Topological quantum phase transition in a non-Hermitian system

Exceptional point (EP) associated with eigenstates coalescence in non-Hermitian systems has many exotic features. In general, EPs are sensitive to the system parameters. Here we show topologically stable EPs in two-dimensional non-Hermitian systems; through mapping the average values of Pauli matrices under the eigenstate of system Bloch Hamiltonian to a real planar vector field, we find that EPs are the topological defects of the vector field and protected by chiral symmetry. Topological quantum phase transitions occur at merging or splitting of EPs. Different EP configurations relate to different topological phases that characterized by two types of vortices. The corresponding winding numbers are topological invariants, being zero or one-half, and reflect the topological charges carried by the EPs and the topological features of two-band coalescences.

In this Letter, we show that topological nature of band degeneracies is determined by the topological defects of a real vector field F(k) mapped from the Bloch Hamiltonian.F(k) is composed by the average values of Pauli matrices in non-Hermitian systems in contrast to the effective magnetic field in Hermitian systems.The topo-logical defects of F(k) determine the EPs in the Brillouin zone.EPs are protected by system symmetries and their configurations characterize different topological phases.DPs split to pairs of EPs when non-Hermiticity is introduced; EPs inherit half-integer vortices from their parent DPs; topological QPTs occur associated with the creation of new configurations when EPs (DPs) merge or split.Topology of EPs (DPs) is characterized through topological invariant: the winding numbers 0 or ±1/2 (0 or ±1).Our findings elucidated in the semi-Dirac systems are revealed in a bilayer square lattice.
Topological QPTs in semi-Dirac systems.-Weconsider a two-dimensional non-Hermitian semi-Dirac Hamiltonian [78][79][80][81][82] that defined in the momentum space H(k) = B(k) • σ describes a spin in an effective twocomponent complex magnetic field B(k) = (B x , B y , 0), where σ = (σ x , σ y , σ z ) is a vector of Pauli matrices.H(k) can be retrieved via a bilayer square lattice with diagonal couplings and staggered losses.V and Γ are real-valued parameters; nonzero Γ introduces the non-Hermiticity.H(k) possesses a chiral symmetry due to {H(k), σ z } = 0 [83]; the band degeneracy occurs at zero energy when where EPs occur.We illustrate the topological phases and the topological QPTs at four Since H(k) includes only two Pauli matrices and commutes with σ z , we employ a planar vector field F(k) = (F x , F y ) to characterize the topology of H(k).The zero energy band degeneracy E ± = ± B 2 x + B 2 y = 0 implies that the expectation value of H(k) under its eigenstate is zero, i.e., H(k) = B x σ x + B y σ y = 0; therefore, B x = B y = 0 ( σ x = σ y = 0) at DPs (EPs) in Hermitian (non-Hermitian) system.The expectation values of the two Pauli matrices compose the planar vector field for non-Hermitian system instead of F(k) = (B x , B y ) for Hermitian system [65,66].EPs are protected by symmetry, exhibiting similar behaviors as DPs in Hermitian lattices and being topologically stable.Moreover, EPs are the topological defects of the manifold F(k), where F(k) approaches zero with vortex or antivortex structures.
A topological invariant characterizes the topological property of F(k) where Fx(y) = F x(y) / F 2 x + F 2 y and ∇ = ∂/∂k.The integral is performed in the k x -k y plane along a closed loop C; 2πw I accounts the varying direction of vector field F(k).The (both eigenstates yielding identical) winding number w I describes the vortices with topological charges at EPs and is in parallel to the winding number that originated from the generalized Berry phase [28-31, 45, 72, 77], which characterizes the topology of EPs on another aspect (cf.Discussion).EPs move along k x axis as V and Γ vary.The EP configurations of w I reflect four distinct topological phases [84]: (i) Semi-Dirac point: at Γ = V = 0, H(k) is Hermitian and the BTP is a semi-Dirac point at (k cx , k cy ) = (0, 0) [Fig.1(a)] with anisotropic dispersion of linear and quadratic dispersions along two orthogonal directions [78][79][80][81][82].The semi-Dirac point has winding number w I = 0 and splits into two Dirac points (w I = ±1) for nonzero V without breaking any symmetries [85].
