Entanglement spectrum and entropy in topological non-Hermitian systems and non-unitary conformal field theories
EEntanglement spectrum and entropy in topological non-Hermitian systems and non-unitaryconformal field theory
Po-Yao Chang,
1, 2, 3
Jhih-Shih You, Xueda Wen, and Shinsei Ryu Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Strasse 38, D-01187 Dresden, Germany Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, NJ 08854 USA Institute for Theoretical Solid State Physics, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA James Franck Institute and Kadanoff Center for Theoretical Physics, University of Chicago, Illinois 60637, USA
We study entanglement properties of free-fermion systems without hermiticity by use of correlation matrixand overlap matrix in the biorthogonal basis. We find at a critical point in the non-Hermitian Su-Schrieffer-Heeger (SSH) model with parity and time-reversal symmetry (PT-symmetry) the entanglement entropy exhibitsa logarithmic scaling with corresponding central charge c = − , signaling the emergence of non-unitary con-formal field theory. In addition, we demonstrate that, in the PT-symmetric SSH model and the non-HermitianChern insulators, the entanglement spectrum characterizes the topological properties in terms of the existenceof mid-gap states. INTRODUCTION
In contemporary condensed matter physics, quantum en-tanglement plays a pivotal role in characterizing and obtain-ing a deeper understanding of many-body quantum systems.For example, the topological entanglement entropy providesa direct method to detect topological order [1, 2]. The sub-system size scaling of the entanglement entropy can be usedto extract useful information of conformally invariant sys-tems [3]. Ground-state properties of topological systems arealso reflected in their entanglement spectra [4–9]. Moreover,quantum entanglement paves a way for a deeper understand-ing of the renormalization group and an emergent holographicspace-time [10–12].In this letter, we aim to extend the quantum-entanglementbased approach to systems that lack hermiticity. Non-Hermitian quantum mechanical systems, in particular, thosewhich host topological phenomena, have attracted a lotof interest recently [13–31]. To be concrete, we take,as paradigmatic examples, the non-Hermitian Su-Schrieffer-Heeger (SSH) model with the combination of parity andtime-reversal symmetry (PT-symmetry) [21–24], and non-Hermitian Chern insulators [32]. These non-Hermitian mod-els host a variety of trivial and topological gapped phases, aswell as gapless phases and critical points.In order to introduce and investigate the entanglement en-tropy/spectrum in these non-Hermitian systems, we use a gen-eralized definition of the (reduced) density matrix in terms ofthe biorthogonal basis [33, 34]. First, we will show that fornon-interacting systems, the entanglement entropy and spec-trum can be efficiently computed from the correlation andoverlap matrices in the biorthogonal basis. Furthermore, wewill demonstrate that the entanglement entropy and spectrumcan be used to detect various universal properties of non-Hermitian gapped (topological) systems. For example, weshow that the topological gapped (trivial) PT-symmetric phasein the non-Hermitian SSH model supports (does not support)robust mid-gap states in the entanglement spectrum. The ex- istence of the mid-gap states in the entanglement spectrum isconcurrent with the existence of protected physical boundarymodes in these PT-symmetric phases [23, 24]: Thus, the en-tanglement spectrum provides a way to characterize the topo-logical non-Hermitian phases.Furthermore, we found, at a critical point appearing inthe non-Hermitian SSH model, the entanglement entropy de-creases logarithmically in the subsystem size – this signalsthe emergence of non-unitary conformal field theory (CFT).More specifically, at the critical point separating the trivialPT-symmetric phase and the spontaneously PT-broken phasein the non-Hermitian SSH model, we show that the entan-glement entropy for the subsystem of length L A scales as S A = ( c/
3) ln L A + · · · with c = − . The negative cen-tral charge c = − can be attributed to the bc -fermionic ghosttheory [35–41].The entanglement entropy in non-unitary CFTs has beenstudied previously [42–48], but in most cases, the entan-glement entropy scaling is captured by the effective centralcharge, c eff > , even when the central charge itself is neg-ative. I.e., the entanglement entropy in these cases increaseslogarithmically in the subsystem size, ∼ ( c eff /
3) ln L A . InRef. [48], the authors studied loop models by using thebiorthogonal reduced density matrix to compute the entangle-ment entropy, and found that the entanglement entropy scal-ing is given by the central charge ∼ ( c/
3) ln L A with c < .In this letter, we will show that the negative central chargecan also occur in non-Hermitial topological PT-symmetricsystems [16, 20–24, 27]. Our findings shed new light onthe nature of critical points appearing in non-Hermitian sys-tems, and in particular those in proximity to topological non-Hermitian phases.We also found, at the other critical point of the PT-symmetric SSH model, i.e., the one which separates the topo-logical PT-symmetric phase and the spontaneously PT-brokenphase, the entanglement entropy/spectrum is somewhat morecomplicated and interesting; There are two additional mid-gapstates in the entanglement spectrum which mimic the physical a r X i v : . [ c ond - m a t . s t r- e l ] A p r boundary modes at this critical point. Nevertheless, by mod-ifying the bipartition, we show that the entanglement scalingagain is given by the logarithmic law with c = − . Thisreminds us of the similar behavior at the symmetry-enrichedcritical point found in a Hermitian system [49].We also demonstrate the quantum entanglement in thespontaneously PT-broken phase can also be studied, once weconsider a proper ”ground state” which is obtained by fill-ing modes with real energy. The entanglement entropy scal-ing gives the central charge c = 1 . We show the Jordanblock form at the exceptional point leads to the ground stateidentical to the free Dirac theory. Finally, as yet anotherexample, we study the entanglement spectrum of the non-Hermitian Chern insulators. The mid-gap states in the entan-glement Hamiltonian mimic the physical boundary modes innon-Hermitian systems. Our method provides an alternativeway to study the entanglement properties in both critical andtopological non-Hermitian systems. BIORTHOGONAL BASIS AND NON-HERMITIANHAMILTONIAN
