-mode quantized field. The changes of Chern number for fixed magnetic flux ratio indicate that the quantum field indeed can induce topological phase transition. Finally, we construct phase diagrams according to Chern number versus the single-mode quantized field to show all topological phases.
The paper is organized as follows. The Hamiltonian of Harper-Hofstadter model coupled to a single-mode quantized field is introduced in Sec. 2. Eigenspectrum of the system in hybrid representation is calculated and discussed in Sec. 3. The Chern number spectrum in periodic conditions is given in Sec. 4. Finally we conclude in Sec. 5.

The Harper-Hofstader model
We consider the Harper-Hofstader model, which describes the nearest-neighbor hopping of atoms in a square lattice. The system forms a cylinder with length N, circumference W and a uniform magnetic field is exerted. The Hamiltonian of the system is given by [44,45] where creates a fermion at site i in one of two internal states, pseudospin labeled by σ = {↑, ↓} , i, j represents nearest neighbors, and t is the hopping amplitude. Î stands for the identity matrix. Lattice plaquette ij depends on artificial vector potentials, and we employ the Landau gauge A = (0, A y , 0) . Here, A y = �x = 2παx , α = p/q (p and q are integers) is the magnetic flux per plaquette in units of flux quantum. The flux β pierce the cylinder along its height, which can be interpreted as an angle twisting the boundaries. Twist angles also can be used to define topological invariants [54]. We will average over β to reduce the effects of finite circumference W along the y-direction, and the boundary conditions are implemented via ĉ N +1,y,σ = 0 and ĉ x,W +1,σ =ĉ x,1,σ (finite sites N along the x-direction).
We consider nearest-neighbor hoppings on a twodimensional square lattice with a cylinder boundary conditions along the y-direction as sketched in Fig. 1a. The horizontal direction is the direction along the cylinder, and the vertical one is around the cylinder. When the magnetic flux per plaquette α is not zero, the magnetic flux accumulates due to jumping around the lattice as shown in Fig. 1b. The spectrum of the system versus lattice momentum k is shown in Fig. 1c. The periodicity of the bulk states is dictated by the denominator q and there are three bands in momentum space with q = 3 . The boundaries of the magnetic Brillouin zone are at k y a ∈ (−π/3, π/3) and k x a ∈ (−π , π) , where a is the lattice constant.
We start by writing the second quantization form of the Hamiltonian in Eq. (1) for a rectangular lattice in real space as Due to the chosen boundary conditions, the momentum along the y-direction is conserved while the momentum along x is not. Therefore, it is convenient to write the Hamiltonian in a mixed-space representation by taking the advantage of the conserved momentum along y. We write the annihilation operator by employing the Fourier transformation along y as where m represents the x coordinate in the two-dimensional lattice, and N y represents the number of sites along y. In mixed-space representation, the Hamiltonian (1) becomes where � † m,k y = (� † m,k y ,↑� † m,k y ,↓ ) creates fermion at (m, k y ) . We write the details as shown in Appendix A (from Eqs. (27) to (30)).

The coupling of the Harper-Hofstader system with a quantized field
In this section, we focus on the coupling of the topological material with quantized fields. The light-matter interaction has been studied for various systems, which is a fundamental tool of quantum physics in applications. For example, topological-enhanced nonreciprocal scattering [55], topological properties in the steady state with dissipation [56] and dissipative topological phase transition [36]. For simplicity, we do not consider decoherence of the single-mode field.
Consider a single-mode field with momentum q and frequency ω q = ν�q� , where ν is the speed of light. The Hamiltonian of the single-mode field is given by where b † q and b q are creation and annihilation operators, respectively. The Hamiltonian that describes the interaction between the system and the quantized fields may take the following form, where g is the effective atom-photon coupling constant [57,58] and q = (k T , 0, 0) represents the nonzero component k T in x-direction. Experimentally, internal transitions between two atomic ground states (pseudospins) can be solved by utilizing Raman scattering processes [49,59].
The coupling of the system (atoms) to the singlemode light field by Eq. (6) can be depicted in Fig. 2. The Hamiltonian in the real space is shown in Appendix B. The green and pink bars in Fig. 2a indicate the atoms in spin-up and spin-down states, respectively. The total transition path of photon (from gray atom to blue atom obliquely) is divided schematically into two continuous processes. The first process is regarded as the turnover of atoms with different pseudospins (from gray atom to blue atom vertically), and the second process is regarded as the momentum transition of atoms with the same pseudospin (from blue atom to blue atom horizontally). The above statements mean that k T can be understood as a spin-dependent momentum transfer in the process.
With this consideration, we might rewrite the total effective Hamiltonian as Ĥ total =Ĥ ↑↑ +Ĥ ↓↓ +Ĥ ↓↑ +Ĥ ↑↓ +Ĥ l , which describes the whole system including lattice atoms and light field, see Fig. 2b. Here, H ↑↓ and Ĥ ↓↑ stand for the processes of spin-flip. Ĥ ↑↑ and Ĥ ↓↓ represent the processes of momentum transfer. And K y ≡ 2παm − β/W = eHx/ c − β/W , eHx/ c stands for the y-component of an artificial vector potential. It may be generated by laser-assisted tunnelings [60,61]. We also give the Hamiltonian of the first four parts in a real-space for more intuitive understanding. For details, see Appendix B (from Eqs. (32) to (35)). We choose the basis as follows, and The implementation to realize a single-photon coupled to a single-site theoretically [62,63]. With these settings, we express the matrix form of the Hamiltonian for the system in a mixed-space representation as where the diagonal elements represent spin-dependent kinetic energies and the off-diagonal terms stand for spin flips. The spin-up (↑) kinetic energy is where θ y = (k y + K y )a y , then Harper's Hamiltonian in mixed-space representation is given by In the next section, we will discuss the effects of magnetic and single-mode fields on the eigenspectrums of the system.

