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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 calcu‑ late 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. Keywords: Quantized fields, Chern number, Topological features, Phase diagrams instance, quantum quench [19, 20], thermalization [21, 1 Introduction 22] and decoherence [23–25]. Since the observation of the quantized Hall effect [1], Manipulating the topological features of matter by cou- electronic topological quantum matter [2] became one pling the systems to electromagnetic fields becomes an of the most active subjects of condensed matter physics. active research area for many years. Various topological The researchers have paid much attention to topological structures coupled to electromagnetic fields are studied for materials [3–7] including but not limited to Chern insu- different issues, including topological phases induced pho - lators (CIs) [8–12] in the past decades. Novel topologi- tocurrent [26–33], topological order by dissipation [34– cal phases that correspond to different conducting edge 36], and optical Hall conductivity [37–40]. Interestingly, or surface states are predicted and observed. Topological classical electromagnetic fields can change the energy band phase of matter exist not only in electronic systems but structure of the topological materials and induce nontrivial also in ultracold atomic gases in optical lattices [13–16]. topological edge states in topological insulators such as The later system increases the modulation flexibility of HgTe/CdTe quantum well [41] or graphene [42]. In addi- topological materials and inspires a wide interest in top- tion, the superradiant phase transition occurs in quantum ological insulators subject to external fields [17, 18], for spin Hall insulator for arbitrary weak coupling between the system and fields [43]. This provides us with a new perspec - tive to study the topological features of topological matter *Correspondence: yixx@nenu.edu.cn coupled to a quantized field. Many problems remain open, Center for Advanced Optoelectronic Functional Materials Research, and Key including how topological features can take place in a sys- Laboratory for UV Light‑Emitting Materials and Technology of Ministry tem where the topological tight-binding system coupled to of Education, Northeast Normal University, 130024 Changchun, People’s Republic of China a single-mode quantized field with momentum, and what is Full list of author information is available at the end of the article © The Author(s) 2022. 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We over β to reduce the effects of finite circumference W will answer these questions in this paper. along the y-direction, and the boundary conditions are In this paper, we first introduce our framework that con - implemented via cˆ = 0 and cˆ =ˆc (finite N+1,y,σ x,W+1,σ x,1,σ sists of the Harper-Hofstadter model and a single-mode sites N along the x-direction). quantized field. The Harper-Hofstadter model contains the We consider nearest-neighbor hoppings on a two- Harpers model [44] and the Hofstadter model [45] for opti- dimensional square lattice with a cylinder boundary con- cal lattices, which is realizable in experiments [46, 47]. With ditions along the y-direction as sketched in Fig. 1a. The the development of the Ultracold atoms, it has become an horizontal direction is the direction along the cylinder, important platform for considering topological matter and the vertical one is around the cylinder. When the coupled to a quantized field, the experimental implemen - magnetic flux per plaquette α is not zero, the magnetic tation of our scheme can be designed with several theoreti- flux accumulates due to jumping around the lattice