Before discussing quantum entanglement of specific non-Hermitian lattice systems, we start by developing a convenientmethod to compute the entanglement spectrum/entropy in thebiorthogonal basis, which is valid for free (quadratic) systems.Specifically, we will show that the generalized reduced den-sity matrix can solely be constructed from the correlation ma-trix or the overlap matrix, much the same way in regular Her-mitian systems. Our method is complementary to the flattensingular-value decomposition recently proposed in [50].Consider a generic quadratic non-Hermitian Hamiltonian, H = (cid:80) ij φ † i H ij φ j , with H (cid:54) = H † and { φ i , φ † j } = δ ij being fermionic operators. The biorthogonal basis is con-structed from the left and right eigenvectors of H , H| R α (cid:105) = (cid:15) α | R α (cid:105) , H † | L α (cid:105) = (cid:15) ∗ α | L α (cid:105) , such that (cid:104) L α | R β (cid:105) = δ αβ [33]. The single-particle Hamiltonian can then be written as H = (cid:80) α (cid:15) α | R α (cid:105)(cid:104) L α | , and, correspondingly, the Hamiltoniancan be written as H = (cid:88) α (cid:15) α (cid:88) i ( R αi φ i ) † (cid:88) j ( L αj φ j ) = (cid:88) α (cid:15) α ψ † Rα ψ Lα , (1)where ψ † Rα and ψ † Lα are the right and left creation operatorssuch that | R α (cid:105) = ψ † Rα | (cid:105) , | L α (cid:105) = ψ † Lα | (cid:105) , and ψ Rα | (cid:105) = ψ L α | (cid:105) = 0 . It should be noted that ψ Lα and ψ Rβ are notordinary fermionic operators, in the sense that they satisfy thecommutation relationship { ψ Lα , ψ † Rβ } = δ αβ . We refer theseoperators the bi-fermionic operators.Many-body eigenstates can be constructed by actingon the vacuum with a set of (”occupied”) bi-fermioniccreation opreators. For example, | G R (cid:105) = (cid:81) α ∈ occ . ψ † Rα | (cid:105) satifies H | G R (cid:105) = (cid:80) α (cid:15) α ψ † Rα ψ Lα (cid:81) β ∈ occ . ψ † Rβ | (cid:105) = (cid:80) β ∈ occ . (cid:15) β | G R (cid:105) . Similarly, a many-body left eigen- state | G L (cid:105) = (cid:81) α ∈ occ . ψ † Lα | (cid:105) satisfies H † | G L (cid:105) = (cid:80) α ∈ occ . (cid:15) ∗ α | G L (cid:105) .From many-body left and right eigenstates, we can con-struct a non-Hermitian density matrix as ρ = | G R (cid:105)(cid:104) G L | suchthat ρ † (cid:54) = ρ and ρ = ρ . With this generalized notion of thedensity matrix, we can introduce measures of quantum entan-glement by partitioning the total system into two subsystems A and B , and then by taking the partial trace over subsystem B , ρ A = Tr B ρ . We can discuss the spectrum of the general-ized reduced density matrix, and the entanglement entropy.In Hermitian systems, if the Hamiltonian has the quadraticform, the reduced density matrix can be constructed eitherfrom the correlation matrix [51, 52] or the overlap matrix[53]. We extended these derivations to non-Hermitian sys-tems with quadratic form in terms of bi-fermionic operatorsas follows[41].– The correlation matrix is defined as C ij = (cid:104) G L | φ † i φ j | G R (cid:105) = (cid:88) α ∈ occ . L αi R † αj . (2)The entanglement Hamiltonian H A can be introducedby C = e −H A / (1 + e −H A ) . The entanglement en-tropy for subsystem A can then be introduced as S A = − (cid:80) δ [ ξ δ ln ξ δ + (1 − ξ δ ) ln(1 − ξ δ )] , where ξ δ are the eigen-values of C ij .– The overlap matrix is defined as M Aαβ = (cid:88) i ∈ A L † αi R βi = (cid:88) δ p δ ( L Aαδ ) † R Aβδ , (3)where L Aαδ and R Aβδ are the corresponding left and right eigen-vectors of M A with eigenvalues p ∗ δ and p δ . The original leftand right many-body wavefucntions as well as the reduceddensity matrix can be expressed in this new basis. In particu-lar, ρ A is given by ρ A = (cid:79) δ [ p δ | L Aδ (cid:105)(cid:104) R Aδ | + (1 − p δ ) | (cid:105)(cid:104) | ] . (4)The entanglement entropy can be directly obtained from thereduced density matrix as S A = − (cid:80) δ [ p δ ln p δ + (1 − p δ ) ln(1 − p δ )] . ENTANGLEMENT ENTROPY AND ENTANGLEMENTSPECTRUM IN NON-HERMITIAN SYSTEMSNon-Hermitian SSH model
We now study the non-Hermitian SSH model [24, 54] withthe PT-symmetry defined in momentum space by H k = (cid:18) iu v k v ∗ k − iu (cid:19) , (5)where v k = we − ik + v with u, v, w ∈ R and k is thesingle-particle momentum. Here the PT-symmetry is defined w v
Now we can com-pute the entanglement entropy and entanglement spectrumfrom the correlation matrix or the overlap matrix. In thisnon-Hermitian SSH model with PT-symmetry, there are two critical points that separate the PT-symmetric phases and thespontaneously PT-broken phase at w − v = ± u . When u = 0 ,the SSH model has only one critical point at w = v and thesystem is a critical free-fermion chain. There, the entangle-ment entropy scales logarithmically in the subsystem size L A , S A ∼ ( c/
3) ln L A with c = 1 [3].For finite u , we first analyze the entanglement entropy atthe critical point w − v = − u which separates the trivial PT-symmetric phase and the spontaneously PT-broken phase. Atthis critical point, we observe that all eigenvalues of the cor-relation matrix are real and they come in pairs, ξ α > and ξ β = 1 − ξ α < . This pair-wise structure of the eigen-values guarantees that the entanglement entropy is real andnegative. As shown in Fig. 2(a), we found from numericalcalculations that the entanglement entropy scales logarithmi-cally S A = ( c/
3) ln[sin( πL A /L )] + const . with the centralcharge c = − . Since the spectrum at this critical pointis linear around k = ± π , one can write down the effec-tive field theory action as S = (cid:82) dxdt ( ψ † b ¯ ∂ψ c + ¯ ψ † b ∂ ¯ ψ c ) ,where ∂ = (1 / ∂ x − i∂ t ) , ¯ ∂ = (1 / ∂ x + i∂ t ) , and ψ b/c ( ¯ ψ b/c ) represent the fermionic fields for the right-moving(left-moving) modes. (We have set the Fermi velocity to beone). We identify ψ † b ( c ) as the right(left) creation operator ψ † R ( L ) defined in Eq. (1). A crucial observation here is thatstates associated with these right and left fermionic operatorsare not normalizable at the critical point [55]: The ill-definednorm of the quantum states can be thought of as a hallmarkof the ghost theory. The above action, with proper assignmentof the conformal dimensions, defines the bc -ghost CFT withcentral charge c = − [35–41]. The entanglement entropydetects the correct central charge as we expect.The appearance of the negative central charge, however, issensitive to the choice of the boundary condition. To obtainthe c = − scaling of the entanglement entropy, we needto choose the periodic boundary condition and consider thehalf-filled ground state, where we fill the (left/right) state atthe crossing point but with a tiny momentum shift [56]. Onthe other hand, imposing anti-periodic, open boundary con-ditions, or simply removing the state at the crossing points k = ± π , we recover the central charge c = 1 in the en-tanglement entropy scaling [57]. The sensitivity to boundaryconditions is consistent with the fact that the ill-defined normoccurs only at the crossing point k = ± π . Furthermore, wemonitor the entanglement entropy scaling as a function of thetwisting angle of the boundary condition (i.e., a shift of single-particle momenta) for very small ( ∼ − ) to large twistingangle ( ∼ π ). We observe that the entanglement entropy scal-ing crosses over from a convex ( c = − ) to concave function( c = 1 ) [41]. This crossover also occurs by going from thecritical point to a PT-symmetric phase with a small gap. Wenote that a similar crossover was also observed in the quantumIsing chain in an imaginary magnetic field [45].We now turn to the other critical point ( w − v = u ) sep-arating the topological PT-symmetric phase and the sponta-neously PT-broken phase. It exhibits a very different entan- (a) (b) Re [ S A ] Im [ S A ]
10 20 30 40 L A - - S A Re [ S A ] Im [ S A ]
10 20 30 40 L A S A FIG. 2. The entanglement entropy as a function of the subsystemsize L A with the total system length L = 100 . (a) At the criticalpoint separating the trivial PT symmetric gapped phase and the PTbroken phase, ( w, v, u ) = (1 . , . , . . The numerical data isfit to S A ( L A ) = − . − .
