Generalized Chern numbers based on open system Green's functions
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Generalized Chern numbers based on open system Green’sfunctions
M. Bel´en Farias, Solofo Groenendijk and Thomas L. Schmidt
Department of Physics and Materials Science, University of Luxembourg, 1511 Luxembourg,Luxembourg
Abstract.
We present an alternative approach to studying topology in open quantum systems, relyingdirectly on Green’s functions and avoiding the need to construct an e ff ective non-HermitianHamiltonian. We define an energy-dependent Chern number based on the eigenstates of theinverse Green’s function matrix of the system which contains, within the self-energy, all theinformation about the influence of the environment, interactions, gain or losses. We explicitlycalculate this topological invariant for a system consisting of a single 2D Dirac cone and findthat it is half-integer quantized when certain assumptions over the damping are made. Awayfrom these conditions, which cannot or are not usually considered within the formalism ofnon-Hermitian Hamiltonians, we find that such a quantization is usually lost and the Chernnumber vanishes, and that in special cases, it can change to integer quantization. Keywords : Open systems, topology, non-Hermitian, Chern number eneralized Chern numbers based on open system Green’s functions
1. Introduction
To study the properties of quantum systems and understand how they manifest themselves inour macroscopic everyday world, it is usually necessary to take into account the interactionof such systems with their surroundings. In condensed-matter systems, the e ff ects of anenvironment, be it the coupling to phonons, the presence of impurities [1, 2], or externalprocesses that produce gain or loss [3, 4] are hard to avoid. As an exact mathematicaltreatment of the degrees of freedom of both the system and its environment is often verycomplex, an open quantum system approach [5] is usually the best strategy to study andunderstand quantum systems in more realistic settings and exploit their unique properties.A relatively new direction is the study of topology in such open systems [6, 7].One approach that has gained traction lately is the use of e ff ective non-Hermitian (nH)Hamiltonians [8, 9, 10, 11, 12]. In this formalism, the environment is accounted for bysupplementing the Hamiltonian of the isolated system under study with non-Hermitian (nH)terms, which can be thought of as arising from tracing out the degrees of freedom of theenvironment. In dissipative systems, the non-Hermitian part is related to the broadening ofthe system’s energy levels [13], i.e., to the inverse life-time of the quasi-particle excitations[11, 8].The nH framework has predicted intriguing phenomena such as the nH bulk-boundarycorrespondence [14], the nH skin e ff ect – a feature of nH Hamiltonians in which the majorityof the eigenstates are localized at the boundaries [15, 10] – and the conversion of bandtouching points into exceptional points at which two eigenstates coalesce and the Hamiltonianbecomes defective [11, 16, 17]. A topological classification of the band structures ofnH systems has been proposed in Refs. [18, 19], and the Altland-Zirnbauer classificationof topological invariants [20] has been recently extended to nH systems [21, 22, 23].However, in contrast to topological invariants in hermitian systems, those constructed fromnH Hamiltonians are less obviously connected to physical quantities.For example, in Ref. [19], Shen et al. have used the left and right eigenstates of anH Hamiltonian to define a unique nH Chern number, which is quantized even though theHall conductivity of the system is not. This result is at first surprising, because topologicalinvariants of interacting systems were in fact defined already much earlier from the interactingGreen’s function and were used to prove the quantization of the Hall conductivity even ininteracting quantum Hall systems [24, 25, 26]. However, for generic nH Hamiltonians theMatsubara Green’s functions used in these works become discontinuous, which makes thisapproach inapplicable and ultimately allows for a loss of quantization of the Hall conductivityin open systems despite the existence of a quantized nH Chern number [27, 28, 29].The main goal of this work is to extend the range of open systems which can be classifiedtopologically. For this purpose, we define a topological invariant directly from the Green’sfunctions. Our approach is general in that we do not assume specific analytic properties of theGreen’s function. It thus applies to the retarded Green’s function, which describes equilibriumproperties in open systems, but also to the Keldysh Green’s function, which can capture non-equilibrium dynamics. Our formalism does not apply to the Matsubara Green’s function as it eneralized Chern numbers based on open system Green’s functions ff erence between thisapproach and nH Chern numbers is that in the latter, the self-energy Σ ( ω, k ) is usually assumedto be ω -independent in order to obtain a time-local Hamiltonian formalism [13, 17, 28, 30, 31].This approximation is valid in many cases. In other cases, however, as for instance forquasi-particle spectra of disordered Weyl semimetals, the frequency dependence of the self-energy plays an important role [16, 32]. It is also worth noting that most studies involvingnH Hamiltonians, particularly those interested in topological classifications, consider onlymomentum-independent nH terms. One notable exception to this rule is the recent preprintby Wang et al. which considers momentum-dependent decay rates [33].A related topological invariant based on Green’s function has been recently proposed byKawabata et al. [34]. They showed that in one and three spatial dimensions, nH topologicalsystems can be described by e ff ective Chern-Simons field theories and are characterized byquantized topological response functions. The latter are in fact constructed from the Green’sfunctions and depend on the energy at which the system is probed. However, this frameworkdoes not apply to two-dimensional open systems.We shall present here a definition of such topological invariants in 2D systems, withouthaving to rely on nH Hamiltonians. We will propose an analogous definition for a energy-dependent nH Chern number that can be obtained directly from the Green’s functions, andshow that this topological invariant is also quantized in many scenarios. Moreover, if thisnumber is constructed based on the retarded Green’s function and if certain assumptionsabout the energy and momentum dependence of the self-energy are fulfilled, we show thatthis invariant coincides with the nH Chern number defined in Ref. [19]. Our definition,however, is more general and provides a topological invariant valid for a larger class of opensystems. Moreover, we shall also delineate more precisely the conditions under which suchquantization is lost, allowing us to shed some light onto the potential limitations of the nHtopology.