Eigenspectrum
In this section, we first find the eigenspectrum of the Hamiltonian matrix in Eq. (16)  The total dimension of the matrix Ĥ total in Eq. (17) is 2N × 2N because the size of the space along the x-direction is N.
In the following, we consider a size of the optical lattice to be 30 sites (a complete lattice period) along the x-direction as well as two spin states per site and periodic boundary conditions along the y-direction, so k y is a good quantum number. The case analysis demonstrates that the eigenvalues E k y are labeled by a discrete band index n and momentum k y , are also functions of the photon momentum k T , electron-light coupling constant g, as well as flux ratio α = �/� 0 .
In Fig. 3, we show the spectrum of α = �/� 0 versus the energy E 0 with k T = 0 . A standard butterfly graph in the case of zero photon momentum k T = 0 and g/t = 0 is shown in Fig. 3a. For photon momentum k T = 1.2 and g/t = 0.5 , the standard butterfly graph split into two (16) parts because the energy corresponding to the spin up and down is not the same as Fig. 3b shows. We next discuss the case with flux ratio α = 1/3 , where 3 is the smallest integer denominator for the Hofstadter model exhibiting topological nontrivial bands [64]. We will analyze the eigenvalues E k y for different photon momentum k T and coupling constant g. Let us first consider the case k T = 0 and g/t = 0 , the results are shown in Fig. 3c. We can clearly see three sets of degenerate bulk bands connected by spin-degenerate edge bands. The dashed lines indicate the edge bands and lines with a band shape indicate the bulk bands in Fig. 3c and d. The periodicity of bulk bands and the edge bands are 2π/3 and 2π , respectively, along the k y -direction as shown in Fig. 3c. The case of k T = 0.84 and g/t = 0.5 as shown in Fig. 3d, it is the same as in Fig. 3b, which shows that the singlemode field leads to the separation of the energy spectrum. From Fig. 3d, we find six sets of bulk bands connected by edge states, which are subjected to photon momentum parameter k T and coupling constant g. All the bulk bands have the same period of 2π/3 , while the edge bands are 2π along the k y -direction. This means the boundaries of the magnetic Brillouin zone are at k y = ±π/3.

