Dissipative spin chain as a non-Hermitian Kitaev ladder
We derive exact results for the Lindblad equation for a quantum spin chain (one-dimensional quantum compass model) with dephasing noise. The system possesses doubly degenerate nonequilibrium steady states due to the presence of a conserved charge commuting with the Hamiltonian and Lindblad operators. We show that the system can be mapped to a non-Hermitian Kitaev model on a two-leg ladder, which is solvable by representing the spins in terms of Majorana fermions. This allows us to study the Liouvillian gap, the inverse of relaxation time, in detail. We find that the Liouvillian gap increases monotonically when the dissipation strength is small, while it decreases monotonically for large , implying a kind of phase transition in the first decay mode. The Liouvillian gap and the transition point are obtained in closed form in the case where the spin chain is critical. We also obtain the explicit expression for the autocorrelator of the edge spin. The result implies the suppression of decoherence when the spin chain is in the topological regime.
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Introduction.—With recent advances in quantum engineering, it becomes increasingly important to study how the coupling to the environment affects a system. The time evolution of such an open system is described by a master equation. Under rather general conditions that the evolution is Markovian and Completely Positive and Trace Preserving (CPTP), one obtains the Lindblad equation Breuer and Petruccione (2002) for the time-dependent density matrix. In the past, this quantum master equation had been mostly used to describe few-particle systems in, e.g., quantum optics. However, recent years have witnessed a growing interest in many-particle systems in the Lindblad setting Prosen and Žnidarič (); Žnidarič (2014, 2015); Medvedyeva et al. (2016).
Although there are several approaches to analyze the Lindblad equation such as perturbation theory Gallis (1996); Li et al. (2014) and numerical methods Prosen and Žnidarič (); Cui et al. (2015); Kshetrimayum et al. (2017); Raghunandan et al. (2018), exact results for the full dynamics is few and far between. In some cases Prosen (2008, 2011), the nonequilibrium steady states (NESSs) can be constructed exactly, but it is more challenging to completely diagonalize the Liouvillian (the generator of the Lindblad equation). In this sense, much fewer cases are known as exactly solvable models Prosen (2008); Medvedyeva et al. (2016). The difficulty lies in dealing with the space of linear operators, the dimension of which grows more rapidly than that of the Hilbert space. This limits the system size amenable to exact numerical diagonalization. To make matters worse, it is often the case that effective interactions arise from dissipation even when the Hamiltonian itself is reducible to that of a free-particle system. This prevents us from understanding the full dynamics of the system.
In this Letter, we present an exactly solvable dissipative model which corresponds to the non-Hermitian many-body quantum system. The model we propose has a conserved charge that leads to two exact NESSs. Moreover, our model can be seen as a non-Hermitian Kitaev model on a two-leg ladder DeGottardi et al. (2011); Wu (2012). Therefore, by applying Kitaev’s technique Kitaev (2006), our model can be mapped to free Majorana fermions in a static gauge field, which allows us to fully diagonalize the Liouvillian. We numerically identify the gauge sectors where the first decay modes live. Then, assuming the flux configurations obtained, we derive the exact Liouvillian gap, the inverse of the relaxation time. We also study the infinite temperature autocorrelator of an edge spin and obtain its exact formula by applying techniques from combinatorics.
Models and NESSs.—We consider the Lindblad equation for the density matrix
where denotes the Hamiltonian of the one-dimensional quantum compass model Brzezicki et al. (2007); Feng et al. (2007); You and Tian (2008); Eriksson and Johannesson (2009); Jafari (2011); Liu et al. (2012) given by
and are Lindblad operators. Here () are the Pauli operators at site , and we assume that the number of sites is even and the open boundary conditions are imposed. It suffices to consider the case , as the other cases can be obtained by an appropriate unitary transformation. The operator is called a Liouvillian or a Lindbladian. A NESS is a fixed point of the dynamics Eq. (1), i.e., an eigenstate of the Liouvillian with eigenvalue . For our model, there are two steady states , where is an identity matrix and is a conserved charge, i.e.,
The proof goes as follows. Because of the Hermiticity of Lindblad operators, there is a trivial NESS, i.e., completely mixed state . One can easily verify that if , then also satisfies . Although itself is not positive semi-definite which is a necessary condition to be a density operator, one can construct the following operators , which are orthogonal projections, and hence positive semi-definite. This then gives (normalized) steady states . We have checked numerically for small that they are the unique NESS of the system.
