Entanglement spectrum and entropy in topological non-Hermitian systems and non-unitary conformal field theories

We propose a method of computing and studying entanglement quantities in non-Hermitian systems by use of a biorthogonal basis. We find that the entanglement spectrum characterizes the topological properties in terms of the existence of mid-gap states in the non-Hermitian Su-Schrieffer-Heeger (SSH) model with parity and time-reversal symmetry (PT symmetry) and the non-Hermitian Chern insulators. In addition, we find that at a critical point in the PT symmetric SSH model, the entanglement entropy has a logarithmic scaling with corresponding central charge $c=-2$. This critical point then is a free-fermion lattice realization of the non-unitary conformal field theory.


INTRODUCTION
In contemporary condensed matter physics, quantum entanglement has been proven to be a key concept to reveal properties of many-body quantum systems.For example, the topological entanglement entropy provides a direct method to detect topological order [1,2].Entanglement spectra can extract the low energy excitation in the topological systems and fractional quantum Hall systems [3][4][5][6][7].Moreover, it paves a way for a deeper understanding of the renormalization group and an emergent holographic space-time [8][9][10].
In this letter, we investigate quantum entanglement in non-Hermitian systems, ranging from non-Hermitian gapped (topological) systems, to critical systems, which can be described by non-unitary conformal field theories (CFTs) at low energies.To investigate these non-Hermitian systems, we first use a generalized definition of the reduced density matrix in terms of the biorthogonal basis [11].We show that this generalized-reduced density matrix can be constructed from the correlation matrix and the overlap matrix in freefermion systems.The early studies of the entanglement in the non-Hermitian systems are mostly restricted in spin chains [12,13], non-unitary CFTs [14,15], and holographic approach [16].Another approach is using the flatten singularvalue decomposition to extract the entanglement spectrum in the topological non-hermitian systems [17].Besides these developments, our methods are applicable to both gapped and gapless phases and can be extended to non-equilibrium setups.
As a particular example, we investigate the non-Hermitian Su-Schrieffer-Heeger (SSH) model with the combination of parity and time-reversal symmetry (PT symmetry).There are two PT symmetric phases, trivial and topological ones, separated by a PT broken phase.In PT symmetric phases, the energy spectrum is fully gapped and the PT symmetry enforces the energy to be real.We first show that the existence of midgap states in the entanglement spectrum is concurrent with the existence of protected physical boundary modes in these PT symmetric phases [18,19]: The topological PT symmetric phase supports the mid-gap states in the entanglement spectrum, while the trivial PT symmetry phase does not.Thus, the entanglement spectrum provides a way to characterize the topological non-Hermitian phases.
Next, we study the entanglement entropy at the critical points which separate one of the PT symmetric phases and the PT broken phase.At the critical point separating the trivial PT symmetric phase and the PT broken phase, we show that the entanglement entropy for the subsystem of length L A scales as S A = (c/3) ln L A +• • • with c = −2.This negative central charge c = −2 can be attributed to the bc-ghost fermionic theory which is a non-unitary CFT [20][21][22][23][24][25][26].On the other hand, at the other critical point separating the topological PT symmetric phase and the PT broken phase, there are two additional mid-gap states in the entanglement spectrum which mimic the physical boundary modes at this critical point.Nevertheless, by modifying the bipartition, we show that the entanglement scaling again is given by the logarithmic law with c = −2.
The quantum entanglement in the PT broken phase can also be studied, once we consider a proper "ground state" which is obtained by filling modes with real energy.The entanglement entropy scaling in this case gives the central charge c = 1.We show the Jordan block form at the exceptional point leads to the ground state identical to the free Dirac theory.
As yet another example, we study the entanglement spectrum of the non-Hermitian Chern insulators.The mid-gap states in the entanglement Hamiltonian mimic the physical boundary modes in non-Hermitian systems.Our method provides an alternative way to study the entanglement properties in both topological or critical non-Hermitian systems.
A many-body right wavefunction can be expressed in terms of the bi-fermionic operators as Using the definition of many-body right and left wave functions, we can construct the non-Hermitian density matrix, ρ = |G R G L | such that ρ † = ρ and ρ 2 = ρ.Once the density matrix of the non-Hermitian system is constructed, we can then introduce measures of quantum entanglement from the reduced density matrix by integrating out the degrees of freedom in subsystem B, ρ A = Tr B ρ.Here the system is partitioned into A and B. In Hermitian systems, if the Hamiltonian has the quadratic form, the reduced density matrix can be constructed from either the correlation matrix [27] or the overlap matrix [28].We extended these derivations to non-Hermitian systems with quadratic form in terms of bi-fermionic operators as follows.(Detailed derivations of the above results are provided in the Supplementary Material [26]).
-The correlation matrix is defined as The entanglement Hamiltonian H A can be introduced by The entanglement entropy for non-Hermitian systems can then be introduced as where ξ δ are the eigenvalues of C ij .
-The overlap matrix is defined as where L A αδ and R A βδ are the corresponding left and right eigenvectors of M A with eigenvalues p * δ and p δ .The original left and right many-body wavefucntions as well as the reduced density matrix can be expressed in this new basis.In particular, ρ A is given by The entanglement entropy can be directly obtained from the reduced density matrix as

