Renormalization group analysis of stability of band crossing in spin-1 spin-orbit coupled degenerate Fermi gas

In one of my recent paper entitled Topological phases in spin-1 Fermi gases with two-dimensional spin-orbit coupling, we study the topological phases of Rashba spin-orbit (SO) coupled three-component Fermi gases. In fact, I also investigated the stability of the band crossings under repulsive $SU(3)$-invariant interactions, which is not included in the paper. In the following, I share my note on this matter.

Abstract

In this note, I unitize mean-field theory (MFT), Renormalization Group (RG) and scaling to study the (in)stability of band touching points including Dirac point and quadratic band touching (QBT) in spin-1 degenerate Fermi gases with Rsahba-like SOC.

Review of system Hamiltonian

The single-particle Hamiltonian in k-space is
$$\begin{pmatrix} \delta_1-2t_s(\cos k_x + \cos k_y) & t_{so}(-\sin k_x + i\sin k_y) & 0 \\ -t_{so}(\sin k_x + i\sin k_y) & 2t_s(\cos k_x + \cos k_y) & t_{so}(-\sin k_x + i\sin k_y) \\ 0 & -t_{so}(\sin k_x + i\sin k_y) & \delta_2-2t_s(\cos k_x + \cos k_y) \end{pmatrix}.$$

If we only consider the touching points at $\Gamma$ point, the effective single-particle Hamiltonian becomes
$$\begin{pmatrix} \delta_1-(k_x^2+k_y^2) & t_{so}(-k_x + ik_y) & 0 \\ -t_{so}(k_x + ik_y) & k_x^2+k_y^2 & t_{so}(-k_x + ik_y) \\ 0 & -t_{so}(k_x + ik_y) & \delta_2-(k_x^2+k_y^2) \end{pmatrix},$$ where we also set $t_s=1$ (this does not affect the scaling since it can be treated as a mass term).

For QBT, we require $\delta_-=0$ and its low-energy Hamiltonian is
$$H_0 = \begin{pmatrix} 0 & (-k_x + ik_y)^2 \\ (k_x + ik_y)^2 & 0 \end{pmatrix}.$$

For the Dirac point, we only consider the cases when it appears between the lower two bands and consequently, $\delta_2=8$ is assumed (one may also set $\delta_1=1$ and investigate the problem, but those two cases are essentially the same). Now, we have
$$H_0 = \begin{pmatrix} 0 & -k_x + ik_y \\ -k_x - ik_y & 0 \end{pmatrix}$$ for the Dirac point.

Scaling and numeric renormalization

Before preceding, we first study the scaling of a massive (quadratic dispersion)
and massless (linear dispersion) Fermion fields respectively. A massive scaler (spinless) Fermi field can be described by the following Lagrange $\mathcal{L} = \phi^\dagger\frac{\partial\phi}{\partial\tau}+\frac{1}{2m}|\nabla\phi|^2 -\mu|\phi|^2-\frac{u}{2}\phi^\dagger\nabla\phi^\dagger\phi\nabla\phi$. At quantum critical point $\mu=0$ and $T=0$, such a a theory is invariant under scaling transformation $x’=xe^{-l}, \tau’=\tau e^{-zl}, \phi’=\phi e^{dl/2}$.

We choose dynamic critical exponent $z=2$ and the mass $m$ is assumed to remain invariant under scaling. Then we also have dim$[\mu]=2$ and dim$[u]=-d$. A similar argument for massless Dirac field $\mathcal{L} = \phi^\dagger_R\left(\frac{\partial}{\partial\tau} - iv_F\nabla\right)\phi_R+\phi^\dagger_L\left(\frac{\partial}{\partial\tau} + iv_F\nabla\right)\phi_L$ would give $x’=xe^{-l}, \tau’=\tau e^{-zl}, \phi_{R,L}’=\phi_{R,L} e^{l/2}$. We also require $z=1$ and dim$[\phi_{R,L}]=1/2$ holds for all dimension $d$.

