The phase diagram of the quantum compass ladder model is investigated through numerical density matrix renormalization group based on infinite matrix product state algorithm and analytic effective perturbation theory. For this model we obtain two symmetry-protected topological phases, protected by a Z2 × Z2 symmetry, and a topologically-trivial Z2-symmetry-breaking phase. The symmetry-protected topological phases--labeled by symmetry fractionalization--belong to different topological classes, where the complex-conjugate symmetry uniquely distinguishes them. An important result of this classification is that, as revealed by the nature of the Z2-symmetry-breaking phase, the associated quantum phase transitions are accompanied by an explicit symmetry breaking, and thus a local-order parameter conclusively identifies the phase diagram of the underlying model. This is in stark contrast to previous studies which require a non-local string order parameter to distinguish the corresponding quantum phase transitions. We numerically examine our results and show that the local-order parameter is related to the magnetization exponent 0.12 ± 0.01.
The phase diagram of the quantum compass ladder model is investigated through numerical density matrix renormalization group based on infinite matrix product state algorithm and analytic effective perturbation theory. For this model we obtain two symmetry-protected topological phases, protected by a Z2 × Z2 symmetry, and a topologically-trivial Z2-symmetry-breaking phase. The symmetry-protected topological phases--labeled by symmetry fractionalization--belong to different topological classes, where the complex-conjugate symmetry uniquely distinguishes them. An important result of this classification is that, as revealed by the nature of the Z2-symmetry-breaking phase, the associated quantum phase transitions are accompanied by an explicit symmetry breaking, and thus a local-order parameter conclusively identifies the phase diagram of the underlying model. This is in stark contrast to previous studies which require a non-local string order parameter to distinguish the corresponding quantum phase transitions. We numerically examine our results and show that the local-order parameter is related to the magnetization exponent 0.12 ± 0.01.
By using the infinite time evolving block decimation in the presentation of infinite matrix product states, we study an extended quantum compass model (EQCM). This model does not only include extremely rich phase diagrams due to competitions of orbital degrees of freedom and anisotropic couplings between pseudospin-1/2 operators but also have the capacity to describe property of protected qubits for quantum computation which leads to lots of attentions paid to the phase boundaries of the EQCM. However, few attentions are paid to long-range topological string correlation order parameters of the EQCM. To study order parameters, one should understand spontaneous symmetry breaking which relates to Landau quantum phase transitions theory. Once spontaneous symmetry breaking happens, there should exist some local order which can be described by a local order parameter. This order parameter can be used to distinguish the phase from others. For continuous quantum phase transitions, in the critical regime, critical exponents can be extracted. Unfortunately, the long-range topological string correlation orders are beyond Landau quantum phase transitions theory, one can not directly use two paradigms of Landau-Ginzburg-Wilson. Usually, one can define a local order parameter by local magnetization. Naturally, one can also refer to this way to define the long-range topological string correlation order parameters by long-range topological string correlations on the following conditions, i.e. the quantum system undergoes a hidden spontaneous symmetry breaking; the long-range topological string correlation order parameter can be used to distinguish the phase from others; for continuous quantum phase transitions, the long-range topological string correlation order parameter satisfies scaling law when control parameter getting close to critical point. Based on above idea, in order to characterize the topological ordered phases and quantum phase transitions in the EQCM, even/odd long-range topological string correlations are introduced based on even/odd bonds. Hereafter, fidelity per lattice site, even/odd long-range topological string correlations, the saturation behavior of odd long-range topological string correlations and order parameters are calculated. The long-range topological string correlations show three distinguished behaviors which include decaying to zero, monotonic saturation and oscillatory saturation. By the above characterizations, oscillatory/monotonic odd long-range topological string correlation order parameter is derived. Then ground-state phase diagram of order parameters is computed which includes oscillatory/monotonic odd long-range topological string correlation order phase and antiferromagnetic phase. In the critical regime, critical exponent β=1/8 extracted from monotonic odd long-range topological string correlation order parameter and local magnetization shows the phase transition belongs to Ising universality. In addition, the phase transition points, the order of the phase transitions of fidelity show consistent with the results of order parameters.
