^{1}, Santanu K. Maiti

^{2,a)}and S. N. Karmakar

^{1}

### Abstract

We investigate numerically the integer quantum Hall effect (IQHE) in a two-dimensional square lattice with non-interacting electrons in presence of disorder and subjected to uniform magnetic field in a direction perpendicular to the lattice plane. We employ nearest-neighbor tight-binding Hamiltonian to describe the system, and obtain the longitudinal and transverse conductivities using Kubo formalism. The interplay between the magnetic field and disorder is also discussed. Our analysis may be helpful in studying IQHE in any discrete lattice model.

We thank Shreekantha Sil for some useful discussion.

I. INTRODUCTION

II. MODEL AND THEORETICAL FORMULATION

A. The model

B. Linear response Kubo formalism

III. NUMERICAL RESULTS AND DISCUSSION

A. Energy spectrum

B. Transverse and longitudinal conductivities

C. Effect of magnetic field and temperature

IV. CONCLUSION

### Key Topics

- Magnetic fields
- 17.0
- Quantum Hall effects
- 10.0
- Magnetic flux
- 6.0
- Eigenvalues
- 5.0
- Numerical modeling
- 5.0

## Figures

Schematic diagram of a square lattice subjected to a perpendicular magnetic field ** B **.

Schematic diagram of a square lattice subjected to a perpendicular magnetic field ** B **.

Energy levels for a square lattice () when is set at 0.1. The black lines represent the locations of the energy levels for the ordered (*W* = 0) case. On the other hand, the blue lines denote the positions of the energy levels for the disordered () case, where (a), (b), and (c) correspond to *W* = 1, 2, and 3, respectively.

Energy levels for a square lattice () when is set at 0.1. The black lines represent the locations of the energy levels for the ordered (*W* = 0) case. On the other hand, the blue lines denote the positions of the energy levels for the disordered () case, where (a), (b), and (c) correspond to *W* = 1, 2, and 3, respectively.

Transverse conductivity (red curve) as a function of Fermi energy for a square lattice () considering , *W* = 1, and . The Landau bands (light green) are superimposed on it.

Transverse conductivity (red curve) as a function of Fermi energy for a square lattice () considering , *W* = 1, and . The Landau bands (light green) are superimposed on it.

Longitudinal conductivity (red curve) as a function of Fermi energy for a square lattice () for the same parameter values given in Fig. 3. The Landau bands (light green) are superimposed on it.

Longitudinal conductivity (red curve) as a function of Fermi energy for a square lattice () for the same parameter values given in Fig. 3. The Landau bands (light green) are superimposed on it.

Probability amplitude (PA) at different lattice sites (N) of a square lattice () for the same parameter values as mentioned in Fig. 3. The top and bottom panels correspond to the results for the eigenstates selected from the left and right edges of the Landau band, respectively, while the middle one is for the eigenstates lie in the centre of the Landau band. The three different columns are associated with the three different Landau bands shown in Fig. 3. To understand the nature of energy eigenstates more clearly, in each figure, we present the results for two states those are chosen from the respective regions.

Probability amplitude (PA) at different lattice sites (N) of a square lattice () for the same parameter values as mentioned in Fig. 3. The top and bottom panels correspond to the results for the eigenstates selected from the left and right edges of the Landau band, respectively, while the middle one is for the eigenstates lie in the centre of the Landau band. The three different columns are associated with the three different Landau bands shown in Fig. 3. To understand the nature of energy eigenstates more clearly, in each figure, we present the results for two states those are chosen from the respective regions.

Hall conductivity as a function of Fermi energy for a square lattice () with *W* = 1 and . The dark-violet, pink and blue lines correspond to *Q* = 5, 10, and 15, respectively.

Hall conductivity as a function of Fermi energy for a square lattice () with *W* = 1 and . The dark-violet, pink and blue lines correspond to *Q* = 5, 10, and 15, respectively.

Hall conductivity as a function of Fermi energy for a square lattice () considering *W* = 1 and . The blue, pink, and dark-violet colors correspond to , 0.05, and 0.1, respectively.

Hall conductivity as a function of Fermi energy for a square lattice () considering *W* = 1 and . The blue, pink, and dark-violet colors correspond to , 0.05, and 0.1, respectively.

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