We show that a class of exactly solvable quantum Ising models, including the transverse-field Ising model and anisotropic XY model, can be characterized as the loops in a two-dimensional auxiliary space. The transverse-field Ising model corresponds to a circle and the XY model corresponds to an ellipse, while other models yield cardioid, limacon, hypocycloid, and Lissajous curves etc. It is shown that the variation of the ground state energy density, which is a function of the loop, experiences a nonanalytical point when the winding number of the corresponding loop changes. The winding number can serve as a topological quantum number of the quantum phases in the extended quantum Ising model, which sheds some light upon the relation between quantum phase transition and the geometrical order parameter characterizing the phase diagram.PACS numbers: 03.65. Vf, 75.10.Jm, 05.70.Fh, Introduction. Characterizing the quantum phase transitions (QPTs) is of central significance to both condensed matter physics and quantum information science. QPTs occur only at zero temperature due to the competition between different parameters describing the interactions of the system. A quantitative understanding of the second-order QPT is that the ground state undergoes qualitative changes when an external parameter passes through quantum critical points (QCPs).There are two prototypical models, Bose-Hubbard model and transverse-field Ising model, based on which the concept and characteristic of QPTs can be well demonstrated. However, among the two, only the transverse-field Ising model is exactly solvable [1], so as to be a unique paradigm for understanding the QPTs. Recently, more attention has been paid to theoretical studies of exactly solvable quantum spin models involving nearest-, next-nearest-neighbor interactions, and multiple spin exchange models, etc [3][4][5][6][7][8][9][10]. Those models are closer to real quasi-one-dimensional magnets [11][12][13] comparing to standard ones with only nearest-neighbor couplings. Furthermore, it has been shown that quantum spin models can be simulated in artificial quantum system with controllable parameters. Quantum simulation of spin chain can be experimentally realized through neutral atoms stored in an optical lattice [14,15], trapped ions [16][17][18][19][20][21][22][23][24] and NMR simulator [25]. This system often serves as a test bed for applying new ideas and methods to quantum phase transitions.A fundamental question is whether QPTs in Ising model can have a connection to some topological characterizations. It is interesting to note in this context that some simple Ising models have been found to exhibit topological characterization [26][27][28][29]. The purpose of the present work is to shed some light upon the relation between QPTs and a geometrical parameter characterizing the phase diagram, through the investigation of a class of quantum Ising models.In this work, we present an extended quantum Ising

We study two coupled Su-Schrieffer-Heeger (SSH) chains system, which is shown to contain rich quantum phases associated with topological invariants protected by symmetries. In the weak coupling region, the system supports two non-trivial topological insulating phases, characterized by winding number N = ±1, and two types of edge states. The boundary between the two topological phases arises from two band closing points, which exhibit topological characteristics in onedimensional k space. By mapping Bloch states on a vector field in k space, the band degenerate points correspond to a pair of kinks of the field, with opposite topological charges. Two topological nodal points move and merge as the inter-chain coupling strength varies. This topological invariant is protected by the translational and inversion symmetries, rather than the antiunitary operation. Furthermore, we find that when a pair of nodal points is created, a second order quantum phase transition (QPT) occurs, associating with a gap closing and spontaneously symmetry breaking. This simple model demonstrates several central concepts in the field of quantum materials and provides a theoretical connection between them.

We systematically study a Kitaev chain with imbalanced pair creation and annihilation, which is introduced by non-Hermitian pairing terms. Exact phase diagram shows that the topological phase is still robust under the influence of the conditional imbalance. The gapped phases are characterized by a topological invariant, the extended Zak phase, which is defined by the biorthonormal inner product. Such phases are destroyed at the points where the coalescence of groundstates occur, associating with the time-reversal symmetry breaking. We find that the Majorana edge modes also exist for the open chain within unbroken time-reversal symmetric region, demonstrating the bulk-edge correspondence in such a non-Hermitian system.

Parity-time symmetry is of great interest. The reciprocal and unidirectional features are intriguing besides the symmetry phase transition. Recently, the reciprocal transmission, unidirectional reflectionless and invisibility are intensively studied. Here, we show the reciprocal reflection/transmission in -symmetric system is closely related to the type of symmetry, that is, the axial (reflection) symmetry leads to reciprocal reflection (transmission). The results are further elucidated by studying the scattering of rhombic ring form coupled resonators with enclosed synthetic magnetic flux. The nonreciprocal phase shift induced by the magnetic flux and gain/loss break the parity and time-reversal symmetry but keep the parity-time symmetry. The reciprocal reflection (transmission) and unidirectional transmission (reflection) are found in the axial (reflection) -symmetric ring centre. The explorations of symmetry and asymmetry from symmetry may shed light on novel one-way optical devices and application of -symmetric metamaterials.

We studied the critical dynamics of spectral singularities. The system investigated is a coupled resonator array with a side-coupled loss (gain) resonator. For a gain resonator, the system acts as a wave emitter at spectral singularities. The reflection probability increased linearly over time. The rate of increase is proportional to the width of the incident wave packet, which served as the spectral singularity observer in the experiment. For a lossy resonator, the system acts as a wave absorber. The emission and absorption states at spectral singularities coalesce in a finite parity-time (PT ) symmetric system that combined by the gain and loss structures cut from corrresponding scattering systems at spectral singularities; in this case, the PT -symmetric system is at an exceptional point with a 2 * 2 Jordan block. The dynamics of the PT -symmetric system exhibit the characteristic of exceptional points and spectral singularities