In order to investigate the quantum phase transition in the one-dimensional quantum compass model, we numerically calculate non-local string correlations, entanglement entropy and fidelity per lattice site by using the infinite matrix product state representation with the infinite time evolving block decimation method. In the whole range of the interaction parameters, we find that four distinct string orders characterize the four different Haldane phases and the topological quantum phase transition occurs between the Haldane phases. The critical exponents of the string order parameters β = 1/8 and the cental charges c = 1/2 at the critical points show that the topological phase transitions between the phases belong to an Ising type of universality classes. In addition to the string order parameters, the singularities of the second derivative of the ground state energies per site, the continuous and singular behaviors of the Von Neumann entropy and the pinch points of the fidelity per lattice site manifest that the phase transitions between the phases are of the second-order, in contrast to the first-order transition suggested in previous studies.
Bose gas on a two-leg ladder exhibits an interesting topological phase. We show the presence of a bosonic symmetry-protected-topological (SPT) phase protected by ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry. This symmetry leads to Z4 fractional quantization of the Z4 Berry phase, which is a topological order parameter to identify the bulk. Using the Z4 Berry phase, we show that the interacting bosonic system possesses rich topological phases depending on the particle density and strength of interaction. Based on the bulk-edge correspondence, each edge state of the SPT phases is discussed in relation to the Z4 Berry phases. In particular, we have found an intermediate phase that is not adiabatically connected to a simple adiabatic limit that possesses unconventional edge states, which we numerically demonstrate by employing the density-matrix renormalization-group algorithm.
Abstract Classifying and understanding phases of matter is an important task in condensed matter physics. The class of ‘conventional’ symmetry-broken phases is well understood in terms of Landau’s theory. A paradigmatic example is the Z2-symmetric Ising model with symmetric (paramagnetic) and symmetry-broken (ferromagnet) phases, which can be distinguished by measuring the magnetization as a local order parameter. Topological phases of matter, however, are less understood and no complete classification is known. These lecture notes discuss schemes that provide an understanding and characterization of certain topological phases of matter in models with local Hamiltonians. The main focus is on symmetry-protected topological (SPT) phases in one-dimensional bosonic systems. On the basis of the entanglement properties of such systems, a motivation is given for the matrix product state (MPS) representation of ground states. Using the MPS framework, it is demonstrated how SPT phases can be classified using projective representations of the symmetries.
We study the interplay between the Kitaev and Ising interactions on a ladder geometry. We show that the ground state of the Kitaev ladder is a symmetry-protected topological (SPT) phase, which is protected by a ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. The nature of the SPT phase is confirmed by degeneracy of the entanglement spectrum. Nonlocal order parameters that indirectly measure phase factors (inequivalent projective representations of the symmetries) of the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry explicitly show the protection of the SPT phase under the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. We derive the effective theory to describe the topological phase transition on the ladder geometry, which is given by a transverse field Ising model with/without next-nearest-neighbor coupling based on the primary Ising configurations. The ladder has three phases, namely, the Kitaev SPT, symmetry-broken ferro/antiferromagnetic order, and classical spin liquid. The nonzero quantum critical point and its corresponding central charge are provided by the effective theory, which are in full agreement with the numerical results, i.e., the divergence of entanglement entropy at the critical point and change of the entanglement spectrum degeneracy. The central charge of the critical points are either c=1 or c=2, with the magnetization and correlation exponents being 1/4 and 1/2, respectively. The transition from the classical spin-liquid phase of the frustrated Ising ladder to the Kitaev SPT phase is mediated by a floating phase, which shows strong finite entanglement scaling.
Topological properties of the spin-1/2 dimerized Heisenberg ladder are investigated, focusing on the plateau phase in the magnetic field whose magnetization is half of the saturation value. Although the applied magnetic field removes most of the symmetries of the system, there is a symmetry-protected topological phase supported by the spatial inversion symmetry. The Z2 Berry phase associated with a symmetry-respecting boundary and quantized into 0 and $\ensuremath{\pi}$ is used as a symmetry-protected topological order parameter. Edge states are also analyzed to confirm the bulk-edge correspondence. In addition, a symmetry-breaking boundary is considered. Then, we observe a different type of quantization of the Berry phase, i.e., a quantization into $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ of the Berry phase. In this case, the bulk-edge correspondence is also different, namely, there emerge ``polarized'' edge states for the case with $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ quantization. We also evaluate the entanglement entropy by the infinite time-evolving block decimation (iTEBD) to complement the Berry-phase-based arguments. Further, a different type of the topological order parameter is extracted from the matrix product state representation of the ground state given by the iTEBD.