Critical theories of phase transition between symmetry protected topological states and their relation to the gapless boundary theories

Symmetry protected topological states (SPTs) have the same symmetry and the phase transition between them are beyond Landau?s symmetry breaking formalism. In this paper we study (1) the critical theory of phase transition between trivial and non-trivial SPTs, and (2) the relation between such critical theory and the gapless boundary theory of SPTs. Based on examples of SO(3)$\mathrm{SO}(3)$ and SU(2)$\mathrm{SU}(2)$ SPTs, we propose that under appropriate boundary condition the critical theory contains the delocalized version of the boundary excitations. In addition, we prove that the boundary theory is the critical theory spatially confined between two SPTs. We expect these conclusions to hold in general and, in particular, for discrete symmetry groups as well.     

Kitaev模型与拓扑量子相变

文章通过在一种准一维路径上引入自旋算符的约当-维格纳（Jordan—Wigner）变换，证明了Kitaev自旋模型完全等价于一个不含任何非物理自由度的自由Majorana费米子模型。通过对偶变换，进一步证明了这个系统中存在的量子相变可用非定域的拓扑序参量来描述；并且，这些非定域的拓扑序参量在对偶空间变成为定域的朗道类型的序参量。文章作者的工作揭示了传统的量子相变和拓扑量子相变的内在关系，扩展了朗道二级相变理论的适用范围。     

The 1D Ising model and the topological phase of the Kitaev chain

It has been noted that the Kitaev chain, a p-wave superconductor with nearest-neighbor pairing amplitude equal to the hopping term Δ=t$\Delta =t$, and chemical potential μ=0$\mu =0$, can be mapped into a nearest neighbor Ising model via a Jordan–Wigner transformation. Starting from the explicit eigenstates of the open Kitaev chain in terms of the original fermion operators, we elaborate that despite this formal equivalence the models are physically inequivalent, and show how the topological phase in the Kitaev chain maps into conventional order in the Ising model.     

Quantum phase transitions in matrix product states of one-dimensional spin-1 chains

We present a new model of quantum phase transitions in matrix product systems of one-dimensional spin-1 chains and study the phases coexistence phenomenon. We find that in the thermodynamic limit the proposed system has three different quantum phases and by adjusting the control parameters we are able to realize any phase, any two phases equal coexistence and the three phases equM coexistence. At every critical point the physical quantities including the entanglement are not discontinuous and the matrix product system has long-range correlation and N-spin maximal entanglement. We believe that our work is helpful for having a comprehensive understanding of quantum phase transitions in matrix product states of one-dimensional spin chains and of certain directive significance to the preparation and control of one-dimensional spin lattice models with stable coherence and N-spin maximal entanglement.     

Picturing quantum phase transitions

We discuss the quantum phase transitions (QPT) in N-spin chains from the point of view of collective observables. We show that the measurement space representation is a convenient tool for the analysis of phase transitions, allowing the determination of an appropriate set of macroscopic order parameters (for a given Hamiltonian). Quantum correlations in the vicinity of the critical points are analyzed both in the ground states and low temperature thermal states.     

Quantum phase transitions in matrix product states of one-dimensional spin-1/2 chains

For the matrix product system of a one-dimensional spin-1/2 chain, we present a new model of quantum2 phase transitions and find that in the thermodynamic limit, both sides of the critical point are respectively described by phases ｜Ψa 〉=｜1··· 1 representing all particles spin up and ｜Ψb 〉=｜0··· 0 representing all particles spin down, while the phase transition point is an isolated intermediate-coupling point where√ the two phases coexist equally, which is2 described by the so-called N-qubit maximally entangled GHZ state ｜Ψpt =√2/2（｜1··· 1 ＋｜0··· 0）. At the critical point,2the physical quantities including the entanglement are not discontinuous and the matrix product system has longrange correlation and N-qubit maximal entanglement. We believe that our work is helpful for having a comprehensive understanding of quantum phase transitions in matrix product states of one-dimensional spin chains and of potential directive significance to the preparation and control of one-dimensional spin lattice models with stable coherence and N-qubit maximal entanglement.     

