Real space topological invariant and higher-order topological Anderson insulator in two-dimensional non-Hermitian systems
Hongfang Liu, Ji-Kun Zhou, Bing-Lan Wu, Zhi-Qiang Zhang, Hua Jiang
RReal space topological invariant and higher-order topological Anderson insulator intwo-dimensional non-Hermitian systems
Hongfang Liu, Ji-Kun Zhou, Bing-Lan Wu, Zhi-Qiang Zhang, ∗ and Hua Jiang
1, 2, † School of Physical Science and Technology, Soochow University, Suzhou, 215006, China Institute for Advanced Study, Soochow University, Suzhou 215006, China (Dated: February 25, 2021)We study the characterization and realization of higher-order topological Anderson insulator (HO-TAI) in non-Hermitian systems, where the non-Hermitian mechanism ensures extra symmetries aswell as gain and loss disorder. We illuminate that the quadrupole moment Q xy can be used as thereal space topological invariant of non-Hermitian higher-order topological insulator (HOTI). Basedon the biorthogonal bases and non-Hermitian symmetries, we prove that Q xy can be quantized to 0or 0 .
5. Considering the disorder effect, we find the disorder-induced phase transition from normalinsulator to non-Hermitian HOTAI. Furthermore, we elucidate that the real space topological in-variant Q xy is also applicable for systems with the non-Hermitian skin effect. Our work enlightensthe study of the combination of disorder and non-Hermitian HOTI. PACS numbers:
I. INTRODUCTION
With the development of topological theory of Her-mitian systems, many kinds of topological states havebeen proposed and realized , among which the non-Hermitian counterparts have attracted great atten-tion. Compared with the Hermitian cases, the non-Hermitian topological states are based on the non-Bloch theory due to the unique features of non-Hermitian Hamiltonian . Soon after the estab-lish of the non-Hermitian topological band theory, thereis a blooming investigations of disorder effect innon-Hermitian systems. By using the non-commutativegeometry method , the disorder-induced phasetransitions of non-Hermitian Chern insulator andnon-Hermitian Su-Schrieffer-Heeger model have beenstudied. The non-Hermitian topological Andersoninsulator is also reported.In these years, the Hermitian higher-order topologicalinsulator (HOTI) is one of the most focus of topolog-ical states. Not only the nested-Wilson-loop method but also the real-space topological invariant werereported to characterize such states. Specially, in two-dimensional systems, the quadrupole moment Q xy was proposed to be the real space topological invariantof HOTI. Initially, the quantization of this topological in-variant was thought to be protected by the point group inthe HOTI which is considered as the topological crystalinsulators . However, Li et al. showed that the quan-tized Q xy could also be protected by chiral symmetry orparticle-hole symmetry in Hermitian systems. Such re-sults still hold even with disorder effects. They also care-fully studied the disorder-induced phase transition in thecorresponding systems and predicted the existence of ahigher-order topological Anderson insulator (HOTAI).Very recently, the interplay of HOTI andnon-Hermitian is proposed and gains extensiveinterests . The topological invariants in momen-tum space to distinguish such topological phases are also investigated . However, the researches referredto real space topological invariant and disorder-inducedtopological phase transition of non-Hermitian HOTI areseldom reported. Since the Hamiltonian H (cid:54) = H † , thechiral symmetry or particle-hole symmetry in Hermitiancases will change into the distinctive symmetries innon-Hermitian cases. Moreover, disorders with gain andloss are also available for non-Hermitian samples. Thesefeatures should affect the disorder effect and the relatedphase transitions, which is unique for non-Hermitiansystems.In this paper, we propose that the quadrupole mo-ment Q xy , defined in the frame of the biorthogonalbases , can be considered as the real space topo-logical invariant of non-Hermitian HOTI. We prove that Q xy is quantized to 0 or 0 . H with − H † , whichis universal for non-Hermitian samples with line gapalong the real axis. Specifically, taking the pseudoanti-Hermiticity symmetry as an example, we demonstratethat the quantized Q xy = 0 . Q xy , the disorder-induced topological phase transition in non-HermitianHOTI is also studied. With the help of the pseudoanti-Hermiticity protected quantization of Q xy , we uncoverthat the non-Hermitian HOTI is robust against disor-der. More importantly, the non-Hermitian HOTAI ispredicted in such a system. Finally, we manifest that Q xy is also applicable for systems with non-Hermitianskin effect (NHSE) .This paper is organized as follows: In Sec. II, wepresent the method and the model. We demonstrate therealization of non-Hermitian HOTAI in Sec. III. Then, abrief discussion and summary are presented in Sec. IV. a r X i v : . [ c ond - m a t . d i s - nn ] F e b II. METHOD AND MODELA. Quantized quadrupole moment protected bypseudoanti-Hermiticity symmetry