cal studies [48, 49]. In order to connect 1D and 2D physics, as shown in Fig. 1b. The spectrum of the system versus we express the model in a mixed real- and momentum- lattice momentum k is shown in Fig. 1c. The periodicity space called mixed-space representation [50–53]. With of the bulk states is dictated by the denominator q and these arrangements, we calculate the energy bands of the there are three bands in momentum space with q = 3 . quantized light-matter interaction system for both open The boundaries of the magnetic Brillouin zone are at and periodic boundary conditions. And then we calculate k a ∈ (−π/3, π/3) and k a ∈ (−π , π) , where a is the lat- y x the Chern number of the system and show the topological tice constant. quantum phase transitions induced by a single-mode quan- We start by writing the second quantization form of tized field. The changes of Chern number for fixed mag - the Hamiltonian in Eq. (1) for a rectangular lattice in real netic flux ratio indicate that the quantum field indeed can space as induce topological phase transition. Finally, we construct ˆ ˆ H = cˆ H cˆ . 0 r r phase diagrams according to Chern number versus the r (2) single-mode quantized field to show all topological phases. The paper is organized as follows. The Hamiltonian of Due to the chosen boundary conditions, the momentum Harper-Hofstadter model coupled to a single-mode quan- along the y-direction is conserved while the momentum tized field is introduced in Sec. 2. Eigenspectrum of the sys- along x is not. Therefore, it is convenient to write the tem in hybrid representation is calculated and discussed in Hamiltonian in a mixed-space representation by taking Sec. 3. The Chern number spectrum in periodic conditions the advantage of the conserved momentum along y. We is given in Sec. 4. Finally we conclude in Sec. 5. write the annihilation operator by employing the Fourier transformation along y as 2 Hamiltonian 2.1 The Harper‑Hofstader model ik y cˆ = e , r m,k (3) We consider the Harper-Hofstader model, which describes N the nearest-neighbor hopping of atoms in a square lattice. The system forms a cylinder with length N, circumference where m represents the x coordinate in the two-dimen- W and a uniform magnetic field is exerted. The Hamilto - sional lattice, and N represents the number of sites along nian of the system is given by [44, 45] y. In mixed-space representation, the Hamiltonian (1) becomes † i[� −β/W ]I ij H = −t cˆ e cˆ , 0 j (1) ˆ ˆ ˆ ˆ H = H , �i,j� 0 m,k m,k y y m,k (4) m,k † † † where cˆ = (cˆ cˆ ) creates a fermion at site i in one of i i,↑ i,↓ † † † ˆ ˆ ˆ two internal states, pseudospin labeled by σ = {↑, ↓} , i, j � = (� � ) where creates fermion at m,k m,k ,↑ m,k ,↓ y y y represents nearest neighbors, and t is the hopping ampli- (m, k ) . We write the details as shown in Appendix A tude. I stands for the identity matrix. Lattice plaquette ij (from Eqs. (27) to (30)). depends on artificial vector potentials, and we employ the Landau gauge A = (0,A ,0) . Here, A = �x = 2παx , y y 2.2 The coupling of the Harper‑Hofstader system α = p/q (p and q are integers) is the magnetic flux per with a quantized field plaquette in units of flux quantum. The flux β pierce the In this section, we focus on the coupling of the topologi- cylinder along its height, which can be interpreted as an cal material with quantized fields. The light-matter inter - angle twisting the boundaries. Twist angles also can be action has been studied for various systems, which is a used to define topological invariants [ 54]. We will average Han et al. AAPPS Bulletin (2023) 33:1 Page 3 of 11 Fig. 1 Nearest‑neighbor hopping of atoms in a square lattice with a uniform magnetic field. a Nearest ‑neighbor hopping on a two ‑ dimensional