Stability of Time-Reversal Symmetry Protected Topological Phases

Time-reversal symmetry (TRS) protected topological phases, such as TRS protected topological insulator (TI) in two- and three-dimension, are intriguing state-of-matters that have been extensively studied in the past two decades. Unlike the quantum Hall effect, the topological classifications of these states require the presence of TRS classification_review . With TRS, the nontrivial topological properties have been firmly established in a closed system review_TI1 ; review_TI2 . A natural question is whether the concept of TRS protected topology, as well as its physical consequences such as quantized conductance, still holds in the presence of coupling to environment. This is certainly a very important issue, because for any practical applications of these topological materials, it is inevitable that the materials should be coupled to external environment.

A natural expectation is that the protection from TRS is guaranteed if the Hamiltonian of the system, the environment, and the coupling between the environment and the system all obey TRS. However, this expectation is recently challenged by McGinley and Cooper Cooper . They show explicitly that the coupling to environment can lead to de-cohernence between two Kramers doublet state even though the coupling and the environment both obey TRS. Their argument is deeply rooted in the fact that even if an isolated system obeys TRS, its subsystem can behavior as seemingly violating the TRS. Actually, this fact plays a key role in thermalization of a closed quantum system. In quantum thermalization, considering a pure state of a closed system whose evolution equations obey the TRS, the evolution of its subsystem can undergo an irreversible process that loses information and reach thermalization with the rest of the system acting as a bath thermalization1 ; thermalization2 .

Without loss of generality, we consider an open quantum system coupled to environment through a pair of operators $\hat{\mathcal{O}}$ and $\hat{\mathcal{O}}^{\dagger}$, and both operators obey TRS but do not have to be hermitian. Nevertheless, the entire Hamiltonian, including the system, the system-environment coupling and the environment itself, is hermitian and obeys TRS. By treating the degree-of-freedoms of the environment with the Markovian approximation, the open quantum system can be described by a non-Hermitian Hamiltonian with a Langevin noise term, which ensures the quantum mechanical commutative relation and preserves the trace of the density matrix supple . This non-Hermitian Hamiltonian can be generally written as

where $\gamma$ is the dissipation strength. $\hat{H}_{0}$ is the Hermitian Hamiltonian of the system itself, and it also obeys TRS. $\hat{\xi}$ is the Langevin noise operators that satisfy $\langle\hat{\xi}(t)\hat{\xi}^{\dagger}(t^{\prime})\rangle=2\gamma\delta(t-t^{\prime})$ and $\langle\hat{\xi}^{\dagger}(t)\hat{\xi}(t^{\prime})\rangle=\langle\hat{\xi}(t)\hat{\xi}(t^{\prime})\rangle=\langle\hat{\xi}^{\dagger}(t)\hat{\xi}^{\dagger}(t^{\prime})\rangle=0$. All the calculation done with this non-Hermitian calculation should be accompanied by averaging over the Langevin noise term in the end. In Ref. Pan , we have developed a non-Hermitian linear response theory. This theory starts with the equilibrium state of $\hat{H}_{0}$ and treats dissipation order by order, which determines how an equilibrium system responds to a weak dissipation. To implement the non-Hermitian linear response theory, we should introduce an interaction picture which separates out the dissipation term from the system term. For instance, in the interaction picture, we should define $\hat{\mathcal{O}}^{I}(t)=e^{i\hat{H}_{0}t}\hat{\mathcal{O}}e^{-i\hat{H}_{0}t}$, and similar definitions for other operators with upper scribe I.

Summary of Results. We consider generally a pair of Kramers degenerate eigen-states of $\hat{H}_{0}$, say $\ket{\Psi}_{1}$ and $\ket{\Psi}_{2}$. When we specifically consider a TRS protected TI, they can be chosen as a pair of degenerate edge states located at the same edge. We denote the Hilbert space formed by these two states as $\mathcal{H}_{\text{K}}$. In this letter, by studying the linear response of the density matrix, the Green’s function and the matrix element of a local impurity potential respectively, we obtain three main results as summarized below:

1. Loss of Coherence. Suppose that initially the quantum state is a pure state in $\mathcal{H}_{\text{K}}$ as $\ket{\Psi}=\alpha_{1}\ket{\Psi_{1}}+\alpha_{2}\ket{\Psi_{2}}$, where $\alpha_{i=1,2}$ are two constants, the initial density matrix is given by $\hat{\rho}_{\text{K}}(0)=|\Psi\rangle\langle\Psi|$. By turning on the dissipation, the quantum state evolves under $\hat{H}$ and the density matrix becomes $\hat{\rho}(t)$. By projecting onto $\mathcal{H}_{\text{K}}$ space by the projection operator $\hat{\Pi}_{\text{K}}$, one can obtain the projected density matrix $\hat{\rho}_{\text{K}}(t)=\frac{1}{\mathcal{N}}\hat{\Pi}_{\text{K}}\hat{\rho}(t)\hat{\Pi}_{\text{K}}$, with normalization factor $\mathcal{N}={\rm Tr}(\hat{\Pi}_{\text{K}}\hat{\rho}(t)\hat{\Pi}_{\text{K}})$. We define $\delta\hat{\rho}_{\text{K}}(t)=\hat{\rho}_{\text{K}}(t)-\hat{\rho}_{\text{K}}(0)$. We show that $\delta\hat{\rho}_{\text{K}}(t)$ is not proportional to $\hat{\rho}_{\text{K}}(0)$.

2. Break of Degeneracy. With dissipation, the retarded Green’s function in $\mathcal{H}_{\text{K}}$ space is a two-by-two matrix $\mathcal{G}$ with the matrix elements defined as

where $\hat{c}_{i}$ and $\hat{c}^{\dagger}_{i}$ are annihilation and creation operators corresponding to eigenstates $\ket{\Psi_{i}}$ of $\hat{H}_{0}$, and $\hat{c}_{i}(t)=e^{i\hat{H}t}\hat{c}_{i}e^{-i\hat{H}t}$. We show that $\mathcal{G}_{ij}$ is no longer proportional to an identity matrix in the $\mathcal{H}_{\text{K}}$ space.

3. Presence of Backscattering. For a local impurity potential $\hat{V}$, we consider the matrix element of this impurity potential between two Kramers states, i.e. $V^{0}_{ij}=\bra{\Psi_{i}}\hat{V}\ket{\Psi_{j}}$. Suppose, without dissipation, this matrix element is identically zero for $i\neq j$. This can be satisfied, for instance, when $\ket{\Psi_{1}}$ and $\ket{\Psi_{2}}$ are respectively the left-moving and the right-moving edge states of a quantum spin Hall state. With dissipation, we need to consider

where $\hat{V}(t)=e^{i\hat{H}t}\hat{V}e^{-i\hat{H}t}$. We show that $V_{ij}(t)\neq 0$ for $i\neq j$.

Result 1 naturally leads to $S_{\text{v}}(t)\neq S_{\text{v}}(t=0)$ and $S_{\text{R}}(t)\neq S_{\text{R}}(t=0)$, where $S_{\text{v}}(t)=-\text{Tr}\hat{\rho}_{\text{K}}(t)\log\hat{\rho}_{\text{K}}(t)$ is the von Neumann entropy and $S_{\text{R}}(t)=-\log\text{Tr}\hat{\rho}^{2}_{\text{K}}(t)$ is the second Renyi entropy. That is to says, for $t>0$, the entropy becomes non-zero and the system loses its phase coherence. This is consistent with the result presented in Ref. Cooper . Result 2 and 3 are the central results of this work. With the Result 2, we can further plot the spectrum function $A(\omega)$, which shows two split peaks. This means the lack of Kramers degeneracy for a non-Hermitian open system even though the coupling to environment also respects the TRS. Result 3 is directly related to the two-dimensional quantum spin Hall. Quantized conductance is the hallmark of quantum spin Hall, due to the forbidden of the backscattering between the left- and the right-moving edge states. Therefore, the presence of backscattering means that the conductance of a quantum spin Hall state is not perfectly quantized. This physics has been qualitatively discussed in Ref. dissipation ; Cooper . This is perhaps one of the reasons that accuracy of quantization observed in quantum spin Hall samples so far exp ; dissipation ; Du_exp is far less than that observed in quantum Hall samples, in addition to other possible explanation such as the inelastic scatterings exp . These results are essentially due to the irreversible nature of the bath, and bare a lot of similarity as the H-theorem in statistical mechanics. In other word, these results can be viewed as the manifestations of the H-theorem in the TRS protected topology.