We suppose a non-Hermitian Hamiltonian H holds thepseudoanti-Hermiticity symmetry η with η = η − and ηHη = − H † . Such symmetry extends the chiral symme-try in Hermitian cases . According to the properties ofHamiltonian in non-Hermitian systems, the eigenvaluesof H and H † are given as follows : H | ψ nr (cid:105) = E n | ψ nr (cid:105) , H † | ψ nl (cid:105) = E ∗ n | ψ nl (cid:105) , (1)where | ψ nr (cid:105) and | ψ nl (cid:105) are the n th standard right and lefteigenvectors, respectively. The spectrum of eigenvaluescan be separated into two parts with Re [ E ] > Re [ E ] <
0, which implies a line gap along Re [ E ]. Re [ E ]represents the real part of the eigenvalue E . We supposeall the occupied states with Re [ E ] < Re [ E ∗ ] <
0) con-struct a matrix U r ( U l ). On the other hand, the matrix V r ( V l ) constructed by the unoccupied states are shownin Fig. 1. Following the previous studies in Hermitiansystems , we define the quadrupole moment in non-Hermitian systems as : Q xy = 12 π Im { log[det( U † l ˆ QU r ) (cid:113) det ( ˆ Q † )] } , (2)with ˆ Q = exp [ i π ˆ x ˆ y/ ( N x N y )]. ˆ x ( N x ) and ˆ y ( N y ) arethe coordinate operator (sample size) along x and y di-rections, respectively. This definition makes use of thebiorthogonal bases , which is different from Her-mitian cases.Next, let us prove that the pseudoanti-Hermiticity [ H = − ηH † η , where the operatormatrix η is a real diagonal matrix with η = η − ]guarantees the quantization of the quadrupole moment Q xy . The quantization of Q xy is protected by thepseudoanti-Hermiticity instead of the chiral symmetryor particle-hole symmetry in Hermitian systems. Similarto the previous study , one obtainsdet( U † l ˆ QU r ) = det[ U † l ( ˆ Q − I + I ) U r ]= det[ I + U † l ( ˆ Q − I ) U r ]= det[ I + ( ˆ Q − I ) U r U † l ] , (3)by using the Sylvester’s determinant identity det( I +AB) = det( I + BA). Biorthogonality-normalization rela-tions indicate RL † = (cid:0) U r V r (cid:1) (cid:18) U † l V † l (cid:19) = I . (4)Thus, one has U r U † l = I − V r V † l . Noticing V † l V r = I , one Re(E) Im(E) En -En * U r V r Re(E) Im(E) -En En * U l V l unoccupiedoccupied pseudoanti-Hermiticity unoccupied occupied(0, 0) (0, 0) Hermitian ConjugationHermitian Conjugation (a): H (b): H † FIG. 1: (Color online). The relationship between n th eigen-values E n , E ∗ n , − E ∗ n , and − E n . The corresponding eigenvec-tors are | ψ nr (cid:105) , | ψ nl (cid:105) , η | ψ nl (cid:105) , and η | ψ nr (cid:105) . (a) E n and − E ∗ n arethe eigenvalues of H . (b) E ∗ n and − E n are the eigenvalues of H † . E n ( − E n ) and E ∗ n ( − E ∗ n ) are connected by the Hermi-tian conjugation. The Hermitian conjugation corresponds tothe transformation between Hamiltonian H and H † , in which | ψ nr (cid:105) will transform into | ψ nl (cid:105) . E n ( E ∗ n ) and − E n ( − E ∗ n ) arecorrelated by pseudoanti-Hermiticity. We define the occupied( Re [ E ] <
0) and unoccupied ( Re [ E ] >
0) conditions with thehelp of line gap along axis Re [ E ]. The matrices constructedby the occupied (unoccupied) standard right and left eigen-vectors are marked as U r ( V r ) and U l ( V l ), respectively. obtains:det( U † l ˆ QU r ) = det[ I + ( ˆ Q − I )( I − V r V † l )]= det[ ˆ Q − ( ˆ Q − I ) V r V † l ]= det[ I + ( ˆ Q † − I ) V r V † l ] det( ˆ Q )= det( V † l ˆ Q † V r ) det( ˆ Q ) . (5)After considering the pseudoanti-Hermiticity ηHη = − H † and Eq. (1), one has − H † [ η | ψ nr (cid:105) ] = E n [ η | ψ nr (cid:105) ] , − H [ η | ψ nl (cid:105) ] = E ∗ n [ η | ψ nl (cid:105) ] . (6)Comparing Eq. (6) with Eq. (1), one can find that η | ψ nr (cid:105) and η | ψ nl (cid:105) are the standard left and right eigenvectorswith the corresponding eigenvalues − E n and − E ∗ n , re-spectively. That is to say, if E n ( E ∗ n ) is the eigenvalueof H ( H † ), η ensures that it has a corresponding part-ner − E n ( − E ∗ n ), which is the eigenvalue of H † ( H ). Therelationships between different eigenvalues and eigenvec-tors of H and H † are illustrated in Fig. 1. We have topoint out that the Hermitian conjugation transformationused in this paper corresponds to the transformation be-tween Hamiltonian H and H † . Thus, | ψ nr (cid:105) ( η | ψ nl (cid:105) ) willtransform into | ψ nl (cid:105) ( η | ψ nr (cid:105) ) under such a transformation,which is called Hermitian conjugation only for simplicity.Provided that Re [ E n ] <