ENTANGLEMENT SPECTRUM AND ENTANGLEMENT ENTROPY IN NON-HERMITIAN SYSTEMS
Non-Hermitian SSH model We now study the non-Hermitian SSH model [19,29] with the PT symmetry defined in moemtnum space by where v k = we −ik + v with u, v, w ∈ R and k is the singleparticle momentum.Here the PT symmetry is defined as where and the corresponding left eigenvectors are with The right and left eigenvectors satisfy the biorthogonal condition, L k,± |R k,± = 1 and L k,∓ |R k,± = 0.
There are three phases in the non-Hermitian SSH model with u = 0 [Fig.1(a) with the parameters w, v, u > 0].In the PT broken phase (|w−v| < u), the energy spctrum is comples, and gapless with two exceptional points.The region with w − v > u realizes one of the PT symmetric phases, where the spectrum is fully gapped, and there is a pair of edge modes with imaginary energy This phase is a topological non-Hermitian phase supporting protected boundary modes and characterized by the non-zero global complex Berry phase [29].Finally, the other gapped PT symmetric phase (w − v < −u) is trivial, and does not support edge mode [Fig.1(b)].
Entanglement spectrum-Now we can compute the entanglement spectrum and entanglement entropy from the correlation matrix and the overlap matrix.First, we consider the PT symmetric phase (fully gapped case), where the ground state is well defined since there is no imaginary energy mode in the fully gapped phase.The entanglement spectrum is obtained from the eigenvalues of the correlation matrix and the overlap matrix with the periodic boundary condition.The entanglement spectra from both methods are identical.In the following, we only present the results from the correlation matrix, l a t e x i t > u < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 0 i e u W 3 5 z l w 9 n B j S 7 A C A E w L z n P I = " > A A A B 6 X i c b V B N S 8 N A E J 3 4 W e t X 1 a O X x S J 4 s S R V 0 G P R i 8 c q 9 g P a U D b b T b t 0 s w m 7 E 6 G E / g M v H h T x 6 j / y 5 r 9