Application of the Schur’s Lemma. Before proceeding into the details of the derivation, we should emphasize that these results essentially rely on the TRS being an anti-unitary symmetry. In other words, if the symmetry that protects the topological phase is a unitary symmetry, the phenomena 1-3 described above should not occur. Mathematically, the difference roots in the celebrated Schur’s Lemma in the group theory Schur . The Schur’s Lemma says that, for a unitary group, if an operator $\hat{M}$ commutes with all elements in the group, then this operator, in an irreducible representation, has to be proportional to an identity matrix. Nevertheless, when the Schur’s Lemma is applied to an anti-unitary group, not only the operator $\hat{M}$ has to commute with all elements in the group, but also the operator $\hat{M}$ has to be a hermitian operator, then this operator is proportional to an identity matrix in an irreducible representation anti-Schur . As we will see below, when the hermitian condition and respecting the anti-unitary symmetry condition cannot be satisfied simultaneously, this operator is generally no longer proportional to identity. This is the key mathematical reason responsible for the difference between the unitary symmetry protection and the anti-unitary symmetry protection.

To be more concrete, we will give two examples that will be repeated used below:

The first example is about $\hat{\Pi}_{\text{K}}\hat{\mathcal{O}}^{\text{I}}(t)\hat{\mathcal{O}}^{{\dagger},\text{I}}(t)\hat{\Pi}_{\text{K}}$. By using the fact that the states in $\mathcal{H}_{\text{K}}$ are degenerate state of $\hat{H}_{0}$, we have $\hat{\Pi}_{\text{K}}e^{\pm i\hat{H}_{0}t}=e^{\pm i\hat{H}_{0}t}\hat{\Pi}_{\text{K}}=e^{\pm iE_{0}t}\hat{\Pi}_{\text{K}}$, and therefore, $\hat{\Pi}_{\text{K}}\hat{\mathcal{O}}^{\text{I}}(t)\hat{\mathcal{O}}^{{\dagger},\text{I}}(t)\hat{\Pi}_{\text{K}}=\hat{\Pi}_{\text{K}}\hat{\mathcal{O}}\hat{\mathcal{O}}^{\dagger}\hat{\Pi}_{\text{K}}$. Note that $\hat{\Pi}_{\text{K}}\hat{\mathcal{O}}\hat{\mathcal{O}}^{\dagger}\hat{\Pi}_{\text{K}}$ is hermitian and time-reversal symmetric. It is also important to note that the Hilbert space $\mathcal{H}_{\text{K}}$ of two Kramers degenerate states forms an irreducible representation of the TRS, thus, the projection $\hat{\Pi}_{\text{K}}$ enforces the restriction to an irreducible space of TRS. Therefore, this term obeys the Schur’s Lamma, and consequently, it is proportional to identity. The same holds for $\hat{\Pi}_{\text{K}}\hat{\mathcal{O}}^{{\dagger},\text{I}}(t)\hat{\mathcal{O}}^{\text{I}}(t)\hat{\Pi}_{\text{K}}$.

The second example is about $\hat{\Pi}_{\text{K}}\hat{\mathcal{O}}^{\text{I}}(t)\hat{\Pi}_{\text{K}}$ and $\hat{\Pi}_{\text{K}}\hat{\mathcal{O}}^{{\dagger},\text{I}}(t)\hat{\Pi}_{\text{K}}$. It can also be shown that $\hat{\Pi}_{\text{K}}\hat{\mathcal{O}}^{\text{I}}(t)\hat{\Pi}_{\text{K}}=\hat{\Pi}_{\text{K}}\hat{\mathcal{O}}\hat{\Pi}_{\text{K}}$. Because $\hat{\mathcal{O}}$ is time-reversal symmetric but is generally not hermitian, this term does not satisfy the Schur’s Lemma for the anti-unitary TRS. The same holds for $\hat{\Pi}_{\text{K}}\hat{\mathcal{O}}^{{\dagger},\text{I}}(t)\hat{\Pi}_{\text{K}}$. However, for unitary symmetry, because the Schur’s Lemma does not require the operator being hermitian, all these operators are proportional to identity in an irreducible space if they obey the unitary symmetry. Hence, the unitary symmetry protected topological states are stable against coupling to environment.

