An attempt is made to show the topological transition on conformal manifold for quantum Ising model with longer range interaction. This model Hamiltonian system has different gapped phases with different topological indices and also different quantum critical lines depending on the presence and absence of the transverse field. We also present the central charge for the different regimes of the parameter space. We show explicitly on the interplay of criticality, topology and central charge (C) in the presence and absence of transverse field and also non-universal character of C. We show how the Lifshitz transition occurs in the presence of transverse field and changes the universality class of the central charge in presence of transverse field. We show explicitly the existence of conformal field theory (CFT) criticality and non-CFT criticality. We have presented an explicit calculations to find the relation between the polynomial function and the Anderson-pseudo spin model Hamiltonian. Our results are far more enrich than the existence results of noninteracting fermionic many body system. This work not only provides a new perspective in topological state of conformal field theory but also for low dimensional quantum many body system.
An attempt is made to show the topological transition on conformal manifold for quantum Ising model with longer range interaction. This model Hamiltonian system has different gapped phases with different topological indices and also different quantum critical lines depending on the presence and absence of the transverse field. We also present the central charge for the different regimes of the parameter space. We show explicitly on the interplay of criticality, topology and central charge (C) in the presence and absence of transverse field and also non-universal character of C. We show how the Lifshitz transition occurs in the presence of transverse field and changes the universality class of the central charge in presence of transverse field. We show explicitly the existence of conformal field theory (CFT) criticality and non-CFT criticality. We have presented an explicit calculations to find the relation between the polynomial function and the Anderson-pseudo spin model Hamiltonian. Our results are far more enrich than the existence results of noninteracting fermionic many body system. This work not only provides a new perspective in topological state of conformal field theory but also for low dimensional quantum many body system.
The investigation and characterization of topological quantum phase transition between gapless phases is one of the recent interest of research in topological states of matter. We consider transverse field Ising model with three spin interaction in one dimension and observe a topological transition between gapless phases on one of the critical lines of this model. We study the distinct nature of these gapless phases and show that they belong to different universality classes. The topological invariant number (winding number) characterize different topological phases for the different regime of parameter space. We observe the evidence of two multi-critical points, one is topologically trivial and the other one is topologically active. Topological quantum phase transition between the gapless phases on the critical line occurs through the non-trivial multi-critical point in the Lifshitz universality class. We calculate and analyze the behavior of Wannier state correlation function close to the multi-critical point and confirm the topological transition between gapless phases. We show the breakdown of Lorentz invariance at this multi-critical point through the energy dispersion analysis. We also show that the scaling theories and curvature function renormalization group can also be effectively used to understand the topological quantum phase transitions between gapless phases. The model Hamiltonian which we study is more applicable for the system with gapless excitations, where the conventional concept of topological quantum phase transition fails.
The thermal dynamics of the two-dimensional Ising model and quantum dynamics of the one-dimensional transverse-field Ising model (TFIM) are mapped to one another through the transfer-matrix formalism. We show that the fermionised TFIM undergoes a Fermi-surface topology-changing Lifshitz transition at its critical point. We identify the degree of freedom which tracks the Lifshitz transition via changes in topological quantum numbers (e.g., Chern number, Berry phase etc.). An emergent SU(2) symmetry at criticality is observed to lead to a topological quantum number different from that which characterises the ordered phase. The topological transition is also understood via a spectral flow thought-experiment in a Thouless charge pump, revealing the bulk-boundary correspondence across the transition. The duality property of the phases and their entanglement content are studied, revealing a holographic relation with the entanglement at criticality. The effects of a non-zero longitudinal field and interactions that scatter across the singular Fermi surface are treated within the renormalisation group (RG) formalism. The analysis reveals that the critical point of the 1D TFIM and the 1D spin-1/2 Heisenberg chain are connected via a line of SU(2)-symmetric theories. We extend our analysis to show that the classical to quantum correspondence links the critical theories of Ising models in various dimensions holographically through the universal effective Hamiltonian that describes the Lifshitz transition of the 1D TFIM. We obtain in this way a unified perspective of transitions in Ising models that lie beyond the traditional Ginzburg-Landau-Wilson paradigm. We discuss the consequences of our results for similar topological transitions observed in classical spin models, topological insulators, superconductors and lattice gauge-field theories which are related to the Ising universality class.