w-v
Re[E] -1 -0.5 0 0.5 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " / y which has real space information for the entangling boundary modes.
In the topological non-Hermitian phase (w − v > u) where the physical edge modes are present, there are two mid-gap states in the entanglement spectrum with exact half unit in the real part Re [ξ] = 0.5 and nonvanishing imaginary part Im [ξ] = 0 [Fig. 2 (a-b)].However, while these two midgap states are localized at the entangling boundaries, there are numerous localized boundary modes which are not the midgap states in the correlation matrix [Fig. 2 (c)].On the other hand, in the trivial phase where no physical edge modes are present, we observe four additional states with non-vanishing imaginary eigenvalues of the correlation matrix [Fig. 2 (d-e)].There are also numerous localized boundary modes which are not the mid-gap states in this phase.These numerous localized modes in the correlation matrix are similar to the non-Hermitian skin effect in non-Hermitian systems with open boundary condition [18,30,31].
Critical points and non-unitary CFT-In this non-Hermitian SSH model with PT symmetry, there are two critical points that separate the PT symmetric phases and PT broken phase at w − v = ±u.When u = 0, the SSH model has only one critical point at w = v and the system is a critical free-fermion chain.The entanglement entropy scales logarithmically S A ∼ (c/3) ln L A , where L A is the subsystem size.This scaling behavior agrees with the conformal field theory prediction and has corresponding central charge c = 1 [32].
For finite u, we first analyze the entanglement entropy at the critical point w − v = −u which separates the trivial PT symmetric phase and the PT broken phase.At this critical point, we observe that all eigenvalues of the correlation matrix are real and they come in pairs, ξ α > 1 and ξ β = 1 − ξ α < 0. This pair-wise structure of the eigenvalues guarantees that the entanglement entropy is real and negative.As shown in Fig. 3(a), the entanglement entropy scales logarithmically Since the spectrum at this critical point is linear at k = ±π, one can write down the effective field action as , and ψ b/c ( ψb/c ) represent the fermionic fields for the right-moving (left-moving) modes.(We have set the Fermi velocity to be one).We identify ψ † b(c) as the right(left) creation operator ψ † R(L) defined in Eq. (1).A crucial observation here is that states associated to these right and left fermionic operators are not normalizable at the critical point [33]: The ill-defined norm of the quantum states can be thought of as an inkling of the ghost theory.The above action, with proper assignment of the conformal dimensions, defines the bc-ghost CFT with central charge c = −2 [20][21][22][23][24][25][26].The entanglement entropy detects the correct central charge as we expect.However, there are subtleties to get the c = −2 scaling of the entanglement entropy.We need to choose the periodic boundary condition and consider the half-filling ground state, where we fill the (left/right) state at the crossing point but with a tiny momentum shift [34].While imposing antiperiodic, open boundary conditions, or not including the state at the crossing point, the scaling of the entanglement entropy gives the central charge c = 1 [35].One possible explanation of the sensitivity of boundary conditions is that the ill-defined norm only occurs at the crossing point k = π, which the bcghost CFT are applicable for.
We now turn to the other critical point (w − v = u) separating the topological PT symmetric phase and the PT broken phase.It exhibits a very different entanglement entropy scaling.At this critical point, we observe an additional pair of eigenvalues ξ ±,α = 0.5 ± iI α , where I α depends on the subsystem size.This pair in the entanglement spectrum mimics the protected boundary modes in the gapless point in the physical spectrum [see Fig. 1(b) at w − v = u and Fig. 5(a) in [26]].Although this pair of eigenvalues contributes to the pure real part of the entanglement entropy, the imaginary part I α depends on the total system size.We find the entanglement entropy has the following scaling S A = α ln[sin(πL A /L)] + const., where α decreases as a function of the total system size [Figs.5(b-c) in [26]].The length-dependent coefficient α is not described by the CFT.However, at the effective field theory level, these two critical points are the same and should have the same scaling behavior in the entanglement entropy.To resolve this discrepancy, we consider another bipartition which cutting through the unit-cell.In this bipartition, it is equivalent to shifting the half of the unit-cell of the original system.Under this shifted unit-cell, the hopping w and v are interchanged and the entanglement spectrum are identical to the critical point that separates the trivial PT symmetric phase and the PT broken phase.We recover the c = −2 entanglement entropy scaling under this bipartition.Recently, in Hermitian systems, the protected boundary modes in the critical systems have also been discussed in Ref. [36].To make comparison, the entanglement spectrum in this model also has boundary modes in the entanglement spectrum reflecting the existence of physical boundary modes.We can also shift the half of the unit-cell and recover the c = 1 scaling of this freefermion system [26].The critical point separating the topological PT symmetric phase and the PT broken phase can also be seen as the symmetry-enriched bc-ghost CFT.
Last, we consider the entanglement properties in the PT broken phase (|w − v| < u), where there are 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 the single-particle state with real energy.As shown in Fig. 3 entropy of this "ground state" follows the CFT scaling behavior, S A = (c/3) ln[sin(πL A /L)] + const.with c = 1.The appearance of the c = 1 CFT behaviour can be understood as follows.At the exceptional points, two eigenstates are coalescing into one and the Hamiltonian cannot be diagonalized.One can expand the Hamiltonian at the exception point with a Jordon block form [37,38], where we linearize the spectrum at the exceptional point k EP , set the velocity to be unity [39], and γ is an arbitrary complex number.At the exceptional point k EP , two eigenstates collapse to one eigenstate (1, 0) with energy −(k − k EP ).The ground state can be expressed as |G = k−kEP>0 (1, 0) T , which has the identical form of the ground state of the free Dirac theory with charge c = 1.
Entanglement spectrum of the non-Hermitian Chern insulators Finally, we consider the non-Hermitian Chern insulator [40] in two dimensions defined in momentum space by where t, m, and γ are real parameters and we assume t > 0.
The complex dispersion relation of the fermions is Exceptional points appear when two bands E ± (k EP ) = 0 at certain momenta k = k EP .In the presence of such a point, the Hamiltonian (9) becomes defective and the eigenstates coalesce and linearly depend on each other.The gapped phases with E + (k) = E − (k) for all k are characterized by the first Chern number and in the case of γ = 0, the model reduces to the well-known Hermitian Chern insulator.The entanglement spectrum of the non-Hermitian Chern insulators is obtained from the eigenvalues of the correlation matrix and the overlap matrix with the periodic boundary conditions in both x and y directions.Since it has been shown that the bulk-edge correspondence in the presence of non-Hermitian physical systems is sensitive to the boundary conditions, in the following we investigate the entanglement spectrum of the non-Hermitian gapped phases by setting entangling boundary either in the y direction or in the x direction.In the Supplementary material [26], we also provide the entanglement spectrum of the non-Hermitian model [Eq.( 9)] with general parameters.
In the gapped phase with the topologically nontrivial bulk, we observe two chiral mid-gap modes localized at the right and left edges, respectively.[Figs.4 (b) and (c)].In particular, for the entangling boundary along the x direction (k x is a good quantum number), the right mid-gap mode has the largest positive imaginary part for γ > 0, whereas the left mid-gap mode has the largest negative imaginary part of ξ [Fig.4 (c)].In contrast, the imaginary parts of the mid-gap states vanish for the entangling boundary along the y direction (k y is a good quantum number) [Fig.4 (b)].This phenomenon implies the amplification of the right mid-gap mode and the attenuation of the left mid-gap mode, similar to the behaviors of the physical edge modes in the non-Hermitian Chern insulator and a topological insulator laser discussed in other contexts [40][41][42].

