Topological invariant for 2D open systems
We study the topology of 2D open systems in terms of the Green’s function. The Ishikawa-Matsuyama formula for the integer topological invariant is applied in open systems, which indicates the number difference of gapless edge bands arisen from the poles and zeros of Green’s function respectively. Meanwhile, we define another topological invariant via the single particle density matrix, which works for general gapped systems and is equivalent to the former for the case of weak coupling to an environment. We also discuss two applications. For time-reversal invariant insulators, the index can be expressed by the invariant of the each spin-subsystem. As a second application, we consider the proximity effect when an ordinary insulator is coupled to a topological insulator.
The study of the quantum Hall effect has led to the new classification paradigm of quantum phases based on topological properties Thouless1982prl (); Wen1990prb (); Hasan2010rmp (); Qi2011rmp (). The corresponding topological invariants have been constructed for various systems with interaction Thouless1982prl (); Ishkawa1987npb (); Wang2010prl (); Gurarie2011prb (); Niu1985prb (), with disorder Niu1985prb (), and with time-reversal symmetry Kane2005prl (); Sheng2006prl (); Lee2008prl (). For the non-interacting case, the topological index is given by the integral of the Berry curvature determined by single-particle wavefunctions of the system Thouless1982prl (); Wang2010prl (). For the interacting case, it is more convenient to use the expression in terms of the single-particle Green’s function Ishkawa1987npb () instead of the many body wavefunction Niu1985prb (). In combination with (dynamical) mean field theory (DMFT) Georges1996rmp (), it is feasible to obtain the topological index through calculating multiple integrals of the interacting Green’s function Wang2012epl (). Meanwhile, two new developments have allowed to simplify the calculation: 1) the topological Hamiltonian method, which captures topological properties of the original interacting Hamiltonian Wang2012prx (); 2) the quasiparticle Berry curvature method, where the Berry curvature is determined by the quasiparticle states if these states have a long lifetime Shindou2008prb ().
However, all of these methods are valid only for closed quantum systems. A realistic system, inevitably, is coupled to an environment. Specially for the non-interacting case, the ground state of an open system is expected to be a reduced density matrix (a mixed state) rather than a pure state. An extension of the Berry phase to mixed states in 1D systems Uhlmann1986 (); Nieuwenburg2014 () and related observables (e.g. Thouless pumping) Linzner2016prb (); Bardyn2017 () has been proposed. For 2D systems, there are several different versions of topological invariants for a general density matrix Diehl2011np (); Rivas2013prb (); Bardyn2013njp (); Huang2014prl (); Budich2015prb (); Grusdt2017 (), and only one of them is corresponding to the holonomy Budich2015prb (). To include interaction effects, as for closed systems, the best option is to express the topological invariant in terms of the Green’s function.
For a general system immersed in an environment , the effective partition function can be obtained by integrating out the degree of freedom of . The resulting effective action retains the symmetry. Accordingly, the first Chern number of the system can be defined by the Ishikawa-Matsuyama formula,
where , , run through , , with being the imaginary frequency, and the Matsubara Green’s function is . Here, and represent the internal degrees of freedom of . The translational symmetry is supposed for both and . The index (1) is a well-defined topological invariant Qi2008prb () iff there are neither poles nor zeros for for all and imaginary frequency .
We develop both the topological Hamiltonian method and the Berry curvature method to evaluate eq. (1). The topological Hamiltonian gives an effective single-particle description of the open system. On the other hand, the Berry curvature method shows that for an open system, besides the bands determined by the poles of the Green’s function (energy spectrum), the ‘bands’ from the zeros (denoted as blind bands here), i.e. , appear and contribute to the topological properties. The value of indicates the number difference of gapless edge modes and gapless edge blind bands.
When the coupling to the environment becomes larger, the blind bands may cross the Fermi surface (Fig.1c), i.e. for some , and thus becomes ill-defined even though the energy spectrum are still gapped. For a general gapped system, we propose a topological invariant based on the single-particle density matrix with , which is equivalent to when the system is weakly coupled to the environment. For a non-interacting system, there is a one-to-one correspondence between the density matrix and the entanglement Hamiltonian Peschel2002 (); Fidkowski2010prl (). Unlike the previous discussions on the relation between entanglement spectrum (entropy) and topology properties of the full system ()Kitaev2006prl (); Hsieh2014prl (); Chandran2014prl (), we focus on the physical implication of the eigenstates of the density matrix for . The index (see eq. (6) below) is determined by the Berry curvature of the dominant eigenstates of the density matrix , as proposed in Ref. Budich2015prb () for non-interacting systems, giving the number of gapless edge modes for .