square lattice with the periodic boundary conditions along the y‑ direction. b A partial view of the system. c The energy bands of the Harper‑Hofstader model in momentum space, where the Chern numbers of three bands are 1,−2, 1 . The parameters are N = 30 , a = a = t = 1 , x y W = 9 , and β = 0.2π fundamental tool of quantum physics in applications. For transition path of photon (from gray atom to blue atom example, topological-enhanced nonreciprocal scattering obliquely) is divided schematically into two continuous [55], topological properties in the steady state with dis- processes. The first process is regarded as the turnover of sipation [56] and dissipative topological phase transition atoms with different pseudospins (from gray atom to blue [36]. For simplicity, we do not consider decoherence of atom vertically), and the second process is regarded as the single-mode field. the momentum transition of atoms with the same pseu- Consider a single-mode field with momentum q and dospin (from blue atom to blue atom horizontally). The frequency ω = ν�q� , where ν is the speed of light. The above statements mean that k can be understood as a q T Hamiltonian of the single-mode field is given by spin-dependent momentum transfer in the process. With this consideration, we might rewrite the total effective ˆ ˆ H = ω b b , ˆ ˆ ˆ ˆ ˆ ˆ l q q (5) q Hamiltonian as H = H + H + H + H + H , ↑↑ ↓↓ ↓↑ ↑↓ total l which describes the whole system including lattice atoms ˆ ˆ where b and b are creation and annihilation operators, and light field, see Fig. 2 b. Here, respectively. The Hamiltonian that describes the inter - ̂ ̂ ̂ ̂ H =−2t {cos[(k + k )a]− cos [(k + K )a]}Φ Φ , action between the system and the quantized fields may ↑↑ x T y y m,k ,↑ m,k ,↑ y (7) m,k take the following form, ∗ † † † ˆ ˆ † ̂ ̂ ̂ ̂ H =−( g cˆ b cˆ + gcˆ b cˆ ), I k+q,↑ q k−q,↓ H =−2t {cos [(k − k )a]− cos [(k + K )a]}Φ Φ , q ↓↓ x T y y m,k ,↓ k,↓ k,↑ m,k ,↓ y (8) m,k k k (6) where g is the effective atom-photon coupling constant ˆ ˆ ˆ ˆ H = −g b , ↑↓ q m,k ,↓ m,k ,↑ y (9) [57, 58] and q = (k ,0,0) represents the nonzero com- m,k ponent k in x-direction. Experimentally, internal tran- sitions between two atomic ground states (pseudospins) ∗ † † ˆ ˆ ˆ H =−g b ↓↑ m,k ,↑ can be solved by utilizing Raman scattering processes q m,k ,↓ y (10) m,k [49, 59]. The coupling of the system (atoms) to the single- ˆ ˆ ˆ H and H stand for the processes of spin-flip. H and ↑↓ ↓↑ ↑↑ mode light field by Eq. (6) can be depicted in Fig. 2. The H represent the processes of momentum transfer. And ↓↓ Hamiltonian in the real space is shown in Appendix B. K ≡ 2παm − β/W = eHx/c − β/W , eHx/c stands The green and pink bars in Fig. 2a indicate the atoms for the y-component of an artificial vector potential. It in spin-up and spin-down states, respectively. The total Han et al. AAPPS Bulletin (2023) 33:1 Page 4 of 11 Fig. 2 a Schematic diagram of the coupling between the field (photon) and the system(atoms in the lattice). The blue and gray spheres represent atoms of spin up and spin down, respectively. b Schematic diagram of the coupling of pseudospins defined by H at α = 1/3 . The dashed total rectangular box represents the coupling(red lines) between the pseudospins inside the lattice. The solid green and blue lines indicate the couplings of spin up and down between different lattices, respectively may be generated by laser- assisted tunnelings [60, 61]. where the diagonal elements represent spin-depend- We also give the Hamiltonian of the first four parts in a ent kinetic energies and the off-diagonal terms real-space for more intuitive understanding. For details, stand for spin flips. The spin-up (↑) kinetic energy is ̂ ̂ 𝜀 (k) =−2t{cos[(k + k )a]+ cos[(k + K )a]}, see