CONCLUSION AND OUTLOOK
In this letter, we propose a method of computing entanglement properties in non-Hermitian systems by use of the biothogonal basis.The entanglement spectrum of the non-Hermitian systems mimics the physical spectrum and thus can detect the topological properties of non-Hermitian systems.In particular, we find a free-fermion lattice model for a realization of a non-unitary CFT with the central charge c = −2.This lattice model is at a critical point separating the PT symmetric phase and the PT broken phase in a PT symmetric SSH model.Our finding provides a new direction for studying various properties in non-Hermitian systems in both topological and critical phases, including non-equilibrium dynamics [43].
Note added-During the preparation of this manuscript, we became aware of partially related work by Herviou et al. [44].Supplementary Material for "Entanglement spectrum and entropy in topological non-Hermitian systems and non-unitary conformal field theories".
In this document we present the detailed derivations of the computations of the entanglement properties from the correlation matrix and the overlap matrix.

CORRELATION MATRIX
We use the correlation matrix [27] to extract eigenvalues of the reduced density matrix in a free fermion system.Since the theory is free, the reduced density matrix has a Gaussian form , where H A αβ refers to the entanglement Hamiltonian.
The correlation matrix is defined where |G R and |G L are the right and left ground states.One can simultaneously diagonalize C ij and H A αβ and find The correlation matrix can be expressed by the left and right eigenvectors as By using the commutation relation {φ i , φ † j } = δ ij and φ i |0 = 0, we have where 12), the correlation matrix is One should notice that where ξ δ are the eigenvalues of C ij .

