Surface topological quantum criticality:
Conformal manifolds and Discrete Strong Coupling Fixed Points

In this section, we begin with a minimal lattice model for a 3D topological insulator and explore its various topological phases. By numerically solving for the surface states, we show explicitly that the same topological bulk can have distinct surface states (see Fig.1).

To realize a lattice model for a 3d topological insulator, we discretize a 3d Dirac Hamiltonian. We consider a cubic lattice for the purpose. The resulting minimal lattice model has the form[53],

	$\displaystyle\mathcal{H}_{0}$	$\displaystyle=$	$\displaystyle\sum_{\textbf{k}}c^{\dagger}_{\textbf{k}}\mathcal{H}_{0}(\textbf{k})c_{\textbf{k}}$		(1)
	$\displaystyle\mathcal{H}(\textbf{k})$	$\displaystyle=$	$\displaystyle\sum_{i=x,y,z}\alpha_{i}\sin k_{i}+\beta\left[1-\sum_{i=x,y,z}t_{i}\cos k_{i}\right]$

where $c^{\dagger}_{\textbf{k}}$ is a four-component spinor defined in the spin and orbital basis with the definition, $c^{\dagger}_{\textbf{k}}=(\begin{array}[]{cc}c^{\dagger}_{1\textbf{k},\uparrow}\,\,c^{\dagger}_{1\textbf{k},\downarrow}\,\,c^{\dagger}_{2\textbf{k},\uparrow}\,\,c^{\dagger}_{2\textbf{k},\downarrow}\end{array})$ where 1(2) stands for the orbital index. $\alpha_{i}(i=x,y,z)$ and $\beta$ are the Dirac matrices that satisfy the following anti-commutation relations:

$$\{\alpha_{i},\alpha_{j}\}=2\delta_{ij},\,\,\{\alpha_{i},\beta\}=0$$

and they have the following definition: $\alpha_{i}=\tau_{x}\otimes s_{i}$ and $\beta=\tau_{z}\otimes s_{0}$. Here $\tau_{i}$ and $s_{i}$ ($i=x,y,z$) are Pauli matrices defined in atomic orbital space and spin space, respectively.

By tuning the parameters $t_{i}(i=x,y,z)$, we can study all 16 distinct topological phases (including weak and strong TI phases). However, our prime objective here is to look for the possibility of distinct surface states for the same bulk topological invariant. Therefore, we reduce the number of tuning parameters by setting $t_{x}=t_{y}=t$. This simplification leaves us with two tuning parameters: $t$ and $t_{z}$.

Now, we explore the various topological phases of the system as a function of the tuning parameters $t$ and $t_{z}$. Since the Hamiltonian has inversion symmetry, calculating the bulk topological invariants $(v_{0};v_{1},v_{2},v_{3})$ is straightforward. The $Z_{2}$ invariants can be calculated directly from the parity eigenvalues of the ground state at the eight TR invariant points in the bulk Brillouin zone(BZ)[54]. Fig.1(a), shows the phase diagram in the $t-t_{z}$ space.

To study the surface states explicitly, we derive the surface spectrum by solving a slab geometry of the lattice with open boundary conditions at the edges. We show the surface spectrum for the surface normal to the $z-$axis (Figs.1(b), 1(c)).

Now consider the blue-colored topological phase identified by the topological invariant $(1;0,0,0)$ in Fig.1. A bulk gap closing separates two regions. The region with $t>0$ has a single Dirac cone phase on the surface normal to the $z-$axis (see Fig.1)(c)). On the other hand, the blue region with $t<0$ has three Dirac cones surface, as evident from the surface spectrum shown in Fig.1(b). The bulk gap closes at the boundary separating these topological phases (the white line in Fig.1), pointing towards a topological phase transition.

Examining Fig.1, we observe that the regions with different surface states are not continuously connected despite belonging to the same bulk topological phase. We must cross a critical point to transition from a surface with a single Dirac cone to one with three Dirac cones.