Appendix B (from Eqs. (32) to (35)). and the spin- ↑ x T y y ̂ ̂ 𝜀 (k) =−2t{cos[(k − k )a]+ We choose the basis as follows, down (↓) kinetic energy is ↓ x T cos[(k + K )a]}, where a is the lattice spacing in the y y | � =| �⊗|{0}�, m,k ,↑ m,k ,↑ y y (11) square lattice. The momentum shift k (− k ) is corre- T T | � | � sponding to the ↑ ( ↓ ) state along the x-direction. The and matrix containing the momentum of light parameter k a | � =| �⊗|{1}�. as phase factors is m,k ,↓ m,k ,↓ y y (12) ik a −t e 0 The implementation to realize a single-photon coupled to x J = . (14) −ik a 0 − t e a single-site theoretically [62, 63]. With these settings, we x express the matrix form of the Hamiltonian for the sys- The matrix indexed by position x = ma is tem in a mixed-space representation as −2t cos θ − g D = , ε (k) − g m ∗ (15) ˆ −g − 2t cos θ + ω y q H = , (13) total −g ε (k) + ω ↓ q where θ = (k + K )a , then Harper’s Hamiltonian in y y y y mixed-space representation is given by Han et al. AAPPS Bulletin (2023) 33:1 Page 5 of 11 † † parts because the energy corresponding to the spin up H = J total m−1,k m,k y and down is not the same as Fig. 3b shows. m,k We next discuss the case with flux ratio α = 1/3 , where + D m m,k y 3 is the smallest integer denominator for the Hofstad- m,k (16) m,k ter model exhibiting topological nontrivial bands [64]. We will analyze the eigenvalues E for different photon + J . m+1,k m,k momentum k and coupling constant g. Let us first con - m,k sider the case k = 0 and g/t = 0 , the results are shown In the next section, we will discuss the effects of mag - in Fig. 3c. We can clearly see three sets of degenerate bulk netic and single-mode fields on the eigenspectrums of bands connected by spin-degenerate edge bands. The the system. dashed lines indicate the edge bands and lines with a band shape indicate the bulk bands in Fig. 3c and d. The perio - dicity of bulk bands and the edge bands are 2π/3 and 2π , 3 Eigenspectrum respectively, along the k -direction as shown in Fig. 3c. In this section, we first find the eigenspectrum of the The case of k = 0.84 and g/t = 0.5 as shown in Fig. 3d, Hamiltonian matrix in Eq. (16) and then express it as a it is the same as in Fig. 3b, which shows that the single- function of the magnetic flux and the momentum of mode field leads to the separation of the energy spectrum. the single-mode field. We work in a cylindrical geom - From Fig. 3d, we find six sets of bulk bands connected by etry with finite sites N along the x-direction as open edge states, which are subjected to photon momentum boundaries, while periodic boundary conditions along parameter k and coupling constant g. All the bulk bands the y-direction are considered. In this situation, the spin- have the same period of 2π/3 , while the edge bands are 2π dependent Harper’s matrix along the k -direction. This means the boundaries of the D J … 0 00 0 magnetic Brillouin zone are at k =±π/3. m−N∕2 ⎛ ⎞ ⎜ J ⋱⋱ 0 00 0 ⎟ ⎜ ⎟ ⋮⋱ D J 00 0 m−1 ⎜ ⎟ H = 0 0J D J0 0 , ⎜ ⎟ total m4 Chern numbers ⎜ ⎟ 0 0 0J D ⋱⋮ m+1 In this section, we convert the cylindrical geometry into ⎜ ⎟ 0 00 0 ⋱⋱ J ⎜ ⎟ a torus one to study the Chern number spectrum, where ⎝ ⎠ 0 00 0 … J D m+N∕2 periodic boundary conditions are imposed along the x (17) and y directions. For rational α = 1/3 , we write the spin- has a tridiagonal block structure. The boundary condi - dependent Harper’s Hamiltonian as a 6 × 6 matrix in tion in the x-direction is open, while there is a discrete momentum (k , k ) space x y translational invariance along the y-direction. The matri - H H ces D , J and the null matrix 0 are square matrices of ++ +− H(k , k ) = , x y (18) H H −+ −− 2 × 2 dimension expanded by internal states |↑� and |↓� . The total dimension of the