The configuration of BTPs identified by the winding numbers characterizes different topological phases; the configuration only changes at topological QPTs of the creation, annihilation, or split of vortex and antivortex.
Topological QPTs in a bilayer square lattice.-Weconsider a tight-binding bilayer square lattice [Fig.2(a)], typically describing the dissipative ultracold atomic gas in optical lattices [101][102][103][104][105][106].The topological features and topological QPTs in the semi-Dirac system are all revealed in the non-Hermitian bilayer square lattice.The Hamiltonian has the form H = 2 λ=1 H λ + H T , where the intralayer term is and the interlayer term is H T = T |1, j, l 2, j, l| + h.c.; λ = 1 (2) is the index that labels the upper (lower) layer, and (j, l) is the in-plane site index; J and T denote the intralayer and interlayer hoppings; t and γ are the diagonal couplings and staggered losses.Different sublattices have different losses γ A and γ B due to the environment interactions.After a removal of universal loss (γ A + γ B )/2, the losses are equivalently described by the balanced gain and loss γ = (γ A − γ B )/2; however, H lacks the paritytime or the charge-conjugation symmetry [42].Discrete symmetries including inversion symmetry, time-reversal symmetry, and particle-hole symmetry usually play a crucial role in characterizing topological phases [107].Here t breaks the inversion symmetry and γ breaks the time-reversal symmetry of the bilayer square lattice, but H has a chiral symmetry (χHχ Twenty-six topological phases exhibit five types of winding number distributions of BTPs as listed in Table I.
or inversion symmetry ensures the Bloch Hamiltonian including only two Pauli matrices [65].
The translational symmetry ensures H is invariant under arbitrary two combinations of the mirror reflection, the translation, the 90 degrees rotation, and the layer interchange.Symmetries protect the Bloch Hamiltonian h(k x , k y ), being invariant under substitution (k x , k y ) → (±k x(y) , ±k y(x) ) [84], and also protect the number of BTPs (DPs and EPs) with vortex structures.The BTPs can move but cannot be removed as system parameters varying except when they merge or split associated with a topological QPT and the creation of a new BTP configuration.The BTP configuration is a topological invariant to characterize the topological phases of bilayer square lattice, while the phase diagram is depicted in Fig. 2(b).
The system is Hermitian at γ = 0, being time-reversal symmetric without inversion symmetry (t = 0).The time-reversal symmetry requires DPs at ±k with identical winding number [62,93].Eight DPs may exist due to the symmetry protection.They locate and move along the k x(y) = π/2 lines; their locations satisfy a fourfold rotational symmetry (C 4 symmetry) with respect to the axis that perpendicular to the k x -k y plane, being mirrorreflection-symmetric about k x(y) = 0 and k x = ±k y [98].Eight DPs merge into four at |T | = 0 and 2J [62,92].Top panel of Fig. 3 shows the DP configurations.
In the presence of non-Hermiticity, each DP splits into two normal EPs and the corresponding Bloch Hamiltonian h(k x , k y ) is defective [89][90][91], generating sixteen EPs at maximum.h(k x , k y ) has chiral symmetry, which requires the EPs (E ′ ± = 0) on the lines of k x(y) = ±π/2 if t = 0.Although EPs are independent of t, nonzero t is crucial for the existence of isolated EPs at For special system parameters, the hybrid EPs appear at high symmetric points in the Brillouin zone, where normal EPs merge and the number of EPs reduces.
w I characterizes the topological features of BTPs as the topological defects of planar vector field F ′ (k).The value of w I is given in Fig. 3(a).Distinct topological phases are created when BTPs merge or split.Topological phases being reflection symmetric about γ = 0 has identical w I configurations, exhibiting seventeen configurations of w I [Fig.3(a)].The vortex is revealed from F ′ (k) in Fig. 4(a) at γ = J/2, T = −3J/2 for a certain w I configuration.