This observation highlights that the same topological bulk can exhibit either a single or three Dirac cones on the surface, as shown in Fig. 1. A critical point separates the two distinct surface states, where the bulk gap closes. This is due to the higher symmetry of the lattice model, which, in addition to time-reversal symmetry (TRS), also possesses crystal symmetries. If the model only has TRS as the protecting symmetry, an adiabatic path connecting the two surface states should exist without closing the bulk gap. This section aims to look for a unitary deformation that connects these distinct surface states.

Even though realizing such a unitary deformation may seem straightforward, at least theoretically, it is tricky. We have to preserve the time-reversal symmetry throughout the unitary transformation. In addition, the Hamiltonian and the unitary operator should be smoothly defined on the surface of the toroid defined by the Brillouin zone. Starting from an effective field theory, we shall derive the explicit form of the unitary operator that could connect the two distinct surfaces.

Here, we study the bulk topological invariants and examine the underlying physics that leads to distinct surface states for the same topological bulk. We demonstrate that the transition between the distinct surface states is made possible by flipping the fermion mass signs $\delta_{\Gamma_{i}}$’s (defined in Eqns.2a,2b) at the TRI points on any one of the three $k_{a}=\pi(a=x,y,z)$ planes in bulk BZ (see Fig.2).

For a 3d lattice, there are eight TRI points in the Brillouin zone. If the lattice is cubic, then we could represent the TRI momenta simply as $\Gamma_{i=(n_{1},n_{2},n_{3})}=\pi(n_{1}\hat{x}+n_{2}\hat{y}+n_{3}\hat{z})$ where $n_{i}$ can be either 0 or 1. From ref.[7] the four $Z_{2}$ invariants $(v_{0};v_{1},v_{2},v_{3})$ have the following definition,

	$\displaystyle(-1)^{v_{0}}$	$\displaystyle=$	$\displaystyle\prod_{n_{i}=0,1}\delta_{n_{1},n_{2},n_{3}}$		(2a)
	$\displaystyle(-1)^{v_{j}}$	$\displaystyle=$	$\displaystyle\prod_{n_{(i\neq j)}=0,1;n_{j}=1}\delta_{n_{1},n_{2},n_{3}}$		(2b)

where $\delta_{n_{1},n_{2},n_{3}}$ can be $\pm 1$. For an inversion symmetric system, $\delta_{\Gamma_{i}}$ is simply the parity eigenvalue of the ground state at the TR-invariant point $\Gamma_{i}$[54].

It is clear from Eqn.2a that the system is topologically non-trivial for an odd number of negative fermion mass signs $\delta_{\Gamma_{i}}$’s; hence, a gapless surface state exists. The other three invariants ($v_{1},v_{2},v_{3}$) correspond to the product of $\delta_{\Gamma_{i}}$’s at the four TR-invariant points on the three distinct $k_{a}=\pi$ planes (here $a=x,y,z$ respectively for $v_{1},v_{2},v_{3}$).

Now consider the TI surface normal to the $z$-axis. The surface BZ is 2-dimensional with four TR-invariant points defined by $\Lambda_{i=(n_{1},n_{2})}=\pi(n_{1}\hat{x}+n_{2}\hat{y})$. To determine whether a Dirac cone exists at the surface TRI point $\Lambda_{i}$, we should calculate the time-reversal (TR) polarization $\pi_{\Lambda_{i}}$ defined at the TRI point. It is given by,

$\displaystyle\pi_{\Lambda_{i}}=\delta_{\Gamma_{i1}}\delta_{\Gamma_{i2}}$

(3)

where $\Gamma_{i1}$ and $\Gamma_{i2}$ have the same $k_{x}$ and $k_{y}$ components as $\Lambda_{i}$ but differ from each other in their $k_{z}$ component, which is $0$ for $\Gamma_{i1}$ and $\pi$ for $\Gamma_{i2}$. If $\pi_{\Lambda_{i}}=-1$, a surface Dirac cone exists at the $\Lambda_{i}$ point. So the bulk mass must change sign as it travels from $\Gamma_{i1}$ to $\Gamma_{i2}$ for a surface Dirac cone to appear at the $\Lambda_{i}$ point.