Topological Hamiltonian method. We suppose that in eq.(1) has neither poles nor zeros, and try to find the corresponding topological Hamiltonian through smooth deformation Wang2012prx (). The deformation path is set to be : for . For , we have , which is presumed to have neither poles nor zeros. For , similar as in Ref. Wang2012prx (), using the Lehmann representation of in the zero-temperature limit jzheng2017 (), it is easy to check that for any vector in the subspace , which implies that (the imaginary part of) every eigenvalue of is nonzero. Note that also cannot diverge for a well-defined . Therefore, during the deformation, remains well-defined and thus the deformation is smooth. It means that the single-particle topological Hamiltonian , whose Green’s function is , has the same topological properties as .
On the other hand, the Green’s function for the full system () can be written as a block matrix,
where , , , and . The indices and refer to the internal degrees of freedom for the environment. Introducing the projection operator for the subspace, we have . Immediately, we obtain the relation , where is the topological Hamiltonian of the full system. In particular, for a non-interacting system, , the topological property for the system is determined by instead of . The former contains the information on the coupling between and .
Berry curvature method. For the integral (1), considering the contour integral in Fig. 1a for , and using the residue theorem, the result only depends on the behavior of the Green’s function at its poles and zeros jzheng2017 (). Generally, the integral (1) gives
where and are the eigenvectors of the Green’s function with divergent eigenvalues (i.e. poles of ) and zero eigenvalues (i.e. zeros of ) for , respectively. Clearly, the value of indicates the difference between the number of gappless edge modes and gappless edge blind bands.
For the full system, the zeros of may appear only for strong interactions with some special symmetries Gurarie2011prb (), and then the Berry curvature method will give a similar result as eq. (3). In most cases, has no zeros. We suppose that there are in total bands for the full system, and the first bands are filled. Then the first Chern number of the full system is described by the filled (quasi-particle) states,
where run through , jzheng2017 (); Shindou2008prb (); Wong2013prl (). Note that the Green’s function is , where with the self-energy . The poles are determined by . The state in eq.(4) is the eigenstate of with the eigenvalue (numerically, is the value of where changes its sign). Here, we have supposed that each band has long lifetime to use the concept of quasiparticle, which breaks down for strong interaction cases.
For the non-interacting case, the states in the set are orthogonal to each other, and eq. (4) is invariant under a transformation for in jzheng2017 (). For the interacting case, these states are not orthogonal anymore, but with the assumption that they are still linearly independent (this is expected for weak and moderate interactions) and using Gram-Schmidt orthogonalization, we prove that any orthogonal basis of the set gives exactly the same Chern number (4) in the supplementary material jzheng2017 (). As a result, the Chern number is only related to the space spanned by the set for both cases. This finding allows us to simplify the problem.
Choosing the contour integral for the Green’s function as shown in Fig. 1a, we get the following relation:
where and includes all of the internal degrees of freedom ( and ). The energy in the denominator is the -th eigenvalue of , which satisfies . Besides, . Thus from eq. (5), for the single-particle density matrix with , the eigenstate set of with nonzero eigenvalues, spans exactly the same space as the set . Consequently, the Chern number (4) equals the sum of the integral of Berry curvature of the all states . This provides a new method for evaluating the topological index beyond the topological Hamiltonian method.
For an open system , in general, at some frequencies. To see the emergence of these blind bands, let us for simplicity focus on the non-interacting case. We still suppose that there are in total bands for the full system, and the first bands are filled. For the case that the system decouples from the environment , we now prove that no blind bands exist. Let the set with (i.e., filled bands and unfilled bands) belong to the system and the other bands with (i.e., filled bands and unfilled bands) belong to the environment. Set , then for and the set forms a complete orthogonal set in the subspace . So the Green’s function becomes . If for a nonzero , this implies are linearly dependent, which contradicts the orthogonality. Therefore, the Green’s function has no zero eigenvalues.
