On the eigenvalues of some non-Hermitian Hamiltonians with space-time symmetry
aa r X i v : . [ qu a n t - ph ] M a r On the eigenvalues of some non-Hermitian Hamiltonians with space-time symmetry
Paolo Amore ∗ Facultad de Ciencias, Universidad de Colima, Bernal D´ıaz del Castillo 340, Colima, Colima, Mexico.
Francisco M. Fern´andez † and Javier Garcia ‡ INIFTA (UNLP, CCT La Plata-CONICET), Divisi´on Qu´ımica Te´orica,Blvd. 113 y 64 S/N, Sucursal 4, Casilla de Correo 16, 1900 La Plata, Argentina
We calculate the eigenvalues of some two-dimensional non-Hermitian Hamiltonians by means of apseudospectral method and straightforward diagonalization of the Hamiltonian matrix in a suitablebasis set. Both sets of results agree remarkably well but differ considerably from the eigenvaluesobtained some time ago by other authors. In particular, we do not observe the multiple phasetransitions claimed to occur in one of the anharmonic oscillators.
PACS numbers: 03.65.-w
I. INTRODUCTION
In a recent paper, Klaiman and Cederbaun[1] studied the spectrum of non-Hermitian Hamiltonians H = H + iλW by means of the point-group symmetries of the Hermitian H and non-Hermitian W parts. They showed that, inprinciple, the symmetry properties of the Hamiltonian are responsible for the appearance of real eigenvalues in thespectrum of the non-Hermitian Hamiltonian H . To this end they constructed an effective energy-dependent HermitianHamiltonian that exhibits the same real spectrum as the non-Hermitian one. They illustrated the main theoreticalresults by means of suitable chosen examples. One of them is of great interest because it exhibits multiple phasetransitions. As the parameter λ increases two real eigenvalues approach each other, coalesce and become a pairof complex conjugate numbers. That is to say, the space-time symmetry is broken beyond the coalescence point.However, on increasing λ those same complex eigenvalues become real again, separate, just to approach each otheragain and coalesce at a larger value of λ .The purpose of this paper is a critical discussion of those results. In Sec. II we outline the point-group symmetry ofthe models considered by Klaiman and Cederbaum[1]. In Sec. III we compare present results with the ones of thoseauthors and briefly discuss an example not considered by them. In Sec. IV we draw conclusions. ∗ Electronic address: [email protected] † Corresponding author: [email protected] ‡ Electronic address: [email protected]
II. THE MODELS
Three of the examples considered by Klaiman and Cederbaun[1] are based on the Hamiltonian H = H + iλW,H = − (cid:0) ∂ x + ∂ y (cid:1) + α x x + α y y , (1)where α x = 1 and α y = √
2. They wrote the eigenvectors of H formally as | n x , n y i = | n x i ⊗ | n y i , (2)where n x , n y = 0 , , . . . and | n x i , | n y i denote the eigenvectors of the x - and y -quartic oscillators, respectively.They described the symmetry of H by means of the point group D h (isomorphic to C v ) with symmetry operations { E, P, P x , P y } that are given by the coordinate transformations E : { x, y } → { x, y } ,P : { x, y } → {− x, − y } ,P x : { x, y } → {− x, y } ,P y : { x, y } → { x, − y } . (3)It follows from E | n x , n y i = | n x , n y i ,P | n x , n y i = ( − n x + n y | n x , n y i ,P x | n x , n y i = ( − n x | n x , n y i ,P y | n x , n y i = ( − n y | n x , n y i , (4)that the eigenvectors are bases for the irreducible representations A g , B g , A u or B u when ( n x , n y ) is (even, even),(odd, odd), (even, odd) or (odd, even), respectively. III. RESULTS
Before discussing the non-Hermitian Hamiltonians considered by Klaiman and Cederbaun[1] we first focus on theHermitian Hamiltonian H . Since α y > α x it is clear that E (0)10 ( B u ) < E (0)01 ( A u ). Surprisingly their figures 3 and4 show exactly the reverse order. Besides, the same level order appears in Fig. 5 where the authors labelled theeigenvalues by means of the point group C i instead of D h .We calculated the lowest eigenvalues E (0) n x n y of H by three completely different approaches: the Riccati-Pad´emethod (RPM)[2, 3], a pseudospectral method[4] and the straightforward diagonalization method (DM) using a basisset of products of eigenfunctions of the harmonic oscillator H HO = p + q . The three methods agree remarkably wellfor λ = 0 and the latter two ones for all λ (the RPM does not apply to nonseparable problems).Table I shows the lowest eigenvalues of H as well as the symmetry of the corresponding eigenfunctions accordingto the point groups C i and D h . By simple inspection it is clear that the results of this table do not agree with thosefor λ = 0 in figures 3, 4, and 5 of Ref.