Kitaev’s Mapping.—A density matrix can be thought of as a -dimensional vector (See Supplemental Material 111See Supplemental Material for details. for details). In this sense, we can identify the Liouvillian for the one-dimensional chain as a non-Hermitian Hamiltonian of a ladder system Žnidarič (2014, 2015) (see Fig. 1):
where the Hilbert space of the RHS is the “ space” and is the Pauli matrix for the th Bra site. The non-unitary terms in the Liouvillian (1) correspond to the non-Hermitian terms in .
We apply to this ladder Hamiltonian (4) the technique by Kitaev Kitaev (2006), which was originally used to solve the quantum spin model on a honeycomb lattice. In order to solve the model, Kitaev developed an elegant technique: substituting Majorana fermion operators for spin operators . Here, and are Majorana operators obeying the Clifford algebra , and with being the anti-commutator. After the mapping, we have a quadratic Hamiltonian of itinerant Majorana fermions (’s) in each sector specified by the static gauge field (i.e., each link has the sign in the hopping amplitude). Thus, we can diagonalize the Hamiltonian and obtain all eigenvalues and eigenstates, sector by sector. Besides the honeycomb lattice, Kitaev’s mapping is applicable to other lattices with a similar Hamiltonian. Examples include the ladder system DeGottardi et al. (2011), which is the Hermitian analogue of our model (4).
Next, we define usual Dirac fermions each of which is made of two Majorana fermions and : . Here, (resp. ) is the Majorana operator for the (resp. ) spin. Then, the model is mapped to the Su-Schrieffer-Heeger (SSH) model Su et al. (1979) with imaginary chemical potential Klett et al. (2017); Lieu (2018). The Hamiltonian reads Note1 ()
where and is a tridiagonal and complex-symmetric matrix given by
’s come from the gauge degree of freedom and take the value . The solution of a non-Hermitian quadratic form of fermions is similar to that of a Hermitian one. One can construct many-body eigenstates of by just filling single-particle energies of the Hamiltonian, which can be obtained by diagonalizing .
Symmetries of the Hamiltonian (5) enable us to restrict the configurations of ’s to consider. First, and have the same spectrum because the flux configuration in the ladder system is invariant under sending . (In view of the SSH model, is transformed to by the charge conjugation .) Second, due to the inversion symmetry, the transformation leaves the spectrum of unchanged. In the following, we only consider the configurations in which the number of positive ’s is not less than the number of negative ’s.
hence, the inverse of the relaxation time. It is clear from Eqs. (4) and (7) that the Liouvillian gap corresponds to the gap between the first and second largest imaginary parts of eigenvalues of , the former of which is . The configuration which gives the eigenvalue is for all . The reason is as follows. In this configuration, the one-particle energy levels are obtained just by shifting those of the original SSH model by . Therefore, we obtain the eigenvalue by filling all the energy levels. Then, the Liouvillian gap is recast as
where denotes the th eigenvalue of , and the number of ’s which are . Since at least one of ’s must be when , we have . One might think that we need to consider the case where for all and some single-particle energy levels are empty. However, this is not the case. The Liouvillian gap in this sector must be greater than or equal to in Eq. (8), as removing one fermion from an occupied state in this case decreases the imaginary part of the eigenvalue of by . Thus, it suffices to consider configurations different from the one with for all .
Fig. 2 shows the numerical results of as a function of for various for a system size . Here, ‘topological’, ‘critical’, and ‘trivial’ cases refer to the regions , , and , respectively, in analogy with the Hermitian SSH model. From this figure, we can see a kind of phase transition of the first decay mode in every (topological, critical, or trivial) case. We also numerically obtained the chemical-potential configurations which give the first decay mode, as also shown in Fig. 2 222We do not show all the configurations which give the same eigenvalues. In Note1 (), up to symmetries mentioned above, every configuration which gives the first decay mode is shown.. In the ‘phase I’ of the critical and topological cases, the gap behaves as exactly . The reason for this behavior becomes clear in Eq. (8). When is small enough, all satisfy , then follows. In the topological case, the situation is slightly different. In this case there exists a which satisfies , and is smaller than . For finite , there is another configuration in the region of Note1 (), but this region shrinks to zero as . In the ‘phase II’, the gap behaves asymptotically in each case. This increase in relaxation time as can be thought of as the Quantum Zeno effect Vasiloiu et al. (2018).
Our extensive numerical calculation suggests that the chemical-potential configurations which give the first decay modes do not depend on the system size. Then, under this assumption, we can obtain the exact formula for the Liouvillian gap and the transition point in the thermodynamic limit of the critical case (see Note1 () for more details):
We have confirmed that this result agrees well with the numerical one for .