Loss of Coherence. Here we consider density matrix in the interaction picture $\hat{\rho}(t)=\hat{\cal U}(t)\hat{\rho}_{\text{K}}(0)\hat{\cal U}^{\dagger}(t)$ with $\hat{\cal U}(t)=e^{i\hat{H}_{0}t}e^{-i\hat{H}t}$, and by expanding $\hat{\rho}(t)$ to the leading order of $\gamma$ and averaging the noise, we can obtain

where the second term results from averaging over the Langevin noise. By projecting back to $\mathcal{H}_{\text{K}}$, the first term in the r.h.s of Eq. (4) can be written as

and the second term in the r.h.s of Eq. (4) can be written as

With the two examples discussed above, we can conclude that Eq. (5) is proportional to $\hat{\rho}_{\text{K}}(0)$ but Eq.(6) is not proportional to $\hat{\rho}_{\text{K}}(0)$. Hence $\delta\hat{\rho}_{\text{K}}(t)$ is not proportional to $\hat{\rho}_{\text{K}}(0)$, and the entropy changes.

Break of Degeneracy. Here we apply the linear response theory to the Green’s function defined in Eq.(2), and consider that the Kramers doublet are both occupied by a pair of fermions. Similar as the discussion above, we consider $\delta G_{ij}=\mathcal{G}_{ij}-\mathcal{G}^{(0)}_{ij}$, where

and $\mathcal{G}^{(0)}_{ij}$ is the Green’s function without dissipation. It is easy to see that $\mathcal{G}^{0}_{ij}\propto\delta_{ij}$. What we need to show is that $\delta\mathcal{G}_{ij}$ is not proportional to $\delta_{ij}$. We can also expand $\delta\mathcal{G}_{ij}$ to the leading order of $\gamma$. We shall not show the full expression of this term here supple . Generally speaking, there are two types of terms in the leading order expansion. One kind of terms include, for instance,

which involve $\hat{\Pi}_{\text{K}}\hat{\mathcal{O}}^{{\dagger},\text{I}}(t_{1})\hat{\mathcal{O}}^{\text{I}}(t_{1})\hat{\Pi}_{\text{K}}$. The other kind of terms include, for instance,

which involve $\hat{\Pi}_{\text{K}}\hat{\mathcal{O}}^{\text{I}}(t_{1})\hat{\Pi}_{\text{K}}$ and $\hat{\Pi}_{\text{K}}\hat{\mathcal{O}}^{{\dagger},\text{I}}(t_{1})\hat{\Pi}_{\text{K}}$. With the two examples discussed above, we can also see that the first kind of term is still proportional to $\delta_{ij}$ but the second kind of term is not. Hence, up to the leading order of $\gamma$, $\mathcal{G}$ is already not an identity matrix and the spectrum is split.

Presence of Backscattering. Here we consider the matrix element defined in Eq. (3). Similarly, we define $\delta V_{ij}=V_{ij}(t)-V^{0}_{ij}$, and we expand $\delta V_{ij}$ to the leading order of $\gamma$ supple . Here, as a typical example, we focus on one of the terms that are similar as the ones discussed above, which reads

Here we should note a difference between the discussion here and the two cases above. Above two results can both be proved within the Kramers degenerate space $\mathcal{H}_{\text{K}}$. However, if in this case we are restricted in the $\mathcal{H}_{\text{K}}$ space, $V^{\text{I}}$ is an identity matrix that commutes with $\hat{H}_{0}$. Thus, Eq. (10) becomes

where $\hat{\Pi}_{\text{K}}\hat{\mathcal{O}}^{{\dagger}}\hat{\mathcal{O}}\hat{\Pi}_{\text{K}}$ is hermitian and obeys TRS. One can show that this holds for other terms in the leading order expansion of $\delta V_{ij}$. Therefore, restricted in $\mathcal{H}_{\text{K}}$ space, it is an identity matrix and cannot induce backscattering.

Hence, we should consider $\hat{V}$ operator out of $\mathcal{H}_{\text{K}}$ space, where $\hat{V}$ is no longer represented as identity and in general does not commute with $\hat{H}_{0}$. Then, it is easy to see that the operator in the $[...]$ of Eq. (10) does not respect TRS, although it is a hermitian one. Therefore, this term does not obey the Schur’s lemma and is not proportional to identity. Once not being identity matrix, nothing guarantees this term to be a diagonal matrix, and generically, the off-diagonal matrix elements exist, which lead to $\delta V_{ij}\neq 0$ for $i\neq j$. Similar discussions can be applied to other terms in the first order expansion of $\delta V_{ij}$. Taking $i$ and $j$ as a pair of degenerate counter-propagating edge states of a quantum spin Hall, we have now established the presence of backscattering and the absence of perfect quantization of conductance.