[1] in agreement with the discussion above.We first consider the non-Hermitian perturbation W = xy that is invariant with respect to P : P W P = W . On theother hand, the whole Hamiltonian (1) is invariant under two antiunitary transformations A x = T P x and A y = T P y ,where T is the time-reversal operator. According to the authors it exhibits two space-time symmetries that are ageneralization of the well known PT symmetry[1]. In this case the states that transform as A g ( A u ) couple to statesthat transform as B g ( B u ). The authors illustrate such couplings in their Fig. 3 but, as discussed above, some of thelabels of the lines E mn ( λ ) appear to exhibit a reverse order and the numerical values of the eigenvalues E mn (0) donot appear to agree with present calculation displayed in Table I.The second example is given by W = x y . In this case the states A g ( B g ) couple with the A u ( B u ) ones as shown inFig. 4 in the paper by Klaiman and Cederbaum[1]. The eigenvalues of H exhibit the discrepancy already discussedabove.The non-Hermitian perturbation W = x y + xy is of special interest because the authors identified pairs of statesthat are real for 0 < λ < λ b , coalesce at λ b and become complex conjugate for λ b < λ < λ c , then real again for λ c < λ < λ f and coalesce again at λ = λ f to become complex conjugate once more. Bender et al[8] have recentlydiscussed such consecutive phase transitions in the case of classical and quantum-mechanical linearly-coupled harmonicoscillators (see also [9]). We calculated the same eigenvalues E mn ( λ ) in the same range of values of λ and did notfind any of the multiple phase transitions mentioned by Klaiman and Cederbaum. Fig. 1 shows present results thatexhibit the customary phase transitions for multidimensional oscillators[5].Finally, we want to discuss a problem that was not considered by Klaiman and Cederbaum[1] but may be ofinterest. When α x = α y = 1 the Hamiltonian H is invariant under the unitary transformations of the point group C v . This group exhibits a degenerate irreducible representation E and, therefore, is beyond the discussion of thepaper of Klaiman and Cederbaum[1]. However, we deem it worth mentioning it here as another example of thosediscussed by Fern´andez and Garcia[6, 7]. In this case the non-Hermitian perturbation W = xy (with point group C v )couples the degenerate eigenvectors | m, n + 1 i and | m + 1 , n i and the ST -symmetric non-Hermitian operator (1)exhibits complex eigenvalues for all λ >
0. More precisely, some of the eigenvectors of H belonging to the irreduciblerepresentation E with real eigenvalues are coupled by the non-Hermitian perturbation and become eigenvectors of H belonging to the irreducible representations B and B with complex eigenvalues. As argued by Fern´andez andGarcia the ST symmetry is not as robust as the P T one (were P is the inversion operation in the point group). IV. CONCLUSIONS
In this paper we carried out three completely different calculations of the eigenvalues and eigenfunctions of theHermitian operator H and two of them for the eigenvalues and eigenfunctions of the non-Hermitian operator (1)with three non-Hermitian perturbations W . The agreement of the results provided by those methods makes usconfident of their accuracy. Present results do not agree with those of Klaiman and Cederbaum[1]. Straightforwardcomparison of the results in Table I with those in figures 3, 4 and 5 of Ref.[1] shows that the magnitude of theeigenvalues and the level ordering are quite different. Present eigenvalues E mn ( λ ) for W = x y + xy displayed inFig. 1 do not exhibit the multiple phase transitions discussed by those authors but the well-known symmetry breakingat exceptional points common to other two-dimensional PT-symmetric oscillators[5].In addition to all that, we have also shown that the ST symmetry proposed by Klaiman and Cederbaum[1] is TABLE I: Lowest eigenvalues E (0) n x n y of H and the symmetry of the eigenvectors according to the point groups C i and D h . n x n y E (0) n x n y C i D h . . . . . . . . . . . . . . . . . . . . . . . not as robust as the P T one[5]. In two recent papers Fern´andez and Garcia[6, 7] have already discussed two other ST -symmetric cases that exhibit phase transitions at the trivial Hermitian limit. Those authors also argued that thecoupling of the degenerate states of H to produce complex eigenvalues will not take place when P W P = − W . [1] S. Klainman and L. S. Cederbaum, Phys. Rev. A , 062113 (2008).[2] F. M. Fern´andez, Q. Ma, and R. H. Tipping, Phys. Rev. A , 1605 (1989).[3] F. M. Fern´andez, Q. Ma, and R. H. Tipping, Phys. Rev. A , 6149 (1989).[4] P. Amore and F. M. Fernandez, Phys. Scr. , 045011 (2010).[5] C. M. Bender and D. J. Weir, J. Phys. A , 425303 (2012).[6] F. M. Fern´andez and J. Garcia, Ann. Phys. , 195 (2014).[7] F. M. Fern´andez and J. Garcia, PT-symmetry broken by point-group symmetry, arXiv:1308.6179v2 [quant-ph][8] C. M. Bender, M. Gianfreda, S. K. ¨Ozdemir, B. Peng, and L. Yang, Phys. Rev. A , 062111 (2013).[9] F. M. Fern´andez, Algebraic treatment of PT-symmetric coupled oscillators, arXiv:1402.4473 [quant-ph] λ R e ( E ) λ I m ( E ) FIG. 1: First eight eigenvalues of the non-Hermitian Hamiltonian (1) with W = xy + x y . The continuous blue lines anddashed red ones indicate states with symmetry A g and A uu