Dynamic phase transitions in the kinetic mixed spin-1/2 and spin-5/2 Ising model under a time-dependent oscillating magnetic field

We calculate the dynamic phase transition (DPT) temperatures and present the dynamic phase diagrams in the kinetic mixed spin-1/2 and spin-5/2 Ising model under the presence of a time-dependent oscillating external magnetic field. We employ the Glauber transition rates to construct the set of mean-field dynamic equations. The time variation of the average magnetizations and the thermal behavior of the dynamic magnetizations are investigated, extensively. The nature (continuous or discontinuous) of the transitions is characterized by studying the thermal behaviors of the dynamic magnetizations. The DPT points are obtained and the phase diagrams are presented in two different planes. Phase diagrams contain four fundamental phases and three coexistence or mixed phases, which strongly depend on interaction parameters. The phase diagrams are discussed and a comparison is made with the results of the other mixed spin Ising systems.     

Dynamic phase transitions in the kinetic spin-1 Blume-Capel model on the Bethe lattice

The stationary states of the kinetic spin-1 Blume-Capel (BC) model on the Bethe lattice are analyzed in detail in terms of recursion relations. The model is described using a Glauber-type stochastic dynamics in the presence of a time-dependent oscillating external magnetic field (h) and crystal field (D) interactions. The dynamic order parameter, the hysteresis loop area and the dynamic correlation are calculated. It is found that the magnetization oscillates around nonzero values at low temperatures (T) for the ferromagnetic (F) phase while it only oscillates around zero values at high temperatures for the paramagnetic (P) phase. There are regions of the phase space where the two solutions coexist. The dynamic phase diagrams are obtained on the (kT/J,h/J) and (kT/J,D/J) planes for the coordination number q=4. In addition to second-order and first-order phase transitions, dynamical tricritical points and triple points are also observed.     

Geometric phases and quantum phase transitions in inhomogeneous XY spin-chains:Effect of the Dzyaloshinski-Moriya interaction

We study geometric phases of the ground states ofinhomogeneous XY spin chains in transverse fields withDzyaloshinski--Moriya (DM) interaction, and investigate the effectof the DM interaction on the quantum phase transition (QPT) of suchspin chains. The results show that the DM interaction couldinfluence the distribution of the regions of QPTs but could notproduce new critical points for the spin-chain. This study extendsthe relation between geometric phases and QPTs.     

Multiple-quantum-coherence dynamics of spin-1 Bose-Einstein condensate during quantum phase transitions

Multiple quantum coherences are often employed to describe quantum many-body dynamics in nuclear spin systems and recently,to characterize quantum phase transitions in trapped ions.Here we investigate the multiple-quantum-coherence dynamics of a spin-1 Bose-Einstein condensate.By adjusting the quadratic Zeeman shift,the condensate exhibits three quantum phases.Our numerical results show that the spectrum of multiple quantum coherence does indeed catch the quantum critical points.More importantly,with only a few low-order multiple quantum coherences,the spin-1 condensate exhibits rich signals of the many-body dynamics,beyond conventional observables.The experimental implementation of such multiple quantum coherence protocol is also discussed.     

Coarsening in fluid phase transitions

We review the understanding of the kinetics of fluid phase separation in various space dimensions. Morphological differences, percolating or disconnected domains, based on overall composition in a binary liquid or on density in a vapor–liquid system, are discussed. Depending upon the morphology, various possible mechanisms for domain growth are pointed out and discussions of corresponding theoretical predictions are provided. On the computational front, useful models and simulation methodologies are presented. Theoretically predicted growth laws have been tested via molecular dynamics simulations of vapor–liquid transitions. In the case of a disconnected structure, the mechanism has been confirmed directly.     