0, the following relations | ψ nr (cid:105) ∈ U r , | ψ nl (cid:105) ∈ U l , η | ψ nr (cid:105) ∈ V l , and η | ψ nl (cid:105) ∈ V r can beobtained. These relations manifest that ηU r = V l as wellas ηU l = V r , and Eq. (5) can be rewritten as:det( U † l ˆ QU r ) = det( V † l ˆ Q † V r ) det( ˆ Q )= det( U † r η ˆ Q † ηU l ) det( ˆ Q ) . (7) gain: +i γ loss: -i γ π J t (b) J γ theory non-trivial trivial Qxy (a)
FIG. 2: (Color online). (a) Schematic diagram of the tight-binding model. The blue square (pink circle) represents thesites with loss − iγ (gain iγ ). The green and black solid linesshow the nearest hopping with strength t and J , respectively.The dashed box is the primitive cell. π flux is also considered.(b) The quadrupole moment Q xy versus J and γ . The redsolid line is obtained based on the theoretical analysis. Inall our calculation of Q xy , the periodic boundary condition isadopted with sample size N = 40. Since η is a real diagonal matrix and ˆ Q is a unitary-diagonal matrix, one obtaines [ η, ˆ Q ] = 0 and ˆ Q † ˆ Q = I .Thus,det( U † l ˆ QU r ) (cid:113) det( ˆ Q † ) = det( U † r ˆ Q † U l ) (cid:113) det( ˆ Q )= [det( U † l ˆ QU r ) (cid:113) det( ˆ Q † ) ] † . (8)Therefore, det( U † l ˆ QU r ) (cid:113) det( ˆ Q † ) is a real number, and Q xy should be quantized to 0 or 0 . B. Model
We consider a square-lattice model protected by thepseudoanti-Hermiticity as an example. The primitive cellcontains 16 sites, as shown in Fig. 2(a). It holds thepseudoanti-Hermiticity η , and the Hamiltonian reads : H = (cid:88) n i ( γ + ε n ) η n a † n a n + (cid:88) (cid:104) nm (cid:105) t nm e iφ nm a † n a m (9)where t nm ( J or t ) represents the nearest-neighbor hop-ping strength between sites n and m with the lattice con-stant a = 1. J ( t ) corresponds to the black (green)solid lines in Fig. 2(a). η n represents the diagonal el-ement η ( n, n ) with the operator matrix of pseudoanti-Hermiticity η = σ z σ z σ z σ z . a † n ( a n ) is the creation (an-nihilation) operator of site n . The magnetic flux isdetermined by φ nm = e (cid:126) (cid:82) A · dl with the vector poten-tial A = ( − By, , dl is the vector between n and m sites. We fix φ = a Be/ (cid:126) = π and t = 1 for simplicity. γ denotes gain or loss strength marked in different colors,which is illustrated in Fig. 2(a).To maintain the pseudoanti-Hermiticity symmetry, thegain and loss disorder in one primitive cell is equivalentwith ε n = ε . For different primitive cells, ε is determinedby Anderson disorder with uniformly distributed ε ∈ [ − W/ , W/ W is the disorder strength. Notethat the system is set under half-filling conditions, i.e,the Fermi energy equals zero hereafter. III. NUMERICAL RESULTS
This section illuminates that the real space topologicalinvariant Q xy is a powerful tool to identify the higher-order topological nontrivial/trivial phases even when thenon-Hermitian appears. We also investigate the disordereffect in the corresponding non-Hermitian systems. A. Q xy as an indicator of non-Hermitian HOTI Based on the quadrupole moment Q xy calculation un-der the periodic boundary condition, we first obtain aphase diagram of non-Hermitian Hamiltonian in Eq. (9)with W = 0. As plotted in Fig. 2(b), Q xy in the green re-gion equals to 0 . Q xy clearly showsthe phase boundary shift between topological nontrivialand trivial phases by varying non-Hermitian strength γ .The red square solid line corresponds to the theoreticallypredicted phase boundary with γ = 2( J − ). Specif-ically, one can easily find that these two-phase boundariesobtained by different methods coincide perfectly.To better understand the nontrivial phase mentionedabove, we analyze Re [ E ] (blue) and Im [ E ] (red) ver-sus J under the periodic boundary condition as well asthe open one for γ = 1. Theoretically, the bulk gapcloses and reopens at J = (cid:112) / ≈ .