2. Non-Hermitian Chern numbers
In particular, our work will extend the definition of a nH Chern number presented in Ref. [19].There, the authors consider a nH Bloch Hamiltonian ˜ H ( k ) with eigenstates˜ H | ψ Rn i = E n | ψ Rn i , (1)˜ H † | ψ Ln i = E ∗ n | ψ Ln i . (2)where n is the band index. Right and left eigenstates are distinct for nH Hamiltonians andare denoted by | ψ R , Ln i here. These eigenstates then allow, a priori, the definition of Berrycurvatures, B α,β n , i j ( k ) ≡ i h ∂ i ψ α n | ∂ j ψ β n i , (3)with α, β ∈ { L , R } and i , j ∈ { k x , k y } . The integral over the corresponding Berry curvature thenleads to a topologically quantized nH Chern number. eneralized Chern numbers based on open system Green’s functions ff ective nH Hamiltonian can be thought of as consisting of two parts, ˜ H = H + Σ ,where H is the Hermitian Hamiltonian of the system without the influence of the environmentor interactions, whereas Σ is a nH correction to it. Such a correction can often be interpretedas arising from the retarded Green’s function, G ret ( k , ω ) = h ω − H ( k ) − Σ ret ( k , ω ) i − , (4)where is the identity matrix. For energy-independent Σ ret , we can see clearly that theeigenstates of the Green’s function will coincide with the ones of the nH Hamiltonian (theeigenvalues will be di ff erent). However, for a general ω -dependent self energy, this will nolonger be the case. In this case, we can still calculate the eigenstates of the Green’s function (orequivalently of its inverse) and thus define a new ω -dependent Chern number which we willcall C ( ω ). Since for ω = C ( ω ) coincides with the nH Chern number defined in Ref. [19],there will exist an interval close to ω = C ( ω ) will be quantized as well.Hence, the first step in defining this new topological invariant is to study the eigenstatesand eigenvalues of the inverse Green’s function. We shall do so in Sec. 3. In Sec. 4 we willdefine the ω -dependent Chern number and calculate it for a specific model. We will show that,for certain types of damping, it is quantized. In Sec. 5 we generalize these results, consideringdi ff erent situations for the influence of the environment, and find that such a quantizationmight change or be lost depending on the form of damping. Lastly, in Sec. 6 we present ourconclusions.
3. Eigenstates and eigenvalues of the inverse Green’s function
As a first step, we are interested in calculating the eigenstates of the Green’s function orequivalently of its inverse. To allow for a nontrivial topology, we assume a multiband systemsuch that the Green’s function becomes a matrix in the number of orbitals. So we are interestedin diagonalizing the operator G − ( k , ω ) = ω − H ( k ) − Σ ( k , ω ) . (5)If we had obtained the Green’s function directly from the e ff ective action of the system, itmight be easier to calculate directly the eigenstates of G ( k , ω ) instead of those of the inverse.But in case of a Green’s function constructed from the self-energy, it is more convenient todiagonalize the inverse (5).As the simplest case of a nontrivial band topology, we will consider a two-band model,for which the most general form of the inverse Green’s function can be cast as G − ( k , ω ) = ǫ ( k , ω ) + d ( k , ω ) · σ , (6)where σ is the vector whose components are the three Pauli matrices. This operator is notHermitian because in general ǫ, d i ∈ C . Its complex eigenvalues are given by g ± ( k , ω ) = ǫ ( k , ω ) ± d ( k , ω ) (7) eneralized Chern numbers based on open system Green’s functions d = q d x + d y + d z . (8)Note that d is complex and in general does not coincide with the norm of d , the latter given by p | d x | + | d y | + | d z | . The (unnormalized) eigenstates of G − , which fulfil G − | χ R ± i = g ± | χ R ± i ,are of the form | χ R ± i = d z ± dd x + id y ! . (9)Since G − is not a Hermitian operator, its right and the left eigenstates do not coincide. So wecan define the left eigenstates such that ( G − ) † | χ L ± i = g ∗± | χ L ± i . A possible choice is | χ L ± i = d ∗ z ± d ∗ d ∗ x + id ∗ y ! . (10)In general, the eigenstates { | χ R ± i , | χ L ± i} form a biorthogonal basis [35], as we can readily checkthat h χ L ± | χ R ∓ i = (cid:16) d z ± d , d x − id y (cid:17) d z ∓ dd x + id y ! = d z − d + d x + d y = . (11)However, for certain values of d x , y , z , the eigenvectors | χ R ± i do not form a basis of the Hilbertspace. Indeed, for d x = d y = d z = − d ( d z = d ), the eigenvector | χ R + i ( | χ R − i ) vanishes.Near this point, one therefore has to resort to a di ff erent choice of eigenvectors, for instance | ˜ χ R ± i = d x − id y − d z ± d ! . (12)Hence, each of the four eigenstates | χ R ± i , | ˜ χ R ± i can vanish at certain points in momentum space.This behavior is in fact essential for the definition of a Chern number [19]. Indeed, let us havea closer look at what happens when d x = d y = d x = d y =