When the system is weakly coupled to the environment, we have and now can be nonzero even for . Hence the new poles from emerge, accompanied by the appearance of the zeros of Gurarie2011prb (). The latter cancels the effects from the new bands (see eq. (3)), which is consistent with the robustness of the topological index. Intuitively, the set is overcomplete and it allows to be an eigenstate of with zero eigenvalue. Note that for the new poles with , one has for the weak coupling case. Thus for being away from (), the weak coupling corrects the Green’s function slightly, and cannot contribute zeros. However, if for , the correction becomes significant. As a result, for the weak coupling case, the blind bands will always be close to those new bands (Fig.1b) and actually they always appear in pairs Gurarie2011prb (). Consequently, below the Fermi surface, the number of the blind bands and that of the new bands are the same. The number of effective bands occupied in the system is , where and are the number of energy bands and blind bands below the Fermi surface, respectively.
Density matrix description. For weak coupling and the non-interacting case, the blind bands are always close to the new poles. This property allows us to smoothly tune for a well-defined , avoiding bands or blind bands crossing the zero point, so that the first poles () are moved together to and the others to , without changing the topological index (1). The final Green’s function becomes , where and . By using the completeness , we find , where is the unit matrix in the subspace . Besides, we have .
The topological Hamiltonian for the final Green’s function is , where . Consequently, the eigenstates of with negative eigenvalues are exactly the eigenstates of with the largest eigenvalues. For the interacting case, using the Lehmann representation of the Green’s function and by properly tuning the position of the many body eigenenergy, a similar discussion can be applied and the same conclusion can be obtained jzheng2017 ().
For strong coupling, the blind bands may cross the zero point even for a gapped system, in which case the Chern number (1) becomes ill-defined. To extend the concept of the topological index to a general gapped system, we propose the following formula as a generalized topological index:
where are the eigenstate of with the largest eigenvalues. From the above discussion, for the weak coupling case, we have . Moreover, the formula (6) is the same as the definition of topology for a ‘density matrix’ in Bardyn2013njp (); Budich2015prb ().
Without interaction and without coupling to the environment, the eigenvalues of are (with degeneracy ) and , which gives the gapped flat band structure. The interaction and coupling will change the single particle distribution, and the null eigenvalues become finite. However, the majority eigenstates are still gapped from the minority eigenstates. The topological phase transition occurs when the -th largest eigenvalue of becomes degenerate with the -th largest eigenvalue of or the Green’s function close the gap (which may induce the energy band inversion for and thus the eigenstates of are also changed suddenly). In contrast, by the definition of the Chern number (1), the topological phase transition can happen when the blind bands close and reopen a gap between them without any energy band inversion Gurarie2011prb ().
topological invariant. For a time reversal () invariant 2D system, the topological index in a form of Green’s function is a five-dimensional integral Wang2010prl (). For the case with conserved spin, the index can be expressed through the topological index of the decoupled spin subsystem (see eq.(1)), i.e. Kane2005prl (); Hohenadler2013jpcm (); Wu2006prl (); Xu2006prb (), which simplifies the calculation largely. Here, the Chern number reflects the number of edge states for each spin. The spin conservation can be violated by different types of spin-orbit coupling (SOC) Kane2005prl (). In the following, we point out that for the weak SOC case, , and for a general gapped case, , relating the topology of the subsystem and that of the full system.
Let and be the Green’s functions for each spin subsystem respectively. Due to -symmetry, we have the following relations jzheng2017 (): , and . The first two relations imply and from eqs. (1) and (6). The last relation shows a close relation between the density of state for each subsystem, which indicates that the subsystems and the full system close and reopen a gap at the same time. The topological state for the full system is changed when the topological properties in the subsystem are changed due to the SOC coupling to the other subsystem. Thus, the index can be obtained from the index of the subsystem.
In Fig. 1 in the supplementary material jzheng2017 (), we plot the numerical results for the topological index of the subsystem of the Kane-Mele model, showing the exact relation to the index, .
Proximity effect. A simple but fundamental question is what happens when an ordinary insulator is coupled to a topological insulator. As an example, here we consider a system on a bi-layer honeycomb lattice, with each layer described by the Haldane model Haldane1988prl (): , where is the nearest-neighbor tunneling, is the next nearest-neighbor tunneling with a phase and , is the staggered potential, and is the layer index. The parameters are , , , and , so that the first layer is a Chern insulator and the second one is an ordinary insulator.
