Excited State Quantum Phase Transitions Studied from a Non-Hermitian Perspective
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Excited State Quantum Phase Transitions Studied from a Non-Hermitian Perspective
Milan ˇSindelka ∗ Institute of Plasma Physics, Academy of Sciences of the Czech Republic, Prague 8, 18200, Czech Republic
Lea F. Santos † Department of Physics, Yeshiva University, New York, New York 10016, USA
Nimrod Moiseyev ‡ Schulich Faculty of Chemistry, Institute of Solid State, and Faculty of Physics,Technion - Israel Institute of Technology, Haifa 32000 Israel (Dated: June 26, 2018)A main distinguishing feature of non-Hermitian quantum mechanics is the presence of exceptional points(EPs). They correspond to the coalescence of two energy levels and their respective eigenvectors. Here, we usethe Lipkin-Meshkov-Glick (LMG) model as a testbed to explore the strong connection between EPs and the on-set of excited state quantum phase transitions (ESQPTs). We show that for finite systems, the exact degeneracies(EPs) obtained with the non-Hermitian LMG Hamiltonian continued into the complex plane are directly linkedwith the avoided crossings that characterize the ESQPTs for the real (physical) LMG Hamiltonian. The valuesof the complex control parameter α that lead to the EPs approach the real axis as the system size N → ∞ . Thishappens for both, the EPs that are close to the separatrix that marks the ESQPT and also for those that are faraway, although in the latter case, the rate the imaginary part of α reduces to zero as N increases is smaller. Withthe method of Pad´e approximants, we can extract the critical value of α . Introduction.–
A quantum phase transition (QPT) corre-sponds to the vanishing of the gap between the ground stateand the first excited state in the thermodynamic limit [1, 2].Excited state quantum phase transitions (ESQPTs) are gener-alizations of QPTs to the excited levels [3, 4]. They emergewhen the QPT is accompanied by the bunching of the eigen-values around the ground state. This divergence in the densitystates at the lowest energy moves to higher energies as thecontrol parameter increases above the QPT critical point. Theenergy value where the density of states peaks marks the pointof the ESQPT.ESQPTs have been analyzed in various theoretical mod-els [4–21] and have also been observed experimentally [22–27]. They have been linked with the bifurcation phe-nomenon [20] and with the exceedingly slow evolution of ini-tial states with energy close to the ESQPT critical point [18–20]. Equivalently to what one encounters in QPTs, the non-analycities associated with ESQPTs occur in the thermody-namic limit. When dealing with finite systems, signaturesof these transitions are usually inferred from scaling analy-sis. There are, however, studies based on new microcanonicaldistributions that claim that QPTs can be predicted withoutconsiderations of thermodynamic limits [28, 29]. One mightexpect analogous results for ESQPTs.In this work, we show that the nonanalycities associatedwith QPTs and ESQPTs can be found in finite systems whenthe control parameter of the Hamiltonian is continued into thecomplex plane. The Hamiltonian that we study, H ( α, N ) = αH I ( N ) + (1 − α ) H II ( N ) , (1) ∗ [email protected] † [email protected] ‡ [email protected] is a linear combination of two noncommuting operators, [ H I , H II ] = 0 , where α is the control parameter and N isthe system size. As the control parameter varies from α = 1 to α = 0 the spectrum of the full Hamiltonian is transformedfrom the spectrum of H I to the spectrum of H II . The tran-sition of the ground and excited states from one symmetry toa mixture of different symmetry solutions is continuous in α ,yet quite sharp. It is only in the limit of N → ∞ that a point ofnonanalyticity appears for the ground state at a critical value α c and for the excited states at values of α ESQPT < α c . Infinite systems, the sharp transition from one type of symme-try adapted solution to another one is associated with avoidedcrossings. We show that these avoided crossings are con-nected with the exceptional points (EPs) of the non-Hermitianform of H ( α, N ) , where α is complex.The association between EPs and avoided crossings wasfirst presented in [30]. Connections have also been madebetween EPs and QPTs [31–34] and between EPs and ES-QPTs [35]. Here, we further elaborate the studies of ESQPTsfrom the perspective of non-Hermitian Hamiltonians takinginto account both EPs close and also far apart from the realaxis.The EPs that we calculate correspond to the exact degen-eracies of the non-Hermitian Hamiltonian found for specificvalues α EP of the complex control parameter. More precisely,they correspond to branch point singularities of the eigenval-ues and eigenvectors [36–40]. We show that as N → ∞ , thecomplex α EP approach and accumulate at the real axis, there-fore coinciding with the QPT and ESQPT critical values of thereal (physical) Hamiltonian. We notice that all EPs approachthe critical values, those close to the separatrix that marks theESQPT and also those far away. However, the distant onesconverge to those values more slowly. Using the Pad´e extrap-olation technique, we demonstrate that these critical valuescan be derived from the EPs obtained with finite system sizes. Model and separatrix.–