666 ln[sin( πL A /L )] (solid line),leading to the identification c = − . (b) In the PT broken phase, ( w, v, u ) = (0 . , . , . . Here, the state is obtained by fillingonly negative real-energy modes. The numerical data can be fit to S A = 1 .
93 + 0 . πL A /L )] , i.e., c = 1 . glement entropy scaling. At this critical point, we observean additional pair of eigenvalues ξ ± ,α = 0 . ± iI α , where I α depends on the subsystem size. This pair in the entan-glement spectrum mimics the protected boundary modes inthe gapless point in the physical spectrum [see Fig. 1(b) at w − v = u and Fig. 7(a) in [41]]. Although this pair of eigen-values contributes to the pure real part of the entanglemententropy, the imaginary part I α depends on the total systemsize. We find the entanglement entropy shows the scaling S A = α ln[sin( πL A /L )] + const . , where α decreases as afunction of the total system size [Figs. 7(b-c) in [41]]. Thelength-dependent coefficient α is not described by CFT. How-ever, these two critical points are seemingly described by thesame effective field theory, and one would expect the samescaling behavior in the entanglement entropy.To shed some light on this issue, we consider another bi-partition which cuts through the unit cell[4]. With this shiftedunit-cell, the hopping w and v are interchanged and the en-tanglement spectra are identical to the critical point that sep-arates the trivial PT-symmetric phase and the spontaneouslyPT-broken phase. We recover the c = − entanglement en-tropy scaling under this bipartition. The above story remindsus of the protected boundary modes (both in the physical andentanglement spectra) in the symmetry-enriched critical pointin the Hermitian system discussed in Ref. [49]. There, by re-defining the unit-cell, we also recover the c = 1 scaling of thefree-fermion system [41]. In this sense, the non-unitary CFTappearing at the critical point separating the topological PT-symmetric phase and the spontaneously PT-broken phase maybe viewed as a non-Hermitian analog of symmetry-enrichedcritical theory.Lastly, let us consider the entanglement properties in thespontaneously PT-broken phase ( | w − v | < u ) , where thereare two exceptional points. In this phase, some of the single-particle energies are purely imaginary and filling such single-particle states would be unphysical. Here, we consider the ”ground state” which is constructed by filling only stateswith real energy. As shown in Fig. 2(b), the entanglemententropy of this ”ground state” follows the CFT scaling be-havior, S A = ( c/
3) ln[sin( πL A /L )] + const . with c = 1 .The appearance of the c = 1 CFT behavior can be un-derstood as follows. At the exceptional points, two eigen-states are coalescing into one and the Hamiltonian cannotbe diagonalized. One can expand the Hamiltonian at theexception point k EP with a Jordon block form [58, 59], H = (cid:82) dk Ψ † k [ σ z ( k EP − k ) + γ ( σ x + iσ y )] Ψ k where Ψ † k =( ψ † ,k , ψ † ,k ) are fermionic fields, and γ is an arbitrary com-plex number. (Once again, we set the Fermi velocity to beunity [60].) At the exceptional point k EP , two eigenstates col-lapse to one eigenstate (1 , with energy − ( k − k EP ) . Theground state can be expressed as | G (cid:105) = (cid:81) k − k EP > (1 , T ,which has the identical form of the ground state of the freeDirac theory with central charge c = 1 . Fully gapped phases and entanglement spectrum—
Next,we study the PT-symmetric, fully gapped phases, where theground state is well defined since there is no imaginary energymode. Once again, the entanglement spectrum is obtainedfrom the eigenvalues of the correlation matrix and the overlapmatrix with the periodic boundary condition. In the topologi-cal non-Hermitian phase ( w − v > u ) where the physical edgemodes are present, there are two mid-gap states in the entan-glement spectrum with Re [ ξ ] = 1 / and non-vanishing imag-inary part Im [ ξ ] (cid:54) = 0 [Fig. 3 (a-b)]. In addition to these twomid-gap states localized at the entangling boundaries, thereare numerous other localized boundary modes which are notthe mid-gap states in the correlation matrix [Fig. 3 (c)]. Onthe other hand, in the trivial phase where no physical edgemodes are present, we observe four additional states withnon-vanishing imaginary eigenvalues of the correlation matrix[Fig. 3 (d-e)]. There are also numerous localized boundarymodes which are not the mid-gap states in this phase. Theselocalized modes in the correlation matrix are similar to thenon-Hermitian skin effect in non-Hermitian systems with theopen boundary condition [17, 23, 61]. Non-Hermitian Chern insulators
Finally, we consider the non-Hermitian Chern insula-tor [32] in two dimensions defined in momentum space by H ( k ) =( m + t cos k x + t cos k y ) σ x + ( iγ + t sin k x ) σ y + ( t sin k y ) σ z , (8)where t, m, and γ are real parameters and we assume t > .The complex dispersion relation is given by E ± ( k ) = ± [( m + t cos k x + t cos k y ) + (cid:112) ( iγ + t sin k x ) + ( t sin k y ) . Excep-tional points appear when the two bands satisfy E ± ( k EP ) = 0 at certain momenta k = k EP , at which the Hamiltonian (8)is defective and the eigenstates coalesce and linearly dependon each other. The gapped phases with E + ( k ) (cid:54) = E − ( k ) forall k are characterized by the first Chern number, and in the (a) (b) (c)(d) (e) (f)
100 200n0.51Re [ ξ ]
100 200n - [ ξ ]
100 200x0.5 | ψ A |
100 200n0.51Re [ ξ ]
100 200x0.5 | ψ A |