Example: the Kane-Mele model with Dissipation. Here we use the celebrated Kane-Mele model on a honeycomb lattice for two-dimensional quantum spin Hall to illustrate these three results more concretely kane-mele . For this model, we have

where $i$ and $j$ are site index and $s$ and $s^{\prime}$ are spin index, and $\bm{\sigma}$ are the Pauli matrices. The first term is the nearest neighbor hopping with strength $J$. The second term is a spin-orbit coupling between second neighbor hopping, with $\nu_{ij}=\pm 1$ and strength $\lambda_{\text{SO}}$. The third term is the nearest Rashba term with strength $\lambda_{\text{R}}$, where $\bm{d}_{ij}$ is the vector connecting $i$- and $j$-sites. The last term is a staggered potential with $\xi_{i}=\pm 1$ for different sublattices and strength $\lambda_{\nu}$. We choose the parameters such that the model is in the topological nontrivial insulator state.

We introduce the coupling operator $\hat{O}$ either defined on site-$i$ as

or defined on a nearest neighboring link $\langle i,j\rangle$ as

It is easy to see that the operators defined above obey TRS and are not hermitian. In the numerical simulation, we include a number of $\hat{O}$ operators defined above located at the edge of an sample, and the number of coupling operators is denoted by $M$. We note that the discussion above can be generalized straightforwardly to the cases with more coupling operators.

Here we numerically diagnolize the Kane-Mele model on a $N_{x}\times N_{y}$ sample, with open boundary condition along $\hat{x}$ and periodical boundary condition along $\hat{y}$. Here we should emphasize that, in order to obey TRS, the operators $\hat{\mathcal{O}}$ have to be a quadratic fermion operator, and therefore, the total Hamiltonian contains four-fermion terms and cannot be solved by diagnolizing a quadratic matrix. Therefore, although the spectrum of $\hat{H}_{0}$ can be obtained exactly, the effects of dissipation still needs to be computed by the non-Hermitian linear response theory, and the numerical results are shown in Fig. 1. We take two edge states of $\hat{H}_{0}$ with same energy and located at the same edge as the Kramers degenerate states $\ket{\Psi_{i}}$ ($i=1,2$). First, starting with an initial pure state $\ket{\Psi}=\frac{1}{\sqrt{2}}\ket{\Psi_{1}}+\frac{1}{\sqrt{2}}\ket{\Psi_{2}}$, we determine the evolution of density matrix, with which we compute the time dependence of the second Renyi entropy $S_{\text{R}}(t)$ shown in Fig. 1(a). One can see that the entropy increases linearly in time, with larger slop for larger $M$. Secondly, the spectral function $A(\omega)$ for these two states are shown in Fig. 1(b). One can see that the peaks of two spectral functions are split. Thirdly, we compute the backscattering matrix element of an on-site impurity potential between these two edge states. As discussed above, this calculation needs to involve the contributions out of the Kramers degenerate states. Here we plot the results with contributions from all states, as well as results with contributions from edge states only, which shows both edge and bulk states contribute to the non-zero matrix elements.

Remarks. In summary, we have discussed how a system responds to dissipations, with TRS imposed on both the system and the environment, as well as the coupling operators between them. The main results are the absence of Kramers degeneracy and the absence of accurate quantization of conductance for TRS protected TI. We also recover the results reported in Ref. Cooper on losing of phase coherence. However, different from Ref. Cooper , we employ a non-Hermitian Hamiltonian formalism for open quantum system with the Markovian approximation to the environment. Different from many works on non-Hermitian physics in recent literatures, our non-Hermitian Hamiltonian contains Langevin noise term to ensure unitarity. In fact, above discussions show that the Langevin noise terms plays a crucial role because most terms violating the Schur’s Lemma are essentially from the Langevin noise average. We should also emphasize that the way we impose the TRS symmetry leads to quartic non-Hermitian terms. This also makes our model different from those considered in recent works on topological classification of non-Hermitian Hamiltonian, where the models under consideration are always quadratic classification_non_Hermitian ; classification_non_Hermitian2 .

Acknowledgment. We thank Zhong Wang, Yun Li, Chaoming Jian, Hong Yao and Pengfei Zhang for helpful discussions. This work is supported by Beijing Outstanding Young Scientist Program (HZ), NSFC Grant No. 11734010 (HZ and YC), NSFC under Grant No. 11604225 (YC), MOST under Grant No. 2016YFA0301600 (HZ) and Beijing Natural Science Foundation (Z180013) (YC).