0, we have d = p d z . By looking at the definitions of the states, we can seethat if d = d z , the states | χ − i and | ˜ χ + i vanish, while if d = − d z , | χ + i and | ˜ χ − i vanish. Letus now formulate in a more concrete way the conditions for which each of these eigenvectorsvanish. To do that, let us recall the definition of a complex square root [36] √ u + iv = ± s u + √ u + v + i sgn( v ) s − u + √ u + v . (13)With this definition, we can easily see that q d z = d z sgn (cid:2) R e( d z ) (cid:3) , (14)which tells us that the sign of the real part of d z determines which is the well-defined set ofeigenstates for the inverse Green’s function. We see that for d x = d y =
0, if R e( d z ) > d = d z .While for R e( d z ) < d x = d y = d = − d z .With this information we can finally write the complete set of eigenstates for our system.These states shall be normalized following the convention for biorthogonal basis for whichwe require that h χ L ± | χ R ± i = eneralized Chern numbers based on open system Green’s functions | ψ R + ( k , ω ) i R e ( d z ) < = p d ( d − d z ) d x − id y d − d z ! , (15) | ψ R − ( k , ω ) i R e ( d z ) < = p d ( d − d z ) d z − dd x + id y ! , (16) | ψ R + ( k , ω ) i R e ( d z ) > = p d ( d + d z ) d z + dd x + id y ! , (17) | ψ R − ( k , ω ) i R e ( d z ) > = p d ( d + d z ) d x − id y − d z − d ! . (18)The corresponding left eigenstates are obtained by changing d → d ∗ . It is easy to see that, fora given value of d z , the set { | ψ R + i , | ψ R − i , | ψ L + i , | ψ L − i} forms a biorthogonal basis. In order to be able to define a Chern number for our system, we need to guarantee thatthe system will remain separable in the sense that there are no touching points betweenthe complex bands determined by the eigenvalues of the inverse Green’s function. Thesebands will cross in any point at which g + ( k , ω ) = g − ( k , ω ), which could hold if and only if d =
0. This extends the notion of separability of non-Hermitian Hamiltonians [19] to Green’sfunctions.It is worth noting that points in which the condition for the bands to cross is fulfilled,when d =
0, are actually exceptional points where the two eigenstates coalesce. For instancefor d = d x + id y ) | ψ R + ( k , ω ) i R e ( d z ) < = − d z | ψ R − ( k , ω ) i R e ( d z ) < . That is, the twostates become linearly dependent, and a complete set of eigenstates can no longer be defined.The only exception to this statement is the point d x = d y = d z =
0. At this point the systembecomes not only Hermitian, but also trivial (i.e., proportional to the identity), and the statesremain well-defined and linearly independent, so that this particular point is not an EP, but aHermitian band crossing.Let us briefly consider the specific system we shall use later on in this paper: a single2D Dirac cone interacting with an environment that has been integrated out, and whose e ff ecton the system will be encoded in four complex functions Γ µ = ( Γ , Γ x , Γ y , Γ z ), leading to aself-energy P i = x , y , z Γ i σ i + Γ σ . So we have ǫ ( k , ω ) = ω − Γ ( k , ω ) , (19) d x ( k , ω ) = − k x v x − Γ x ( k , ω ) , (20) d y ( k , ω ) = − k y v y − Γ y ( k , ω ) , (21) d z ( k , ω ) = − m − Γ z ( k , ω ) . (22)The previous equations are in fact completely general and can describe any two-band model,as long as the functions Γ µ are arbitrary functions of ω and k .The first conclusion we can draw is that the damping through the ’0-channel’, that is, thedamping kernel that is proportional to the identity matrix, can never close the gap, regardless eneralized Chern numbers based on open system Green’s functions ǫ ( k , ω ), any damping introduced through that channel won’t a ff ect the Chern number. Assuch, if it were the sole source of damping ( Γ ∈ C but Γ i ∈ R ), the Chern number would beindistinguishable from the Hermitian case (Im Γ =
0) and hence would be quantized to halfinteger [37, 38].Another particular case worth considering is where the imaginary part of only one of thethree Γ i is non-vanishing. Consider for instance, Γ x , Γ y ∈ R , but Γ z = i γ where γ is a realconstant. In such a case d x ( k , ω ) + d y ( k , ω ) + d z ( k , ω ) = , (23)( k x v x − Γ x ( k , ω )) + ( k y v y − Γ y ( k , ω )) + m − γ ( k , ω ) + im γ ( k , ω ) = , (24)and we can see from the last equation that, if the gap is originally open ( m , Γ x = i γ but Γ y , z ∈ R , there are two exceptional points atthe momenta ( k x , k y ) = (0 , ± p γ − m / v y ) if | γ | > m . In such a case, the Chern number wouldbecome ill-defined, so we will assume | Γ x | < m when considering the case of finite Γ x . Thecase Γ y = i γ but Γ x , z ∈ R is analogous.
