# Generating Many Majorana Corner Modes and Multiple Phase Transitions in Floquet Second-Order Topological Superconductors

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Results

#### 3.1. Case 1: $\mu ={J}^{\prime}={\Delta}_{1}=0$

#### 3.2. Case 2: ${J}^{\prime}={\Delta}_{1}=0$

#### 3.3. Case 3: ${J}^{\prime}=0$

#### 3.4. Case 4: General Situation

## 4. Discussion and Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**An illustration of the 2D lattice model. The chains A (blue balls) and B (orange balls) are related to the operators ${\widehat{a}}^{(\u2020)}$ and ${\widehat{b}}^{(\u2020)}$ in $\widehat{H}$. A uniform chemical potential $\mu $ is applied to each site. In the Floquet model, the superconducting pairing amplitude $\Delta $ is replaced by $\Delta \left(t\right)={\sum}_{\ell}\delta (t-\ell )$.

**Figure 2.**Typical spectra and Majorana corner modes of the static SOTSC under open boundary conditions along x and y directions. (

**a**,

**b**) show the energy spectra E of $\widehat{H}$ versus J and $\Delta $, with other system parameters chosen to be $(\mu ,{J}^{\prime},{\Delta}_{1},{\Delta}_{2})=(0.25\pi ,0.05\pi ,0.2\pi ,0.4\pi )$. (

**c**,

**d**) show the probability distributions of the four Majorana corner modes with $E=0$ in (

**a**,

**b**) at $J=1.5\pi $ and $\Delta =1.5\pi $, respectively. The lattice sizes are set as ${N}_{x}={N}_{y}=60$ for all panels.

**Figure 3.**Gap functions and Floquet spectrum versus J under different boundary conditions in Case 1. (

**a**) Gap functions under PBCXY. (

**b**) Floquet spectrum under PBCXY. (

**c**) Floquet spectrum under OBCX, PBCY. (

**d**) Floquet spectrum under PBCX, OBCY. The crossing points between the vertical dotted lines and the horizontal axis show the bulk gap-closing points predicted by Equation (31). Other system parameters are set as $(\Delta ,{\Delta}_{2})=(0.5\pi ,0.2\pi )$. The number of cells along x and y directions are ${N}_{x}={N}_{y}=60$.

**Figure 4.**Floquet spectrum versus J under OBCs in Case 1. (

**a**) Quasienergies at different J under OBCs along both x and y directions (OBCXY). The crossing points between the vertical dotted lines and the horizontal axis show the bulk gap-closing points predicted by Equation (31). The numbers in red denote the numbers of Floquet corner modes at zero and $\pi $ quasienergies. (

**b**,

**c**) show the absolute values of quasienergies of the first fifteen (in stars) and last eleven (in circles) Floquet eigenstates indexed by j at $J=2.5\pi $. Other system parameters are $(\Delta ,{\Delta}_{2})=(0.5\pi ,0.2\pi )$. The number of cells along x and y directions are ${N}_{x}={N}_{y}=60$. The total number of Floquet eigenstates is $N=$ 14,400.

**Figure 5.**Probability distributions of Floquet corner modes in Case 1 with zero and $\pi $ quasienergies in panels (

**a**–

**c**) and panels (

**d**,

**e**), respectively. Other system parameters are chosen to be $(J,\Delta ,{\Delta}_{2})=(2.5\pi ,0.5\pi ,0.2\pi )$. The number of cells along x and y directions are ${N}_{x}={N}_{y}=60$.

**Figure 6.**Floquet spectrum versus ${k}_{x}$ (${k}_{y}$) in Case 1 under PBCX, OBCY (OBCX, PBCY) in panels (

**a**–

**c**) [(

**d**–

**f**)]. The gray dots, red circles and blue stars highlight the bulk states, and states localized around the left edge and the right edge of the lattice. The value of hopping amplitude is set to $J=0$ for panels (