100 200n - [ ξ ] FIG. 3. The real part (a) and imaginary part (b) of the eigenvaluesof the correlation matrix C A with ( w − v, u ) = (0 . , . . Thereare two mid-gap states. (c) The wavefunction amplitude of the lo-calized states including two mid-gap states. The real part (d) andimaginary part (e) of the eigenvalues of the correlation matrix C A with ( w − v, u ) = ( − . , . . There are four states with imaginaryeigenvalues. (f) The wavefunction amplitude of the localized stateswhere the total system is two hundreds sites and n is the eigen-levelindex. case of γ = 0 , the model reduces to a Hermitian Chern in-sulator. We computed the entanglement spectrum of the non-Hermitian Chern insulator model (8) from the correlation ma-trix and/or the overlap matrix with the periodic boundary con-ditions in both x and y -directions. Since it has been knownthat the existence of the bulk-edge correspondence in (phys-ical) non-Hermitian systems is sensitive to boundary condi-tions, in the following we investigate the entanglement spec-trum of the non-Hermitian gapped phases by setting entan-gling boundary either in the y direction or in the x direction.In the topologically non-trivial gapped phase, we observetwo chiral mid-gap modes localized at the right and left edges,respectively [Figs. 4 (b) and (c)]. In particular, for the entan-gling boundary running along the x -direction,the right mid-gap mode has the largest positive imaginary part for γ > ,whereas the left mid-gap mode has the largest negative imag-inary part [Fig. 4 (c)]. In contrast, the imaginary parts of themid-gap states vanish for the entangling boundary along the y -direction[Fig. 4 (b)]. This phenomenon implies the amplifi-cation of the right mid-gap mode and the attenuation of the leftmid-gap mode, similar to the behaviors of the physical edgemodes in the non-Hermitian Chern insulator and a topologicalinsulator laser discussed in other contexts [32, 62, 63]. CONCLUSION AND OUTLOOK
Many critical points/phases in Hermitian quantum many-body systems are described by unitary CFTs. For example,the Tomonaga-Luttinger liquid phase with an integral centralcharge is rather ubiquitous in one dimensional many-bodysystems. In contrast, virtually nothing has been known onthe many-body aspects of critical points/phases appearing in
FIG. 4. Topologically nontrivial gapped phases with nonzero Chernnumber C = − ( t = 1 . , m = − . , γ = 0 . ). (a) Complex-bandstructures of the non-Hermitian Chern insulator E ± = E ± ( k x , k y ) ,where the orange and blue bands represent E + and E − , respectively.(b) Complex entanglement spectrum as a function of k y , and the mid-gap modes at k y = 0 and ξ = 0 . along x direction. (c) Complexentanglement spectrum as a function of k x , and the mid-gap modesat k x = 0 for ξ = 0 . − . i (yellow squares) and for ξ =0 . . i (gray dots) along y direction. non-Hermitian systems, in particular in terms of their (confor-mal) field theory description. In this letter, from the entangle-ment entropy scaling, we identified the non-unitary CFT with c = − in a PT-symmetric non-Hermitian hosting topologicalphases. It deserves further investigation as to why this partic-ular CFT is realized. We also demonstrated the entanglementspectrum can detect the topological properties in the gappedphase of the PT-symmetry SSH mode and the non-HermitianChern insulator. Note added – During the preparation of this manuscript, webecame aware of a partially related work by Herviou et al.[64].
ACKNOWLEDGMENTS
P.-Y.C. thanks Natan Andrei, Pochung Chen, MarkusHeyl, Yi-Ping Huang, Elio K¨onig, Yashar Komijani, SarangGopalakrishnan, and Inti Sodemann for valuable discussions.We are thankful to Benoit Estienne for discussion about ex-tracting the real and effective central charges, and thankfulto Hassan Shapourian for pointing out the scaling of the en-tanglement entropy is sensitive for the boundary conditions.P.-Y.C. was supported by the Young Scholar Fellowship Pro-gram by Ministry of Science and Technology (MOST) in Tai-wan, under MOST Grant for the Einstein Program MOST108-2636-M-007-004. X.W. is supported by the Gordon andBetty Moore Foundations EPiQS initiative through Grant No.GBMF 4303 at MIT. S.R. is supported in part by the NationalScience Foundation under Grant No. DMR-1455296, and bya Simons Investigator Grant from the Simons Foundation. [1] A. Kitaev and J. Preskill, Phys. Rev. Lett. , 110404 (2006).[2] M. Levin and X.-G. Wen, Phys. Rev. Lett. , 110405 (2006).[3] P. 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2) sin( φ ∗ k /
2) +cos( φ k /
2) cos( φ ∗ k / diverges at the critical point v − w = u ,which is ill-define.[56] The (left/right) state is divergent at the crossing point ( k = π ).When computing the entanglement entropy, we shift the mo-mentum by a small small amount δ = 0 . which avoidsthe divergence at the crossing point..[57] Hassan Shapourian (private communication).[58] K. Kanki, S. Garmon, S. Tanaka, and T. Petrosky,Journal of Mathematical Physics , 092101 (2017),https://doi.org/10.1063/1.5002689.[59] G. Demange and E.-M. Graefe, Journal of Physics A: Mathe-matical and Theoretical , 025303 (2011).[60] The velocity at the exceptional point is infinity. However, thecentral charge will not depend on the velocity of the effective theory, we take the velocity to be unity.[61] Y. Xiong, Journal of Physics Communications , 035043(2018).[62] Y. Xu, S.-T. Wang, and L.-M. Duan, Phys. Rev. Lett. ,045701 (2017).[63] G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D.Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev,Science (2018), 10.1126/science.aar4003.[64] L. Herviou, N. Regnault, and J. H. Bardarson, arXiv e-prints, arXiv:1908.09852 (2019), arXiv:1908.09852 [cond-mat.mes-hall].[65] R. Verresen, N. G. Jones, and F. Pollmann, Phys. Rev. Lett. , 057001 (2018).[66] C. Itzykson, H. Saleur, and J.-B. Zuber, Europhysics Letters(EPL) , 91 (1986).[67] J. Cardy and E. Tonni, Journal of Statistical Mechanics: Theoryand Experiment , 123103 (2016). Supplementary Material for “Entanglement spectrum and entropy in topologicalnon-Hermitian systems and non-unitary conformal field theory”.