OVERLAP MATRIX
Alternatively, we can use the overlap matrix the extract the entanglement properties.The basic idea is rotate the reduced density matrix in a new biothogonal basis in the subsystem A, {|L The rotation matrix is constructed from the overlap matrix [28], where L A iβ and R A iα are the corresponding left and right eigenvectors of M A with eigenvalues p * i and p i .Notice that the overlap matrix is non-Hermitian So in the diagonal basis, the non-Hermitian overlap matrix is Thus we can normalized the left and right eigenvectors of Here we express  18), we can construct the local biorthogonal basis as From the property of overlap matrix M A + M B = I, M A and M B can be spontaneously diagonolized with the corresponding eigenvalues p i and 1 − p i .
Thus the biorthogonal basis {|R B i , |L B i } in subsystem B can also be constructed from the rotation matrices R A αi and L A βj .The right vector |R α in the total system can be rotated by Now we consider many-body wave function after the rotation where . Then the density matrix in the rotated many-body wave function has a tensor product form where Then the reduced density matrix has a tensor product form Now we can compute the entanglement entropy If we define , the entanglement entropy is And also the eigen-spectrum of ρ A which is the entanglement spectrum.

ENTANGLEMENT PROPERTIES AT THE CRITICAL POINT WHICH SEPARATES THE TOPOLOGICAL PT SYMMETRIC PHASE AND THE PT BROKEN PHASE
In the SSH model with the PT symmetry, the critical point which separates the topological PT symmetric phase and the PT broken phase has a pair of "boundary modes" in the entanglement spectrum has the eigenvalue ξ ±,α = 0.5 ± iI α [see Fig. 5(a)].Since ξ ±,α and ξ ∓,α = 1 − ξ ±,α are complex conjugation, 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 L ]] + const.as shown in Fig. 5(b).The coefficient α(L) depends on the total system size L is shown in Fig. 5(c).At thermodynamic limit 1/L → 0, we expect this coefficient vanishes.
< l a t e x i t s h a 1 _ b a s e 6 4 = " v r j z K / W q 5 a E J 6 c s 9 s 9 z Z j S z I The "boundary modes" in the entanglement spectrum are not exponentially localized but are power law decay at the boundaries [Fig.6(a)].Thus, these boundary modes can give subsystem-size dependent contribution to the entanglement entropy.If we naively subtract out the contribution of boundary modes in the entanglement entropy, as shown in Fig. 6(b), the effective central charge we extract from the coefficient still depends on the total system size.
We observe the eigenvalues of the boundary modes are ξ ±,α = 0.5 ± iI α with I α depends on the subsystem size.If we shift the half of the unit-cell, the boundary modes in the entanglement spectrum can be removed and the spectrum is identical to the critical point that separates the trivial PT symmetric phase and the PT broken phase.The entanglement entropy scaling gives the central charge c = −2.

SYMMETRY ENRICHED CFT
In this section, we present the entanglement spectrum and entanglement entropy analysis on the free-fermion version of the symmetry enriched CFT studied in Ref. [36].The Hamiltonian for this critical chain [show in the left panel in Fig. 7(a)] is One can immediately see there will be two isolated sites when the open boundary condition is introduced.It is shown these boundary modes are exponentially localized as long as the parity and time-reversal symmetries are preserved in the bulk [45].On the other hand, if we are allow to introduce boundaries by cutting through the unit-cell, or equivalently by shift the half of the unit-cell of the original chain, the system become the regular critical chain [see right panel in Fig. 7(a)].
We compute the entanglement spectrum and entropy in this symmetry enriched CFT.There are two mid-gap states which in localized at the boundaries of the entanglement Hamiltonian [Fig.7(b)].Due to these boundary modes, the entanglement entropy does not have logarithmic scaling and we cannot extract the central charge.
However, if we bipartite the system such that the entangling boundaries cutting through the unit-cell, there is no boundary mode in the entanglement spectrum and the central charge c = 1 can be directly extract from the entanglement entropy scaling [Fig.7(c)].