In Fig. 3 Discussion.-Anotherwinding number is defined as with E(k) = (E x , E y ) being the band energy E in the complex plane, and Êx =Re(E)/|E|, Êy =Im(E)/|E|.Both two bands yield identical w II due to chiral symmetry.w II differs from w I and characterizes the topology of band coalescence [45,52]; w II reflects the chirality of normal EPs [84] that associated with the coalescence eigenstate [109], the generalized Berry phase in non-Hermitian system [28][29][30], and substantially the topological structure of complex band Riemann surface [23][24][25].Each of the two bands accumulates Berry phase ±π when encircling EPs for two circles; the ± sign in front determines the opposite chiralities of EPs [14,15,21,28,30] as a result of the square-root type Riemann sheet.The merge of EPs with opposite (identical) chiralities creates DP or hybrid EP [31,52].The winding number is w II = ±1/2 (0) for normal EP (DP and hybrid EP).
All the DP and EP configurations of w II are shown in Fig. 3(b).The winding numbers w I and w II capture two topological aspects of non-Hermitian systems.w I is equivalent to the winding number originated from the non-Hermitian generalization of Berry phase that characterizing the varying direction of the effective magnetic field [45] and the Chern number [52,77].w II reveals the vorticity of Riemann sheet (energy band), which coincides shown by the white arrows on the top, and the background density plot is with Refs.[45,52] and the generalized Berry phase for two circles of EP encircling in the parameter space [77].Ultracold atomic gas in optical lattices [101][102][103][104][105][106][110][111][112][113][114][115][116][117], photonic crystals [118,119], and coupled resonators [45,[120][121][122][123][124] provide versatile optical platforms for the study of topological physics.The fine tuned system parameters are particularly beneficial for the realization of topological systems that are hard to be realized in condensed matter physics.Trapping cold 6 Li or 40 K atoms in a dissipative spin-dependent optical lattice is a candidate for realizing the bilayer square lattice [65,77,101,102,105], and next-nearest-neighbour (diagonal) coupling is introduced by applying additional laser beams.Optical lattice of coupled resonators (waveguides) is another candidate [45,[120][121][122][123][124].The π phase difference between the diagonal couplings t of sublattices A and B is controlled by the optical path length of auxiliary linking resonators (waveguides).
The diagonal coupling term t breaks the inversion symmetry and plays a key role in the existence of isolated topologically stable EPs.When t = 0, the EPs are no longer isolated points but rings at cos k cx + cos k cy = (−T ± γ) / (2J) [22,75,76,[125][126][127].However, when t = 0 and |γ| = |T − 4J| (|T + 4J|), the EP rings [22] reduce to isolated points at (k cx , k cy ) of k cx = k cy = ±π (0) with trivial topological features w I = w II = 0; the real or imaginary part of spectrum in the momentum space possesses a Dirac-cone band structure for Recently, the topological features of EP rings are revealed in threedimensional topological Weyl semimetals [77].
Conclusion.-In summary, the topological stable EPs in non-Hermitian systems are studied.EPs correspond to the topological defects of a real valued vector field with fractional topological charges inherited from their parent DPs in Hermitian systems.The number of EPs is protected by the discrete symmetries and being invariant until EPs merging or splitting accompanied with topological QPT.A pair of half-integer topological charged EPs can merge into a hybrid EP with vanishing topological charge or a DP with integer topological charge zero or one.Another winding number is introduced to characterize the topology of energy coalescence (Riemann sheet band structure).The various configurations of DPs and EPs outline distinct topological phases.Our findings extend the understanding of topological aspects of EPs in non-Hermitian systems, which may stimulate interest in chasing for topologically stable EPs in other physical systems and in experiments.
We acknowledge the support of Chinese Natural Science Foundation (Grant Nos.11374163 and 11605094).