Quantum criticality in one-dimensional quantum systems is characterized by emergent conformal field theories (CFTs), whose central charge counts independent gapless degrees of freedom. Establishing a microscopic connection between this universal conformal structure and the momentum-space topology of the underlying quasiparticle spectrum remains challenging. Here, we uncover a direct correspondence between Fermi-point topology, conformal criticality, and quantum entanglement in an extended quantum spin chain with competing cluster interactions, exchange anisotropy, and a transverse magnetic field. We show that interaction- and field-driven Lifshitz transitions generate conformal critical phases with effective central charges ceff=1/2, 1, 3/2, 2, and 3, including a multicritical point where Ising and Luttinger-liquid sectors coexist. Importantly, the central charge is not determined simply by the number of lattice gap closings or Fermi points, but by the number and conformal content of independent low-energy continuum sectors after accounting for lattice symmetries, reciprocal-lattice identifications, and mode equivalences. Thus, Lifshitz transitions may leave the conformal anomaly unchanged or modify the central charge depending on whether spectral reconstruction generates new independent continuum sectors. Real- and momentum-space entanglement spectra provide complementary microscopic signatures, revealing both the conformal content and momentum-space organization of critical modes. Our results establish a microscopic framework linking Fermi-point topology to emergent CFTs and show how interaction-driven spectral reconstruction can generate higher-central-charge criticality and unconventional multicritical behavior.
By constructing an exactly solvable spin model, we investigate the critical behaviors of transverse-field Ising chains interpolated with cluster interactions, which exhibit various types of topologically distinct Ising critical points. Using fidelity susceptibility as an indicator, we establish the global phase diagram, including ferromagnetic, trivial paramagnetic, and symmetry-protected topological phases. Different types of critical points exist between these phases, encompassing both topologically trivial and nontrivial Ising critical points, as well as Gaussian critical points. Importantly, we demonstrate the existence of a Lifshitz transition between these topologically distinct Ising critical points, with central charge and critical exponents determined through finite-size scaling. This work serves as a valuable reference for further research on phase transitions within the gapless quantum phase of matter.
Despite great successes in the study of gapped phases, a comprehensive understanding of the gapless phases and their transitions is still under development.In this paper, we study a general phenomenon in the space of (1 + 1)-dimensional critical phases with fermionic degrees of freedom described by a continuous family of conformal field theories (CFTs), also known as the conformal manifold.Along a one-dimensional locus on the conformal manifold, there can be a transition point, across which the fermionic CFTs on the two sides differ by stacking an invertible fermionic topological order (IFTO), point by point along the locus.At every point on the conformal manifold, the order and disorder operators have power-law two-point functions, but their critical exponents cross over with each other at the transition point, where stacking the IFTO leaves the fermionic CFT unchanged.We call this continuous transition on the fermionic conformal manifold a topological transition.By gauging the fermion parity, the IFTO stacking becomes a Kramers-Wannier duality between the corresponding bosonic CFTs.Both the IFTO stacking and the Kramers-Wannier duality are induced by the electromagnetic duality of the (2 + 1)-dimensional Z 2 topological order.We provide several examples of topological transitions, including the familiar Luttinger model of spinless fermions (i.e., the c = 1 massless Dirac fermion with the Thirring interaction) and a class of c = 2 examples describing U(1) × SU(2)-protected gapless phases.
The quantum phase transition (QPT) of the one-dimensional (1D) quantum compass model in a transverse magnetic field is studied in this paper. An exact solution is obtained by using an extended Jordan and Wigner transformation to the pseudo-spin operators. The fidelity susceptibility, the concurrence, the block-block entanglement entropy, and the pseudo-spin correlation functions are calculated with antiperiodic boundary conditions. The QPT driven by the transverse field only emerges at zero field and is of the second-order. Several critical exponents obtained by finite size scaling analysis are the same as those in the 1D transverse field Ising model, suggesting the same universality class. A logarithmic divergence of the entanglement entropy of a block at the quantum critical point is also observed. From the calculated coefficient connected to the central charge of the conformal field theory, it is suggested that the block entanglement depends crucially on the detailed topological structure of a system.