Violation and Revival of Kramers' Degeneracy in Open Quantum Systems

Kramers' theorem ensures double degeneracy in the energy spectrum of a time-reversal symmetric fermionic system with half-integer total spin. Here we are now trying to go beyond the closed system and discuss Kramers' degeneracy in open systems out of equilibrium. In this letter, we prove that the Kramers' degeneracy in interacting fermionic systems is equivalent to the degeneracy in the spectra of different spins together with the vanishing of the inter-spin spectrum. We find the violation of Kramers' degeneracy in time-reversal symmetric open quantum systems is locked with whether the system reaches thermal equilibrium. After a sudden coupling to an environment in a time-reversal symmetry preserving way, the Kramers doublet experiences an energy splitting at a short time and then a recovery process. We verified the violation and revival of Kramers' degeneracy in a concrete model of interacting fermions and we find Kramers' degeneracy is restored after the local thermalization time. By contrast, for time-reversal symmetry $\tilde{\cal T}$ with $\tilde{\cal T}^2=1$, we find although there is a violation and revival of spectral degeneracy for different spins, the inter-spin spectral function is always nonzero. We also prove that the degeneracy in spectral function protected by unitary symmetry can be maintained always.

Introduction. Kramers' degeneracy theorem tells us for fermionic systems with half-integer total spin where timereversal symmetry (TRS) is presented, all energy levels are doubly degenerate [1,2]. This theorem plays a vital role in the quantum-spin Hall effect [3,4] as well as in the stability of the superconducting phase with disorder [5].
Kramers' theorem is expected for thermal equilibrium systems since the grand canonical distribution is only related to the Hamiltonian where the double degeneracy is presented. This can be proved straightforwardly in equilibrium systems [6]. Recently, topological states protected by TRS in non-Hermitian systems and Floquet systems are widely discussed in varieties of systems [7][8][9][10][11]. However, we must be cautious because anti-unitary symmetry in effective Hamiltonian is fragile in open quantum systems. A later discovery made by McGinley and Cooper says that TRS is not stable in open quantum systems even with a TRS preserving interaction [12]. Any pure state as a superposition of Kramers' states can not maintain its coherence after coupling to the environment. Its consequences for symmetry-protected topological states are later studied [13,14]. Even more surprising, by non-Hermitian linear response theory [15], it is discovered that after coupling to the environment in a TRS preserving way, the Kramers' degeneracy is lifted and two helical topological edge states in topological insulator will mix with each other [16]. Since previous results are mostly based on perturbations, a thorough study of Kramers' degeneracy in open quantum systems at full time-scale is needed. * ychen@gscaep.ac.cn In this work, we go beyond the perturbation theory and study the whole dynamical process, focusing on the Kramers' degeneracy for a time-reversal invariant (TRI) interacting fermion system with a TRS preserving interaction between the system and the bath. We find that the Kramers' degeneracy is locked with thermal equilibrium. We find a violation of Kramers' degeneracy after a sudden coupling to an environment, and the violation enlarges with time while it shrinks afterward as is shown in Fig. 1. The Kramers' degeneracy revives as the system gradually reaches new thermal equilibrium. Due to the connection between Kramers' degeneracy and local thermalization, we also find the Kramers' degeneracy in the spectrum can experience a violation and revival process in isolated systems that satisfy the eigenstate thermalization hypothesis (ETH).
In the following, we first establish a general relation between Kramers' degeneracy and single-particle spectral functions. Then with the help of solvable interacting models, we can calculate the quench dynamics of the spectral function and the distribution function with a sudden coupling to a bath. By checking the time scale of distribution function reaching thermal equilibrium and Kramers' degeneracy signal in spectral functions, we can draw our conclusion. We also discussed how spectral functions evolute for system-to-bath interaction baring TRST withT 2 = 1 and unitary symmetry S as a comparison. We stress that our conclusion applies to general interacting fermion systems and is not restricted by the Markovian approximation. By this study, we find Kramers' degeneracy only emerges in thermal equilibrium systems, which implies TRS is only a good symmetry in equilibrium systems and the breaking extent of Kramers' degeneracy can be set as a measure of the extent of the system being away from equilibrium.
However, for general interacting quantum systems, it is hard to probe a specific eigenstate experimentally. It is Green's function of local operators that can be measured by various experimental protocols [18], which in general give the spectrum and distribution function information of the quasi-particles. Here we are trying to discuss the Kramers' degeneracy in open quantum systems, where even no eigenstates of the system are well defined. Therefore we have to introduce Green's function form of the Kramers' degeneracy. First, we introduce real-time Green functions G ≷ (t, t ), which are defined as whereĉ j,σ (t ) = e iĤ i tĉ j,σ e −iĤ i t is fermion annihilation operator in Heisenberg picture. Here tr is over the Hilbert space of the system. Then it is straightforward to show that there is an [6]. Here, , as an analogy of ψ 1 |ψ 2 = 0, is a signature of havingT 2 = (−1) N s . Using their relation to the spectral function In this work, we are interested in understanding the Kramers' degeneracy in the quench dynamics from the Green's function perspective. At t = 0, we change the Hamiltonian fromĤ i toĤ f , which also satisfiesTĤ fT −1 =Ĥ f . In particular, we couple the original system to an additional bath by OperatorsÔ j andξ j are TRI operators in the system and the environment. Generally, using the time-reversal transformation, one can show that for t, t > 0 we have Here is the quantum expectation on the bath density matrix, which is assumed to be thermal with inverse temperature β B . Since the bath contains a much larger degree of freedom, we assume that it does not evolve when coupled to the small system. Similar relations hold for G < σ σ and thus A σ σ .