We use this procedure to locate the gapless points on the surface BZ in Fig. 2. In Fig. 2(a), there is a single negative sign at the $(\pi,\pi,\pi)$ point in the BZ. The bulk topological invariant is $(1;1,1,1)$. Using Eqn.3, we find that the TR polarization value $\pi_{\Lambda_{i}}=-1$ at the surface TRI point $\Lambda_{i}=(\pi,\pi)$ as shown in Fig.2(b). As a result, there must be a Dirac cone at the $(\pi,\pi)$ point in the surface BZ.

In Fig. 2(c), we flipped the bulk mass signs at the four TR points in the bulk BZ’s $k_{z}=\pi$ plane. The four topological invariants did not change under this transformation and remained $(1;1,1,1)$. However, in Fig. 2(d), we see that the projected surface BZ has its TR polarization value $\pi_{\Lambda_{i}}$ negative at three of the four TR invariant points. The TI surface now has three Dirac cones at these three TR invariant points. Note again that the bulk topology has not changed under this transformation.

Thus, deforming the TI surface from a single Dirac cone state to a three Dirac cone state or vice versa is fundamentally connected to the flipping of the sign of $\delta_{\Gamma_{i}}$’s at the four TRI points on any one of the three distinct $k_{a}=\pi$($a=x,y,z$) planes in bulk BZ. In the following, we develop a unitary operation that could flip the fermion mass signs at one of the three planes, thereby deforming the surface states.

Here, we shall develop an explicit unitary operator that can smoothly connect the two distinct surface states of a TI with the same bulk topological invariant. We consider the 3D TI to have inversion symmetry so that $\delta_{\Gamma_{i}}$ (see Eqn.2a) is just the sign of the mass gap (parity eigenvalue) at the TRI point $\Gamma_{i=(n_{1},n_{2},n_{3})}$.

Before introducing interactions, we emphasize a property of the gapless surface fermions that distinguishes them from the gapless Dirac fermions in a 2D lattice. To begin with, let us define a ’pseudo-chirality’ operator as

	$\displaystyle\Gamma_{s}$	$\displaystyle=$	$\displaystyle\beta\alpha_{z},\,\,\,\Gamma^{2}_{s}=1;$
	$\displaystyle\left[\Gamma_{s},\alpha_{x}\right]$	$\displaystyle=$	$\displaystyle[\Gamma_{s},\alpha_{y}]=0.$		(20)

In the $\alpha-\beta$ representation introduced in this article, $\Gamma_{s}=\tau_{y}s_{z}$, where $\tau_{y}$ is the Pauli matrix for the orbital part and $s_{z}$ is the Pauli matrix for spins (see Eqn.1).

The structure of the $\Gamma_{s}$ operator suggests that the eigenvalue can only be $\pm 1$. We call this term ’pseudo-chirality’ because the conventional chirality operator is usually not defined for even space dimensions. The important message here is that the surface fermions of a 3d TI must be $eigenstates$ of this ’pseudo-chirality’ operator. Since a strong TI has an odd number of Dirac cones, there would be an odd number of eigenstates for a specific eigenvalue of $\Gamma_{s}$ operator at a given surface. The majority of the eigenstates would have a specific pseudo-chirality so that the TI surface can be associated with that specific pseudo-chirality value. The location of the surface TRI point at which the Dirac cone is defined is irrelevant here.

This is rather an intrinsic property of the topological insulator, disregarding the detailed structure of the surface Brillouin zone. Now let us compare it with a gapless Dirac fermion in a bulk 2d lattice. Each Dirac point will always have both positive and negative pseudo-chiralities. Therefore, a 2d lattice would always have an equal number of eigenstates for each eigenvalue of the $\Gamma_{s}$ operator. So a bulk 2d lattice cannot be associated with a specific chirality. This is an intrinsic property of lattices due to the well-known fermion-doubling theory[40, 41].