CORRELATION MATRIX
In this supplementary material, we present the detailed derivations of the entanglement properties from the correlation matrixand the overlap matrix.We first use the correlation matrix [52] to extract eigenvalues of the reduced density matrix in a free fermion system. Sincethe theory is free, the reduced density matrix has a Gaussian form ρ A = N exp( − (cid:80) α,β H Eαβ φ † α φ β ) , where H Eαβ refers to theentanglement Hamiltonian. The correlation matrix is defined as C ij = (cid:104) G L | φ † i φ j | G R (cid:105) = Tr ρ A φ † i φ j , (9)where | G R (cid:105) and | G L (cid:105) are the right and left ground states. One can simultaneously diagonalize C ij and H Eαβ and find C = e −H E e −H E . The correlation matrix can be expressed by the left and right eigenvectors as C ij = (cid:104) G L | φ † i φ j | G R (cid:105) = (cid:104) | (cid:89) i (cid:48) ∈ occ . ( (cid:88) α L αi (cid:48) φ α ) φ † i φ j (cid:89) j (cid:48) ( (cid:88) β R † βj (cid:48) φ † β ) | (cid:105) . (10)By using the commutation relation { φ i , φ † j } = δ ij and φ i | (cid:105) = 0 , we have φ j (cid:89) j (cid:48) ( (cid:88) β R † βj (cid:48) φ † β ) | (cid:105) = R † j (cid:89) j (cid:48) (cid:54) =1 ψ † Rj (cid:48) | (cid:105) − R † j (cid:89) j (cid:48) (cid:54) =2 ψ † Rj (cid:48) | (cid:105) + R † j (cid:89) j (cid:48) (cid:54) =3 ψ † Rj (cid:48) | (cid:105) − · · · φ i (cid:89) i (cid:48) ( (cid:88) α L † αi (cid:48) φ † α ) | (cid:105) = L † i (cid:89) i (cid:48) (cid:54) =1 ψ † Li (cid:48) | (cid:105) − L † i (cid:89) i (cid:48) (cid:54) =2 ψ † Li (cid:48) | (cid:105) + L † i (cid:89) i (cid:48) (cid:54) =3 ψ † Li (cid:48) | (cid:105) − · · · , (11)where ψ † Rj (cid:48) = (cid:80) β R † βj (cid:48) φ † β and ψ † Li (cid:48) = (cid:80) α L † αi (cid:48) φ † α , which satisfy { ψ Li , ψ † Rj } = δ ij . Using Eq. (11), the correlation matrix is C ij = (cid:88) α L αi R † αj . (12)One should notice that C ∗ ji = (cid:80) α R αi L † αj (cid:54) = C ij is not Hermitian. The entanglement entropy is S A = − Tr ρ A ln ρ A = − Tr[ e −H E e −H E ln e −H E e −H E + 11 + e −H E ln 11 + e −H E ]= − (cid:88) δ [ ξ δ ln ξ δ + (1 − ξ δ ) ln(1 − ξ δ )] , (13)where ξ δ are the eigenvalues of C ij . OVERLAP MATRIX
Alternatively, we can use the overlap matrix to extract the entanglement properties. The basic idea is to rotate the reduceddensity matrix in a new biothogonal basis in the subsystem A , {| L A ( B ) i (cid:105) , | R A ( B ) i (cid:105)} such that (cid:104) L A ( B ) i | R A ( B ) i (cid:105) = δ ij . The rotationmatrix is constructed from the overlap matrix [53], M Aαβ = (cid:90) x ∈ A dx (cid:104) L α | R β (cid:105) = (cid:88) i p i ( L Aiβ ) † R Aiα , (14)where L Aiβ and R Aiα are the corresponding left and right eigenvectors of M A with eigenvalues p ∗ i and p i . Notice that the overlapmatrix is non-Hermitian, [ M Aαβ ] † = [ M Aβα ] ∗ = [ (cid:90) x ∈ A dx (cid:104) L β | R α (cid:105) ] ∗ (cid:54) = (cid:90) x ∈ A dx (cid:104) L α | R β (cid:105) . (15)So in the diagonal basis, the non-Hermitian overlap matrix is (cid:88) αβ ( L Aiα ) † M Aαβ R Aβi = p i . (16)Thus we can normalized the left and right eigenvectors of M A as (cid:88) αβ ( L Aiα ) † √ p i (cid:104) ˜ L Aα | ˜ R Aβ (cid:105) R Aβj √ p j = δ ij . (17)Here we express M Aαβ = (cid:104) ˜ L Aα | ˜ R Aβ (cid:105) , where | ˜ L Aα (cid:105) and | ˜ R Aα (cid:105) are left and right vectors span only in subsystem A . These left andright vectors are not yet biorthogonal, but from Eq. (17), we can construct the local biorthogonal basis as | R Ai (cid:105) = (cid:88) α R Aiα √ p i | ˜ R Aα (cid:105) , | L Ai (cid:105) = (cid:88) α L Aiα √ p i | ˜ L Aα (cid:105) , (cid:104) L Ai | R Aj (cid:105) = δ ij . (18)From the property of the overlap matrix M A + M B = I , M A and M B can be simultaneously diagonolized with the correspond-ing eigenvalues p i and − p i , (cid:88) αβ ( L Aiα ) † M Aαβ R Aβj + (cid:88) αβ ( L Aiα ) † M Bαβ R Aβj = p i δ ij + (1 − p i ) δ ij = δ ij . (19)Thus the biorthogonal basis {| R Bi (cid:105) , | L Bi (cid:105)} in subsystem B can also be constructed from the rotation matrices R Aαi and L Aβj . Theright vector | R α (cid:105) in the total system can be rotated by R Aαi such that (cid:88) α R Aαi | R α (cid:105) = √ p i | R Ai (cid:105) + (cid:112) − p i | R Bi (cid:105) . (20)Now we consider many-body wave function after the rotation | ˜ G R (cid:105) = (cid:89) i ( √ p i ψ A † Ri + (cid:112) − p i ψ B † Ri ) | (cid:105) , (21)where ψ A ( B ) † Ri | (cid:105) = | R A ( B ) i (cid:105) . Then the density matrix in the rotated many-body wave function has a tensor product form ρ = | ˜ G R (cid:105)(cid:104) ˜ G L | = (cid:79) i | ν Ri (cid:105)(cid:104) ν Li | , (22)where | ν Ri (cid:105) = √ p i | R Ai (cid:105) A | (cid:105) B + √ − p i | (cid:105) A | R Bi (cid:105) B . Then the reduced density matrix has a tensor product form ρ A = (cid:88) i (cid:104) L Bi | ρ | R Bi (cid:105) = (cid:79) i ( p i | L Ai (cid:105)(cid:104) R Ai | + (1 − p i ) | (cid:105)(cid:104) | ) . (23)Now we can compute the entanglement entropy as S A = − (cid:88) i [ p i ln p i + (1 − p i ) ln(1 − p i )] . (24)Furthermore, if we define p i := q i q i + q − i and − p i := q − i q i + q − i , the entanglement entropy is S A = − (cid:88) i [ q i q i + q − i ln q i q i + q − i + q − i q i + q − i ln q − i q i + q − i ]= (cid:88) i [ln( q i + q − i ) − q i − q − i q i + q − i ln q i ] . (25)The eigen-spectrum of ρ A gives the entanglement spectrum.0
25 50 75 100 l A - - S A
25 50 75 100 l A - - S A
25 50 75 100 l A - - - S A
25 50 75 100 l A - - - S A
25 50 75 100 l A S A
25 50 75 100 l A - - - S A (a) (b) (c)(d) (e) (f) FIG. 5. The entanglement entropy scaling as a function of subsystem size l A with fixed total system size L = 200 and different momentumshifts δ . (a) δ = 0 . , (b) δ = 0 . , (c) δ = 0 . , (d) δ = 0 . , (e) δ = 0 . , (f) δ = 0 . . The corresponding c/ for(a),(b),(c), and (f) are . , . , . and . , respectively.