matrix H in Eq. (17) is total which defines 3 × 3 block matrices H or H , where −− ++ 2N × 2N because the size of the space along the x-direc- spin states |↑� = |+� and |↓� = |−� . The spin-diagonal tion is N. 3 × 3 block matrices In the following, we consider a size of the optical lat- tice to be 30 sites (a complete lattice period) along the i(k +k )a −i(k +k )a x T x T Γ − e − e ⎛ ⎞ −i(k +k )a i(k +k )a x-direction as well as two spin states per site and periodic ⎜ x T x T ⎟ H = −e Γ − e , (19) ++ 2 ⎜ ⎟ i(k +k )a −i(k +k )a x T x T boundary conditions along the y-direction, so k is a good y ⎝ −e − e Γ ⎠ quantum number. The case analysis demonstrates that the eigenvalues E are labeled by a discrete band index k i(k −k )a −i(k −k )a y x T x T ⎛ Γ + − e − e ⎞ n and momentum k , are also functions of the photon y ⎜ −i(k −k )a i(k −k )a ⎟ x T x T H = , −e Γ + − e −− (20) ⎜ ⎟ momentum k , electron-light coupling constant g, as well i(k −k )a −i(k −k )a T x T x T −e − e Γ + ⎝ ⎠ as flux ratio α = �/� . where k + k and k − k describe respectively the In Fig. 3, we show the spectrum of α = �/� versus 0 x T x T spin states |↑� = |+� and |↓� = |−� momentum trans- the energy E with k = 0 . A standard buttery fl graph in 0 T fer along the x direction. The kinetic energy terms are the case of zero photon momentum k = 0 and g/t = 0 Ŵ =−2 cos(k a − 2παm) , with the magnetic flux ratio is shown in Fig. 3a. For photon momentum k = 1.2 and T m y α = 1/3 and m values (0, 1, 2). g/t = 0.5 , the standard buttery fl graph split into two Han et al. AAPPS Bulletin (2023) 33:1 Page 6 of 11 Fig. 3 The spectrum of the system with α = �/� . a and b are for E (in units of t). Eigenvalues E (in units of t) of the Harper’s matrix Eq. (17) 0 0 k vs k for magnetic flux ratio α = 1/3 , β = 0.2π , a = 1 , and W = 9 are shown in c and d. The parameters are a k = 0 and g/t = 0 , b k = 0.84 and y T T g/t = 0.5 , c k = 0 and g/t = 0 , and d k = 0.84 and g/t = 0.5 T T The spin-off-diagonal 3 × 3 block matrices are 2 (m ) C = d kF (k), m (22) σ xy 2πi −g 00 H = 0 − g 0 , +− (21) where the domain of integration is the mag- 00 − g netic Brillouin zone. Namely, � = [−π ,π] and � = [−π/q, π/q] . The function and H = H . They describe spin-flip processes −+ +− (m ) (m ) induced by the independently tunable single-mode field. σ σ ∂A (k) ∂A (k) y x (m ) F (k) = − , (23) The energy spectrum is shown in Fig. 5, which is similar xy ∂ ∂ x y to the cases with open boundary conditions, but there are no boundary states with periodic boundary conditions is the Berry curvature expressed in terms of the Berry (m ) along the x and y directions. As a consequence, there is connection A (k) =�u (k)|∂ |u (k)� , where m j m σ σ no edge state in the later case. |u (k)� are the eigenstates of the Hamiltonian H(k , k ) m x y Next, we analyze the Chern spectrum with a single- defined in Eq. (18). In the limit of no electron-light cou - mode quantized field and fixed flux ratio α = 1/3 . The pling, k = 0 and g = 0 , the energy spectrum for flux energy spectrum associated with the Hamiltonian ratio α = p/q is doubly degenerate with q magnetic H(k , k ) in Eq. (18) has six bands E (k) labeled by the x y m σ bands and (q − 1) gaps, such that the Chern number from band index m ( σ labels the spin). There are at least 2 Eq. (22) reduces to the standard form in the literature band gaps when the bands are double degenerate, and [64, 65]. To compute the Chern number C , we general- there are at most 5 gaps when the system has no degen- ize the discretization method used in the quantum Hall eracy. In the absence of overlapping regions between system [66, 67] without electron-light coupling k = 0 th the energy bands E (k) , the Chern number for the m σ σ and g = 0 . For this purpose, we