For a weak TI, there is an even number of surface Dirac cones ($\mathcal{N}=2$). In general, there could be an equal number of eigenstates of positive and negative pseudo-chiralities. However, each Dirac cone is uniquely associated with a specific pseudo-chirality. From this perspective, the TI surface fundamentally differs from a bulk 2D lattice when considering individual Dirac cones. However, it is also possible that all (or a majority of) the surface Dirac cones share the same pseudo-chirality eigenvalue, giving the surface a definite pseudo-chirality. Therefore, one can assert that the TI surface is always $distinct$ from the bulk 2d lattice, even if the state is a weak TI.

Given this unique property of the topological surface fermions, it would be illuminating to study the effect of interactions between them which would be fundamentally different from the case of a 2d lattice. Here, we shall look at the dynamics of the surface subject to strong, attractive interactions. We are mainly interested in the singlet-pairing interactions that could result in a phase transition to a superconductor. We shall construct an effective 2-dimensional interaction action that could capture the essential low-energy dynamics of the system. We then apply the renormalization group analysis to study the effect of these four-fermion interactions on the TI surface. Since the four-fermion interaction is marginal in $1+1D$, we perform a $(2+\epsilon)$ expansion to study the RGE flow. In the coming sections, we explore the fixed point manifolds in the case of $\mathcal{N}=2$ followed by $\mathcal{N}=3$.

In section III, we started with a 3d Dirac lattice model suitable for a cubic lattice and solved it numerically to determine the surface states. However, since we want to study the effects of interactions on the TI surface, we need to better understand the analytical form of the surface states, at least near the corresponding TRI points. As discussed at the beginning of this section, we cannot simply start with a 2d lattice model to discuss surface fermions.

Here, we begin with a bulk 3d Hamiltonian, expand around those TRI points to first order in crystal momentum where the sign of mass gap is negative, and solve for the surface fermions analytically. We then end up with a 2-component spinor state localized in the direction normal to the surface. This is equivalent to the domain-wall construction typically employed in field theory to avoid the fermion-doubling problem[55, 56, 57, 58, 40, 41]. The details of the derivation are moved to the appendix A.

Now we are ready to formulate the effective 2D low-energy theory for the TI surface. As discussed in the previous section, the surface BZ contains four TRI points, allowing the surface to host up to four ($\mathcal{N}\leq 4$) Dirac cones at these TRI points, depending on the bulk topology. We assign a flavor index ${}^{\prime}i^{\prime}$ ($i=1,2,3,4$) to differentiate them. A surface fermion of flavor ${}^{\prime}i^{\prime}$ belongs to the Dirac cone at the TRI point $\textbf{Q}_{i}$ (refer to Fig.5 for $\textbf{Q}_{i}$’s definition).

Utilizing this framework, we present the Lagrangian governing the free-fermion dynamics on the TI surface.

	$\displaystyle\mathcal{L}_{f}$	$\displaystyle=$	$\displaystyle\sum_{i=1}^{4}\varepsilon_{i}\psi^{\dagger}_{i}\left[\partial_{\tau}+\hat{h}_{i}\right]\psi_{i}$		(21)
	$\displaystyle\text{where}\,\,\hat{h}_{1}$	$\displaystyle=$	$\displaystyle-is_{x}\partial_{y}+is_{y}\partial_{x}\,,\,\,\,\hat{h}_{2}=-is_{x}\partial_{y}-is_{y}\partial_{x}$
	$\displaystyle\hat{h}_{3}$	$\displaystyle=$	$\displaystyle is_{x}\partial_{y}-is_{y}\partial_{x}\,\,,\,\,\,\,\hat{h}_{4}=is_{x}\partial_{y}+is_{y}\partial_{x}$
	$\displaystyle\text{and}\,\,\varepsilon_{i}$	$\displaystyle=$	$\displaystyle\begin{cases}1,\,\,\,\text{Dirac cone present at }Q_{i}\\ 0\,\,\,\text{No Dirac cone at }Q_{i}\end{cases}$