SUPPLEMENTARY MATERIAL
The vortices and winding numbers of the semi-Dirac model The two-dimensional non-Hermitian semi-Dirac Hamiltonian in the momentum space is where the effective two components complex magnetic field is and σ = (σ x , σ y , σ z ) is a vector of Pauli matrices.
The eigen energy E of H(k) is and the corresponding eigenstate is where The average values of the Pauli matrices under the eigenstates are The winding numbers w I and w II are associated with the vortices of the planar real vector fields F(k) = (F x , F y ) and the energy band E(k) = (E x , E y ) = (Re(E), Im(E)), respectively.F(k) = (B x , B y ) in Hermitian system; in contrast, F(k) = ( σ x , σ y ) in non-Hermitian system.E(k) is trivial in Hermitian system but nontrivial in non-Hermitian system.From the definitions of winding numbers with Fx(y) = F x(y) / F 2 x + F 2 y and with Êx =Re(E)/|E|, Êy =Im(E)/|E|, we notice that both two eigenstates (eigen energies) yield identical winding numbers w I (w II ).Supplementary Figure 5 depicts the vortices characterized by the two types of winding numbers for the four distinct topological phases in different regions of parameter space V -Γ.The planar vector field F(k) = (B x , B y ) is shown in Hermitian cases (i) and (ii) of Γ = 0 in Supplementary Figures 5(a) and 5(b).While, the planar vector fields F(k) = ( σ x + , σ y + ) and E(k) = (Re(E + ), Im(E + )) are shown in non-Hermitian cases (iii) and (iv) of Γ = 0 in Supplementary Figures 5(c)-5(f).The vortices of the vector fields F(k) and E(k) reveal the corresponding winding numbers w I and w II , respectively.(i) At Γ = V = 0, H(k) is Hermitian and the band touching occurs at a zero-energy semi-Dirac point k c = (0, 0).The winding numbers are w I = w II = 0.The vortex and winding number is reflected from the planar vector field shown in Supplementary Figure 5(a).
(ii) At Γ = 0 = V , H(k) is Hermitian and the band touching points (BTPs) are Dirac points at k c = (± √ V , 0).The winding numbers are w I = ±1 and w II = 0. Similarly, these two diabolic points (DPs) degeneracy can be regarded as the mergence of two normal exceptional points (EPs) in case (iii) with the same (opposite) half-integer winding numbers w I(II) .The vortices and winding numbers are reflected from the planar vector field shown in Supplementary Figure 5(b).
(iv) At V = |Γ| = 0, there are three EPs at k c = (0, 0) and (± √ 2V , 0).The first type of winding number is w I = 0 for the hybrid EP at origin (0, 0), which can be regarded as the mergence of the two central normal EPs in (iii) with the opposite half-integer winding numbers w I(II) ; while the other two normal EPs at sides remain unchanged, being w I = ±1/2 for (± √ 2V , 0).For the three EPs at {(− √ 2V , 0), (0, 0), ( √ 2V , 0)}, the second type of winding number is w II = {−1/2, 0, +1/2} when Γ > 0, but is w II = {+1/2, 0, −1/2} when Γ < 0. The first type of vortices and winding numbers (w I ) for Γ > 0 are reflected from the planar vector fields shown in Supplementary Figure 5(d); the second type of vortices and winding numbers (w II ) for Γ > 0 are reflected from the planar vector fields shown in Supplementary Figure 5(f).Two types of vortices for the planar vector fields F (k) and E (k) in the bilayer lattice are shown in Supplementary Figure 7. Three typical EP configurations are de-picted, including both the zero and half-integer vortices of w I,II = 0 and ±1/2 for the EPs.The winding numbers w I and w II reflect the varying directions of planar vector fields F (k) and E (k), respectively.Supplementary Figure 7(a, b) includes sixteen one-half vortices (normal EP w I,II = ±1/2).Supplementary Figure 7(c, d) includes eight zero vortices (hybrid EP w I,II = 0).