25 50 75 100 l A - - - S A
25 50 75 100 l A - - S A
25 50 75 100 l A - - S A
25 50 75 100 l A S A (a) (b)(c) (d) FIG. 6. The entanglement entropy scaling as a function of subsystem size l A with fixed total system size L = 200 , momentum shift δ =0 . , and different gap ∆ = 2 (cid:112) | v k | − u . Here we fix ( w, v ) = (1 . , . and change u . (a) ∆ = 0 . , (b) ∆ = 0 . , (c) ∆ = 0 . , and (d) ∆ = 0 . . CROSSOVER OF THE ENTANGLEMENT ENTROPY SCALING FROM c = − TO c = 1 As we mentioned in the main text, both left and right eigenvectors are singular at the crossing point ( k = π ). To get the c = − entanglement entropy scaling behavior, we introduce a tiny momentum shift δ to avoid directly taking the crossingpoint.We observe that if the momentum shift is comparable to the size of the discrete momentum π/L , the entanglement entropyscaling gives the central charge c = 1 . The crossover from c = − to c = 1 entanglement entropy scaling is shown in Fig. 5.The entanglement entropy as a function of the subsystem size goes from a convex function to a concave function. This crossover1behavior can also be seen when we are slightly away from the critical point [6]. In other words, a small finite momentumshift can be seen as introducing a small gap at the crossing point. We can linearize the spectrum at the crossing point and thecorresponding velocity is v eff . = √ wv . We can convert the momentum shift to the corresponding gap ∆ ∼ v eff . δ which is thesame order of the crossover behavior shown in Fig. 6. This crossover is also observed in the quantum Ising chain in an imaginarymagnetic field [45]. ENTANGLEMENT PROPERTIES AT THE CRITICAL POINT SEPARATING THE TOPOLOGICAL PT SYMMETRIC PHASEAND THE SPONTANEOUSLY PT BROKEN PHASE
In the SSH model with PT symmetry, the critical point which separates the topological PT symmetric phase and the PTbroken phase has a pair of ”boundary modes” in the entanglement spectrum ξ ± ,α = 0 . ± iI α [Fig. 7(a)]. Since ξ ± ,α and ξ ∓ ,α = 1 − ξ ± ,α are complex conjugate to each other, this pair does not generate the imaginary part of the entanglement entropy,i.e., the entanglement entropy is still real. The scaling behavior of the entanglement entropy is S A = α ( L ) ln[sin[ πl A L ]] + const . as shown in Fig. 7(b). The coefficient α ( L ) depends on the total system size L as shown in Fig. 7(c). At thermodynamic limit /L → , we expect this coefficient vanishes. - - - L α [ ξ ] - [ ξ ] (a) (b) (c)
10 20 30 40 L A S A L = 80
10 40 80x | ψ A |
120 160 200L - - - C eff
120 160 200L - - - C eff (a) (b) (c) FIG. 8. (a) The boundary modes in the entanglement spectrum at the critical point w − v = u . (b) The effective central charge c eff = 3 ∗ α aftersubtracting out the contributions of the boundary modes as a function of total system size L . Here α being the coefficient in the logarithmicscaling of the entanglement entropy, S A = α ln[sin[ πL A L ]] + const . . The effective central charge c eff as a function of total system size L foronly including the imaginary part of the eigenvalues of the boundary modes in the entanglement spectrum. We observe the eigenvalues of the boundary modes are ξ ± ,α = 0 . ± iI α with I α depending on the subsystem size. If we shift2the unit-cell by half of the lattice constant, the boundary modes in the entanglement spectrum can be removed and the spectrumis identical to the critical point that separates the trivial PT symmetric phase and the PT broken phase. The entanglement entropyscaling gives the central charge c = − . SYMMETRY ENRICHED CFT
In this section, we present the entanglement spectrum and entanglement entropy analysis on the free-fermion version of thesymmetry enriched CFT studied in Ref. [49]. The Hamiltonian for this critical chain [show in the left panel in Fig. 9(a)] is H ( k ) = (cid:18) e ik + e i k e − ik + e − i k (cid:19) . (26)One can immediately see that there will be two isolated sites when the open boundary condition is imposed. It is shown thatthese boundary modes are exponentially localized as long as the parity and time-reversal symmetries are preserved in the bulk[65]. On the other hand, if we introduce boundaries by cutting through the unit-cell, or equivalently shifting the half of theunit-cell of the original chain, the system becomes the regular critical chain [see right panel in Fig. 9(a)].We compute the entanglement spectrum and entropy in this symmetry-enriched CFT. There are two mid-gap states whichare localized at the boundaries of the entanglement Hamiltonian [Fig. 9(b)]. Due to these boundary modes, the entanglemententropy does not have logarithmic scaling and we cannot extract the central charge.However, if we bipartite the system such that the entangling boundaries cut through the unit-cell, there is no boundary modein the entanglement spectrum and the central charge c = 1 can be directly extracted from the entanglement entropy scaling [Fig.9(c)]. Shift half the unit-cell n0.51 ξ n0.51 ξ
10 20 30 40 50 L A S A
10 20 30 40 50 L A S A (a)(b) (c) FIG. 9. (a) The free-fermion version of symmetry enriched CFT, which has boundary modes in the critical chain as shown in left panel. Aftershifting half of the unit-cell, it has the regular critical chain configuration and has no boundary modes (right panel). (b) The entanglementspectrum (left) and the entanglement entropy scaling (right) in the symmetry enriched CFT. There are two mid-gap states in the entanglementspectrum corresponding to two boundary modes in the entanglement Hamiltonian. The entanglement entropy scaling does not satisfy thelogarithmic scaling. (c) The entanglement spectrum (left) and the entanglement entropy scaling (right) in the critical chain case. There are nomid-gap states in the entanglement spectrum and the entanglement entropy scaling gives the central charge c = 1 . BC -GHOST CFT In this section, we briefly review the bc -ghost CFT [35–40]. The effective action for the bc -ghost theory is S = (cid:90) d z ( ψ b ¯ ∂ψ c + ¯ ψ b ∂ ¯ ψ c ) , (27)3where ψ b/c are the fermionic