Autocorrelator at .—
In this section, we study the ‘infinite temperature’ autocorrelator of the edge spin
where is the adjoint operator of the Liouvillian, which describes the time evolution of an operator as follows:
In other words, it corresponds to the Heisenberg picture of the open quantum system. A fundamental motivation in quantum engineering is for localized degrees of freedom to maintain coherence over long times. In Kemp et al. (), this quantity for the transverse-field Ising and XYZ chains without dissipation (i.e., closed system) has been studied as the witness of long coherence times for edge spins. The autocorrelator for the dissipative transverse-field Ising model was also studied in Vasiloiu et al. (2018). Here, we study for our model with . Let us consider how the adjoint Liouvillian acts on . For notational simplicity, we set and . In this case, one finds
and for ,
It is important to note that ’s are Hermitian and form an orthonormal set, i.e.,
The inner product for matrices is called the Hilbert-Schmidt inner product, with which Eq. (11) takes the form
Now we compute by considering the so called ‘Riordan paths’ Chen et al. (2008); Cohen et al. (2016) (Motzkin paths Stanley (1999) with no horizontal steps at the bottom line) weighted through and . (see Fig. 3). To this end, it is useful to consider the generating function of the weighted Riordan paths, which can be obtained by the so called ‘Kernel method’ Prodinger (0304). The autocorrelator in terms of reads
where denotes the coefficient of in . This can be rewritten as a contour integral
where . Here we have chosen the contour in the -plane so that it surrounds the origin and is sufficiently small. As a result, the contour in the -plane is sufficiently large. The final explicit expression for is cumbersome and is not shown here Note1 (). But from the result, we can derive the inverse of the decay time of the autocorrelator: for , one has
and for ,
where . In particular, we find that the decay is suppressed in the topological regime (), although goes to as in both regime.
Conclusions.—We have studied the one-dimensional quantum compass model with dephasing noise and obtained the exact steady states. We showed that the model can be mapped to the non-Hermitian Kitaev model on a ladder, which is solvable by representing the spins in terms of Majorana fermions. This technique allows us to study the Liouvillian gap exactly. In particular, in the critical case where , the gap in the thermodynamic limit is obtained analytically. We have also studied the autocorrelator of the edge spin and obtained its exact formula using the technique of combinatorics.
Acknowledgments.—We would like to thank T. Prosen and M. Žnidarič for valuable comments. H.K. was supported in part by JSPS KAKENHI Grant Nos. JP18H04478 and JP18K03445. N.S. acknowledges support of the Materials Education program for the future leaders in Research, Industry, and Technology (MERIT).
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Supplemental Materials for: Dissipative spin chain as a non-Hermitian Kitaev ladder
Appendix A Mapping of the Liouvillian to the Non-Hermitian Hamiltonian
In this section, we explain more details on identifying the Liouvillian as a non-Hermitian Hamiltonian. Let be a linear operator on Hilbert space (i.e., ). Assuming , there is a complete orthonormal basis of , and can be regarded as an matrix
where a sans-serif style and its element is just a c-number. Now, we introduce a new space of dimension by the following linear map :
Note that depends on the choice of the basis , but after fixing the basis, is an isomorphism, i.e., and are in one-to-one correspondence.
Let us consider how superoperators () look like in the space. All superoperators in the Lindbladian have the form
This map is rewritten in the language of matrices as
for , and . In the space, the superoperator is seen as the following map:
Therefore, this superoperator can be thought of as the tensor product of two matrix . (Note that this matrix is basis-dependent because it is not .) Then, one can identify the Liouvillian in Eq. (1) as
Appendix B Non-Hermitian Quadratic Form of Fermions
In a usual closed free fermionic system, we can generally write the Hamiltonian as follows:
where , and is an Hermitian matrix. In general, all eigenvalues of a Hermitian matrix are real, and eigenvectors can form an orthonormal basis (after normalization and orthogonalization in degenerate spaces). In other words, can be diagonalized by a unitary matrix , and the Hamiltonian is rewritten as
where are the eigenvalues of . Then, we can define new operators , and it is easily verified that also satisfy anticommutation relations. Therefore, we obtain
and all eigenvalues of are obtained by with arbitrary choice of each .