define the link function band is Han et al. AAPPS Bulletin (2023) 33:1 Page 7 of 11 maximum of 2q non-overlapping bands and a maximum �u (k)|u (k + δk )� m m j (m ) σ σ L (k) = , of 2q Chern numbers. |�u (k)|u (k + δk )�| m m j σ σ Chern numbers are properties of bands E (k) or band and obtain the Berry curvature bundles with degeneracy V and are independent of the location of the chemical potential µ. However, Chern σ m L (k)L (k + δk ) x x y numbers are defined only within band gaps and their F (k) = ln , (24) m m xy σ σ L (k + δk )L (k) values are dependent on the gap where the chemical x y y potential locates. If the chemical potential µ is located in which is a purely imaginary number defined in the range a band gap corresponding to the filling factor ν = r/2q , of −π ≤ I F (k) ≤ π . The Chern number becomes then the Chern number is the sum of Chern numbers of xy bands with energies E <µ (m ) C = F (k). σ xy (25) ν=r/2q 2πi C = C . r m (26) m ,E<µ When the energy bands E (k) overlap, we need to rede- σ fine the link variable of the degenerate bundle with Furthermore, via the bulk-edge correspondence [67], the degeneracy V via the multiplet Chern number C calculated from the torus geometry (V ) (1) (V ) |ψ (k)�= |u (k)�, . . . , |u (k)� , leading to m m m σ σ σ (bulk system without edges) measures the total chirality of edge states that are present in the gap for the cylindri- (V) (V) Det�ψ (k)|ψ (k + δk )� m m j (m ) σ σ cal geometry. L (k) = , (V) (V) |Det�ψ (k)|ψ (k + δk )�| m m j σ σ In Fig. 4, Chern number C calculated from the torus geometry are shown as a function of photon momentum with these definitions, the expression for the Berry cur - parameter k . It can be seen that the Chern numbers for vature defined in Eq. (24) remains valid when written in r = 1 and r = 5 possess the same dependence on k . On terms of the new link functions defined above. For two the contrary, the Chern numbers for r = 2 shows the internal states and magnetic flux ratio α = p/q , there is a opposite dependence with respect to the case for r = 4 . Fig. 4 Chern numbers C calculated from the torus geometry are shown for different filling factors ν = r/2q (r =1,2,3,4,5,6). Taking r = 2 as an example, it can be seen that a step change occurs at the critical point k ∼ 0.84 . The other parameters chosen are α = 1/3 , β = 0.2π , a = 1 , g = 0.5 , and W = 9 Han et al. AAPPS Bulletin (2023) 33:1 Page 8 of 11 Considering the case of r = 2 , the Chern number has a We show the enlarged energy corresponding to the step change at the critical photon momentum k ∼ 0.84 . edge states in the band gap between the second and This indicates that the topology of the system changes at the third bulk bands marked by the blue lines in Fig. 5d this point, and the system goes from one non-trivial top- and e. Open boundaries are considered along the x ological phase to the other non-trivial topological phase direction, with L and R denoting the left and right as the momentum k increases. boundaries, respectively. The green region with two pairs Now, we are in order to analyze the connection of chiral edge states and the Chern number 2 are shown between the band structures (Fig. 5) and the phases in respectively in Fig. 5d and 4c. The pink region with a pair Fig. 4. Energy bands as a function of k are shown in of chiral edge states and Chern number − 1 are shown in Figs. 5a–c, where Fig. 5a, b, and c correspond to differ - Fig. 5e and 4c, respectively. ent photon momentum k = k , k = k , and k = k , The energy dispersions in x direction are similar to the T i T c T f respectively. Taking r = 2 as an example, we need to case of periodic boundaries, the spectra (Fig. 5a and c) examine only the second and the third bulk bands are plotted for comparison