4. Berry curvature and Chern number
With the states defined in Equation (15) we can define the following ω -dependent Berrycurvature B ± , i j ( k , ω ) ≡ i h ∂ i ψ L ± ( k , ω ) | ∂ j ψ R ± ( k , ω ) i , (25)giving rise to an ω -dependent Chern number defined as C ± ( ω ) = π Z d k ǫ i j B LR ± , i j ( k , ω ) , (26)where ǫ i j denotes the Levi-Civita symbol in two dimensions and the summation over i and j is implied. The last expression can be rewritten as C ± ( ω ) = i π Z d k h ǫ i j ∂ i h ψ L ± ( k , ω ) | ∂ j ψ R ± ( k , ω ) i (27) − ǫ i j h ψ L ± ( k , ω ) | ∂ i ∂ j ψ R ± ( k , ω ) i | {z } = i , (28)where the last term vanishes because it is the contraction of a symmetric and an antisymmetricquantity. In vectorial notation, we have C ± ( ω ) = i π Z d S · ∇ × h ψ L ± ( k , ω ) |∇ ψ R ± ( k , ω ) i , (29)where d S = d k ˆ z . The surface over which the integral is performed is, in principle, the whole( k x , k y ) plane ( R ). We can consider it (going over to polar coordinates) as a disk of radius eneralized Chern numbers based on open system Green’s functions ρ → ∞ , and use Stokes’s theorem to write the integral as a line integral over the boundary C of this disk, C ± ( ω ) = i π Z C d ℓ |{z} ρ d θ ˆ θ ·h ψ L ± ( ρ, θ, ω ) | ∇ |{z} ρ ∂ θ ˆ θ + ... | ψ R ± ( ρ, θ, ω ) i (30) = lim ρ →∞ i π Z π d θ h ψ L ± ( ρ, θ, ω ) | ∂ θ ψ R ± ( ρ, θ, ω ) i . (31)This last expression remains valid for any two-band model, and for any type of Green’sfunction (advanced, retarded and Keldysh) except the Matsubara Green’s function. As wementioned earlier, even though a Chern number can be constructed from the MatsubaraGreen’s function [26], it is, due to possible discontinuities, in general not a quantizedtopological invariant [27]. In the following, we will specify a concrete system and explicitlycalculate this non-Hermitian Chern number. From now on, we will consider a single 2D Dirac cone interacting with an environment thathas been already integrated out. The e ff ect of the environment is contained in the four Γ µ functions, and the whole system is defined by the expressions in Eqs. (19)-(22).In principle, these Γ µ functions are arbitrary functions of ω and k . Depending on thesymmetry of the system, some of them may vanish while others remain nonzero, dependingon the type of damping (or gain) present in the system. To be specific, we will begin byconsidering the case when Γ z ( ω ) , Γ ( ω ) ,
0, while Γ x = Γ y =
0. This case correspondsfor example to the self-energy arising from electron-phonon scattering to the lowest order inthe coupling constants [30]. Later on, we will extend our results to the case where Γ z is afunction of momentum as well, and to the cases where damping is present in other channels.We will also assume that the system has rotational symmetry, v x = v y = v , but we will relaxthis assumption later.Under these assumptions, the parameters of the system become ǫ ( k , ω ) = ω − Γ ( ω ) , (32) d x = − v ρ cos θ , (33) d y = − v ρ sin θ , (34) d z = − m − Γ z ( ω ) , (35) d = p v ρ + ( m + Γ z ) , (36)where we have used polar coordinates such that k x = ρ cos θ and k y = ρ sin θ . In this case, thechoice of eigenstates presented in Equation (15) will be determined by sgn[ R e( m + Γ z ( ω )] . ω -dependent Chern number We are now in a position to explicitly calculate the ω -dependent Chern number defined inEquation (31). In the case considered here, with rotational symmetry and Γ z ( ω ) ,
0, the eneralized Chern numbers based on open system Green’s functions | ψ R + ( ρ, θ, ω ) i R e ( d z ) < = p d ( d − d z ) − v ρ e − i θ d − d z ! , (37) | ψ R − ( ρ, θ, ω ) i R e ( d z ) < = p d ( d − d z ) d z − d − v ρ e i θ ! , (38) | ψ R + ( ρ, θ, ω ) i R e ( d z ) > = p d ( d + d z ) d z + d − v ρ e i θ ! , (39) | ψ R − ( ρ, θ, ω ) i R e ( d z ) > = p d ( d + d z ) − v ρ e − i θ − d z − d ! , (40)where d and d z are independent of θ .For each band ( ± ), we have two di ff erent states, depending on the sign of R e( d z ). Sincethe shape of the two states corresponding to each band is quite di ff erent, we will do thecalculation independently.For C + and R e( d z ) <