However, some care must be taken when is non-Hermitian. First, eigenvalues of are not necessarily real. Second, eigenvectors with different eigenvalues are in general not orthogonal. Third, left and right eigenvectors with the same eigenvalue are in general not a Hermitian-conjugate to each other. We briefly explain the general prescription for treating non-Hermitian matrices according to Ref. Brody (2014). A similar discussion can be found in Ref. Sternheim and Walker (1972).
Let be a non-Hermitian and non-degenerate matrix. Remember that if a matrix is non-degenerate, then it is diagonalizable. We can easily generalize the following discussion to the case where is degenerate but diagonalizable. However, some degenerate matrices cannot be diagonalized, and such situations are sometimes called “coalescence” instead of “degeneracy”. We do not consider “coalescence” here. See, e.g., Ref. Heiss (2012) for more details about coalescence. Then, has left/right eigenvectors and which satisfy
( and are column vectors.) From these, we obtain
where is the standard inner product of . Because is diagonalizable, each of and is a basis of . Therefore, fixing index and making index run, at least one satisfies . Then, we can assume after relabeling ’s. It follows that if and . Therefore, after “normalization” of and/or so as to satisfy , we obtain
Then, we define
and is diagonalized as
Now, let us return to Eq. (28) with non-Hermitian . By diagonalizing as above, we obtain
and we define
then, the Hamiltonian can be written in the form similar to Eq. (30),
One can easily verify the following anticommutation relations
Therefore, and can be seen as an annihilation and an creation operator of new fermions respectively, although is not equal to unlike Hermitian cases Bilstein and Wehefritz (1998). From these anticommutation relations, it follows that has eigenvalues and , although it is not Hermitian. Then we obtain all eigenvalues of as with arbitrary choice of each .
Appendix C From Kitaev Ladder to Ssh Model
By Kitaev’s mapping, a non-Hermitian Hamiltonian can be seen as a model of free Majorana fermions in a static gauge field. Introducing Majorana fermion operators as and (), we obtain
where the operators, , , , , and commute with the Hamiltonian and their eigenvalues are . Therefore, the Hilbert space splits into sectors labeled by the eigenvalues of these operators. One can define the flux through each plaquette by the eigenvalue of the product of and operators around it. The spectrum of the Hamiltonian depends only on the set of fluxes. Thus, we can fix all but signs of the links, as we have plaquettes in our model. We fix them as
and define to recast as
Next, we define usual Dirac fermions and , consisting of two Majorana fermions and :
It is easy to verify that they satisfy the anti-commutation relations
We then have
After the unitary transformation , we obtain Eq. (5).
Appendix D The First Decay Modes’ Configurations
In Fig. 2, we show for each phase only one example of the configurations where the first decay mode lives, but numerical calculation reveals that there are other such configurations. Tab. 1 shows the numerical results of all such configurations up to symmetries mentioned in the main text.
|very small- phase||phase I||phase II|
Appendix E Derivation of Eq. (9)
We call the pattern of which satisfies “” “pattern 1”, and that which satisfies “” “pattern 2”. (resp. ) denotes the matrix with the pattern 1 (resp. pattern 2). The (unnormalized) eigenstates of whose eigenvalues have negative imaginary parts are obtained by the following ansatz
where and . Letting be the eigenvalue of corresponding to , we obtain the following conditions on this ansatz
Here, we neglect the right boundary condition that is justified in the thermodynamic limit. The solution of is
In the critical case of , has negative imaginary part when , and the gap for pattern 1 (i.e., the argument in parentheses in Eq. (8) for pattern 1) is
In a similar way, we obtain the localized solution for pattern 2 by the ansatz
From these conditions, we obtain in the critical case the gap for pattern 2 as
Finally, we obtain the global gap as
Therefore, the transition point for the critical case is
One may guess that even for topological or trivial case, we can obtain the exact results for the gap in a similar way. However, it would be impossible to obtain the algebraic solutions because in these cases, we need to deal with equations of degree greater than four.
Appendix F Exact Formula of the Autocorrelator With Dissipation
The generating function for the weighted Riordan path is obtained as
by the Kernel method. Then, the autocorrelator is recast as
The contour of is chosen as shown in Fig. 4, and the final results are
In general, it does not have a simpler form. However, in the absence of dissipation, i.e., when , it can be simplified:
We have confirmed numerically that the integral goes to zero as in either case. Therefore, the autocorrelator is non-vanishing as in the topological phase, whereas vanishing in the trivial phase. Moreover, when , it takes a simpler form:
where is the Bessel function of the first kind. From the asymptotic behavior of the Bessel function, we obtain that decays as for large .