where $\hat{h}_{i}$ could effectively describe the free-fermion dynamics near the TRI point $\textbf{Q}_{i}$. Here $\psi_{i}$ is a 2D 2-component fermion field operator with the definition $\psi^{\dagger}_{i}=(\psi^{\dagger}_{i,\uparrow}\,\,\,\psi^{\dagger}_{i,\downarrow})$ that represents the low-energy excitations near the TRI point $\textbf{Q}_{i}$. Here we inserted a term $\varepsilon_{i}$ artificially to take into account the presence or absence of the Dirac cone at the TRI point $\textbf{Q}_{i}$.

Whether a Dirac cone is present or absent at the given TRI point is solely determined by the TI bulk, as we saw in detail in section III. Therefore, one must start with the bulk topological phase and find out the precise value of $\varepsilon_{i},(i=1,2,3,4)$, and the locations of Dirac cones in the surface BZ to construct the effective surface theory.

Let’s consider a scenario where a net attractive interaction exists between electrons in a topological insulator. We’ll assume that the ultraviolet (UV) cut-off of this interaction is smaller than the bulk mass gaps at all eight time-reversal invariant (TRI) points, allowing us to treat the bulk fermions as non-interacting. We further assume that the pairing is of singlet order, intra-orbital, and is between time-reversal Kramer doublets. We start with a 3d interacting theory for the domain-wall fermions. We then integrate out the degrees of freedom normal to the surface to arrive at an effective 2d interacting theory[59]. The details of the derivation are presented in the appendix A for brevity.

Finally, we arrive at the following interaction Lagrangian,

$\displaystyle\mathcal{L}_{\text{I}}$

$\displaystyle=$

$\displaystyle-\sum^{4}_{i,j=1}\varepsilon_{i}\varepsilon_{j}V_{ij}\,\,\psi^{\dagger}_{i}(\textbf{r},\tau)s_{y}\psi^{\dagger T}_{i}(\textbf{r},\tau)\psi^{T}_{j}(\textbf{r},\tau)s_{y}\psi_{j}(\textbf{r},\tau)$

where,

$\displaystyle V_{ij}=V\frac{m_{i}m_{j}}{m_{i}+m_{j}}.$

Here $V$ is the attractive interaction strength in the 3D bulk. It is non-zero within the energy scale $[-\Lambda,\Lambda]$, where $\Lambda$ is the UV cut-off scale of the interaction. Here $i,j$ runs over the surface fermion flavors, subject to their presence. $m_{i}$ and $m_{j}$ are the bulk mass gaps of the Dirac cones between which the scattering of the singlet pair of fermions occurs due to effective attractive interaction. In the effective theory, the singlet pair of fermions from a surface Dirac cone can scatter to either the same or to another Dirac cone (if present), the strength of which is determined by the respective bulk mass gaps assigned to each of the Dirac cones. See Fig.5 for a schematic picture of the scattering.

Let us now study the symmetries of the non-interacting part of the theory defined in Eqn.LABEL:Lfrfmn. A surface with $\mathcal{N}$ Dirac cones can be pictured as a theory of $\mathcal{N}$ distinct flavors of 2-component Dirac fermions. We could define an internal flavor space of $\mathcal{N}$-dimensions. In this flavor space, we define a $\mathcal{N}$ component spinor as $\Psi^{\dagger}=\left(\begin{array}[]{cccc}\psi^{\dagger}_{1}&\psi^{\dagger}_{2}&\psi^{\dagger}_{3}&...\end{array}\right)$. Here $\psi^{\dagger}_{i}(i=1,2,..\mathcal{N})$ is a 2-component spinor (see Eqn.21). In this section, we shall show that the free fermions can be $invariant$ under the $SU(\mathcal{N})$ rotations of the Dirac spinor $\Psi$ in the flavor space if a proper spinor-base is chosen.

