Modelling quantum black holes

Govindarajan, T. R.; Munõz-Castañeda, J. M.

doi:10.1142/s0217732316502102

Cited by 4 publications

(9 citation statements)

References 33 publications

(39 reference statements)

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“…In Fig.1 we plot two configurations in two dimensions which illustrate the results of Propositions 1 and 2. The results obtained in [37] for d = 2 and d = 3 are recovered when |w 0 |x 0 |w 1 | (for d = 3 there is minus sign and the integer part missing). To end this section, let us briefly study the behaviour of the number of negative energy eigenvalues as a function of the dimension d and the angular momentum .…”

Section: On the Number Of Bound Statesmentioning

confidence: 84%

“…We have introduced a rigorous and consistent definition of the potential V δ-δ = w 0 δ(x − x 0 ) + 2w 1 δ (x − x 0 ) in arbitrary dimension, characterizing a selfadjoint extension of the Hamiltonian H 0 (14) defined on R x 0 . In doing so, we have corrected the matching conditons in [37] for the two and three dimensional V δ-δ potential. We have shown that the Dirac-δ coupling requires a re-definition which also depends on the radius x 0 and the δ coupling w 1 .…”

Section: Discussionmentioning

confidence: 99%

“…The former, bearing in mind the original ideas by G. 't Hooft [3,4], would correspond to the states of quantum particles falling into a black hole that would be observed by a distant observer. Indeed, the amount of bound states for two and three dimensions is proportional to x 0 and x 2 0 respectively and as it is mentioned 3 in [37] these bound states would give an area law for the corresponding entropy in quantum field theory when they are interpreted as micro-states of the black hole horizon.…”

Section: The Mean Value Of the Position Operatormentioning

confidence: 94%

“…where w± 0 = w 0 ± 2(1 − d)/x 0 . Recently the potential (11) was studied for two and three dimensions in [37] where the matching conditions used for R λ are those in (16) instead of (18), an approximation which is only satisfied if…”

Section: The δ-δ Interaction In the D-dimensional Schrödinger Equationmentioning

confidence: 99%

“…where y 0 ≡ κx 0 and the right hand side is independent of the energy and the angular momentum. For d = 2, 3 the results of [37] are obtained as a limiting case (x 0 |w 0 | |w 1 |). In particular, the secular equation for the δ-potential (α = 1 and β = w 0 ) is…”

Section: Bound States With the Free Hamiltonian And The Singular Inte...mentioning

confidence: 99%

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Hyperspherical δ-δ′ potentials

2019

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The spherically symmetric potential a δ(r − r 0 ) + b δ (r − r 0 ) is generalised for the ddimensional space as a characterisation of a unique selfadjoint extension of the free Hamiltonian. For this extension of the Dirac delta, the spectrum of negative, zero and positive energy states is studied in d ≥ 2, providing numerical results for the expectation value of the radius as a function of the free parameters of the potential. Remarkably, only if d = 2 the δ-δ potential for arbitrary a > 0 admits a bound state with zero angular momentum.

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Section: On the Number Of Bound Statesmentioning

confidence: 84%

Section: Discussionmentioning

confidence: 99%

Section: The Mean Value Of the Position Operatormentioning

confidence: 94%

Section: The δ-δ Interaction In the D-dimensional Schrödinger Equationmentioning

confidence: 99%

Section: Bound States With the Free Hamiltonian And The Singular Inte...mentioning

confidence: 99%

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Hyperspherical δ-δ′ potentials

2019

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Effective theories for quantum spin clusters: Geometric phases and state selection by singularity

2019

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Magnetic systems with frustration often have large classical degeneracy. We show that their low-energy physics can be understood as dynamics within the space of classical ground states. We demonstrate this mapping in a family of quantum spin clusters where every pair of spins is connected by an XY antiferromagnetic bond. The dimer with two spin-S spins provides the simplest example -it maps to a quantum particle on a ring (S 1 ). The trimer is more complex, equivalent to a particle that lives on two disjoint rings (S 1 ⊗ Z2). It has an additional subtlety for half-integer S values, wherein both rings must be threaded by π-fluxes to obtain a satisfactory mapping. This is a consequence of the geometric phase incurred by spins. For both the dimer and the trimer, the validity of the effective theory can be seen from a path-integral-based derivation. This approach cannot be extended to the quadrumer which has a non-manifold ground state space, consisting of three tori that touch pairwise along lines. In order to understand the dynamics of a particle in this space, we develop a tight-binding model with this connectivity. Remarkably, this successfully reproduces the low-energy spectrum of the quadrumer. For half-integer spins, a geometric phase emerges which can be mapped to two π-flux tubes that reside in the space between the tori. The non-manifold character of the space leads to a remarkable effect -the dynamics at low energies is not ergodic as the particle is localized around singular lines of the ground state space. The low-energy spectrum consists of an extensive number of bound states formed around singularities. Physically, this manifests as an order-by-disorder-like preference for collinear ground states. However, unlike order-by-disorder, this 'order by singularity' persists even in the classical limit. We discuss consequences for field theoretic studies of magnets.

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