In the short time limit, we have σ,−σ = 0 and the Kramers' degeneracy is broken. This generalizes the previous analysis using Markovian baths [16]. As a comparison, for unitary symmetries interchanging + and −, one finds a similar relation It means the degeneracy protected by unitary symmetry is stable in dynamical evolutions. We also notice that the condition that ρ(t, t ) = 0 is very general and has a more natural understanding than the extended Schur Lemma argument given previously [12].
On the other hand, for t, t t th , where t th is the thermalization time, we expect the total system to thermalize ] under simple probes. HereĤ B is the Hamiltonian of the bath. Consequently, we find G ≷ ++ = G ≷ −− , G ≷ σ σ = 0, and the Kramers' degeneracy is restored. For a large system-bath coupling, the characteristic time scale t th resembles its counterpart in isolated quantum systems, where t th ∼ β f for strongly interacting models and t th ∼ 1/ with quasi-particle decay rate for weakly interacting models [19][20][21]. Before turning to concrete examples, we add a few comments. First, the violation and revival of Kramers' degeneracy also exists in isolated quantum systems that satisfy ETH, whereĤ f is only different fromĤ i by certain parameters [22].
In this case, (2) still works, without the average over bath density matrix. In the long-time limit, although the fine-grained density matrix ρ(t ) may differ from the thermal density matrix since the unitary evolution preserves the total entropy, we expect local thermalizationρ(−t ) ∼ρ(t ) ∼ e −β fĤ f /tr[e −β fĤ f ] [23,24]. Here, ∼ means the equivalence under measurement of local operators. As a comparison, the violation can not be restored if the system is many-body localized [22].
Second, it is helpful to compare the above results with TRS withT 2 = 1 or unitary symmetry S. We choose the singleparticle transformation to beT = σ x K or S = iσ y . Here K is the complex conjugate operator. In both cases, the symmetry imposes G ++ = G −− in thermal equilibrium, while generally G +− = 0. When coupled to the bath, for TRST , the G ++ = G −− firstly breaks and then gets restored. While for S, the G ++ = G −− is always preserved during the evolution.
Concrete Model.-We now verify our predictions in interacting fermions using a concrete solvable model. Generally, the simulation of quench dynamics of chaotic quantum systems is hampered by the exponential growth of the Hilbert space dimension. Here, we overcome this difficulty by constructing a TRI SYK model [25][26][27][28][29] by coupling different complex SYK sites [30][31][32][33][34], which is solvable in the large-N expansion. The initial Hamiltonian H i reads (3) Here j/ j a = 1, 2, ...N labels different fermion flavors, σ is a spin index and τ = ± is an additional pseudo-spin index. J τ σ j 1 j 2 j 3 j 4 describes intraspecies' random interaction between fermions, which satisfies independent Gaussian distribution with Under the time-reversal transformation, we find J τ σ j 1 j 2 j 3 j 4 → (J τ −σ j 1 j 2 j 3 j 4 ) * . As a result, the interaction term is TRI after ensemble average. The single-particle Hamiltonian h τ σ τσ is diagonal in the flavor space. Imposing the single-particle TRS with T = iσ y K, we find generally where μ is the chemical potential, K x , K z , J x,y,z are real parameters. Nontrivial terms correspond to celebrated γ matrices wildly used in both condensed matter [17] and high-energy physics [35]. As a result, the singleparticle eigenstates show pairwise degeneracy at energy μ ± √ K 2 x + K 2 y + J 2 x + J 2 y + J 2 z , consistent with Kramers' theorem.