Now, we consider an uniform $SU(3)$ invariant two-body contact interaction
$$V \sum_{i,\sigma\neq\sigma’} \hat{n}_{\sigma} \hat{n}_{\sigma’},$$
then we have dim$[V]=2-d$ and dim$[V]=d-1$ for a massive band touching and Dirac point respectively. Given our case $d=2$, $V$ is marginal (massive spinless fermion) or relevant (Dirac fermion).

Since the QBT is marginal, we calculate the RG beta function to one-loop order. Although it’s an effective two-level model, the third component is hardly mixed near the QBT. Giving the single particle Green function
$$G_0 = \begin{pmatrix} E-\sigma_k & \eta_k \\ \eta_k^* & E+\sigma_k \\ \end{pmatrix} ^{-1},$$
the one-loop correction counts two Feynman diagrams $(-1)\frac{i}{2\pi}\int dE\int\frac{d^2k}{(2\pi)^2}G_{0,ab}G_{0,ba}+(-1)^2\frac{i}{2\pi}\int dE\int\frac{d^2k}{(2\pi)^2}G_{0,aa}G_{0,bb} = \int\frac{d^2k}{(2\pi)^2}\frac{1}{2E_k}$, where $E_k=\sqrt{|\eta_k|^2+\sigma_k^2}$ is the single-particle energy.

Inserting back the effective Hamiltonian for QBT, we have $\beta(V)=\frac{dV}{dl}=\frac{1}{4\pi}V^2+O(V^3)$, where $l$ is momentum rescaling $k’=ke^{-l}$. Note that, this recovers exactly the QBT with $C_6$ symmetry and for $V>0$, the effective coupling would flow to strong coupling. One may also verify that the beta function vanishes at second order for Dirac points but $V$ is always relevant at second order for even higher band touching points with order $n>2$.

Mean-field theory

For the two-level model, we have the energy functional $E_{MFT}=\int dx\left(\psi^\dagger H_0\psi+Vn_an_b\right)$, where $\psi$ is the spinor wave function and we use $a,b$ to denote each charged Fermion field. The energy functional can be rewritten as $E_{MFT}=\int dx\left(\psi^\dagger H_0\psi - \frac{V}{4}(n_a-n_b)^2\right)$ to some constant with the particle number conservation assumption. Defining the order parameter $\delta=\langle \frac{V}{2}(n_a-n_b)\rangle$, and write the energy in terms of $\delta$, we have $E_{MFT}=\int dx\left(\psi^\dagger H_0\psi - 2\delta(n_a-n_b)+\frac{\delta^2}{V}\right)$.

Now, redefine the single-particle Hamiltonian $H_0=H_0-2\delta\sigma_z$, we can solve the following self-consistent equation $\delta=\frac{V}{2}\int dE\int\frac{d^2k}{(2\pi)^2}(G_{0,aa}(k)-G_{0,bb}(k))$, to obtain the order parameter, where $G_0$ is single-particle Green function. This can be easily solved and we obtain $\delta=-\frac{2\pi}{V}+\frac{V}{8\pi}\Lambda^2$ and $\delta=\frac{\Lambda^2}{2}\text{csch}(\frac{4\pi}{V})$ ($\Lambda$ is energy cutoff) for Dirac point and QBT respectively.

Order parameter versus interaction strength.

The result are plotted in the above figure and one can clearly observe that the Dirac point would achieve a finite mass term $-2\delta$ for arbitrarily weak interaction, thus it is instable when $V>0$. On the other hand, $\delta$ exponentially approaches $0$ when $V<\sqrt{2\pi}$ and is proportional to $V$ when $V\gg\sqrt{2\pi}$. This indicates that the QBT is stable against a finite interaction strength and can only be broken with sufficiently large interaction. We also remark that the instability (stability) of Dirac point (QBT) is independent of energy cutoff.