BC-GHOST CFT
In this section, we briefly review the bc-ghost CFT [20][21][22][23][24][25].The effective action for the bc-ghost theory is where ψ b/c are the fermionic ghost fields for the right moving mode and ψb/c denotes the anti-holomophic fermionic ghost fields which correspond to the left moving mode.We use the shorthand notations, z 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.The equations of motion give The above equations imply the operator product expansion (OPE) and the two-point function are The other two-point functions do not have singularity, The Noether's theory gives the normal-ordered holomorphic part of the energy-momentum tensor, The OPEs of T , ψ b , and ψ c are which gives the conformal dimension ∆ b = λ.
The central charge can be obtained from the OPE of T (z)T (w): We can identify the central charge c = −12λ 2 +12λ−2.In the non-Hermitian SSH model at the critical point, (∆ b , ∆ c ) = (1, 0), which gives c = −2.

TWO-POINT FUNCTIONS
We compute the two-point functions in the non-Hermitian SSH model at the critical point v − w = u.We first compute the correlation function ψ † b (x)ψ c (y) with ψ † b (x) is the right creation operator and ψ c (y) is the left annihilation operator.We refer these fields the ghost fields.
This two-point function gives the correct conformal dimensions of the ghost fields, λ b + λ c = 1 [Fig.8(a)].
We can also compute the other two-point functions ψ † b (x)ψ b (y) and ψ † c (x)ψ c (y) where As shown in Fig. 8(b), there is no power law decay as expected in the CFT.

ENTANGLEMENT ENTROPY IN A (1+1)D NON-UNITARY CFT
The field theory approach to entanglement entropy in non-unitary CFTs has previously been studied in, e.g., Refs.14-16,where the twist operator and replica method are used.In Ref.14, it was found that for a non-unitary CFT in which the physical ground state is different from the conformal vacuum, the entanglement entropy has the form S A ∼ c eff log l , where c eff is the effective central charge, l is the length of subsystem A, and is a UV cutoff.In particular, it is found that c eff = c − 24∆, where ∆ < 0 is the lowest conformal dimension of operator in the theory.This result is a reminiscent of the work of by Itzykson, Saleur and Zuber [46], where it was found that the central charge c is replaced by the effective central charge c eff in the expression of the ground state free energy.Later in Ref. 16, it was found that the entanglement entropy in the ghost bc CFT with c = −2 has the form S A ∼ c log l with c = −2.The underlying reason is that in this case the physical ground state is the same as the conformal 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 approach introduced by Cardy and Tonni [47].Let us first introduce the possible difference between the physical ground state and the conformal vacuum.In (1+1)D conformal field theory, the conformal vacuum |0 is defined as the state invariant under all regular conformal transformations, L n |0 = 0, where n ≥ −1 (This results from the requirement that the stress-energy tensor T (z) is regular at z = 0 in the conformal vacuum).Therefore, for the conformal vacuum, we always have L 0 |0 = 0. On the other where we have neglected the O(1) terms which are contributed by the boundaries.In particular, we have c eff = c for unitary CFTs, and c eff = c − 24∆ for non-unitary CFTs.
For the bc ghost CFT with c = −2 as studied in the main text, it is noted that the physical ground state |G is the same as the conformal vacuum |0 , i.e., |G = |0 , and we have ∆ = 0 in Eq.( 37) [16].Then based on Eq.( 44), we have This agrees with the result obtained in Ref. 16 based on the the twist operator approach, with appropriate generalizations of the standard CFT replica technique.Furthermore, for the ghost bc CFTs with λ > 1 (see previous sections), one has ∆ = λ(1−λ) which reduces to c eff = c = −2 for λ = 1.