To begin with, we notice here that the free electron dynamics at different Dirac cones in a TI are usually not identical. The helicity of the Dirac cone at each of the TRI points differs. Here we define helicity as the orientation of the fermion spin relative to its linear momentum. For instance, the expectation value of the spin operator for the surface fermions at the TRI point $\textbf{Q}_{1}$ reads as $\bra{\psi_{1}}\hat{\textbf{s}}\ket{\psi_{1}}=\frac{1}{|\textbf{k}|}\left(k_{y}\hat{x}-k_{x}\hat{y}\right)$, while for the Dirac cone at $\textbf{Q}_{2}$, we see that $\bra{\psi_{2}}\hat{\textbf{s}}\ket{\psi_{2}}=\frac{1}{|\textbf{k}|}\left(k_{y}\hat{x}+k_{x}\hat{y}\right)$ in the momentum space.

However, they are all related by a mirror reflection. Therefore, the helicity of all the Dirac cones could be made equivalent to each other by applying a simple $SU(2)$ $\pi$-rotation (clockwise or anticlockwise) in the spinor space at each Dirac cone. After such a helicity transformation, $SU(\mathcal{N})$ flavor symmetry naturally arises in free fermions.

Consider a Dirac cone at the surface TRI point $\textbf{Q}_{i}$ (see Fig.5 for its definition). We look for a possible $SU(2)$ rotation in the spinor space such that the helicity of the Dirac cone at $\textbf{Q}_{i},i=1,2,3,4$ is made equal to $\textbf{Q}_{1}$. Let $U_{i}$ be the unitary matrix defined for the Dirac cone at $\textbf{Q}_{i}$ which realizes this transformation. $U_{i}$’s would have the following definitions,

$\displaystyle U_{1}$

$\displaystyle=$

$\displaystyle\hat{I},\,\,\,U_{2}=is_{x},\,\,\,U_{3}=is_{z},\,\,\,U_{4}=is_{y}$

(23)

In general, a unitary matrx $U_{i}$, $i=2,3,4$ or $is_{\alpha}(\alpha=x,y,z)$ realizes an $SU(2)$ spinor rotation about the $\alpha$-direction by an angle $\pi$ for a surface fermion at the TRI point $\textbf{Q}_{i}$. Rotating each flavor of the surface fermions independently as $\psi_{i}\rightarrow U_{i}\psi_{i}$, the free-fermion dynamics of all the Dirac cones appear to be identical.

Now we shall examine the interaction term defined in Eqn.LABEL:Lfrfmn under this helicity transformation. For this purpose, it would suffice to study how the singlet pair annihilation operator $\psi^{T}_{i}is_{y}\psi_{i}$ transforms under helicity transformation.

	$\displaystyle\psi^{T}_{i}is_{y}\psi_{i}$	$\displaystyle\rightarrow$	$\displaystyle\psi^{T}_{i}U^{T}_{i}is_{y}U_{i}\psi_{i}$
		$\displaystyle=$	$\displaystyle\psi^{T}_{i}is_{y}\psi_{i}$

We find that the pair annihilation operator is always invariant under the helicity transformation for all $U_{i}$. Hence the effective action defined by Eqns.LABEL:Lfrfmn is invariant under the helicity transformation. To study the effect of these interactions, we find it most convenient to work with a basis where different Dirac cones have the same helicity and appear to be identical.

As a result, we express the free theory of surface fermions in Eqn.21 in terms of the $\mathcal{N}$-component spinor field $\Psi(\textbf{x})$ as,

$\displaystyle\mathcal{L}_{f}$

$\displaystyle=$

$\displaystyle\Psi^{\dagger}\left[\left(\partial_{\tau}+\hat{h}_{1}\right)\otimes\hat{I}_{\mathcal{N}}\right]\Psi$

(24)

Here $\hat{I}_{\mathcal{N}}$ is an identity matrix in the flavor space. $\hat{h}_{1}$ is defined in Eqn.21. Also, notice that we haven’t used the term $\varepsilon_{i}$ here. This is because the helicity invariance implies the location of the surface Dirac cones is not relevant to the theory. Only the number of surface Dirac cones $\mathcal{N}$ matters.

