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Register a new userFurther studies on holographic insulator/superconductor phase transitions from Sturm-Liouville eigenvalue problems
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Huai-Fan Li
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We take advantage of the Sturm-Liouville eigenvalue problem to analytically study the holographic insulator/superconductor phase transition in the probe limit. The interesting point is that this analytical method can not only estimate the most stable mode of the phase transition, but also the second stable mode. We find that this analytical method perfectly matches with other numerical methods, such as the shooting method. Besides, we argue that only Dirichlet boundary condition of the trial function is enough under certain circumstances, which will lead to a more precise estimation. This relaxation for the boundary condition of the trial function is first observed in this paper as far as we know.
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We construct a family of solutions of the holographic insulator/superconductor phase transitions with the excited states in the AdS soliton background by using both the numerical and analytical methods. The interesting point is that the improved Sturm-Liouville method can not only analytically investigate the properties of the phase transition with the excited states, but also the distributions of the condensed fields in the vicinity of the critical point. We observe that, regardless of the type of the holographic model, the excited state has a higher critical chemical potential than the corresponding ground state, and the difference of the dimensionless critical chemical potential between the consecutive states is around 2.4, which is different from the finding of the metal/superconductor phase transition in the AdS black hole background. Furthermore, near the critical point, we find that the phase transition of the systems is of the second order and a linear relationship exists between the charge density and chemical potential for all the excited states in both s-wave and p-wave insulator/superconductor models.
We introduce and present the general solution of three two-term fractional differential equations of mixed Caputo/Riemann Liouville type. We then solve a Dirichlet type Sturm-Liouville eigenvalue problem for a fractional differential equation derived from a special composition of a Caputo and a Riemann-Liouville operator on a finite interval where the boundary conditions are induced by evaluating Riemann-Liouville integrals at those end-points. For each $1/2<alpha<1$ it is shown that there is a finite number of real eigenvalues, an infinite number of non-real eigenvalues, that the number of such real eigenvalues grows without bound as $alpha to 1^-$, and that the fractional operator converges to an ordinary two term Sturm-Liouville operator as $alpha to 1^-$ with Dirichlet boundary conditions. Finally, two-sided estimates as to their location are provided as is their asymptotic behavior as a function of $alpha$.
We continue the study of a non self-adjoint fractional three-term Sturm-Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left-Riemann-Liouville fractional integral under {it Dirichlet type} boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter $alpha$, $0<alpha<1$, there is a finite set of real eigenvalues and that, for $alpha$ near $1/2$, there may be none at all. As $alpha to 1^-$ we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm-Liouville problem with the composition of the operators becoming the operator of second order differentiation.
We show that our Universe lives in a topological and non-perturbative vacuum state full of a large amount of hidden quantum hairs, the hairons. We will discuss and elaborate on theoretical evidences that the quantum hairs are related to the gravitational topological winding number in vacuo. Thus, hairons are originated from topological degrees of freedom, holographically stored in the de Sitter area. The hierarchy of the Planck scale over the Cosmological Constant (CC) is understood as an effect of a Topological Memory intrinsically stored in the space-time geometry. Any UV quantum destabilizations of the CC are re-interpreted as Topological Phase Transitions, related to the desapparence of a large ensamble of topological hairs. This process is entropically suppressed, as a tunneling probability from the N- to the 0-states. Therefore, the tiny CC in our Universe is a manifestation of the rich topological structure of the space-time. In this portrait, a tiny neutrino mass can be generated by quantum gravity anomalies and accommodated into a large N-vacuum state. We will re-interpret the CC stabilization from the point of view of Topological Quantum Computing. An exponential degeneracy of topological hairs non-locally protects the space-time memory from quantum fluctuations as in Topological Quantum Computers.
Sturm-Liouville spectral problem for equation $-(y/r)+qy=lambda py$ with generalized functions $rge 0$, $q$ and $p$ is considered. It is shown that the problem may be reduced to analogous problem with $requiv 1$. The case of $q=0$ and self-similar $r$ and $p$ is considered as an example.
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