ghost fields for the right moving mode and ¯ ψ b/c denotes the anti-holomophic fermionic ghost fieldswhich correspond to the left moving mode. We use the shorthand notations, z = x + it , ¯ z = x − it , ∂ = ( ∂ x − i∂ t ) and ¯ ∂ = ( ∂ x + i∂ t ) . Here we have the anti-commutation relationship { ψ b ( z ) , ψ c ( w ) } = δ ( z − w ) . From dimensional analysis,the conformal dimension of the ghost fields must be ∆ b + ∆ c = 1 . We can parametrize them by ∆ b = λ and ∆ c = 1 − λ . Theequations of motion give ¯ ∂ψ b ( z ) = ¯ ∂ψ c ( z ) = 0 , ¯ ∂ψ c ( z ) ψ b (0) = 2 πδ (2) ( z, ¯ z ) . (28)The above equations imply the operator product expansion (OPE) and the two-point function are ψ b ( z ) ψ c ( w ) ∼ z − w , (cid:104) ψ b ( z ) ψ c ( w ) (cid:105) ∼ z − w . (29)The other two-point functions do not have singularity, (cid:104) ψ b ( z ) ψ b ( w ) (cid:105) ∼ (cid:104) ψ c ( z ) ψ c ( w ) (cid:105) ∼ O ( | z − w | ) .The Noether’s theory gives the normal-ordered holomorphic part of the energy-momentum tensor, T ( z ) =: ( ∂ψ b ) ψ c : − λ∂ : ( ψ b ψ c ) : . (30)The OPEs between T and ψ b/c are T ( z ) ψ b ( w ) ∼ ( ∂ψ b ( z )) ψ c ( z ) ψ b ( w ) − λ∂ ( ψ b ( z ) ψ c ( z ) ψ b ( w )) ∼ z − w ∂ψ b ( z ) − λ∂ ( ψ b ( z ) 1 z − w ) ∼ λ ( z − w ) ψ b ( w ) + 1 − λz − w ∂ψ b ( w ) ,T ( z ) ψ c ( w ) ∼ (1 − λ ) ∂ ( ψ b ( z ) ψ c ( z ) ψ c ( w )) − ψ b ( z )( ∂ψ c ( z )) ψ c ( w ) ∼ (1 − λ ) ∂ ( − z − w ψ c ( z )) + 1 z − w ∂ψ c ( z ) ∼ − λ ( z − w ) ψ c ( w ) + λz − w ∂ψ c ( w ) , (31)which give the conformal dimensions ∆ b = λ , and ∆ c = 1 − λ , respectively.The central charge can be obtained from the OPE of T ( z ) T ( w ) : T ( z ) T ( w ) ∼ [: ( ∂ψ b ) ψ c : − λ∂ : ( ψ b ψ c ) :]( z )[: ( ∂ψ b ) ψ c : − λ∂ : ( ψ b ψ c ) :]( w ) ∼ − z − w ) ∂ z ψ b ( z ) ψ c ( w ) + 6 λ (1 − λ ) 1( z − w ) ∼ ( − λ + 6 λ −
1) 1( z − w ) . (32)We can identify the central charge c = − λ +12 λ − . In the non-Hermitian SSH model at the critical point, (∆ b , ∆ c ) = (1 , ,which gives c = − . TWO-POINT FUNCTIONS
Here, we compute the two-point functions in the non-Hermitian SSH model at the critical point v − w = u . We first computethe correlation function (cid:104) ψ † b ( x ) ψ c ( y ) (cid:105) with ψ † b ( x ) is the right creation operator and ψ c ( y ) is the left annihilation operator. Werefer these fields the ghost fields. (cid:104) ψ † b ( x ) ψ c ( y ) (cid:105) = Tr (cid:88) k L e ik ( x − y ) | R k, − (cid:105)(cid:104) L k, − | = (cid:88) k e ik ( x − y ) L = 1 π sin π ( x − y ) | x − y | (33)This two-point function gives the correct conformal dimensions of the ghost fields, λ b + λ c = 1 [Fig. 10(a)].4We can also compute the other two-point functions (cid:104) ψ † b ( x ) ψ b ( y ) (cid:105) and (cid:104) ψ † c ( x ) ψ c ( y ) (cid:105)(cid:104) ψ † b ( x ) ψ b ( y ) (cid:105) = Tr (cid:88) k L e ik ( x − y ) | R k, − (cid:105)(cid:104) R k, − | = (cid:88) k e ik ( x − y ) L (sin φ k φ ∗ k φ k φ ∗ k , (cid:104) ψ † c ( x ) ψ c ( y ) (cid:105) = Tr (cid:88) k L e ik ( x − y ) | L k, − (cid:105)(cid:104) L k, − | = (cid:88) k e ik ( x − y ) L (sin φ k φ ∗ k φ k φ ∗ k , (34)where φ k = tan − [ | we − ik + v | iu ]. As shown in Fig. 10(b), there is no power law decay as expected in the CFT. | x - y |- - < ψ † b ( x ) ψ c ( y )> | x - y |- < ψ † b / c ( x ) ψ b / c ( y )> (a) (b) FIG. 10. Two-point functions of (a) (cid:104) ψ † b ( x ) ψ c ( y ) (cid:105) and (b) (cid:104) ψ † b/c ( x ) ψ b/c ( y ) (cid:105) as a function of | x − y | . ENTANGLEMENT ENTROPY IN A (1+1)D NON-UNITARY CFT
The field theory approach to the entanglement entropy in non-unitary CFTs has previously been studied in, e.g., Refs. 42–44,where the twist operator and replica method are used. In Ref. 43, it was found that for a non-unitary CFT in which the physicalground state is different from the conformal vacuum, the entanglement entropy has the form S A ∼ c eff log l(cid:15) , where c eff is theeffective central charge, l is the length of subsystem A , and (cid:15) is a UV cutoff. In particular, it is found that c eff = c − , where ∆ < is the lowest conformal dimension of operator in the theory. This result is a reminiscent of the work of by Itzykson, Saleurand Zuber [66], where it was found that the central charge c is replaced by the effective central charge c eff in the expression ofthe ground state free energy. Later in Ref. 42, it was found that the entanglement entropy in the ghost bc CFT with c = − has the form S A ∼ c log l(cid:15) with c = − . The underlying reason is that in this case the physical ground state is the same as theconformal vacuum (see more details in the following discussions).Here we give a brief review of these results on the entanglement entropy in non-unitary CFTs by utilizing the approachintroduced by Cardy and Tonni [67]. Let us first introduce the possible difference between the physical ground state and theconformal vacuum. In (1+1)D conformal field theory, the conformal vacuum | (cid:105) is defined as the state invariant under all regularconformal transformations, L n | (cid:105) = 0 , where n ≥ − (This results from the requirement that the stress-energy tensor T ( z ) isregular at z = 0 in the conformal vacuum). Therefore, for the conformal vacuum, we always have L | (cid:105) = 0 . On the otherhand, the physical vacuum | G (cid:105) , or the physical ground state, is defined as the lowest eigenstate of L . In non-unitary CFTs, wehave L | G (cid:105) = ∆ | G (cid:105) , L | G (cid:105) = ∆ | G (cid:105) , (35)where ∆ ≤ in general. Here L and L have the same lowest eigenvalue because the ground state is translation invariant. In aunitary CFT, we always have | G (cid:105) = | (cid:105) , which is not necessarily true for a non-unitary CFT.In the following derivation of the entanglement entropy, we will only use the conformal symmetry as well as the definition ofphysical ground state | G (cid:105) of a generic non-unitary CFT in Eq. (35).For simplicity, we consider a finite interval A = [ − l/ , l/ in an infinite system in the ground state | G (cid:105) . The path-integral5representation of the reduced density ρ A = Tr B ρ can be expressed as [67] zA BB (36)where z = x + iτ , and two small discs of radius (cid:15) have been removed at the two entanglement cuts x = ± R = ± l as the UVcutoff. The rows and columns of the reduced density matrix ρ A are labelled by the values of the fields on the upper and loweredges of the slit along A . Along the two small discs, the conformal boundary conditions | a (cid:105) and | b (cid:105) are imposed. Then, byconsidering the conformal transformation w = f ( z ) = log z + RR − z , ρ A in Eq. (36) is mapped to the following cylinder in w -plane: uvw (37)One can find that the length of the cylinder in w -plane is W = 2 log (cid:18) l(cid:15) (cid:19) + O ( (cid:15) ) , (38)and the circumference is π in v direction. Then we have Tr A ρ nA = Z n , where Z n is the path integral over the manifold obtainedby gluing n cylinders in (37) along the (gray) edges one by one. Z n can be explicitly evaluated as follows Z n = (cid:104) a | e − H CFT · W | b (cid:105) = (cid:104) a | e − π πn ( L +¯ L − c ) · W | b (cid:105) = (cid:88) k (cid:104) a | k (cid:105) · (cid:104) k | b (cid:105) · e − n (∆ k + ¯∆ k − c ) · W , (39)where in the second step we have inserted a complete basis vectors | k (cid:105) . Considering W (cid:29) , only the lowest weight ∆ k + ¯∆ k dominate in Z n . Now the difference between unitary and non-unitary CFTs comes in. For a unitary CFT, the conformal vacuumis the same as the physical ground state, i.e., | (cid:105) = | G (cid:105) . The term with ∆ k = ¯∆ k = 0 dominates, and therefore Z n (cid:39) (cid:104) a | (cid:105) · (cid:104) | b (cid:105) · e c n · W , for unitary CFTs (40)On the other hand, for a non-unitary CFT, as seen in Eq. (35), the ground state has a possibly negative eigen-energy ∆ ≤ .Then Z n is dominated by the term with ∆ k = ¯∆ k = ∆ : Z n (cid:39) (cid:104) a | G (cid:105) · (cid:104) G | b (cid:105) · e c − n · W =: (cid:104) a | G (cid:105) · (cid:104) G | b (cid:105) · e c eff n · W , for non-unitary CFTs (41)where we have defined c eff = c − . (42)As a remark, the procedure of evaluating Z n here is essentially the same as the calculation of the free energy in the ground stateof a non-unitary CFT as studied in Ref. 66. Then the n -th Renyi entropy and von Neumann entropy can be expressed as S ( n ) A := 11 − n log Tr ( ρ nA )( Tr ρ A ) n = 11 − n log Z n ( Z ) n (cid:39) c eff · nn · W (cid:39) c eff · nn log (cid:18) l(cid:15) (cid:19) ,S A = lim n → S ( n ) A (cid:39) c eff W = c eff (cid:18) l(cid:15) (cid:19) , (43)where we have neglected the O (1) terms which are contributed by the boundaries. In particular, we have c eff = c for unitaryCFTs, and c eff = c − for non-unitary CFTs.For the bc ghost CFT with c = − as studied in the main text, it is noted that the physical ground state | G (cid:105) is the same as theconformal vacuum | (cid:105) , i.e., | G (cid:105) = | (cid:105) , and we have ∆ = 0 in Eq. (35)[42]. Then based on Eq. (42), we have c eff = c = − . (44)This agrees with the result obtained in Ref. 42 based on the the twist operator approach, with appropriate generalizations of thestandard CFT replica technique. Furthermore, for the ghost bc CFTs with λ > (see previous sections), one has ∆ = λ (1 − λ )2 and the central charge c = − λ + 12 λ − . The effective central charge has the expression: c eff = c − · λ (1 − λ )2 , (45)which reduces to c eff = c = − for λ = 1 .6 ENTANGLEMENT SPECTRUM OF THE NON-HERMITIAN 2D MODEL
The non-Hermitian Chern insulator [32] defined by the Bloch Hamiltonian H ( k ) = ( m + t cos k x + t cos k y ) σ x + ( iγ + t sin k x ) σ y + ( t sin k y ) σ z , (46)has complex dispersion relations as E ± ( k ) = ± [( m + t cos k x + t cos k y ) + ( iγ + t sin k x ) + ( t sin k y ) ] / . (47)Exceptional points appear when the two bands satisfy E ± ( k EP ) = 0 . This condition demands sin( k EP ,x ) = 0 . We can find thegapless phases where pairs of exceptional points appear on k x = 0 , k x = ± π , and both k x = 0 and k x = ± π. In the gapped phase with the topologically trivial bulk, no mid-gap modes appear between the gapped complex entanglementbands, as shown in Fig. 11. In the gapless phases, there appear exceptional points on k x = 0 and/or k x = ± π . Complexentanglement spectra in the presence of exceptional points are shown in Fig. 12 and Fig. 13. FIG. 11. The topologically trivial bands with zero Chern number ( t = 1 . , m = − , γ = 0 . ). (a) The orange and blue bands correspondto E + and E − , relatively. (b) Complex entanglement spectrum which is labeled by k y and amplitude of the eigen-modes of k y = 0 for twodegenerate ξ = 0 . (yellow squares) and two degenerate ξ = 0 . (gray dots) along x direction. (c) Complex entanglement spectrumlabeled by k x and the amplitude of the eigen-modes of k x = 0 for ξ = 0 . − . i, . − . i (yellow squares) and for ξ =0 . . i, .
982 + 0 . i (gray dots) along y direction. FIG. 12. The gapless phase with two pairs of exceptional points on both k x = 0 ( t = 1 . , m = − , γ = 1 ). (a) The orange and blue bandscorrespond to E + and E − , relatively. (b) Complex entanglement spectrum which is labeled by k y and the eigen-modes of k y = 0 and for ξ = 0 . − . i (yellow squares) and for ξ = 0 .
339 + 0 . i (gray dots) along x direction. (c) Complex entanglement spectrum labeledby k x and the eigen-modes of k x = 0 for ξ = 0 .
147 + 0 . i (yellow squares) and for . − . i (gray dots) along y direction. FIG. 13. The gapless phase with two pairs of exceptional points on both k x = 0 and k x = ± π ( t = 1 . , m = − . , γ = 1 ). (a) The orangeand blue bands correspond to E + and E − , relatively. (b) Complex entanglement spectrum which is labeled by k y and the eigen-modes of k y = 0 for ξ = 0 . − . i (yellow squares) and for ξ = 0 .
347 + 0 . i (gray dots) along x direction. (c) Complex entanglementspectrum labeled by k x and the eigen-modes of k x = 0 for ξ = 0 . . i (yellow squares) and for ξ = 0 . − . i (gray dots) along yy