225 for γ = 1. Itconsists with the plot of Re [ E ] versus J under the pe-riodic boundary condition (see Fig. 3(a)). Meanwhile, Q xy versus J curve (see Fig. 3(b)) shows that Q xy jumps J I m [ E ] / R e [ E ] -505 J J Q xy (a) (b)(c) periodic boundaryopen boundary Re[E]Im[E] I m [ E ] / R e [ E ] (d) y x FIG. 3: (Color online). (a) The plot of the real Re [ E ] (im-age Im [ E ]) part of the eigenvalue E versus J , marked in blue(red). (b) Q xy versus J . (c) The same with (a), except theopen boundary is adopted. (d) A typical plot of the wave-function, which is marked by black square shown in (c). Wefix γ = 1 in our calculation. from 0 . J ≈ (cid:112) /
2, which signals that the bulkgap closing induces a topological phase transition. Inorder to clarify that Q xy = 0 . Re [ E ] (blue) and Im [ E ] (red) versus J under the open boundary condi-tion. Fig. 3(c) suggests that the zero-energy modes ap-pear when J < (cid:112) /
2, and the fourfold-degeneracy cornerstates emerge [see Fig. 3(d)]. The corner states’ energydeviation from Re [ E ] = 0 for J ≈ (cid:112) / J < (cid:112) /
2, and Q xy clearly distinguishes sucha phase from the trivial one.To sum up, we show that the pseudoanti-Hermiticitysymmetry protected quadrupole moment Q xy (definedas Eq. (2)) can be utilized to distinguish non-HermitianHOTI. B. Non-Hermitian HOTAI
Subsequently, by considering disorder with pseudoanti-Hermiticity symmetry, we investigate the disorder-induced phase transition in non-Hermitian systems. No-tably, we examine the existence of the non-HermitianHOTAI.The evolution of Q xy by increasing disorder strength W with J = 0 . W = 0, the system is non-Hermitian HOTI with Q xy = 0 .
5. Unless otherwise specified, the disorder innumerical calculations holds the pseudoanti-Hermiticitysymmetry η . We find that the average quadrupole mo-ment Q xy (solid blue line) is quantized to 0 . W < Q xy .Next, we investigate the disorder-induced topologicalphase transition for J = 1 .
26, as shown in Figs. 4(d)-(f).The sample is normal insulator (NI) with Q xy = 0 andwithout the zero-energy states under clean limit since1 . > (cid:112) /
2. With the increasing of disorder strength W , the average quadrupole moment Q xy jumps from 0 to0 . Q xy = 0 . W ≈ .
5. Taking W = 2 . Q xy = 0 . W Q xy x y number
10 20 30 40-1-0.500.51 W=2 (a) (b) (c)
10 20 30 40 〈 R e [ E ] 〉 -0.3-0.10.10.3 y xW Q xy number (d) (e) (f) W=2.5 〈 R e [ E ] 〉 〈 | w a v e f un c ti on | 〉〈 | w a v e f un c ti on | 〉 FIG. 4: (Color online). (a) The evolution of quadrupole mo-ment Q xy by increasing disorder strength W with J = 0 . Q xy for each disordered sample, and theblue solid line is the average. (b) and (c) are the averageenergy Re [ E ] and wavefunction [red dots marked in (b)] for W = 2. (d)-(f) is similar to (a)-(c) except J = 1 .