Self-energy correction

Now, we proceed to compute the self-energy correction using Hartree approximation $G^{-1}=G_0^{-1}+\Sigma_H$. The one-loop correction is given by
$$\begin{align}\Sigma_{H,ij}(k)&=(-1)\frac{i}{2\pi}\int dE\int\frac{d^2p}{(2\pi)^2}\big(G_{0,ia}(k)G_{0,bb}(p)G_{0,aj}(k)+G_{0,ia}(k)G_{0,ab}(p)G_{0,bj}(k)\\ &+G_{0,ib}(k)G_{0,ba}(p)G_{0,aj}(k)+G_{0,ib}(k)G_{0,aa}(p)G_{0,bj}(k)\big)\end{align},$$
where $i,j=a,b$.

Specifically, we have $$\Sigma_{H,aa}=\Sigma_{H,bb}=(-1)\frac{i}{2\pi}\int dE\int\frac{d^2p}{(2\pi)^2}\frac{E^3+E(\eta_k^*\eta_p+\eta_k\eta_p^*+\vert\eta_k\vert^2)}{(E^2-\vert\eta_p\vert^2)(E^2-\vert\eta_k\vert^2)^2}=0,$$ and

$$\begin{align} \Sigma_{H,ab}=\Sigma_{H,ba}^* &=\frac{i}{2\pi}\int dE\int\frac{d^2p}{(2\pi)^2}\frac{\eta_k^2\eta_p^*+E^2(2\eta_k+\eta_p)}{(E^2-\vert\eta_p\vert^2)(E^2-\vert\eta_k\vert^2)^2}\\ &=(-1)\int\frac{d^2p}{(2\pi)^2}\left(\frac{2\vert\eta_k\vert+\vert\eta_p\vert}{4\vert\eta_k\vert^3\vert\eta_p\vert(\vert\eta_k\vert+\vert\eta_p\vert)^2} \eta_k^2\eta_p^*-\frac{1}{4\vert\eta_k\vert(\vert\eta_k\vert+\vert\eta_p\vert)^2}(2\eta_k+\eta_p)\right)\\ &=\int\frac{d^2p}{(2\pi)^2}\frac{\eta_k}{2\vert\eta_k\vert(\vert\eta_k\vert+\vert\eta_p\vert)^2} \end{align}.$$

There are two distinct two-loop corrections and we first consider the one without momentum transfer, which is to count one-loop correction for the $G_0(p)$ terms
$$\begin{align} \Sigma_{H,aa} &=\Sigma_{H,bb}=(-1)\frac{i}{2\pi}\int dE\int\frac{d^2p}{(2\pi)}^2\int\frac{d^2q}{(2\pi)^2}\big(E^5+(E^3+E\vert\eta_k\vert^2)(\vert\eta_p\vert^2+\frac{1}{2}Re(\eta_q\eta_p^*))\\ &+E^3\vert\eta_k\vert^2+\frac{1}{2}Re(E^3(2\eta_p+\eta_q)\eta_k^*+E\eta_p^2\eta_q^*\eta_k^*)\big)/\left((E^2-\vert\eta_q\vert^2)(E^2-\vert\eta_p\vert^2)^2(E^2-\vert\eta_k\vert^2)^2\right) \end{align}$$
and it’s obviously trivial.

Acknowledgement

I thank Dr. T. Zeng from Westlake University for helpful discussions and conducting Exact Diagonalization (ED) numerics. The above results are consistent with ED.

Reference

• J. Hou, H. Hu, C. Zhang, Topological phases in spin-1 Fermi gases with two-dimensional spin-orbit coupling, arXiv 1809.04537.
• S. Sachdev, Quantum Phase Transition, Cambridge University Press, Cambridge, UK (1999).
• K. Sun, H. Yao, E. Fradkin, and S. A. Kivelson, Topological Insulators and Nematic Phases from Spontaneous Symmetry Breaking in 2D Fermi Systems with a Quadratic Band Crossing, Phys. Rev. Lett 103, 046811 (2009).