ESQPTs have been extensivelystudied in Hamiltonians with a U ( n + 1) algebraic struc-ture given by H U ( n +1) = αH U ( n ) − (1 − α ) N − H SO ( n +1) .They are composed of two limiting dynamical symmetries, the U ( n ) and the SO ( n + 1) . In the bosonic form, these Hamil-tonians represent limits of the vibron model [41–44], whichis used to characterize the vibrational spectra of molecules.The U ( n ) dynamical symmetry ( α = 1 ) is described by aone-body operator and the SO ( n + 1) dynamical symmetry( α = 0 ) by a two-body operator, so the latter needs to berescaled by the system size N .These U ( n + 1) Hamiltonians show a second-order groundstate QPT at α c = 0 . and ESQPTs for α ESQPT < α c . Ouranalysis is illustrated for the U (2) Hamiltonian, which repre-sents one of the spin versions of the LMG model [8, 45]. TheHamiltonian is written as [13, 20], H U (2) = α (cid:18) N S z (cid:19) − − α ) N S x , (2)where S z = P Ni =1 S zi is the total spin in the z -direction and S x = P Ni =1 S xi is the total spin in the x -direction. The firstterm favors the alignment of the spins in the z direction andthe second term in the x direction.For α = 1 , all eigenvalues of H U (2) are positive. For α = 0 , the eigenvalues are negative and the eigenstates formpairs of degenerate states, one with positive and the other withnegative total magnetization in x . In Fig. 1 (a), we show theeigenvalues versus the control parameter for N = 50 . Theground state QPT occurs at α c = 0 . for E c ≃ . For α < α c ,the lowest energies become smaller than zero, while the stateswith energy close to E c cluster together. The bunching of theenergy levels at E c characterizes the ESQPT.The solid (nearly) horizontal line in Figs. 1 (a) and (b) isthe separatrix that marks the ESQPT. It can be obtained froma semiclassical analysis. The normalized energy differencebetween E c and the ground state eigenvalue E GS is the criticalexcitation energy of the ESQPT. Its equation is given by [4, 5,10] E ESQPT ( α ) = E c − E GS N = [1 − − α )] − α ) . (3)In Fig. 1, the line for the separatrix corresponds to the valueof E c obtained using E ESQPT from Eq. (3) and the numericaldata for E GS .In Fig. 1 (b), we consider N = 100 and zoom in the data for . ≤ α ≤ . . This figure makes clear the effect of the phasetransition on the structure of the eigenstates. The eigenstateswith energy E < E c are almost doubly degenerate. These arethe states with structures closer to the SO (2) symmetry. Thedegeneracy is lifted for E > E c , where the eigenstates havestructure closer to the U (1) symmetry. Quantities such as theparticipation ratio [18, 20] and the fidelity [19] have been usedto capture the abrupt changes in the structures of eigenstatescaused by ESQPTs. Non-Hermitian Formalism and EP.–
In the vicinity of a crit-ical point of a finite system described by a Hermitian Hamilto-nian H ( α, N ) , the crossings of the energy levels are avoided. α -40-2002040 E α -1010.7 0.72 0.74 0.76 0.78 0.80 QPTESQPT (a) (b)
FIG. 1: (Color online) Energy levels vs α for N = 50 (a) and N =100 (b). Eigenstates of one parity are indicated with black solid linesand from the other with dashed red lines. The horizontal green line isthe separatrix, it indicates E c obtained from Eq. (3). Arbitrary units. In contrast, the complex eigenvalues of the correspondingnon-Hermitian Hamiltonian, obtained by continuing the con-trol parameter α into the complex plane, can cross. This de-generacy, accompanied by the coalescence of the correspon-dent eigenvectors, is the EP. We find various EPs for differentcomplex values α EP ( N ) of the control parameter. It has in factbeen proven that in the case of a Hamiltonian such as that inEq. (1), where the two Hermitian operators H I and H II donot commute, there always exists a complex linear pre-factorfor which the EPs are obtained [36]. Our results below sub-stantiate the strong relationship between critical points and theappearance of EPs.Sufficiently close to α EP ( N ) the energy spectrum of thenon-Hermitian Hamiltonian contains two almost degeneratevalues given by E ± ( α, N ) ∼ = E EP ( α EP ( N )) ± C ( N ) p α − α EP ( N ) , (4)where C ( N ) is a function of the system size. The two eigen-vectors corresponding to these energies are | ψ ± ( α, N ) i ∼ = | ψ EP ( N ) i ± | χ ( N ) i p α − α EP ( N ) . (5)The orthogonality condition, which can be extended to sym-metric non-Hermitian Hamiltonians, implies that the innerproduct h ψ ∗∓ ( N ) | ψ ± ( N ) i = 0 . At the critical complex value α EP of the control parameter, the degenerate states becomeself-orthogonal, that is [37] | ψ + ( α EP , N ) i = | ψ − ( α EP , N ) i = | ψ EP ( N ) i , (6) h ψ ∗ EP ( N ) | ψ EP ( N ) i = 0 . Because of the self-orthogonality at α EP , the quantum fluctu-ations at this point become infinitely large if associated withthe expectation value of ∂H ( α, N ) /∂α . This gives furthersupport to associating QPT and ESQPT with EPs.In Fig. 2, we use circles to represent the EPs of the complexLMG Hamiltonian, which is obtained from Eq. (2) by contin-uing α in the complex plane. The real part of the energies E EP are shown in the top panel and the imaginary part in the bot-tom panel. The EPs with the lowest imaginary part of α EP areindicated with a light (red) color. They have Im( α EP ) almostconstant and close to 0.0115 for N = 100 . This array of EPsalso has the lowest Im( E EP ) . For higher Im( E EP ) , we findother rows of EPs also with approximately constant values of Im( α EP ) . FIG. 2: Exceptional points (circles) of the complex dilated LMGHamiltonian (2) for N = 100 . Top panel: the real part of E EP , andbottom panel: the imaginary part of E EP . The EPs with the lowest Im( α EP ) are indicated with a light (red) color. They have almost realvalued energies and Im( α EP ) ∼ . ; the latter is shown with asolid line on the α plane. ESQPT vs. EP.–
To unveil the connection between ESQPTsand EPs, we now compare the results from the Hermitian andnon-Hermitian approaches. In Fig. 3, the thin lines give thereal part of the eigenvalues of the complex LMG Hamiltonianas a function of
Re( α ) , the circles are the EPs as in Fig. 2,and the thick nearly horizontal line is the separatrix. In eachpanel, Re( α ) varies from 0.7 to 0.8, while Im( α ) is held at aconstant value.In Fig. 3 (a), Im( α ) = 0 , so the plot is the same as in Fig. 1(b), but now with the EPs added to it. This figure alreadysuggests a strong link between the EPs and the ESQPT. Asone sees, for E < E c , where we have pairs of degeneratestates, there are no EPs. As the energies increase, they firstappear very close to the point where the degeneracy is liftedand in the vicinity of the separatrix.To better support this relationship, we increase the value of Im( α ) from Fig. 3 (a) to (c) up to Im( α ) = 0 . . The latteris the value of the sequence of EPs with the lowest Im( α EP ) , asshown in Fig. 2. By increasing Im( α ) , the values of Re( E ) ofthe non-Hermitian Hamiltonian change, while the EPs and theseparatrix naturally remain the same. The thin solid lines arecontinuously deformed from Fig. 3 (a) to (c) until the avoidedcrossings become true crossings. They happen right at theEPs with the lowest Re( E EP ) . These EPs are located on thebifurcating branches of the spectrum. Lines intersecting at theEPs exhibit cusps, which is consistent with Eq. (4). Theseobservations indicate that the ESQPT in Fig. 1 is inherentlycaused by the non-Hermitian crossings (cusps). We can thusinterpret ESQPTs as phenomena arising due to the presenceof EPs.In Fig. 3 (d), we choose Im( α ) = 0 . , which is close to -1010-101 R e ( E ) Re( α) -101 R e ( E ) Re( α) -1010.7 0.72 0.74 0.76 0.78 0.80 (c)(a) (b)(d) FIG. 3: Real part of the eigenvalues of the complex dilated LMGHamiltonian (2) vs. the real part of the complex α (thin solid lines)for N = 100 and Im( α ) = 0 (a), . (b), . (c), . (d).Circles are the EPs, they correspond to Re( E EP ) versus Re( α EP ) .The thick nearly horizontal line is the separatrix. the value for the second row of EPs. Analogously to Fig. 3 (c),there are true crossings (cusps) coinciding with the locationsof these EPs. As Im( α ) further increases, the crossings hap-pen for sequences of EPs with higher and higher Re( E ) . Weshow next that as N increases, one by one, these sequences ofEPs approach and accumulate on the real axis. Close to thecritical points, there is a high density of EPs. Thermodynamic limit.–
