Excitations gapless

We recall that we can saturate the t Hooft anomaly conditions either with massless fermionic degrees of freedom or with gapless bosonic excitations. However in absence of Lorentz covariance the bosonic excitations are not restricted to be fluctuations related to scalar condensates but may be associated, for example, to vector condensates [51]. 

When the external magnetic field exceeds a critical value Hc in gapped lowdimensional spin systems it closes the gap and drives the system into a new critical phase with finite magnetization and gapless excitations. When the field is further increased, the system may stay in this critical phase up to the saturation field, above which the system is in a saturated FM state. Under certain conditions, however, the excitations in this high-field phase may again acquire a gap,... 

The excitation spectrum is gapless and has a sound-like behavior [18]. The... 

As far as the continuous symmetry breaking is concerned, the Goldstone theorem states [4] that this will generate hydrodynamic modes, that is, gapless excitations. The order parameter is multicomponent (n > 1) a vector i for magnetism breaking the rotational symmetry, a complex variable i j = ip e/e for charge-density waves and for superconductivity which... 

The physics of free carriers is dominated by the Fermi surface. Looking at the linearized spectrum of Fig. 2, one realizes that electron-hole or electron-electron excitations involving quasiparticles (electrons or holes) on each side of the Fermi surface are gapless. This greatly influences the response functions to external fields. Let an external field Fa(q) couple to the operator Oa(q), where... 

For U < 0, the analysis of Emery [16] captures the essence. The attractive interaction favors singlet pairing and the system develops superconducting fluctuations. The charge- or pair-density fluctuations are gapless in the absence of umklapp processes. The spin excitations develop a gap of order U since singlet pairs need to be broken to create spin excitations. 

If the system is nearly commensurate, discommensurations will naturally form and separate ordered commensurate phases. These discommensurations have a gapless excitation spectrum [60]. This subject has been reviewed in Ref. 61. 

To elucidate the rdle of long-wavelength excitations in the asymptotic behavior of correlation functions, the authors write down a semiphenomeno-logical Hamiltonian describing long-wave gapless excitations,... 

The mean field theory of the paired holon superconductor and its predictions are reviewed for the case in which charged holons on different magnetic sublattices in a doped Cu02 layer interact via a weak attractive pairing potential V. The physical properties of this superconductor often reflect the essential gaplessness of its excitation spectrum. 

Fig. 7. Frequency dependence of the conductivity of the weakly coupled paired holon superconductor (X = 0.1). The excitation spectrum is gapless at T = 0.

In summary, we have reviewed the predicted mean field properties of the paired holon superconductor. Mostly, they differ from the BCS case only in the details of their T-dependences. The microwave absorption, however, is sharply different and directly reflects the essential gaplessness of the excitation spectrum. 

In an undimerized chain a spin-density wave exhibits gapless spin excitations and gapped charged excitations. However, in a dimerized chain all three types of order exhibit gapped spin and charge excitations. In fact, for a dimerized chain the spin-density and bond order waves coexist. Mazumdar and Campbell have shown (Mazumdar and Campbell 1985) that the Pariser-Pople-Parr model will exhibit a broken-symmetry ground state provided that. 

Once the specimen turns to a superconducting state, the obtained superconductor-insulator-normal metal (SIN) spectrum probes the quasiparticle excitation in the superconductor, which directly reflects the symmetry of the order parameter A(k). If A(k) has simple s-wave symmetry, as is realized in conventional low-temperature superconductors, one expects a finite gap of A with overshooting peaks just outside the gap in N(E), as illustrated in fig. 6. Even if A(k) possesses anisotropic s-wave symmetry, a finite gap, corresponding to the minimum gap, appears. In dx2-yi superconductors with A(k) = coslkx - cos 2, in contrast, N(E) is gapless with linear N(E) for E A. It is noted that the extended-s wave A(x) = cos 2kx + cos 2ky is also characterized to possess a gapless feature with two singularities bX E = A and A2. 

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