We consider the relative dynamics -- the dynamics modulo rotational symmetry in this particular context -- of $N$ vortices in confined Bose--Einstein Condensates (BEC) using a finite-dimensional vortex approximation to the two-dimensional Gross--Pitaevskii equation. We give a Hamiltonian formulation of the relative dynamics by showing that it is an instance of the Lie--Poisson equation on the dual of a certain Lie algebra. Just as in our accompanying work on vortex dynamics with the Euclidean symmetry, the relative dynamics possesses a Casimir invariant and evolves in an invariant set, yielding an Energy--Casimir-type stability condition. We consider three examples of relative equilibria -- those solutions that are undergoing rigid rotations about the origin -- with $N=2, 3, 4$, and investigate their stability using the stability condition.
{"title":"Relative Dynamics of Vortices in Confined Bose--Einstein Condensates","authors":"Tomoki Ohsawa","doi":"arxiv-2409.07657","DOIUrl":"https://doi.org/arxiv-2409.07657","url":null,"abstract":"We consider the relative dynamics -- the dynamics modulo rotational symmetry\u0000in this particular context -- of $N$ vortices in confined Bose--Einstein\u0000Condensates (BEC) using a finite-dimensional vortex approximation to the\u0000two-dimensional Gross--Pitaevskii equation. We give a Hamiltonian formulation\u0000of the relative dynamics by showing that it is an instance of the Lie--Poisson\u0000equation on the dual of a certain Lie algebra. Just as in our accompanying work\u0000on vortex dynamics with the Euclidean symmetry, the relative dynamics possesses\u0000a Casimir invariant and evolves in an invariant set, yielding an\u0000Energy--Casimir-type stability condition. We consider three examples of\u0000relative equilibria -- those solutions that are undergoing rigid rotations\u0000about the origin -- with $N=2, 3, 4$, and investigate their stability using the\u0000stability condition.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"22 3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. G. Edge, E. Paul, K. H. Madine, D. J. Colquitt, T. A. Starkey, G. J. Chaplain
The propagation of elastic waves on discrete periodic Euler-Bernoulli mass-beam lattices is characterised by the competition between coupled translational and rotational degrees-of-freedom at the mass-beam junctions. We influence the dynamics of this system by coupling junctions with beyond-nearest-neighbour spatial connections, affording freedom over the locality of dispersion extrema in reciprocal space, facilitating the emergence of interesting dispersion relations. A generalised dispersion relation for an infinite monatomic mass-beam chain, with any integer order combination of non-local spatial connections, is presented. We demonstrate that competing power channels, between mass and rotational inertia, drive the position and existence of zero group velocity modes within the first Brillouin zone.
{"title":"Discrete Euler-Bernoulli Beam Lattices with Beyond Nearest Connections","authors":"R. G. Edge, E. Paul, K. H. Madine, D. J. Colquitt, T. A. Starkey, G. J. Chaplain","doi":"arxiv-2409.07173","DOIUrl":"https://doi.org/arxiv-2409.07173","url":null,"abstract":"The propagation of elastic waves on discrete periodic Euler-Bernoulli\u0000mass-beam lattices is characterised by the competition between coupled\u0000translational and rotational degrees-of-freedom at the mass-beam junctions. We\u0000influence the dynamics of this system by coupling junctions with\u0000beyond-nearest-neighbour spatial connections, affording freedom over the\u0000locality of dispersion extrema in reciprocal space, facilitating the emergence\u0000of interesting dispersion relations. A generalised dispersion relation for an\u0000infinite monatomic mass-beam chain, with any integer order combination of\u0000non-local spatial connections, is presented. We demonstrate that competing\u0000power channels, between mass and rotational inertia, drive the position and\u0000existence of zero group velocity modes within the first Brillouin zone.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"37 8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider mid-spectrum eigenstates of the Sachdev-Ye-Kiteav (SYK) model. We prove that for subsystems whose size is a constant fraction of the system size, the entanglement entropy deviates from the maximum entropy by at least a positive constant. This result highlights the difference between the entanglement entropy of mid-spectrum eigenstates of the SYK model and that of random states.