with the corresponding peri- (arranged from the bottom to the top). Noticing that the odic cases (Fig. 5d and e). As the increase of the photon energy of six bulk bands with k = k possess touching momentum k , the gap is closed and two pairs of chiral T c T points at the second and third bulk bands, i.e., the band edge states disappear (Fig. 3d) at critical point k . This gap is closed at k = k for the second and third bulk means the chiral edge states are merged into the bulk T c bands, we then claim that the system with k is a non-triv- eigenstates, the gap opens, and a pair of chiral edge ial gaped phase, labeled by C = 2 (see the green region state appears again. In addition, the gapped phases for in Fig. 5). While the system with k is a non-trivial phase all k are characterized by the Chern number that deter- f T which can be labeled by its Chern number C =−1 (see mines the chiral edge states in agreement with the phase the pink region in Fig. 5). diagram. Fig. 5 Energy dispersions vs Bloch vector k with photon momentum k . k , k , and k are three special k chosen for a, b, and c, respectively. d and T i c f T e are enlarged energy dispersion vs photon momentum k in order to illustrate edge bands and their location along x direction. The left and right boundaries are marked as L and R, respectively. The photon momentum chosen are in a and d k = 0.3 , in b k = 0.84 , and in c and e k = 1.2 . T T T Other parameters chosen are α = 1/3 , β = 0.2π , a = 1 , and W = 9 Han et al. AAPPS Bulletin (2023) 33:1 Page 9 of 11 Lastly, we discuss the mechanism of the topological We might rewrite the total Hamiltonian ˆ ˆ ˆ H = H + H 0 ↑↑ ↓↓ phase transition induced by a quantized field in Fig. 2b. as The coupling of the internal states of the site to the extra i(2𝜋 m𝛼 −𝛽 ∕W ) H =− tĉ (ĉ + e ĉ )+ h.c. ↑↑ m+1,n,↑ m,n+1,↑ m,n,↑ quantum field are characterized by the coupling strength (28) m,n g, this coupling leads to the splitting of a site on the lat- tice. The hopping between spin up and down in differ - i(2𝜋 m𝛼 −𝛽 ∕W ) H =− tĉ (ĉ + e ĉ )+ h.c. ik a −ik a T T ↓↓ m+1,n,↓ m,n+1,↓ m,n,↑ ent lattices are described by a phase te (or te ). (29) m,n In such a lattice model, modulating the momentum k By employing the Fourier transformation along y-direc- is equivalent to adjusting the coupling strength of next- tion, the Eq. (27) in mixed-space representation as nearest neighbor hopping in the lattices, so that the topo- logical structure of the lattice changes, which induces † † ̂ ̂ ̂ Φ Φ H =− 2t × m,k ,↑ m,k ,↓ y y topological phase transition characterized by the chang- m,k ing of topological invariants (Chern number). In this ̂ ̂ cos k a − cos [(k + K )a] 0 Φ m,k ,↑ x y y sense, we conclude that the quantum field indeed can 0 cos k a − cos [(k + K )a] Φ x y y m,k ,↓ induce topological phase transition. (30) where K = 2παm − β/W . 5 Conclusion In conclusion, we have studied the topological features of an extended Harper-Hofstadter model which describes Appendix B: The Hamiltonian H total atoms in square optical lattices coupled to a single-mode In this Appendix, we derive the Hamiltonian matrix quantized field. We have manipulated the topological given in Eq. 16. Firstly, we write the Eq. (6) in real space features of matter by coupling the systems to quantized as fields. The quantum light field is actually used to cou - † † ∗ ik m † −ik m ̂ T ̂ T ̂ ple different atomic internal states to induce topological H = −(g e c ̂ b c ̂ + g e c ̂ b c ̂ ), I m,n,↑ q m,n,↓ m,n,↓ q m,n,↑ (31) m,n m,n phase transition, which is different from the previous lit - erature on coupling the internal states of atoms [68–70]. Then, we divide the jumping path after adding photons We calculated the energy band structure of the system into two simultaneous