## 1 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 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 momentum-space 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.