Chern numbers
In this section, we convert the cylindrical geometry into a torus one to study the Chern number spectrum, where periodic boundary conditions are imposed along the x and y directions. For rational α = 1/3 , we write the spindependent Harper's Hamiltonian as a 6 × 6 matrix in momentum (k x , k y ) space The spin-off-diagonal 3 × 3 block matrices are and H −+ = H * +− . They describe spin-flip processes induced by the independently tunable single-mode field. The energy spectrum is shown in Fig. 5, which is similar to the cases with open boundary conditions, but there are no boundary states with periodic boundary conditions along the x and y directions. As a consequence, there is no edge state in the later case.
Next, we analyze the Chern spectrum with a singlemode quantized field and fixed flux ratio α = 1/3 . The energy spectrum associated with the Hamiltonian H(k x , k y ) in Eq. (18) has six bands E m σ (k) labeled by the band index m σ ( σ labels the spin). There are at least 2 band gaps when the bands are double degenerate, and there are at most 5 gaps when the system has no degeneracy. In the absence of overlapping regions between the energy bands E m σ (k) , the Chern number for the m th σ band is (21) where the domain of integration is the magnetic Brillouin zone. Namely, � x = [−π , π] and � y = [−π/q, π/q] . The function is the Berry curvature expressed in terms of the Berry connection where |u m σ (k)� are the eigenstates of the Hamiltonian H(k x , k y ) defined in Eq. (18). In the limit of no electron-light coupling, k T = 0 and g = 0 , the energy spectrum for flux ratio α = p/q is doubly degenerate with q magnetic bands and (q − 1) gaps, such that the Chern number from Eq. (22) reduces to the standard form in the literature [64,65]. To compute the Chern number C m σ , we generalize the discretization method used in the quantum Hall system [66,67] without electron-light coupling k T = 0 and g = 0 . For this purpose, we define the link function (22) , maximum of 2q non-overlapping bands and a maximum of 2q Chern numbers.
Chern numbers are properties of bands E m σ (k) or band bundles with degeneracy V and are independent of the location of the chemical potential µ. However, Chern numbers are defined only within band gaps and their values are dependent on the gap where the chemical potential locates. If the chemical potential µ is located in a band gap corresponding to the filling factor ν = r/2q , then the Chern number is the sum of Chern numbers of bands with energies E < µ Furthermore, via the bulk-edge correspondence [67], the Chern number C r calculated from the torus geometry (bulk system without edges) measures the total chirality of edge states that are present in the gap for the cylindrical geometry.
In Fig. 4, Chern number C r calculated from the torus geometry are shown as a function of photon momentum parameter k T . It can be seen that the Chern numbers for r = 1 and r = 5 possess the same dependence on k T . On the contrary, the Chern numbers for r = 2 shows the opposite dependence with respect to the case for r = 4 . Considering the case of r = 2 , the Chern number has a step change at the critical photon momentum k T ∼ 0.84 . This indicates that the topology of the system changes at this point, and the system goes from one non-trivial topological phase to the other non-trivial topological phase as the momentum k T increases. Now, we are in order to analyze the connection between the band structures (Fig. 5) and the phases in Fig. 4. Energy bands as a function of k T are shown in Figs. 5a-c, where Fig. 5a, b, and c correspond to different photon momentum k T = k i , k T = k c , and k T = k f , respectively. Taking r = 2 as an example, we need to examine only the second and the third bulk bands (arranged from the bottom to the top). Noticing that the energy of six bulk bands with k T = k c possess touching points at the second and third bulk bands, i.e., the band gap is closed at k T = k c for the second and third bulk bands, we then claim that the system with k i is a non-trivial gaped phase, labeled by C 2 = 2 (see the green region in Fig. 5). While the system with k f is a non-trivial phase which can be labeled by its Chern number C 2 = −1 (see the pink region in Fig. 5).
We show the enlarged energy corresponding to the edge states in the band gap between the second and the third bulk bands marked by the blue lines in Fig. 5d and e. Open boundaries are considered along the x direction, with L and R denoting the left and right boundaries, respectively. The green region with two pairs of chiral edge states and the Chern number 2 are shown respectively in Fig. 5d and 4c. The pink region with a pair of chiral edge states and Chern number − 1 are shown in Fig. 5e and 4c, respectively.
The energy dispersions in x direction are similar to the case of periodic boundaries, the spectra (Fig. 5a and c) are plotted for comparison with the corresponding periodic cases (Fig. 5d and e). As the increase of the photon momentum k T , the gap is closed and two pairs of chiral edge states disappear (Fig. 3d) at critical point k c . This means the chiral edge states are merged into the bulk eigenstates, the gap opens, and a pair of chiral edge state appears again. In addition, the gapped phases for all k T are characterized by the Chern number that determines the chiral edge states in agreement with the phase diagram. Lastly, we discuss the mechanism of the topological phase transition induced by a quantized field in Fig. 2b. The coupling of the internal states of the site to the extra quantum field are characterized by the coupling strength g, this coupling leads to the splitting of a site on the lattice. The hopping between spin up and down in different lattices are described by a phase te ik T a (or te −ik T a ).
In such a lattice model, modulating the momentum k T is equivalent to adjusting the coupling strength of nextnearest neighbor hopping in the lattices, so that the topological structure of the lattice changes, which induces topological phase transition characterized by the changing of topological invariants (Chern number). In this sense, we conclude that the quantum field indeed can induce topological phase transition.

Conclusion
In conclusion, we have studied the topological features of an extended Harper-Hofstadter model which describes atoms in square optical lattices coupled to a single-mode quantized field. We have manipulated the topological features of matter by coupling the systems to quantized fields. The quantum light field is actually used to couple different atomic internal states to induce topological phase transition, which is different from the previous literature on coupling the internal states of atoms [68][69][70].
We calculated the energy band structure of the system and showed it as a function of the magnetic and singlemode quantized field. We find that the topological properties might be modulated by the single-mode quantized field, and topological quantum phase transitions could be induced by the single-mode quantized field. We have performed this analysis by computing the Chern number and comparing it with the case without quantized fields. Differences in the topological features of the system with and without a single-mode quantized field for fixed magnetic flux ratio are found and discussed. In addition, constructing phase diagrams by the Chern number and analyzing its dependence on the filling factor and the momentum of a single-mode quantized field, we found that Chern numbers are dramatically modified by the momentum of the single-mode quantized field. and the spin-down (↓) kinetic energy is It is easily to find that Ĥ consists of two matrices T and D m . The matrix indexed by position x = ma is where the kinetic energy operator of the system is given by according to Euler's formula with and The matrix containing the momentum of light parameter k T a x as phase factors is In mixed-space representation, we finally write Harper's Hamiltonian