One can see that the free theory of surface fermions in Eqn.24 is invariant under $SU(\mathcal{N})$ flavor rotations of the Dirac spinor. The single-particle Green’s function in the flavor space has the simple form of

$\displaystyle G(\textbf{k},i\omega_{n})$

$\displaystyle=$

$\displaystyle\frac{1}{i\omega_{n}-\left(s_{x}k_{y}-s_{y}k_{x}\right)}\otimes\hat{I}_{\mathcal{N}}.$

(25)

The Green’s function here is proportional to the identity operator in the flavor space, i.e. it is an $SU(\mathcal{N})$ singlet. However, the interaction part in Eqns.LABEL:Lfrfmn does not have such an $SU(\mathcal{N})$ flavor symmetry.

As noticed before in Eq.25 , the non-interacting fermion Green’s functions are $SU(\mathcal{N})$ invariant. The same can be said about products of multiple Green’s functions that don’t involve flavor symmetry-breaking terms. These properties of Green’s functions play an important role in the discussions of the renormalization group equations (RGEs) and their solutions. Following the free theory symmetry above, it turns out that the RGE (see next section) can have an emergent $SO(\mathcal{N})$ symmetry, i.e. RGEs are invariant under an $SO(\mathcal{N})$ flavor rotation of the interaction matrix $V_{ij},i,j=1,...,\mathcal{N}$ which is real and Hermitian. That is $V_{ij}=V_{ji}=V^{*}_{ij}$.

We emphasize here that the generic structure of $V_{ij}$ for surface fermions of the topological bulk always breaks the $SO(\mathcal{N})$ flavor symmetry. That can be explicitly seen in the Eqn.LABEL:Lfrfmn specifically suitable for phonon-mediated interactions([59]). At least two mechanisms lead to the breaking of $SO(\mathcal{N})$ symmetry. One is the surface Dirac cone of different flavors that have flavor-dependent localization lengths or are induced by different bulk mass gaps $m_{i}$. That leads to differences among the diagonal components of $V_{ij}$. The second mechanism is that the bulk interactions with gaped bulk fermions can induce interactions between different flavors of surface fermions. This results in non-zero off-diagonal components of $V_{ij}$. These will effectively break the symmetry of interactions in the flavor space.

However, we shall see in the next subsection that $SO(\mathcal{N})$ flavor rotation symmetry appears as an emergent symmetry in the renormalization group equations (RGEs) when the renormalization effect of interactions is studied up to the one-loop level. This emergent $SO(\mathcal{N})$ symmetry in the RGEs plays a crucial role in forming conformal manifolds in the parameter space, which we shall explore in subsequent sections.

So far, we have developed an effective interacting model for the TI surface. We are now ready to study the surface quantum criticality using the renormalization-group-equation (RGE) technique. In Eqn.LABEL:Lfrfmn, we have an interacting theory of $\mathcal{N}$ flavors of 2-component Dirac fermions in $(2+1)D$. Dimensional analysis suggests that the interaction term in Eqn.LABEL:Lfrfmn is marginal in $1+1$D dimensions. So in $(2+1)D$, the interaction term is irrelevant in the infrared (IR) limit if the interaction is weak. However, in the case of $\mathcal{N}=1$, it had been demonstrated that the interacting theory has an interacting UV fixed point [60, 61, 15, 62, 16, 17, 18]. Here, we study surface quantum criticality phenomena in a more general case with $\mathcal{N}>1$ surface Dirac cones and search for strong coupling conformal-field-theory fixed points that shall define possible universality classes of surface topological quantum criticality for a given $\mathcal{N}$.

Fig.6 illustrates the diagrams contributing to the renormalization of the interaction matrix element $V_{ij}$. As pointed out in the previous subsection, the effective field theory is not invariant under the $SO(\mathcal{N})$ flavor rotation. Nevertheless, we find here that there can be an emergent dynamic symmetry of $SO(\mathcal{N})$ in RGEs. For instance, the contribution from the one-loop diagram (Fig.6(a)), which is proportional to $\sum_{k}V_{ik}V_{kj}$, transforms as a rank-2 tensor when $V_{ij}$ is rotated in the flavor space.