26 and W = 2 . Q xy is calculated under periodic boundary condi-tion. Only several eigenvalues close to Re [ E ] = 0 are plottedin (b) and (e). We also notice that Q xy cannot be quantized understrong disorder W > Q xy for each sample seems random, and theaverage quadrupole moment approximately equals 0.25.This behavior is also different from that in HermitianHOTAI cases , where topological invariant Q xy de-creases to zero under strong disorder. In order to explainthe unquantized Q xy and further ensure the existenceof the non-Hermitian HOTAI, we plot the evolution of (cid:104) Re [ E ] (cid:105) with the increase of W under the periodic bound-ary condition [see Fig. 5(a) and (b)]. Here, (cid:104) Re [ E ] (cid:105) is theaverage of Re [ E ].For J = 0 . W [see Fig. 5(a)]. Q xy is unquantized when W >
3. Similarly, as shown in Fig. 5(b), we considertrivial case with J = 1 .
26. The bulk gap closes andreopens with an increase of disorder strength. Mean-while, Q xy jumps from 0 to 0 .
5, which strongly sug-gests the occurrence of the phase transition from NI tonon-Hermitian HOTAI. Continuing to increase disorderstrength for
W >
3, the bulk gap gradually reduces andcloses. The quadrupole moment Q xy deviates from thequantized value accordingly.In order to better understand the deviation of Q xy from the quantized value, we concentrate on the average Re [ E ] as well as the average wavefunction with W = 4 . J = 1 .
26, the bulk gap disappears, and the av-erage wavefunction extends into the bulk [see Figs. 5(c)-(d)]. Further, a series of (cid:104) Re [ E ] (cid:105) = 0 states appear,which are named as the gapless phase . The existenceof states with (cid:104) Re [ E ] (cid:105) = 0 for the gapless phase is reason-able. Taking H = (cid:20) i ( γ + W ) tt − i ( γ + W ) (cid:21) as an intu- (c)
50 100 150 200-0.4-0.200.20.4 number
W=4.5 〈 R e [ E ] 〉 -3 y x 〈 | w a v e f un c ti on | 〉 〈 R e [ E ] 〉 -1.501.5 Q xy -0.500.50 2 4 6 W/t 〈 R e [ E ] 〉 -1.501.5 Q xy -0.500.50 2 4 6 W/t (a) (b)(d)
FIG. 5: (Color online). (a) The evolution of Re [ E ] with theincrease of W . The red line shows Q xy for each disorderedsample. Here, J = 0 .
5. (b) is similar to (a) with J = 1 .
26. (c)shows Re [ E ] with disorder strength W = 4 .
5. Only severaleigenvalues close to Re [ E ] = 0 are plotted. (d) is the averagewavefunction of the red dots in (c). (a) and (b) are calculatedunder periodic boundary condition with sample size N = 40.(c)-(d) are calculated under open boundary condition with J = 1 .
26 and N = 120. itive example, its eigenvalues are E = ± (cid:112) t − ( γ + W ) .For fixed t and γ , (cid:104) Re [ E ] (cid:105) tends to zero when W is strongenough. For such a case, we have to emphasize thatthe half-filling condition is not well-defined because ofthe existence of states with Re [ E ] = 0 [see Fig. 5(c)].However, the accuracy limitation in the numerical calcu-lation slightly lifts these zero modes’ degeneracy, whichleads to an incorrect half-filling condition. Thus, the spe-cific transformation between the occupied ( U r , U l ) andunoccupied states ( V r , V l ) cannot hold [see Fig. 1]. Theunquantized Q xy can be considered as a signal for theexistence of the gapless phase.These findings indicate that the quantized Q xy requiresnot only symmetry but also the existence of bulk gapalong Re [ E ]. When such a bulk gap disappears, thequantization of Q xy is not available. Significantly, theabsence of bulk gap distinguishes the zero-energy modesin Fig. 5(c) from the non-Hermitian HOTAI shown inFig. 4(e). Moreover, the evolution of eigenvalues versus W also strengthens the existence of non-Hermitian HO-TAI shown in Figs. 4(d)-(f). IV. SUMMARY AND DISCUSSION
In summary, based on the real space topological invari-ant [quadrupole moment in the frame of non-HermitianHamiltonian], we studied the disorder-induced HOTAIin non-Hermitian systems, which is protected by thepseudoanti-Hermiticity. First, we demonstrated that thequadrupole moment Q xy can be quantized to 0 or 0 . Q xy can be utilized to distin-guish non-Hermitian HOTI ( Q xy = 0 .
