## What is/are Breaking Phase?

Breaking Phase - Our results show that although topological phase transitions do not follow a symmetry-breaking mechanism, it might be still possible to use concepts of symmetry-breaking phase transitions for topological ones by finding suitable mappings.^{[1]}The simulation covers the initial development of the wind profile, the growth of the modulational instability, the breaking and post breaking phases.

^{[2]}We do this numerically by using the symmetry-breaking phase transition line of EQCD as a line of constant physics.

^{[3]}We study the resulting chiral symmetry breaking phase transition for models with $N_F=3$ and $N_F=4$ light flavors using the linear sigma model.

^{[4]}In particular, critical sizes and shapes are identified, providing quantitative guidelines for sample fabrication in the experimental hunt for symmetry-breaking phases.

^{[5]}For unmodulated case with inhomogeneous inter- and intra- quadrimer coupling strength $\kappa_1\neq\kappa$, in addition to conventional global $PT$-symmetric phase and $PT$-symmetry-breaking phase, we find that there is exotic phase where global $PT$ symmetry is broken under open boundary condition, whereas it still is unbroken under periodical boundary condition.

^{[6]}Specifically, we investigate the quantum phase transition between Ising ferromagnetic and valence bond solid (VBS) symmetry-breaking phases.

^{[7]}We also give signatures of the $\mathcal {PT}$-symmetric and the symmetry breaking phases for the interacting bosons in experiment.

^{[8]}We do this numerically by using the symmetry-breaking phase transition point of the theory as a line of constant physics.

^{[9]}We theoretically and numerically demonstrate that the spontaneous parity-time (PT) symmetry breaking phase transition can be realized respectively by using two independent tuning ways in a tri-layered metamaterial that consists of periodic array of metal-semiconductor Schottky junctions.

^{[10]}By carefully examining active electronic degrees of freedom based on the lattice symmetry, we propose that two parity-breaking phases at ambient pressure are described by unconventional multipoles, electric toroidal quadrupoles (ETQs) with different components, x^{2}-y^{2} and 3z^{2}-r^{2}, in the pyrochlore tetrahedral unit.

^{[11]}Key pointsThe quantum spin liquid is an exotic state of matter in which interacting spins avoid symmetry-breaking phase transitions and form a ground state exhibiting long-range entanglement and topological order.

^{[12]}Here we consider a parity-time ($\mathcal{PT}$) symmetric extension of the Aubry-Andre-Harper model, undergoing a topological metal-insulator phase transition, and provide rigorous analytical results of energy spectrum, symmetry breaking phase transition and localization length.

^{[13]}First of all, the symmetry can be spontaneously broken in the ground state leading to symmetry-breaking phases.

^{[14]}Resonances, spectral singularities and eigenvalues are analyzed in detail and discussed in the context of the associated laser-absorber modes and $\mathcal{PT}$-symmetry breaking phase transition.

^{[15]}Coarsening dynamics, the canonical theory of phase ordering following a quench across a symmetry breaking phase transition, is thought to be driven by the annihilation of topological defects.

^{[16]}For this model, the dissipative phase transition (DPT), accompanying a parity-time (PT) symmetry-breaking phase transition, occurs when the imaginary field changes through the YLES.

^{[17]}A replica-symmetry-breaking phase transition is predicted in a host of disordered media.

^{[18]}Our purpose is discovering, if any, the relation between the second-order Z_{2}-symmetry-breaking phase transition and the geometric entities mentioned above.

^{[19]}We consider the properties of the small-signal modulation response of symmetry-breaking phase-locked states of twin coupled semiconductor lasers.

^{[20]}The deconfined quantum critical point (DQCP) was originally proposed as a continuous transition between two spontaneous symmetry breaking phases in 2D spin-1/2 systems.

^{[21]}Across the Lifshitz transition point, the sign of the relative phase between the Cooper-pair components drastically changes, leading to the emergence of the time-reversal breaking phase with complex gap functions.

^{[22]}In non-Hermitian crystals showing the non-Hermitian skin effect, ordinary Bloch band theory and Bloch topological invariants fail to correctly predict energy spectra, topological boundary states, and symmetry breaking phase transitions in systems with open boundaries.

^{[23]}Influences of breaking phase angels, velocity and gas pressure were discussed at the same time.

^{[24]}In this paper, we report distinct symmetry-breaking phase transitions in molecules based on oligo-para-phenylene rods and a poly(ethylene oxide) coil depending on the length of the aromatic rod block The observed phase transitions were characterized by differential scanning calorimetry (DSC) and X-ray scattering methods.

^{[25]}Here, we unveil that non-unitary discrete-time quantum walks of photons in systems involving gain and loss show rather generally non-Bloch parity-time symmetry-breaking phase transitions and suggest a bulk probing method to detect such boundary-driven phase transitions.

^{[26]}We infer the existence of a symmetry-breaking phase transition at some finite coupling, and combine this with previous simulation results to deduce that the critical number of flavors for the existence of a quantum critical point in the Thirring model satisfies $0 1$.

^{[27]}We do this numerically by using the symmetry-breaking phase transition point of the theory as a line of constant physics.

^{[28]}In most unconventional and high-temperature superconductors, superconductivity emerges as a nearby symmetry-breaking phase is suppressed by chemical doping or pressure 1 – 7.

^{[29]}The origin of the symmetry-breaking phase transition is associated with the motion or reorientation of the DMEA cations, accompanied by the crystal structures from orthorhombic Pnma to monoclinic P21/c with the temperature decreases.

^{[30]}In this talk we will cover the properties of the small-signal modulation response of symmetry-breaking phase-locked states of twin coupled semiconductor lasers.