The two-loop contributions are due to the fermion line renormalization, shown in Fig.6(c). Its contribution to the RGE flow of the matrix element $V_{ij}$ does not behave as a rank-2 tensor as it has the form of $V_{ij}\sum V_{ik}V_{ki}$. In our model, the fermion field renormalization has a flavor dependence. Hence, the resulting RGE with these two-loop contributions does not exhibit an $emergent$ $SO(\mathcal{N})$ symmetry.

Furthermore, the contribution from the three-loop diagram (Fig.6(b)) which is proportional to $\sum_{k}V_{ik}V^{2}_{kk}V_{kj}$ also does not transform as a tensor of any rank. So the emergent $SO(\mathcal{N})$ symmetry is broken only if we include the two-loop and other higher-loop contributions. One shall also notice that the contributions from Fig.6 (a) and (b) are the typical one-particle irreducible diagrams while (c) and (d) both are related to the standard fermion-field renormalization effect. The ones that are due to the field renormalization shown here do break the emergent $SO(\mathcal{N})$ symmetry.

The renormalization group study is performed in the $2+\epsilon$ space-time dimensions. Restricting ourselves to the one-loop order in the spirit of $\epsilon$ expansion, we can anticipate an emergent $SO(\mathcal{N})$ symmetry in the RG equations. Then we obtain the following RG equation[63, 64, 65],

$\displaystyle\frac{dV_{ij}}{dl}=-\epsilon V_{ij}+\frac{1}{c^{2}}\sum_{k}V_{ik}V_{kj}$

(26)

where $\epsilon=D-2$ and $D$ is the spacetime dimensions. Note that by increasing $l$, we are moving towards the IR limit of the theory. The numerical constant $c^{2}=\frac{\pi}{8}$. $V_{ij}$ is the renormalized dimensionless interaction matrix element. The derivation details of Eqn.26 are moved to the appendix.B for brevity.

As discussed before, limiting our calculation to one loop allows us to observe an emergent $SO(\mathcal{N})$ symmetry in the RG equation. Hence, it is possible to write the RG equation as a matrix differential equation. We could express $V_{ij}$ as an $\mathcal{N}\times\mathcal{N}$ dimensional symmetric matrix $\hat{V}$. The second term in the RHS is simply the square of this matrix. Let us also absorb the coefficient $c^{2}$ into the interaction strength by rescaling $\hat{V}\rightarrow\frac{\hat{V}}{c^{2}}$. So we could re-write this RG equation as,

$\displaystyle\frac{d\hat{V}}{dl}=-\epsilon\hat{V}+\hat{V}^{2}$

(27)

Here, we have expressed the interaction strength as an $\mathcal{N}\times\mathcal{N}$ symmetric matrix.

One can easily verify that Eq.27 has an emergent $SO(\mathcal{N})$ symmetry. That is, the RGEs are invariant under the following transformation.

$\displaystyle\hat{V}(l)\rightarrow\hat{R}^{-1}\hat{V}(l)\hat{R}$

(28)

if $\hat{R}$ is in an $SO(\mathcal{N})$ rotation group.

This dynamic symmetry of RGEs further implies that for any solution $\hat{V}_{c}(l)$ that is not an $SO(\mathcal{N})$ singlet, one can generate a smooth manifold of solutions by simply applying an $SO(\mathcal{N})$ rotation. This plays an important role in understanding the conformal manifolds below.

Note that the $SO(\mathcal{N})$ symmetry of the RG equation in Eqn.27 is $emergent$ and not fundamental. The theory does not have flavor rotation symmetry. The RG equation has this symmetry because we limited our calculations to one-loop contributions. A higher-loop contribution explicitly breaks the flavor rotation symmetry, as evident from the two- or three-loop diagrams shown in Fig.6 (b), (c), and (d).