0, the derivative appearing in Equation (31) gives rise to | ∂ θ ψ R + ( ρ, θ, ω ) i R e ( d z ) < = p d ( d − d z ) iv ρ e − i θ ! , (41)so that C R e ( d z ) < + ( ω ) = lim ρ →∞ i π Z π d θ ( − iv ρ )2 d ( d − d z ) (42) = lim ρ →∞ v ρ p v ρ + ( m + Γ z ) ( p v ρ + ( m + Γ z ) + m + Γ z ) = . (43)When R e( d z ) <
0, the derivative becomes | ∂ θ ψ R + ( ρ, θ, ω ) i R e ( d z ) > = p d ( d + d z ) − iv ρ e − i θ ! , (44)and thus C R e ( d z ) > + ( ω ) = lim ρ →∞ i π Z π d θ iv ρ d ( d + d z ) = − . (45)Then, recalling that d z = − m − Γ z ( ω ), we summarize these two results as C + ( ω ) =
12 sgn (cid:2) R e( m + Γ z ( ω )) (cid:3) . (46)Repeating the same procedure for C − , we arrive to the final expression for the ω -dependent Chern number in our model: C ± ( ω ) = ±
12 sgn (cid:2) R e( m + Γ z ( ω )) (cid:3) . (47)We recall at this point that for a Hermitian single massive 2D Dirac cone, the Chern number is C = sgn( m ) / ω -dependent Chern number thus reflects the fact thatthe real part of Γ z can be interpreted as a renormalization of the mass m . So far, the assumptionwe made is that the influence of such an environment, i.e., the Γ µ functions, depend only on ω , and not on k . We have found that this topological invariant is half-quantized for all values eneralized Chern numbers based on open system Green’s functions ω , though it might flip the sign at some value depending on the explicit form of Γ z ( ω ).We have also found that the value of this ω − dependent Chern number is independent of theimaginary part of Γ z , that is, of the damping.In this Section we have explicitly calculated the ω -dependent Chern number for the casewith rotational symmetry v x = v y = v since it provides us with the key results without gettinginto algebraically complicated steps. However, these results can be extended to the case v x , v y in a straightforward though cumbersome way, and the Chern number calculated inthis way is exactly the same shown in Equation (47). In the next Section, we will studyhow this result is a ff ected for a more general damping: allowing for a dependence on themomentum, and on di ff erent channels. ω -dependent Chern number for more general environments Γ z depends on the wavevector So far we have considered the case in which Γ z ( ω ) did not depend on k , and Γ x = Γ y = Γ z = Γ z ( ω, k ) depend on the momentum as well, in order tounderstand how its a ff ects the Chern number. This will give us further information into thelimitations of the nH Hamiltonian formalism, where usually the Γ µ are set to be constants. Forsimplicity, we will again consider the case with rotational symmetry in which v x = v y , andwrite d x = − v ρ cos θ , (48) d y = − v ρ sin θ , (49) d z = − m − Γ z ( ω, ρ, θ ) , (50) d = p v ρ + ( m + Γ z ( ω, ρ, θ )) . (51)Then the states can again be written as in Equations (37)-(40), but now d and d z mightdepend on θ as well. Because of this, the derivative with respect to θ is of course muchmore complicated. We will show only the calculations for the lower band and R e( d z ) < | ∂ θ ψ R − ( ρ, θ, ω ) i R e ( d z ) < = − (2 d ( d − d z )) − " d − m + Γ z d (2 d − d z ) ∂ θ Γ z d z − d − v ρ e i θ ! (52) + (2 d ( d − d z )) − ∂ θ Γ z (cid:16) m +Γ z d (cid:17) − iv ρ e i θ , which leads to the following expression for the Chern number: C R e ( d z ) < − ( ω ) = lim ρ →∞ i π Z π d θ ( − ∂ θ Γ z (2 d ( d − d z )) " d − m + Γ z d (2 d − d z ) × h ( d z − d ) + v ρ i + d ( d − d z ) " ∂ θ Γ z ( d z − d ) m + Γ z d ! + iv ρ . (53) eneralized Chern numbers based on open system Green’s functions Γ z is a separable function of ρ and θ and can be written as Γ z ( ρ, θ ) = f ( ρ ) g ( θ ) for some arbitrary complex functions f and g .What we shall see is that the result will strongly depend on whether f ( ρ ) grows rapidly with ρ , or not. So let us analyze each case separately. ( ρ ) Let us first consider the case in which f ( ρ ) does not grow with ρ , or grows very slowly, so thatlim ρ →∞ f ( ρ ) ρ = . (54)In this case, d z /ρ , f ( ρ ) / d , and d z / d all vanish when ρ → ∞ , while d /ρ → | v | . So it’s easy tosee that C R e ( d z ) < − ( ω ) = lim ρ →∞ i π Z π d θ " − f ( ρ )4 d g ′ ( θ ) ( − md + f ( ρ ) d g ( θ ) ! − d z d !) × ( + v ρ ( d − d z ) ) + f ( ρ )2 d g ′ ( θ ) md + f ( ρ ) d g ( θ ) ! + iv ρ d ( d − d z ) = i π Z π d θ iv v = − . (55)Which is the same result obtained for Γ z independent of k , so we retain the half-quantizationwe have encountered. It is easy to see that this result is independent of whether Γ z dependson θ or not. The condition (54) thus determines the range of validity of the approximation oftreating the self-energy as k -independent in nH Hamiltonian. ( ρ ) Let us now consider the case of a rapidly growing f ( ρ ), suchthat lim ρ →∞ f ( ρ ) ρ = ∞ . (56)In this case, we can see that d z /ρ → ∞ and d /ρ → ∞ when ρ → ∞ , as well aslim ρ →∞ f ( ρ ) d = p g ( θ ) = g sgn( R e( g )) , (57)lim ρ →∞ d z d = − g p g = − gg sgn( R e( g )) = − sgn( R e( g )) . (58)To calculate these limits, we have assumed that f ∈ R but g ∈ C , and made use of Equation(14). On the other hand, as f ( ρ ) → ∞ when ρ → ∞ , we can see that both d , d z → ∞ .All these considerations allow us to take the limit ρ → ∞ in the Berry curvaturelim ρ → h ψ L − ( ρ, θ, ω ) | ∂ θ ψ R − ( ρ, θ, ω ) i R e ( d z ) < = g ′ ( θ ) g ( θ ) . (59)With this result we can write the Chern number as C R e ( d z ) < − ( ω ) = i π Z π d θ g ′ ( θ ) g ( θ ) = i π (cid:8) log (cid:2) g (2 π ) (cid:3) − log (cid:2) g (0) (cid:3)(cid:9) . (60)From the last expression, we can see that this quantity is always quantized. In the case of real g we see that the Chern number vanishes. However, in some other cases, the Chern number eneralized Chern numbers based on open system Green’s functions Γ z = f ( ρ ) g ( θ ) = f ( ρ ) e i θ one finds C R e ( d z ) < − ( ω ) = −
1. Hence,in this case, the presence of a self-energy can give rise to an integer
Chern number in a systemwhere the Chern number only takes the values ± / Γ z ( ρ, θ ) is not possible because the retarded self-energy must havea negative imaginary part for all θ . For the Keldysh self-energy, in contrast, such a constraintdoes not apply, so such a nontrivial change of Chern number from half-integral quantized tointeger quantized may arise in nonequilibrium systems.With this result we come to the conclusion that the half-quantization found in the caseswhere Γ z does not depend on ρ or grows slowly with it, does not hold in the case of rapidlygrowing Γ z . In this case, the Chern number is found to be quantized to an integer. The caseof rapidly growing Γ z is likely to be the quite common when considering realistic models forthe environment, since the Γ µ functions can be thought of as the decay rates of quasiparticles.Since the latter should be small compared to the quasiparticle energy for small k , the self-energies usually grow with a higher power of k than linearly. As we showed above, in mostcases the Chern number will then vanish, revealing that in these cases the damping rendersthe topology of the system trivial. ff erent channel Lastly we will extend our results to the case in which the damping, instead of acting in the z channel, is in the x channel. Due to symmetry, this is equivalent to taking it in the y channel.Then, our system will be defined by ǫ ( k , ω ) = ω − Γ ( ω ) , (61) d x = − v ρ cos θ − Γ x , (62) d y = − v ρ sin θ , (63) d z = − m , (64)where d = q v ρ + m + Γ x + v Γ x ρ cos θ (65)and Γ x = Γ x ( ω, ρ, θ ). Note that in this case, we have to make the additional assumption that | Γ x | < m for all k and ω to avoid exceptional points.We will again calculate only one of the four possible Chern numbers, since the remainingthree are obtained in an analogous way. This time, let us have a look at C R e ( d z ) < + ( ω ) (which,in the present case, corresponds to m > + eigenstate of the Green’s function isthen written as | ψ R + ( ρ, θ, ω ) i R e ( d z ) < = √ d ( d − m ) − v ρ e − i θ − Γ x d − m ! , (66)and its derivative with respect to θ is given by | ∂ θ ψ R + ( ρ, θ, ω ) i R e ( d z ) < = − (4 d − m ) ∂ θ d d − md ) / − v ρ e − i θ − Γ x d − m ! + √ d ( d − m ) iv ρ e − i θ − ∂ θ Γ x ∂ θ d ! , (67) eneralized Chern numbers based on open system Green’s functions ∂ θ d = Γ x ∂ θ Γ x + v ρ ( ∂ θ Γ x cos θ − Γ x sin θ )2 p v ρ + Γ x + m + v Γ x ρ cos θ , (68)and both Γ x and ∂ θ Γ x depend on the explicit model for the damping. The Berry curvature thenreads h ψ L + ( ρ, θ, ω ) | ∂ θ ψ R + ( ρ, θ, ω ) i R e ( d z ) < = (69) = − (2 d − m ) ∂ θ d [2 d ( d − m )] n v ρ + Γ x + v ρ cos θ + ( d − m ) o + ∂ θ d ( d − m ) − iv ρ + Γ x ∂ θ Γ x + ∂ θ Γ x v ρ e i θ − i Γ x v ρ e − i θ d ( d − m ) . As we did when the damping was allocated in the z channel, we will first consider thecase in which Γ x ( ω ) does not depend on k , and then see how such a dependence a ff ects theresults. Γ x independent of k In this case, the limit ρ → ∞ , necessary to calculate the Chernnumber, becomes straightforward. To take such limit it is useful to see thatlim ρ →∞ d ρ = lim ρ →∞ p v ρ + m + Γ x + v Γ x ρ cos θρ = | v | (70)lim ρ →∞ ∂ θ d ρ = lim ρ →∞ − v ρ Γ x sin θ ρ p v ρ + Γ x + m + v Γ x ρ cos θ = . (71)The Chern number, then, reads C R e ( d z ) < + ( ω ) = lim ρ →∞ i π Z π d θ " − (4 d − m ) ∂ θ d d ( d − m )] ×× n v ρ + Γ x + v ρ cos θ + ( d − m ) o (72) + d ( d − m ) n ∂ θ d ( d − m ) − iv ρ − i Γ x v ρ e − i θ o = . This result shows that the Chern number remains half-quantized when the damping isallocated in a di ff erent channel ( Γ x , Γ z = Γ y =
0, or Γ y , Γ z = Γ x = Γ x growing slowly with ρ If now we allow Γ x to be a function of both frequency andmomentum, we have to again consider two di ff erent cases, depending on its behaviour with ρ . First we consider a slowly growing (or decreasing) Γ x = f ( ρ ) g ( θ ), such that f ( ρ ) /ρ → ρ → ∞ .Then the Berry curvature reads: h ψ L + ( ρ, θ, ω ) | ∂ θ ψ R + ( ρ, θ, ω ) i R e ( d z ) < = (73) = − (2 d − m ) ∂ θ d [2 d ( d − m )] n v ρ + f g + v ρ cos θ + ( d − m ) o + ∂ θ d ( d − m ) − iv ρ + f gg ′ + f g ′ v ρ e i θ − i f gv ρ e − i θ d ( d − m ) , eneralized Chern numbers based on open system Green’s functions d = p v ρ + m + f g + v Γ x ρ cos θ , (74) ∂ θ d = f gg ′ + v ρ f ( g ′ cos θ − g sin θ )2 p v ρ + f g + m + v f g ρ cos θ . (75)To take the limit ρ → ∞ , we uselim ρ →∞ d ρ = | v | (76)lim ρ →∞ ∂ θ d ρ = . (77)With these results, it is easy to see that C R e ( d z ) < + ( ω ) = lim ρ →∞ i π Z π d θ h ψ L + ( ρ, θ, ω ) | ∂ θ ψ R + ( ρ, θ, ω ) i = , (78)and we recover the half-quantized Chern number, for any θ dependence, as long as Γ x growsslowly with ρ . Γ x Now we take again Γ x = f ( ρ ) g ( θ ), but this time we have f ( ρ ) /ρ → ∞ when ρ → ∞ . In this case, we get the following limits:lim ρ →∞ d ρ = g sgn( R e ( g )) , (79)lim ρ →∞ ∂ θ d ρ = g ′ g , (80)lim ρ →∞ ρ d = . (81)Then, we can see that the Berry curvature in the limit of large ρ vanishes:lim ρ →∞ h ψ L + ( ρ, θ, ω ) | ∂ θ ψ R + ( ρ, θ, ω ) i R e ( d z ) < = (82) = lim ρ →∞ − (2 − md ) ∂ θ dd [2(1 − md )] ( v ρ d + f d g + vd ρ d cos θ + (1 − md ) ) + − md ) ( ∂ θ dd (1 − md ) − iv ρ d + f d gg ′ + fd g ′ vd ρ e i θ − i fd gv ρ d e − i θ ) = − g ′ g g sgn ( R e( g )) g + ! + ( g ′ g + gg ′ g sgn ( R e( g )) ) = , which implies that the Chern number vanishes as well. The main di ff erence between this resultand Eq. (60) is that, when the damping was allocated in the Γ z channel, certain functionalforms of Γ z ( ρ, θ ) can result in a non-vanishing, integer-quantized Chern number. In the casein which the damping is allocated in the Γ x or Γ y channels, however, this does not occur andthe Chern number vanishes for any functional form of g ( θ ).In any case, several conclusions can be extracted from the cases analyzed through thissection. We can see that in many cases (when Γ i does not depend on k , or when it doesn’tgrow rapidly with ρ ), we recover the half-quantized result that was observed while using nH eneralized Chern numbers based on open system Green’s functions Γ i functions, this half-quantization is lost, which shows that thenH Hamiltonian formalism might have some limitations for specific environments.