path contributions as shown in and showed it as a function of the magnetic and single- Fig. 2. And we obtain the the following five parts in a mode quantized field. We find that the topological prop - real-space representation erties might be modulated by the single-mode quantized † ik a i(2𝜋 m𝛼 −𝛽 ∕W ) T x H =− tĉ (e ĉ + e ĉ )+ h.c. field, and topological quantum phase transitions could ↑↑ m+1,n,↑ m,n+1,↑ m,n,↑ (32) m,n be induced by the single-mode quantized field. We have performed this analysis by computing the Chern number −ik a i(2𝜋 m𝛼 −𝛽 ∕W ) ̂ T x H =− tĉ (e ĉ + e ĉ )+ h.c. ↓↓ m,n,↓ m+1,n,↓ m,n+1,↓ and comparing it with the case without quantized fields. (33) m,n Differences in the topological features of the system with and without a single-mode quantized field for fixed H =− gcˆ b cˆ , ↑↓ q m,n,↓ m,n,↑ magnetic flux ratio are found and discussed. In addition, (34) m,n constructing phase diagrams by the Chern number and analyzing its dependence on the filling factor and the ∗ † † ˆ ˆ momentum of a single-mode quantized field, we found H =− g cˆ b cˆ , ↓↑ m,n,↑ m,n,↓ q (35) m,n that Chern numbers are dramatically modified by the momentum of the single-mode quantized field. ˆ ˆ ˆ H = ω b b . (36) l q q Then we write the Hamiltonian matrix for ultracold Appendix A: The Hamiltonian H atoms in a mixed-space representation as In this Appendix, we give the specific second quantiza - tion form of the Hamiltonian Eq. (1) in real space repre- ε (k) − g sentation as H = , (37) −g ε (k) + ω ↓ q i(2𝜋 m𝛼 −𝛽 ∕W ) e 0 ĉ j,↑ † † H =−t ĉ ĉ , 0 (27) i,↑ i,↓ i(2𝜋 m𝛼 −𝛽 ∕W ) where the kinetic energy operator along the x direction is 0 e ĉ j,↓ <i,j> Han et al. AAPPS Bulletin (2023) 33:1 Page 10 of 11 Availability of data and materials ˆ ˆ ε (k) =−2t{cos[(k + k )a]+ cos[(k + K )a]}, ↑ x T y y Data sharing not applicable to this article as no datasets were generated or (38) analyzed during the current study. and the spin-down (↓) kinetic energy is Declarations ˆ ˆ ε (k) =−2t{cos[(k − k )a]+ cos[(k + K )a]}. ↓ x T y y Ethics approval and consent to participate (39) Not applicable. It is easily to find that H consists of two matrices T and Consent for publication D . The matrix indexed by position x = ma is Not applicable. −2t cos θ − g Competing interests D = , (40) m ∗ The authors declare that they have no competing interests. −g − 2t cos θ + ω y q Author details where the kinetic energy operator of the system is given 1 Center for Quantum Sciences and School of Physics, Northeast Normal Uni‑ by according to Euler’s formula versity, 130024 Changchun, People’s Republic of China. Center for Advanced Optoelectronic Functional Materials Research, and Key Laboratory for UV Light‑Emitting Materials and Technology of Ministry of Education, Northeast ˆ ˆ T = T (k ) + T (k ), (41) + x − x Normal University, 130024 Changchun, People’s Republic of China. with Received: 11 September 2022 Accepted: 28 November 2022 ik a ik a −t e e 0 T (k ) = , + x (42) −ik a ik a T x 0 − t e e References 1. K.V. Klitzing, G. Dorda, M. Pepper, New Method for High‑Accuracy and Determination of the Fine‑Structure Constant Based on Quantized Hall Resistance. Phys. Rev. Lett. 45, 494 (1980) −ik a −ik a T x −t e e 0 x 2. C.‑K. Chiu, J.C.Y. Teo, A.P. Schnyder, S. Ryu, Classification of topological T (k ) = . − x ik a −ik a quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016) T x 0 − t e e 3. C.L. Kane, E.J. Mele, Quantum Spin Hall Eec ff t in Graphene. Phys. Rev. Lett. (43) 95, 226801 (2005) 4. C.L. Kane, E.J. 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AAPPS Bulletin – Springer Journals
Published: Jan 2, 2023
Keywords: Quantized fields; Chern number; Topological features; Phase diagrams
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