**a**,

**d**), to the first bulk gap-closing point at $E=\pm \pi $ for panels (

**b**,

**e**), and to the first bulk gap-closing point at $E=0$ for panels (

**c**,

**f**). Other system parameters are set as $(\Delta ,{\Delta}_{2})=(0.5\pi ,0.2\pi )$.

**Figure 7.**Gap functions and Floquet spectrum versus J under different boundary conditions in Case 2. (

**a**) Gap functions under PBCXY. (

**b**) Floquet spectrum under PBCXY. (

**c**) Floquet spectrum under OBCX, PBCY. (

**d**) Floquet spectrum under PBCX, OBCY. The crossing points between the vertical dotted lines and the horizontal axis show the bulk gap-closing points predicted by Equation (36). Other system parameters are set as $(\mu ,\Delta ,{\Delta}_{2})=(0.25\pi ,0.5\pi ,0.2\pi )$. The number of cells along x and y directions are ${N}_{x}={N}_{y}=60$.

**Figure 8.**Floquet spectrum versus J under OBCs in Case 2. (

**a**) Quasienergies at different J under OBCXY. The crossing points between the vertical dotted lines and the horizontal axis show the bulk gap-closing points predicted by Equation (36). The numbers in red denote the numbers of Floquet corner modes at zero and $\pi $ quasienergies. (

**b**,

**c**) show the absolute values of quasienergies of the first twelve (in stars) and last eleven (in circles) Floquet eigenstates indexed by j at $J=2\pi $. Other system parameters are $(\mu ,\Delta ,{\Delta}_{2})=(0.25\pi ,0.5\pi ,0.2\pi )$. The number of cells along x and y directions are ${N}_{x}={N}_{y}=60$. The total number of Floquet eigenstates is $N=$ 14,400.

**Figure 9.**Probability distributions of Floquet corner modes in Case 2 with quasienergies zero and $\pi $ in panels (

**a**,

**b**) and panels (

**c**,

**d**), respectively. Other system parameters are set as $(J,\mu ,\Delta ,{\Delta}_{2})=(2\pi ,0.25\pi ,0.5\pi ,0.2\pi )$. The number of cells along x and y directions of the lattice are ${N}_{x}={N}_{y}=60$.

**Figure 10.**Floquet spectrum versus ${k}_{x}$ (${k}_{y}$) in Case 2 under PBCX, OBCY (OBCX, PBCY) in panels (

**a**–

**c**) [(

**d**–

**f**)]. The gray dots, red circles and blue stars highlight the bulk states, and states localized around the left edge and the right edge of the lattice. The value of hopping amplitude is set to $J=\mu $ for panels (

**a**,

**d**), to the first bulk gap-closing point at $E=\pm \pi $ for panels (

**b**,

**e**), and to the first bulk gap-closing point at $E=0$ for panels (

**c**,

**f**). Other system parameters are chosen to be $(\mu ,\Delta ,{\Delta}_{2})=(0.25\pi ,0.5\pi ,0.2\pi )$.

**Figure 11.**Gap functions and Floquet spectrum versus J under different boundary conditions in Case 3. (

**a**) Gap functions under PBCXY. (

**b**) Floquet spectrum under PBCXY. (

**c**) Floquet spectrum under OBCX, PBCY. (

**d**) Floquet spectrum under PBCX, OBCY. The crossing points between the vertical dotted lines and the horizontal axis show the bulk gap-closing points predicted by Equation (40). Other system parameters are set as $(\mu ,\Delta ,{\Delta}_{1},{\Delta}_{2})=(0.25\pi ,0.5\pi ,0.2\pi ,0.4\pi )$. The number of cells along x and y directions are ${N}_{x}={N}_{y}=60$.

**Figure 12.**Floquet spectrum versus J under OBCs in Case 3. (

**a**) Quasienergies at different J under OBCXY. The crossing points between the vertical dotted lines and the horizontal axis show the bulk gap-closing points predicted by Equation (40). The numbers in red denote the numbers of Floquet corner modes at zero and $\pi $ quasienergies. (

**b**,

**c**) show the absolute values of quasienergies of the first sixteen (in stars) and last sixteen (in circles) Floquet eigenstates indexed by j at $J=3\pi $. Other system parameters are $(\mu ,\Delta ,{\Delta}_{1},{\Delta}_{2})=(0.25\pi ,0.5\pi ,0.2\pi ,0.4\pi )$. The number of cells along x and y directions are ${N}_{x}={N}_{y}=60$. The total number of Floquet eigenstates is $N=$ 14,400.