Dispersion near the BTPs of the bilayer square lattice
In this section, we discuss the dispersions near the BTPs (EPs and DPs) with different winding numbers.Here we only discuss the dispersion around EPs at (k cx , ±π/2), and the situations of (±π/2, k cy ) can be obtained by interchanging k x and k y .Set q x = k x − k cx and q y = k y − π/2, the Taylor expansion of the B ′ (k) field to the second order near (k cx , ±π/2) yields x , σ ′ z ) in the momentum space for the EP configuration of phase marked on the bottom left is shown by the white arrow, and the background density plot is F and the dispersion is In the following, we discuss the dispersion near the EPs and DPs in detail.Firstly, we consider the dispersion near normal EPs with w I = ±1/2.Near (k cx , ±π/2) with k cx = 0, π, ±π/2, we approximately have the dispersion along the k y = ξk x (k x = −ξk y ) direction as where ξ is real.Notably, the dispersion along any direction is square-root near normal EPs with w I = ±1/2.

FIG. 2 .
FIG. 2. (a) Schematic of the bilayer square lattice.The intrasublattice hoppings have opposite signs t (red) and −t (blue).The shadowed lattice indicates the lower layer with (γ, t) → (−γ, −t) in contrast to the upper layer.(b) Phase diagram of the bilayer square lattice in the γ-T plane for |T ± γ| ≤ 2. Twenty-six topological phases exhibit five types of winding number distributions of BTPs as listed in TableI.

FIG. 3 .
FIG.3.Schematic of BTP configurations of (a) wI and (b) wII for the topological phases in Fig.2(b).The horizontal (vertical) axis is kx (ky).The blue arrows and red circles represent wI,II = ±1/2, ±1 and 0 as those in Fig.1.wI of the Hermitian (non-Hermitian) bilayer lattice is shown in the bottom left corner of (a); the red single and blue double lines represent wI = +1 and −1 (wI = +1/2 and −1/2), and the yellow filled circles represent wI = 0 (wI = 0).Any pair of nearest neighbor normal EPs have opposite wII.

1 FIG. 5 .
FIG. 5. Two types of vortices in the momentum space of the semi-Dirac model.The planar vector field F(k) = (Fx, Fy) is shown for cases (i-iv).(a, b) The planar vector field is (Fx, Fy) = (Bx, By) for the Hermitian systems in cases (i, ii) as shown by the black arrow.(c, d) The planar vector field is (Fx, Fy) = ( σx , σy ) for the non-Hermitian systems in cases (iii, iv) as shown by the white arrow on the top and the background density plot is F 2 x + F 2 y .(e, f) Another planar vector field E(k) = (Re(E), Im(E)) for the cases (iii, iv).The plots are for the upper level E+.The winding numbers of BTPs (DPs and EPs) inside the rectangles are schematically illustrated at the bottom of each plot as Fig. 1 in the Letter.The system parameters are (a) Γ = V = 0. (b) Γ = 0, V = 4. (c, e) Γ = 3/2, V = 5/2.(d, f) Γ = V = 2.

2 y
; the planar vector E ′ (k) = E ′ x , E ′ y = (Re(E ′ ), Im(E ′ )) is depicted by the black arrow in (b, d), which reflects the winding number wII.The system parameters are γ = J/2, T = −J in (a, b), and γ = T = −J in (c, d).The winding numbers of EPs inside the rectangles are schematically illustrated at the bottom of each plot as Fig. 1 in the Letter.The plots are for the upper level E ′ + .
(8). 6.(a)The phase diagram in the parameter space γ-T .16(8)EPsappear in the colored area inside (outside) the region |T ± γ| ≤ 2, and the system is gapped in the white region.Excepted for the twenty-six BTP configurations shown in Fig.3of the Letter, other BTP configurations outside the region |T ± γ| ≤ 2 are illustrated in (b-e).The upper (lower) panel is the configurations of wI (wII).
I w II w II w I w FIG. 7. Two types of vortices in the bilayer lattice.(a, c) The winding number wI relevant planar vector F ′