6. Conclusions
In this work we have presented an alternative approach for the study and topologicalclassification of open quantum systems. Throughout this paper we have investigated asystem interacting with an environment whose influence on the system is encoded in ageneral self-energy. Moreover, such self-energies can arise not only from the coupling to anenvironment but also as an e ff ective description of e ff ects such as interactions which cannotbe treated exactly. Such systems have often been topologically classified using e ff ective nHHamiltonians and we have proposed Green’s function as an alternative classification method.By relying on the system’s Green’s function instead of the construction of a non-Hermitian (nH) Hamiltonian, we render our approach more general and applicable in principleto any gapped 2D model. The formalism applies to advanced, retarded, as well as KeldyshGreen’s functions, and can thus capture e ff ects due to environments, non-equilibrium,interactions, gain and losses, which can be expressed through the self-energy.Using the eigenstates of the inverse Green’s function, we were able to define a topologicalinvariant, the ω -dependent Chern number. This quantity is analogous to the nH Chern numberdefined in the literature, with two main di ff erences: firstly, it does not require the constructionof a nH Hamiltonian and hence the assumptions implied by such a construction. Secondly,this topological invariant is energy-dependent, and the value of this Chern number can indeedchange at certain energies, even in the case in which it is quantized or half-quantized. This isin line with recent field-theoretical approaches which have also argued that since nH phasesarise in open systems or systems out of equilibrium, the momentum and frequency degrees offreedom have to be treated on di ff erent footings [34].We have considered then in our work a concrete system: a continuum 2D Dirac model,consisting of a single gapped Dirac cone. The influence of the environment, damping,interactions, gain, losses, etc., was encoded within four functions (damping channels) denoted Γ µ = ( Γ , Γ x , Γ y , Γ z ). One of the results of our work is that the outcome significantly dependson the damping channel. Our first result is that the damping allocated in the Γ channel, whichcan be thought of as a (complex) renormalization of the energy (see Equation (61)), does notimpact the Chern number, independently of its functional form. We have considered the casein which the Γ µ functions depend solely on ω , being independent of k . If only one of the Γ i with i = x , y , z was non-vanishing, the Chern number is half-quantized to values ± / ff ectivelyrenormalized by Γ z , we found that its real part might induce a jump in the Chern number.Since the Γ i functions are frequency dependent, so is the change in the Chern number. TheChern number might be di ff erent at di ff erent energies.When we allowed the Γ µ functions to depend as well on the momentum, we found eneralized Chern numbers based on open system Green’s functions Γ µ at large momentum. For slowly-growing functions, the half-quantized result was recovered. In the case of rapidly growing Γ µ ,however, we found that in most cases the ω -dependent Chern number vanished. Since the Γ µ functions can be interpreted as the decay rates of quasiparticles, this result can be understoodas the damping-induced loss of topological quantization. We found, however, that for someparticular functional forms of Γ z , a non-vanishing, integer-quantized Chern number could beobtained.In summary, our results go beyond what has been considered so far in terms of topologyof open quantum systems, particularly for gapped two-band systems. We have made fewerassumptions on the influence of the environment on the system, and we have found thatcertain environment properties that cannot easily be taken into account when using the nHHamiltonian formalism, might indeed produce significant changes to the topology of thesystem. We believe our results will help to broaden the understanding we have so far oftopology in open systems, shedding some light into the regions of validity and the impliedassumptions that underlie the use of nH Hamiltonians, as well as allowing for a broader rangeof systems to be studied. Acknowledgements
We acknowledge financial support from the National Research Fund Luxembourg undergrants ATTRACT A14 / MS / / MoMeSys, CORE C16 / MS / / PARTI, andCORE C20 / MS / / OpenTop
References [1] Mahan G D 2000
Many-Particle Physics (Springer US) ISBN 0306463385[2] Bruus H and Flensberg K 2004
Many-body quantum theory in condensed matter physics: an introduction (Oxford university press)[3] Bruch A, Thomas M, Kusminskiy S V, von Oppen F and Nitzan A 2016
Physical Review B [4] Haughian P, Esposito M and Schmidt T L 2018 Physical Review B [5] Breuer H P and Petruccione F 2002 The theory of open quantum systems (Oxford University Press onDemand)[6] Diehl S, Rico E, Baranov M A and Zoller P 2011
Nature Physics New J. Phys. . Eur. Phys. J. Spec. Top. Journal of Physics: Materials Phys. Rev. Lett. (17) 170401[11] Bergholtz E J, Budich J C and Kunst F K 2019 arXiv preprint arXiv:1912.10048 [12] Rotter I 2009
Journal of Physics A: Mathematical and Theoretical Physical Review B [14] Kunst F K, Edvardsson E, Budich J C and Bergholtz E J 2018 Physical Review Letters [15] Longhi S 2019
Phys. Rev. Research (2) 023013[16] Moors K, Zyuzin A A, Zyuzin A Y, Tiwari R P and Schmidt T L 2019 Phys. Rev. B Phys. Rev. A (4) 042114[18] Wojcik C C, Sun X Q, Bzduˇsek T and Fan S 2020 Physical Review B eneralized Chern numbers based on open system Green’s functions [19] Shen H, Zhen B and Fu L 2018 Phys. Rev. Lett. (14) 146402[20] Altland A and Zirnbauer M R 1997
Phys. Rev. B Physical Review X Physical Review X [23] Liu C H, Jiang H and Chen S 2019 Physical Review B [24] Ishikawa K and Matsuyama T 1986 Zeitschrift f¨ur Physik C Particles and Fields Nuclear Physics B
Physical Review X Phys. Rev. B (8) 081104[28] Philip T M, Hirsbrunner M R and Gilbert M J 2018
Phys. Rev. B arXiv preprint arXiv:2009.10455v1 [30] Kozii V and Fu L 2017 arXiv preprint arXiv:1708.05841 [31] Yoshida T, Peters R, Kawakami N and Hatsugai Y 2019 Physical Review B [32] Zyuzin A A and Zyuzin A Y 2018 Phys. Rev. B arXiv preprint arXiv:2101.02393 [34] Kawabata K, Shiozaki K and Ryu S 2020 arXiv preprint arXiv:2011.11449 [35] Brody D C 2013 Journal of Physics A: Mathematical and Theoretical Basic complex analysis (New York: W.H. Freeman and Company) ISBN 0716718146[37] Qi X L and Zhang S C 2011