**Figure 13.**Probability distributions of Floquet corner modes in Case 3 with quasienergies zero and $\pi $ in panels (

**a**–

**c**) and panels (

**d**–

**f**), respectively. Other system parameters are set as $(J,\mu ,\Delta ,{\Delta}_{1},{\Delta}_{2})=(3\pi ,0.25\pi ,0.5\pi ,0.2\pi ,0.4\pi )$. The number of cells along x and y directions are ${N}_{x}={N}_{y}=60$.

**Figure 14.**Floquet spectrum versus ${k}_{x}$ (${k}_{y}$) in Case 3 under PBCX, OBCY (OBCX, PBCX) in panels (

**a**–

**c**) [(

**d**–

**f**)]. The gray dots, red circles and blue stars highlight the bulk states, and states localized around the left edge and the right edge of the lattice. The value of hopping amplitude is set to $J=\mu $ for panels (

**a**,

**d**), to the first bulk gap-closing point at $E=\pm \pi $ for panels (

**b**,

**e**), and to the first bulk gap-closing point at $E=0$ for panels (

**c**,

**f**). Other system parameters are set as $(\mu ,\Delta ,{\Delta}_{1},{\Delta}_{2})=(0.25\pi ,0.5\pi ,0.2\pi ,0.4\pi )$.

**Figure 15.**Gap functions and Floquet spectrum versus J under different boundary conditions in Case 4. (

**a**) Gap functions under PBCXY. (

**b**) Floquet spectrum under PBCXY. (

**c**) Floquet spectrum under OBCX, PBCY. (

**d**) Floquet spectrum under PBCX, OBCY. Other system parameters are set as $(\mu ,\Delta ,{\Delta}_{1},{\Delta}_{2},{J}^{\prime})=(0.25\pi ,0.5\pi ,0.2\pi ,0.4\pi ,0.05\pi )$. The number of cells along x and y directions are ${N}_{x}={N}_{y}=60$.

**Figure 16.**Floquet spectrum versus J under OBCs in Case 4. (

**a**) Quasienergies at different J under OBCXY. (

**b**,

**c**) show the absolute values of quasienergies of the first sixteen (in stars) and last sixteen (in circles) Floquet eigenstates indexed by j at $J=3\pi $. Other system parameters are $(\mu ,\Delta ,{\Delta}_{1},{\Delta}_{2},{J}^{\prime})=(0.25\pi ,0.5\pi ,0.2\pi ,0.4\pi ,0.05\pi )$. The number of cells along the x and y directions are ${N}_{x}={N}_{y}=60$. The total number of Floquet eigenstates is $N=$ 14,400.

**Figure 17.**Probability distributions of Floquet corner modes in Case 4 with quasienergies at or close to zero and $\pi $ in panels (

**a**–

**c**) and panels (

**d**–

**f**), respectively. Other system parameters are set as $(J,\mu ,\Delta ,{\Delta}_{1},{\Delta}_{2},{J}^{\prime})=(3\pi ,0.25\pi ,0.5\pi ,0.2\pi ,0.4\pi ,0.05\pi )$. The number of cells along x and y directions are ${N}_{x}={N}_{y}=60$.

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## Share and Cite

**MDPI and ACS Style**

Zhou, L.
Generating Many Majorana Corner Modes and Multiple Phase Transitions in Floquet Second-Order Topological Superconductors. *Symmetry* **2022**, *14*, 2546.
https://doi.org/10.3390/sym14122546

**AMA Style**

Zhou L.
Generating Many Majorana Corner Modes and Multiple Phase Transitions in Floquet Second-Order Topological Superconductors. *Symmetry*. 2022; 14(12):2546.
https://doi.org/10.3390/sym14122546

**Chicago/Turabian Style**

Zhou, Longwen.
2022. "Generating Many Majorana Corner Modes and Multiple Phase Transitions in Floquet Second-Order Topological Superconductors" *Symmetry* 14, no. 12: 2546.
https://doi.org/10.3390/sym14122546