{"title":"Deviations from maximal entanglement for eigenstates of the Sachdev-Ye-Kitaev model","authors":"Yichen Huang, Yi Tan, Norman Y. Yao","doi":"arxiv-2409.07043","DOIUrl":"https://doi.org/arxiv-2409.07043","url":null,"abstract":"We consider mid-spectrum eigenstates of the Sachdev-Ye-Kiteav (SYK) model. We\u0000prove that for subsystems whose size is a constant fraction of the system size,\u0000the entanglement entropy deviates from the maximum entropy by at least a\u0000positive constant. This result highlights the difference between the\u0000entanglement entropy of mid-spectrum eigenstates of the SYK model and that of\u0000random states.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"419 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cyrill B. Muratov, Theresa M. Simon, Valeriy V. Slastikov
We demonstrate existence of topologically nontrivial energy minimizing maps of a given positive degree from bounded domains in the plane to $mathbb S^2$ in a variational model describing magnetizations in ultrathin ferromagnetic films with Dzyaloshinskii-Moriya interaction. Our strategy is to insert tiny truncated Belavin-Polyakov profiles in carefully chosen locations of lower degree objects such that the total energy increase lies strictly below the expected Dirichlet energy contribution, ruling out loss of degree in the limits of minimizing sequences. The argument requires that the domain be either sufficiently large or sufficiently slender to accommodate a prescribed degree. We also show that these higher degree minimizers concentrate on point-like skyrmionic configurations in a suitable parameter regime.
{"title":"Existence of higher degree minimizers in the magnetic skyrmion problem","authors":"Cyrill B. Muratov, Theresa M. Simon, Valeriy V. Slastikov","doi":"arxiv-2409.07205","DOIUrl":"https://doi.org/arxiv-2409.07205","url":null,"abstract":"We demonstrate existence of topologically nontrivial energy minimizing maps\u0000of a given positive degree from bounded domains in the plane to $mathbb S^2$\u0000in a variational model describing magnetizations in ultrathin ferromagnetic\u0000films with Dzyaloshinskii-Moriya interaction. Our strategy is to insert tiny\u0000truncated Belavin-Polyakov profiles in carefully chosen locations of lower\u0000degree objects such that the total energy increase lies strictly below the\u0000expected Dirichlet energy contribution, ruling out loss of degree in the limits\u0000of minimizing sequences. The argument requires that the domain be either\u0000sufficiently large or sufficiently slender to accommodate a prescribed degree.\u0000We also show that these higher degree minimizers concentrate on point-like\u0000skyrmionic configurations in a suitable parameter regime.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
T. M. Lawrie, T. A. Starkey, G. Tanner, D. B. Moore, P. Savage, G. J. Chaplain
We leverage quantum graph theory to quickly and accurately characterise acoustic metamaterials comprising networks of interconnected pipes. Anisotropic bond lengths are incorporated in the model that correspond to space-coiled acoustic structures to exhibit dispersion spectra reminiscent of hyperbolic metamaterials. We construct two metasurfaces with embedded graph structure and, motivated by the graph theory, infer and fine-tune their dispersive properties to engineer non-resonant negative refraction of acoustic surface waves at their interface. Agreement between the graph model, full wave simulations, and experiments bolsters quantum graph theory as a new paradigm for metamaterial design.
{"title":"Application of Quantum Graph Theory to Metamaterial Design: Negative Refraction of Acoustic Waveguide Modes","authors":"T. M. Lawrie, T. A. Starkey, G. Tanner, D. B. Moore, P. Savage, G. J. Chaplain","doi":"arxiv-2409.07133","DOIUrl":"https://doi.org/arxiv-2409.07133","url":null,"abstract":"We leverage quantum graph theory to quickly and accurately characterise\u0000acoustic metamaterials comprising networks of interconnected pipes. Anisotropic\u0000bond lengths are incorporated in the model that correspond to space-coiled\u0000acoustic structures to exhibit dispersion spectra reminiscent of hyperbolic\u0000metamaterials. We construct two metasurfaces with embedded graph structure and,\u0000motivated by the graph theory, infer and fine-tune their dispersive properties\u0000to engineer non-resonant negative refraction of acoustic surface waves at their\u0000interface. Agreement between the graph model, full wave simulations, and\u0000experiments bolsters quantum graph theory as a new paradigm for metamaterial\u0000design.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A pedagogical introduction to Bell's inequality in Quantum Mechanics is presented. Several examples, ranging from spin $1/2$ to coherent and squeezed states are worked out. The generalization to Mermin's inequalities and to GHZ states is also outlined.