We consider the quench by coupling the system to an external bath at t = 0 (here we choose the quench as an example. Similar results hold if the coupling is turned on slowly [22]), with system-bath coupling Here, to be concrete, we choose the bath to be an additional SYK model with M N fermion modes (b 1 , b 2 = 1, 2, ..., M) [36][37][38][39]. This corresponds to takes the similar form as (4), with N replaced by M. Generalizations to other bath models are straightforward. The coupling strength satisfies This guarantees that H SB does not affect the evolution of the bath [36][37][38][39], consistent with our previous assumption. We also choose theh to take general form whereμ,K x ,K y ,J x,y,z are independent parameters. The form ofh ensures the ensemble of couplings are also invariant under the TRST . Here we have extendedT to the full system by definingTψ bT −1 =ψ b . In the large-N limit, the Green's functions G ≷ of SYK-like models satisfy the Schwinger-Dyson equation on the Keldysh contour, and the quench dynamics can be simulated by solving corresponding integral equations. Explicitly, we have Here • includes the convolution in real-time, as well as multiplication in σ and τ space. The self-energy is given by melon diagrams shown in Fig. 2, which leads to t ) is the bath correlation function. The retarded/advanced Green's functions are related to G ≷ by G R (t 1 , t 2 ) = θ (t 12 )(G > (t 1 , t 2 ) − G < (t 1 , t 2 )) and t 1 , t 2 )). Similar relations hold for self-energies. Using these relations, (9) and (10) become closed. The numerical approach for solving (9) and (10) with discretized time has been well explained in previous works [37,[40][41][42].
Numerical Results. We now present numerical results of the quench dynamics. Results for slow couplings and periodic couplings are given in Supplemental Material [22]. We choose β i = β f , V = J, and arbitrarily chosen parameters in h andh. Given the real-time Green's function G ≷ (t, t ), we define the temporal Green's functionG ≷ (t r , t ) at time t bỹ This definition preserves the causality of the unitary evolution. HereG ≷ and G ≷ are in matrix form, and the sub-indices are omitted. We define the Fourier transform with respect to t r . The temporal spectral function A(ω, t ) then reads In numerics, we focus on the first site with τ = + and drop the corresponding pseudospin indices for conciseness. The results for A ++ (ω, t ), A −− (ω, t ), and Re A +− (ω, t ) are shown in Fig. 3(a). Before the quench, the system is in thermal equilibrium and the Kramers' theorem ensures A ++ (ω, 0) = A −− (ω, 0) and A +− (ω, 0) = 0. After we couple the system to the bath (t > 0), the degeneracy is lifted. As an example, we find a large discrepancy between A ++ and A −− , as well as a non-vanishing A +− at t = 5. When the time t becomes longer, A ++ − A −− and A +− decays, and becomes almost invisible at t = 30. Our previous analysis shows the revival of Kramers' degeneracy happens when the system arrives at equilibrium with the bath. In a quantum many-body system, the local thermalization can be diagnosed by quantum distribution function F (ω, t ) at time t. It can then be defined as dt r e iωt r (G > (t r , t ) +G < (t r , t )).
We further introduce 1 and 2 to quantify the strength of the Kramers' degeneracy breaking as The numerical results are shown Fig. 3(c). We find both 1 and 2 decay exponentially in the long time. As a comparison, we plot the equilibrium Green's function |G > ++ (t, ∞)|, the decay rate of which corresponds to the local thermalization time [43]. We find their decay rates match, with additional oscillations due to the peaks in the quasi-particle spectral A.
As a comparison, we also plot results for models with symmetryT (T 2 = 1) [44] in Fig. 3(d) and 3(e). We find although the diagonal components of the spectral function show similar behaviors as systems with the symmetryT , we have A +− = 0 at any time. This is consistent with the absence of the Kramers' degeneracy for systems withT 2 = 1.
Summary and Outlook. To summarize, we find for TRS T satisfying T 2 = −1, Kramers' degeneracy in open quantum interacting fermionic systems is equivalent to A ++ (ω) = A −− (ω) together with A +− (ω) = 0. After a sudden coupling to an environment in a time-reversal symmetric way, the Kramers' degeneracy experienced a breaking and restoring process. We further show it works for more general coupling schemes. We find the revival of Kramers' degeneracy happens after the local thermalization time t th . Similar results can be obtained for TRS T with T 2 = 1. But distinctively, A +− (ω) = 0 is not satisfied at all the time. It also means A ++ (ω) = A −− (ω) alone can not be seen as the condition for Kramers' degeneracy. It is also verified that for systems where local thermalization is hard to establish, the violation of Kramers' degeneracy will not recover.
Further, as we have seen, after coupling to a bath, although Kramers' degeneracy can be recovered, there is always a large portion of time Kramers' degeneracy is violated. For this reason, if we start from a pure state in Kramers' space in the initial Hamiltonian, decoherence will happen and be maintained. The decoherence in the final state can be partially implied by the line shape change in the final state spectrum compared with the initial state spectrum. In this sense, we find different respects in TRS of open systems. If a physi-cal result is more sensitive to phase coherence, such as the quantization of the conductance in topological insulators, we argue that these results can not be protected by the revival of Kramers' degeneracy [16]. On the other hand, like in superconductors, the pairing is more relevant to the energy degeneracy of the Kramers' pair. Therefore the superconducting phenomenon may be more stable against the environment. Furthermore, as we see that equilibrium or not is very important for time-reversal symmetric systems, but many transport theories are based on linear response theory, which attributes transport properties as a manifestation of equilibrium correlations. We leave a careful study in these directions to future works.