^{[31]}In this chapter, we first introduce the basic properties of tensor product states and then proceed to discuss how it represents symmetry-breaking phases and topological phases with explicit examples.

^{[32]}We start by showing that any contest that satisfies our assumptions decomposes into two phases, a principal phase (in which states cannot be revisited) and a concluding tie-breaking phase (in which all non-terminal states can be revisited).

^{[33]}Our RG analysis leads to the qualitative conclusion that the emergent interaction-induced symmetry-breaking phases in this model system, and perhaps therefore by extension in twisted bilayer graphene, are generically unstable and fragile, and may thus manifest strong sample dependence.

^{[34]}

## Symmetry Breaking Phase

We study the resulting chiral symmetry breaking phase transition for models with $N_F=3$ and $N_F=4$ light flavors using the linear sigma model.^{[1]}We also give signatures of the $\mathcal {PT}$-symmetric and the symmetry breaking phases for the interacting bosons in experiment.

^{[2]}We theoretically and numerically demonstrate that the spontaneous parity-time (PT) symmetry breaking phase transition can be realized respectively by using two independent tuning ways in a tri-layered metamaterial that consists of periodic array of metal-semiconductor Schottky junctions.

^{[3]}Here we consider a parity-time ($\mathcal{PT}$) symmetric extension of the Aubry-Andre-Harper model, undergoing a topological metal-insulator phase transition, and provide rigorous analytical results of energy spectrum, symmetry breaking phase transition and localization length.

^{[4]}Resonances, spectral singularities and eigenvalues are analyzed in detail and discussed in the context of the associated laser-absorber modes and $\mathcal{PT}$-symmetry breaking phase transition.

^{[5]}Coarsening dynamics, the canonical theory of phase ordering following a quench across a symmetry breaking phase transition, is thought to be driven by the annihilation of topological defects.

^{[6]}The deconfined quantum critical point (DQCP) was originally proposed as a continuous transition between two spontaneous symmetry breaking phases in 2D spin-1/2 systems.

^{[7]}In non-Hermitian crystals showing the non-Hermitian skin effect, ordinary Bloch band theory and Bloch topological invariants fail to correctly predict energy spectra, topological boundary states, and symmetry breaking phase transitions in systems with open boundaries.

^{[8]}

## breaking phase transition

Our results show that although topological phase transitions do not follow a symmetry-breaking mechanism, it might be still possible to use concepts of symmetry-breaking phase transitions for topological ones by finding suitable mappings.^{[1]}We do this numerically by using the symmetry-breaking phase transition line of EQCD as a line of constant physics.

^{[2]}We study the resulting chiral symmetry breaking phase transition for models with $N_F=3$ and $N_F=4$ light flavors using the linear sigma model.

^{[3]}We do this numerically by using the symmetry-breaking phase transition point of the theory as a line of constant physics.

^{[4]}We theoretically and numerically demonstrate that the spontaneous parity-time (PT) symmetry breaking phase transition can be realized respectively by using two independent tuning ways in a tri-layered metamaterial that consists of periodic array of metal-semiconductor Schottky junctions.

^{[5]}Key pointsThe quantum spin liquid is an exotic state of matter in which interacting spins avoid symmetry-breaking phase transitions and form a ground state exhibiting long-range entanglement and topological order.

^{[6]}Here we consider a parity-time ($\mathcal{PT}$) symmetric extension of the Aubry-Andre-Harper model, undergoing a topological metal-insulator phase transition, and provide rigorous analytical results of energy spectrum, symmetry breaking phase transition and localization length.

^{[7]}Resonances, spectral singularities and eigenvalues are analyzed in detail and discussed in the context of the associated laser-absorber modes and $\mathcal{PT}$-symmetry breaking phase transition.

^{[8]}Coarsening dynamics, the canonical theory of phase ordering following a quench across a symmetry breaking phase transition, is thought to be driven by the annihilation of topological defects.

^{[9]}For this model, the dissipative phase transition (DPT), accompanying a parity-time (PT) symmetry-breaking phase transition, occurs when the imaginary field changes through the YLES.

^{[10]}A replica-symmetry-breaking phase transition is predicted in a host of disordered media.

^{[11]}Our purpose is discovering, if any, the relation between the second-order Z_{2}-symmetry-breaking phase transition and the geometric entities mentioned above.

^{[12]}In non-Hermitian crystals showing the non-Hermitian skin effect, ordinary Bloch band theory and Bloch topological invariants fail to correctly predict energy spectra, topological boundary states, and symmetry breaking phase transitions in systems with open boundaries.

^{[13]}In this paper, we report distinct symmetry-breaking phase transitions in molecules based on oligo-para-phenylene rods and a poly(ethylene oxide) coil depending on the length of the aromatic rod block The observed phase transitions were characterized by differential scanning calorimetry (DSC) and X-ray scattering methods.

^{[14]}Here, we unveil that non-unitary discrete-time quantum walks of photons in systems involving gain and loss show rather generally non-Bloch parity-time symmetry-breaking phase transitions and suggest a bulk probing method to detect such boundary-driven phase transitions.

^{[15]}We infer the existence of a symmetry-breaking phase transition at some finite coupling, and combine this with previous simulation results to deduce that the critical number of flavors for the existence of a quantum critical point in the Thirring model satisfies $0 1$.

^{[16]}We do this numerically by using the symmetry-breaking phase transition point of the theory as a line of constant physics.

^{[17]}The origin of the symmetry-breaking phase transition is associated with the motion or reorientation of the DMEA cations, accompanied by the crystal structures from orthorhombic Pnma to monoclinic P21/c with the temperature decreases.

^{[18]}