5) from the trivialones ( Q xy (cid:54) = 0 . Q xy , we un-covered the disorder-induced phase transition in such anon-Hermitian system and demonstrated the existence ofnon-Hermitian HOTAI.The real space topological invariant Q xy is a pow-erful tool to characterize non-Hermitian HOTI. In theappendix, we give another non-Hermitian HOTI with-out pseudoanti-Hermiticity symmetry and with NHSEto further declare the scope of application of Q xy . Weshow that the quantized Q xy is also available for sampleswith sublattice symmetry combined with a non-unitarytransformation. Besides, NHSE has no specific influenceon the topological invariant Q xy , except by consider-ing the non-Bloch band theory . Theoretically, thechiral symmetry and particle-hole symmetry in Altland-Zirnbauer classification for Hermitian cases should be re-placed by specific symmetries in 38-fold classification fornon-Hermitian cases . In addition to the pseudoanti-Hermiticity and sublattice symmetry reported in this pa-per, other symmetries in non-Hermitian systems may alsobe able to be utilized to ensure the quantization of Q xy .Further, the line gap along Im [ E ] will also influence thecharacterization of Q xy . A much careful investigationcan complete the understanding of topological featuresof non-Hermitian systems. acknowledgement We are grateful to Yue-Ran Ding, Zibo Wang, MingGong, and especially Qiang Wei for insightful discussion.This work is financially supported by the National Ba-sic Research Program of China No. 2019YFA0308403,National Natural Science Foundation of China No.11822407, and a Project Funded by the Priority Aca-demic Program Development of Jiangsu Higher Educa-tion Institutions.
Appendix A: Influence of NHSE and the sublatticesymmetry protected quadrupole moment innon-Hermitian systems
In the appendix, we investigate applicability of thequadrupole moment Q xy for other symmetries, and theNHSE is also considered. As an example, we focus onthe sublattice symmetry protected non-Hermitian HOTIwith NHSE. Although sublattice symmetry itself can-not ensure the quantization of Q xy , we prove that thereis another transformation, which connects H to H † .The combination of these two transformations is simi-lar to pseudoanti-Hermiticity, which connects H to − H † .Therefore, the quantization of Q xy is still available evenin the presence of disorder. Significantly, the quadrupolemoment Q xy defined in real space is also applicable forsystems with NHSE.
1. Quantization of Q xy for systems with sublatticesymmetry and NHSE Following the previous study, the Hamiltonian for non-Hermitian HOTI with NHSE reads : H ( k ) = H + H ( k x ) + H ( k y )= t ( τ x σ + τ y σ y ) + iγ ( τ y σ x − τ y σ z )+ λ cos ( k x ) τ x σ − λ sin ( k x ) τ y σ z + λ cos ( k y ) τ y σ y + λ sin ( k y ) τ y σ x , (A1)where τ x/y/z and σ x/y/z are the Pauli matrices. τ and σ are the identity matrices. γ denotes the non-Hermitianstrength. For simplicity, we fix γ = 0 . H representsthe intra primitive cell hopping matrix. H ( k x ) and H ( k y )show the hopping matrices between the nearest neighborprimitive cells along x and y directions, respectively.Eq. (A1) has the sublattice symmetry SH ( k ) S = − H ( k ) with S = τ z σ . Sublattice symmetry operator S connects U l/r to V l/r and vice versa. Eq. (5) impliesthat S itself cannot ensure the quantization of Q xy since U † l U l (cid:54) = I in non-Hermitian systems . The quantizationof Q xy demands another matrix A which connects H with H † . In addition, A has to commutate with ˆ Q and satis-fies A † = A . Moreover, the unbalanced hoppings alongboth x and y directions suggest the existence of NHSE.In terms of two requirements above, we prove that thereis a matrix, which transforms H to H † after eliminatingthe NHSE. Thus, the quantized Q xy is ensured.In the following, we detail the derivation. The matrixforms of H , H ( k x ), and H ( k y ) are: H ≡ T = t − γ − t + γ t + γ t + γt + γ t − γ − t − γ t − γ ,H ( k x ) = T x e ik x + T † x e − ik x ,H ( k y ) = T y e ik y + T † y e − ik y , (A2)with T x = λ