For a given system size, we havea discrete collection of EPs. As the system size increases,the number of EPs increases and they approach the separatrix(which in turn approaches zero, E c /N → ). This is illus-trated in Figs. 4 (a) and (b) for the EPs with the lowest val-ues of Re( E EP ) and Im( E EP ) . As N increases, Im( E EP ) /N and Im( α EP ) go to zero [the same occurs for Re( E EP ) /N (not shown)] . In the thermodynamic limit, Im( α EP ) → , Im( E EP ) /N → , Re( E EP ) /N → E c /N , and Re( α EP ) co-incides with α ESQPT .In Figs. 4 (c) and (d), we compare the EPs with the low-est (filled symbols) and the second lowest (empty symbols)energies for N = 200 and . The second lowest EPs alsoapproach the separatrix as the system size increases, but at asmaller rate than the lowest EPs. The sequences of the lowestEPs for the two system sizes are much closer than the two se-quences of the second lowest EPs. This pattern propagates tohigher energies.In Fig. 4 (e), we select a specific pair ( j , j ) of eigen-states that coalesce and study α EP ( N, j , j ) as a functionof the system size. For large N , α EP ( N, j , j ) changes al-most continuously in the complex α -plane. Using the Pad´eextrapolation method, we can obtain numerically the limit of α EP ( N, j , j ) for /N → . This method avoids the calcu-lation of high order derivatives of α EP with respect to /N ,as needed in Taylor and similar expansions [46, 47]. Thelimit for α EP ( N → ∞ , j , j ) exists and equals the real value α c = 0 . , as confirmed in Fig. 4 (e) for any of the chosenpairs. The convergence is faster for the EPs of lower ener-gies. This shows that the critical point for the QPT can be I m ( E E P ) / N I m ( α E P ) Re( α EP ) I m ( α E P ) Re( α EP ) I m ( α E P ) Re( α EP ) I m ( E E P ) / N (a) (b)(c) (d)(e) FIG. 4: EPs with the lowest and the second lowest values of
Re( E EP ) for different N ’s (a)–(d). Extrapolation towards N → ∞ carried outwith the method of Pad´e approximants (e). The lowest EPs in (a)and (b): N = 100 (circles), N = 150 (squares), and N = 200 (up triangles), N = 250 (down triangles). In (c) and (d): the lowestEPs (filled symbols) and the second lowest EPs (empty symbols) for N = 200 (up triangles) and N = 250 (down triangles). In (e):the circles correspond to 22 system sizes between N = 20 and N =250 . Each line is a specific pair of states ( j , j ) forming an EP, frombottom to top: (0 , , (1 , , (1 , , (1 , , (1 , . The solid lines areobtained via the Pad´e method, leading to the extrapolated points at Im( α EP ) → , where Re( α EP ) → α c = 0 . . obtained from non-Hermitian calculations considering finitesystem sizes.As for the critical points of the ESQPT, we verified thatthe extrapolations of the vertical progressions of the EPs inFig. 3 (a) touch the curves of real eigenvalues (thin solid lines)and this happens very close to where these curves split. Theline made of the intersection points between extrapolated EPsand real eigenvalues is nearly parallel to the separatrix and approaches it as the system size increases.We detect the effects of EPs also in physical observables.For the real Hermitian Hamiltonian of finite systems, the be-havior of quantities such as the total magnetization in the z and in the x direction changes abruptly, yet smoothly, closeto the QPT and ESQPT critical values of the control parame-ter [20]. In the non-Hermitian approach, we find that by keep-ing Im( α ) fixed and varying Re( α ) , a sudden non-analyticaldiscontinuity in the values of those observables occur exactlywhen we reach the associated EP. Contrary to the Hermitiantreatment, where non-analycities occur only in the thermo-dynamic limit, here they appear already for finite N . In fi-nite system sizes, sharp non-analytical transitions associatedwith eigenvalues and eigenvectors can happen only in non-Hermitian quantum mechanics [37]. In the thermodynamiclimit, where the EPs fall into the real α -axis, the results fromthe two approaches, Hermitian and non-Hermitian, coincide. Conclusions. –
Using a finite system described by the LMGmodel with the control parameter α continued into the com-plex plane, we showed that the EPs are linked with the avoidedcrossings that characterize the ground state QPT and ESQPTsobtained for the real (physical) LMG Hamiltonian. These EPsapproach the axis of real α in the thermodynamic limit. Theeigenvalues pertaining to such EPs indicate the position of theseparatrix that marks the ESQPT.The approach presented here can be used for studying phasetransitions in systems other than the LMG model. It should beof particular interest to models where the critical values areunknown and difficult to accurately determine from Hermitianmethods. Acknowledgments
This research was supported by the I-Core: the Israeli Ex-cellence Center Circle of Light, by the Israel Science Founda-tion grants No. 298/11 and No. 1530/15, and by the AmericanNational Science Foundation grant No. DMR-1147430 andNo. DMR-1603418. We thank Ofir Alon, Francisco P´erez-Bernal, and Saar Rahav for fruitful discussions. [1] L. D. Carr,
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