To observe the interference effect, we further introduce a tunneling term between layers, . The topological index with respect to the tunneling coefficient is shown in Fig. 2a. In the whole region of , the index for the first layer is unchanged. For the second layer, there are no edge modes or edge blind bands when . For a small finite , since the full system has gapless poles on the edge, the second layer also has a gapless edge state and simultaneously a gapless blind state appear. The Chern number is still zero, as the new bulk-boundary correspondence we claimed. The second layer subsystem obtains a nonzero topological index when , while the full system displays a transition from a Chern insulator to a trivial one through band inversion. For , the stable edge state disappears, but the gapless blind band remains, and thus for the second layer. For , the spectrum of the full system with zigzag edges is shown in Fig. 2b, lacking edge states. However, the spectrum of the density matrix for each layer contains a gapless edge state, which is shown in Fig. 2c. For the second layer, the “blind” bands become gapless at the edge (Fig. 2d).
Conclusion. We have systemically discussed the topology of 2D open systems. The invariant given by the Ishikawa-Matsuyama formula reflects the the number difference of gapless edge modes and gapless edge blind bands. Moreover, we define another topological invariant in terms of the single particle density matrix, which is applicable for general gapped systems and is equivalent to the former invariant for the case of weak coupling to the environment. Two applications are discussed. For time-reversal invariant insulators, we explain that the relation of the Chern invariant for each spin-subsystem and the index of the full system is given by , which highly simplifies the calculation for the index. Besides, we consider the proximity effect when an ordinary insulator is coupled to a topological insulator, which shows an inverse topological invariant is induced by the nontrivial part Hsieh2016prl ().
The examples given here are for non-interacting cases. The method can applied in the interacting system, also at finite temperature, and the single-particle density matrix an be obtained for example by using dynamical mean field theory.
Acknowledge. This work was supported by the Deutsche Forschungsgemeinschaft via DFG FOR 2414 and the high-performance computing center LOEWE-CSC.
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Supplementary Materials for “Topological invariant for 2D open systems”
Appendix A Berry curvature method
In this section, we would like to derive the expression of the first Chern number in terms of Berry curvature (i.e., the eqs. (3) and (5) in the main text), following the method developed in xShindou2008prb-1_sm (). To calculate the integral,
it is helpful to introduce a similarity transformation , so that , where is diagonalized. Substituting it into eq. (7), we obtain
There are 27 terms in total in eq. (8). 12 terms of them give total derivatives, 3 terms vanish due to asymmetrization, and 6 terms cancel with each other. The remaining terms are
Here the contour integral in complex plane of (denoted by here) is shown in Fig.1a in the main text. The poles and zeros of the Green’s function appear only at real frequencies. For real frequency, the Green’s function is hermitian, and the elements of are , where is the -th eigenstate of with eigenvalue and is the internal degree of freedom. Explicitly, the equation (9) becomes
If has no zeros, but has poles below the Fermi energy, then around the pole , we have
Thus, for each pole, using the residue theorem for eq. (10), we obtain
Defining the quasiparticle band , we can simplify eq. (13) to be . Summing all of these quasiparticle bands below Fermi energy, we get the result (eq.(3)) given in the main text.
Appendix B U(n) symmetry and Gram-Schmidt orthonormalization
For the non-interacting case, are orthogonal with each other. In the following, we prove that in this case, (eq. (3) in the main text) is invariant under a smooth transformation (note that can be chosen to be well-defined in the whole Brillouin zone). Substituting the transformation into the Berry curvature, we obtain
Using the unitary of , we get
Combining eq. (18) with the third term of eq. (16), and using the antisymmetry for exchange of and , we find that the second and third terms of eq. (16) will not contribute to . The last term in eq. (16) is
the first term of which vanishes due to the periodic boundary condition and the second term will be canceled by the antisymmetry in . Finally, we have
showing the symmetry.
Before going to the interacting case, we want to stress that
is quantized and it is stable for a smooth deformation of Bernevig2013book_sm ().
For the interacting case, are not orthogonal to each other. In the following, we prove that these states can be smoothly deformed to an orthogonal basis without changing , supposing that they are still linear independent. Using Gram-Schmidt orthogonalization, we define an orthogonal orthogonal basis: , and , . It is easy to check that for . The smooth deformation then is defined by ():
This deformation is well-defined, since . Thus, we have defined a smooth deformation, so that , , , goes to , , , without changing the Chern number. And using the symmetry of the Chern number for orthogonal basis, we get the conclusion that, the Chern number is only related to the space spanned by .
Appendix C Time reversal invariant system
For a -symmetric system, it is easy to check that if is a set of completed orthogonal basis, then is too. On the other hand, for any and , we have . Using the definitions and , and , where is the many-body Hamitonian, and the index includes lattice index and other internal degrees of freedom besides spin, we have