00 0 0 00 0 0 00 λ and T y = − λ λ . T † x/y isthe hermitian conjugation of T x/y . Due to the NHSE, thecorrect phase diagram is available by combining the non-Bloch band theory under periodic boundary condition.The replacement of momentum k x/y → ˜ k x/y + ik leadsto H (cid:16) ˜ k x (cid:17) = βT x e i ˜ k x + β − T † x e − i ˜ k x ,H (cid:16) ˜ k y (cid:17) = βT y e i ˜ k y + β − T † y e − i ˜ k y ,H † (cid:16) ˜ k x (cid:17) = βT † x e − i ˜ k x + β − T x e i ˜ k x ,H † (cid:16) ˜ k y (cid:17) = βT † y e − i ˜ k y + β − T y e i ˜ k y , (A3) where β = e − k = (cid:112) | ( t − γ ) / ( t + γ ) | . Such replacementleaves H unchanged.Matrix A should connect H (˜ k ) with H † (˜ k ). Thus: A = t + γt − γ t − γt + γ . (A4)It transforms H , H (cid:16) ˜ k x (cid:17) and H (cid:16) ˜ k y (cid:17) as follows: AH A − = H † ,AH (cid:16) ˜ k x (cid:17) A − = a βT x e i ˜ k x + a β − T † x e − i ˜ k x ,AH (cid:16) ˜ k y (cid:17) A − = a βT y e i ˜ k y + a β − T † y e − i ˜ k y , (A5)with a = ( t + γ ) / ( t − γ ), and a = ( t − γ ) / ( t + γ ). Since β = (cid:112) | ( t − γ ) / ( t + γ ) | , a β and a β − have two condi-tions to eliminate the absolute value sign of β . Two casesare analyzed as below: case 1: For t − γt + γ > β = (cid:113) | t − γt + γ | = (cid:113) t − γt + γ . The multi-plication is available a β = β − , a β − = β. (A6)It leads to AH (cid:16) ˜ k x , ˜ k y (cid:17) A − = H † (cid:16) ˜ k x , ˜ k y (cid:17) . (A7) case 2: For t − γt + γ < β = (cid:113) | t − γt + γ | = (cid:113) γ − tt + γ . Thus, a β = − β − , a β − = − β. (A8)One obtains AH (cid:16) ˜ k x , ˜ k y (cid:17) A − = H † (cid:16) ˜ k x + π, ˜ k y + π (cid:17) . (A9)To make it more intuitive, let’s analyze the transforma-tion of H (cid:16) ˜ k x , ˜ k y (cid:17) in the real space for a square samplewith sample size N × N . We suppose there is a diagonalmatrix A R = diag ( diag ( A ) , diag ( A ) , · · · , diag ( A i ) , · · · ) , (A10)with A i = diag ( diag ( A i )). diag represents the diagonal. A i is the construction block of A R , which satisfies A i = ± A . To ensure A R HA − R = H † , the following relationsshould hold A i T A − i = T † ; A i βT x A − j = β − T x , A i βT y A − j = β − T y ; A i β − T † x A − j = βT † x , A i β − T † y A − j = βT † y ; (A11)where the corresponding hopping matrix between i and j primitive cells is clarified in the equation. For all the i ∈ [1 , N ], A i = A when t − γt + γ >
0. It ensures A R HA − R = H † .When t − γt + γ <
0, we assume A i ∈ odd = − A j ∈ even with i, j ∈ [1 , N ] + kN and k ∈ [0 , N − i ∈ [1 , N ] + kN and j ∈ [ N + 1 , N ] + kN with k ∈ [0 , N − A i = − A j . The above limits also preserve A R HA − R = H † . Furthermore, we notice that A R is a diagonal matrixas expected for these two cases, and it commutates withˆ Q . Significantly, A † R = A R is also satisfied.Again, we take Eq. (1) as an example and suppose Re ( E n ) < A R satisfies A R HA − R = H † , which means: H † [ A − R | ψ nr (cid:105) ] = E n [ A − R | ψ nr (cid:105) ] ,H [ A R | ψ nl (cid:105) ] = E ∗ n [ A R | ψ nl (cid:105) ] . (A12)one obtains A − R | ψ nr (cid:105) ∈ U l and A R | ψ nl (cid:105) ∈ U r . Noticingthat the replacement k → ˜ k + ik maintains the sublatticesymmetry S , the relationship SHS = − H still holds with S = S − . Thus: − H [ S | ψ nr (cid:105) ] = E n [ S | ψ nr (cid:105) ] , − H † [ S | ψ nl (cid:105) ] = E ∗ n [ S | ψ nl (cid:105) ] . (A13)It is obvious that S | ψ nr (cid:105) ∈ V r and S | ψ nl (cid:105) ∈ V l .The corresponding relations between different statesare shown in Fig. A1(a) and (b). For state | ψ nr (cid:105) with eigenvalue E n , A R will transform | ψ nr (cid:105) ∈ U r to A − R | ψ nr (cid:105) ∈ U l with E n unchanged [the black solid linein Fig. A1(a) and (b)]. Then, the sublattice symmetry S will transform A − R | ψ nr (cid:105) ∈ U l to SA − R | ψ nr (cid:105) ∈ V l with E n → − E n [the cyan solid line]. The combination of S and A R will connect | ψ nr (cid:105) ∈ U r to SA − R | ψ nr (cid:105) ∈ V l [thepink solid line], which means SA − R connects H to − H † .We emphasize that the Hermitian-conjugation corre-lates E n (eigenvalue of H ) with E ∗ n (eigenvalue of H † ),which is different from the transformation A R . A R cor-relates E n (eigenvalue of H ) with E n (eigenvalue of H † ).It means that the n th eigenvalue of H † ( E n ) should be areal number. Otherwise, E n must have a complex conju-gate partner E ∗ n , as shown in Fig. A1(b).To sum up, we obtain the following transformations: (cid:26) SHS = − HA R HA − R = H † ⇒ V l/r = SU l/r U l = A − R U r U r = A R U l . (A14)For non-Hermitian systems with sublattice symmetry,the quantization of Q xy is not guaranteed since V l/r = SU l/r cannot ensure that det( U † r ˆ Q † U l ) (cid:113) det( ˆ Q ) is a realnumber. However, one can prove the quantization of Q xy is confirmed after combining S and A R . Following -1 0 1-2-1012 -1 0 1-2-1012 -1 0 1-2-1012 Qxy t t t λ λ λ theory (c) W=0 (d) W=1 (e) W=1 with sublattic symmetry without sublattic symmetrywithout disorder
Re(E) Im(E) En U r V r Re(E) Im(E) -En En U l V l unoccupiedoccupied SA R unoccupied occupied (0, 0) (0, 0) (a): H (b): H † A R S -EnHermitian Conjugation En* S FIG. A1: (Color online). (a) and (b) are similar to those inFig. 1, except that the transformation operator is changed. S corresponds to the sublattice symmetry. A R is a matrixdefined in the appendix. (c)-(e) The phase diagram of non-Hermitian Hamiltonian Eq. (A1) with sublattice symmetry.The quadrupole moment Q xy versus t and λ . The blue re-gions represent the topologically trivial phase with Q xy = 0,while the green regions represent the non-Hermitian HOTIwith Q xy = 0 .
5. The middle regions insides the ellipse rep-resents the gapless states [ Q xy lies between 0 and 0.5]. Thephase boundaries (red line) are determined by t = γ + λ and t = γ − λ with γ = 0 .
4. (c) with sublattice symmetryand without disorder W = 0. (d) with sublattice symmetryand disorder strength W = 1. (e) is similar to (d), exceptthat the sublattice symmetry broken disorder is considered. Eq. (5), we have:det( U † l ˆ QU r ) = det( V † l ˆ QV r ) det( ˆ Q )= det( U † l S † ˆ Q † SU r ) det( ˆ Q )= det( U † l ˆ Q † U r ) det( ˆ Q )= det( U † r A − † R ˆ Q † A R U l ) det( ˆ Q )= det( U † r A − R ˆ Q † A R U l ) det( ˆ Q )= det( U † r ˆ Q † U l ) det( ˆ Q ) , (A15)which meansdet( U † l ˆ QU r ) (cid:113) det( ˆ Q † ) = [det( U † l ˆ QU r ) (cid:113) det( ˆ Q † ) ] † . (A16)Thus, similar as Eq. (8), the quantization of Q xy to 0 or0 .
2. Numerical results
We check that the quantized Q xy can be used to predicta correct phase diagram of such a non-Hermitian HOTI.The plots of Q xy versus γ and t are given in Fig. A1.To eliminate the influence of NHSE, the numerical cal-culation is based on H (˜ k x , ˜ k y ) instead of H ( k x , k y ). Theperiodic boundary condition is also adopted.When the disorder is absent, the quantized Q xy per-fectly captures the existence of non-Hermitian HOTI. Asshown in Fig. A1(c), the solid red line is the theoreticallypredicted phase boundary. The regions with Q xy = 0 . Q xy shows different behaviors for different parameter re-gions. When t > γ + λ , one has Q xy = 0, which isa NI. However, Q xy is unquantized when t < γ − λ ,and it seems to contradict with the symmetry-protected quantization of Q xy . Actually, it corresponds to the gap-less phase.Next, we study the influence of disorder effect on thequantization of Q xy . 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