Symmetry breaking at the phase transitions in ferroelectric pyridinium salts

The phase transitions in ferroelectric pyridinium tetrafluoroborate (PyBF₄) and pyridinium perchlorate (PyClO₄) have been characterized by structural studies. The continuous ferroelectric phase transition at 238.7 K in PyBF₄ appears to be a unique case for multiaxial ferroelectrics, while the first order phase transition in PyClO₄ at 248 K is consistent with the Landau theorem for deducing the character of phase transitions from symmetry considerations. The phase transition in both materials is caused by ordering of the pyridinium cation as well as tetrahedral anions.     

Symmetry breaking,phase transition and gravity

We discuss some recent work in which the non-minimal coupling of gravity with a self-interacting scalar field in the presence of matter can lead to a phase transition when the sign of gravitational interaction changes. It is found that gravity becomes repulsive above a critical temperature which may lie in the range 10²⁴ to 10³²K which obtains in the very early universe (10⁻³⁵ to 10⁻⁴³ sec) of the standard model. The results are intimately connected with big bang and possible removal of singularity. Based on the talk given at the International Symposium on Theoretical Physics, Bangalore, November 1984.     

Thermodynamics’ third law and quantum phase transitions

We report on the fact that microscopically enforcing fulfillment of thermodynamics’ third law on a system of fermions automatically yields the values of the external parameter (here coupling strengths in the pertinent Hamiltonian) at which quantum phase transitions take place. Our considerations are illustrated via an exactly solvable model of Plastino and Moszkowski [Il Nuovo Cimento 47, 470 (1978)].     

Symmetry relations for the phase matrix of horizontally oriented particles

Seven symmetry relations, with a wide range of validity, are presented for the phase matrix of horizontally oriented particles. Three of these relations constitute a fundamental basis from which four others can be derived by making combinations. All seven relations can be used for many types of particles, including hexagonal plates and columns as well as spheroids, cylinders and cubes. Some applications are also pointed out.     

Acoustic dispersion near structural phase transitions

We show on various examples, that elastic constant measurements provide a valuable tool for studying the dynamics of solids near phase transitions. Many experimental methods used in the investigation of phase transitions (NMR, neutron scattering, etc.) are able to give information on the dynamics, but in different regions of q- and ω-space, thereby probing different dynamical processes.

A comparison of the dynamics obtained from the elastic measurements with other techniques (NMR) demonstrates this very clearly for the cases of KSCN and C_60.     

Finite-size,magnetic and chemical-potential effects on first-order phase transitions

We perform a study about effects of an applied magnetic field and a finite chemical potential on the size-dependent phase structure of a first-order transition. These effects are introduced by using methods of quantum fields defined on toroidal spaces, and we study in particular the case of two compactified dimensions, imaginary time and a spatial one (a heated film). It is found that for any value of the applied field, there is a minimal size of the system, independent of the chemical potential, below which the transition disappears.     

Structural phase transitions in crystals. I. Database

The database on structural phase transitions is presented in the simple and easy accessible form of table. The symmetry changes and temperature for 3446 phase transitions and the references for each of 2242 registered crystalline materials with well defined stoichoimetry are the main data in this table. The rules for constructing and reading the database table are also presented.     

Geometric phase for degenerate states of spin-1 and spin-1/2 pair

The geometric phase of a bi-particle model is discussed. One can drive the system to evolve by applying an external magnetic field, thereby controlling the geometric phase. The model has degenerate lowest-energy eigenvectors. The initial state is assumed to be the linear superposition or mixture of the eigenvectors. The relationship between the geometric phase and the structures of the initial state is considered, and the results are extended to a more general model.      

Crystal imperfections and structural phase transitions in perovskites

High‐resolution X-ray scattering measurements of the antiferrodistortive phase transitions in the perovskites SrTiO₃, RbCaF₃ and KMnF₃ have recently revealed the existence of two length scales in the critical scattering above T_e . Here we review these observations and discuss how they might be related to the well known existence of two time scales (the "central peak" problem) in the critical dynamic response function above T_e . We reach the tentative conclusion that within a few degrees of T_e both the time and length scales of the fluctuations are strongly influenced by defects and that the intrinsic critical behaviour is swamped by these effects in most crystals.