ENTANGLEMENT SPECTRUM OF THE NON-HERMITIAN 2D MODEL
The non-Hermitian Chern insulator [40] described 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 , has complex dispersion relations as Exceptional points appear when two bands E ± (k EP ) = 0.This condition demands sin(k EP,x ) = 0. We can find the gapless 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 entanglement bands, as shown in Fig. 9.In the gapless phases, there appear exceptional points on k x = 0 and/or k x = ±π.Complex entanglement spectrums in the presence of exceptional points are shown in Fig. 10 and Fig. 11.
t e x i t s h a 1 _ b a s e 6 4 = " 8 Q t 0 n K m U Y A e m N F V 8 I 8 e 9 C T 3 c d Z c = " > A A A B 6 n i c b V D L S g N B E O z 1 G e M r 6 t H L Y B C 8 G H a j o M e g F 4 8 R z Q O S J c x O e p M h s 7 P L z G w k h H y C F w + K e P W L v P k 3 T p I 9 a G J B Q 1 H V T X d X k A i u j e t + O y u r a + s b m 7 m t / P b O 7 t 5 + 4 e C w r u N U M a y x W z s e 8 d c X J Z 4 7 g D 5 z P H 0 x P j T Q = < / l a t e x i t > FIG. 1.(a) The phase diagram of the non-Hermitian SSH model [Eq.(5)] as a function of w − v with fixed (u, v) = (0.5, 0.8).The upper panels are the corresponding real and imaginary parts of the bulk dispersions.(b) The real and imaginary parts of the energy dispersion in the open boundary condition.When w − v > u, there is a pair of edge modes with imaginary energies E edge = ±iu.

FIG. 2 .
FIG.2.The real part (a) and imaginary part (b) of the eigenvalues of the correlation matrix CA with (w − v, u) = (0.9, 0.5).There are two mid-gap states.(c) The wavefunction amplitude of ten localized states including two mid-gap states.The real part (d) and imaginary part (e) of the eigenvalues of the correlation matrix CA with (w − v, u) = (−0.9,0.5).There are four states with imaginary eigenvalues.(f) The wavefunction amplitude of ten localized states.The total system is two hundreds sites and n is the eigen-level index.
t e x i t s h a 1 _ b a s e 6 4 = " P 0 2 n Z E P w l l 9 e J e 1 a 1 b u o 1 h 4 u K 4 2 b P I 4 i n M A p n I M H V 9 C A O 2 h C C w i M 4 R l e 4 c 0 R z o v z 7 n w s W g t O P n M M f + B 8 / g A P U Y 2 X < / l a t e x i t > L = 160 < l a t e x i t s h a 1 _ b a s e 6 4 = " U Y t + c 7 h 9 e 8 n v M f b j o 7 O C U a k 5 6 y k FIG. 5. (a) Entanglement spectrum at the critical point which separates the topological PT symmetric phase and the PT broken phase.The parameters are (w, v, u) = (1.8,1.3.0,5).(b) The entanglement entropy as a function of the subsystem size lA with different total system size L. The scaling of the entanglement entropy satisfies SA(lA) = α(L) ln[sin( πl A L )] + const.with α(L) depending on the total system size.(c) The log-log plot of the α(L) as a function of 1/L.

FIG. 6 .
FIG. 6.(a) The boundary modes in the entanglement spectrum at the critical point w −v = u.(b) The effective central charge c eff = 3 * α after subtracting out the contributions of the boundary modes as a function of total system size L.Here α being the coefficient in the logarithmic scaling of the entanglement entropy, SA = α ln[sin[ πl A L ]] + const..The effective central charge c eff as a function of total system size L for only including the imaginary part of the eigenvalues of the boundary modes in the entanglement spectrum.

FIG. 7 .
FIG. 7. (a) The free-fermion version of symmetry enriched CFT, which has boundary modes in the critical chain as shown in left panel.After shift half of the unit-cell, it has the regular critical chain configuration and has no boundary modes (right panel).(b) The entanglement spectrum (left) and the entanglement entropy scaling (right) in the symmetry enriched CFT.There are two mid-gap states in the entanglement spectrum corresponding to two boundary modes in the entanglement Hamiltonian.The entanglement entropy scaling does not satisfy the logarithmic scaling.(c) The entanglement spectrum (left) and the entanglement entropy scaling (right) in the critical chain case.There are no mid-gap states in the entanglement spectrum and the entanglement entropy scaling gives the central charge c = 1.

2 and the central charge c = −12λ 2 + 12λ − 2 .
The effective central charge has the expression:
where | LA α and | RA α are left and right vectors span only in subsystem A. These left and right vectors are not yet biorthogonal but from Eq. (