{"title":"Introduction to Bell's inequality in Quantum Mechanics","authors":"M. S. Guimaraes, I. Roditi, S. P. Sorella","doi":"arxiv-2409.07597","DOIUrl":"https://doi.org/arxiv-2409.07597","url":null,"abstract":"A pedagogical introduction to Bell's inequality in Quantum Mechanics is\u0000presented. Several examples, ranging from spin $1/2$ to coherent and squeezed\u0000states are worked out. The generalization to Mermin's inequalities and to GHZ\u0000states is also outlined.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Grothedieck bound formalism is studied using `rescaling transformations', in the context of a single quantum system. The rescaling transformations enlarge the set of unitary transformations (which apply to isolated systems), with transformations that change not only the phase but also the absolute value of the wavefunction, and can be linked to irreversible phenomena (e.g., quantum tunnelling, damping and amplification, etc). A special case of rescaling transformations are the dequantisation transformations, which map a Hilbert space formalism into a formalism of scalars. The Grothendieck formalism considers a `classical' quadratic form ${cal C}(theta)$ which takes values less than $1$, and the corresponding `quantum' quadratic form ${cal Q}(theta)$ which takes values greater than $1$, up to the complex Grothendieck constant $k_G$. It is shown that ${cal Q}(theta)$ can be expressed as the trace of the product of $theta$ with two rescaling matrices, and ${cal C}(theta)$ can be expressed as the trace of the product of $theta$ with two dequantisation matrices. Values of ${cal Q}(theta)$ in the `ultra-quantum' region $(1,k_G)$ are very important, because this region is classically forbidden (${cal C}(theta)$ cannot take values in it). An example with ${cal Q}(theta)in (1,k_G)$ is given, which is related to phenomena where classically isolated by high potentials regions of space, communicate through quantum tunnelling. Other examples show that `ultra-quantumness' according to the Grothendieck formalism (${cal Q}(theta)in (1,k_G)$), is different from quantumness according to other criteria (like quantum interference or the uncertainty principle).
{"title":"Rescaling transformations and the Grothendieck bound formalism in a single quantum system","authors":"A. Vourdas","doi":"arxiv-2409.07270","DOIUrl":"https://doi.org/arxiv-2409.07270","url":null,"abstract":"The Grothedieck bound formalism is studied using `rescaling transformations',\u0000in the context of a single quantum system. The rescaling transformations\u0000enlarge the set of unitary transformations (which apply to isolated systems),\u0000with transformations that change not only the phase but also the absolute value\u0000of the wavefunction, and can be linked to irreversible phenomena (e.g., quantum\u0000tunnelling, damping and amplification, etc). A special case of rescaling\u0000transformations are the dequantisation transformations, which map a Hilbert\u0000space formalism into a formalism of scalars. The Grothendieck formalism\u0000considers a `classical' quadratic form ${cal C}(theta)$ which takes values\u0000less than $1$, and the corresponding `quantum' quadratic form ${cal\u0000Q}(theta)$ which takes values greater than $1$, up to the complex Grothendieck\u0000constant $k_G$. It is shown that ${cal Q}(theta)$ can be expressed as the\u0000trace of the product of $theta$ with two rescaling matrices, and ${cal\u0000C}(theta)$ can be expressed as the trace of the product of $theta$ with two\u0000dequantisation matrices. Values of ${cal Q}(theta)$ in the `ultra-quantum'\u0000region $(1,k_G)$ are very important, because this region is classically\u0000forbidden (${cal C}(theta)$ cannot take values in it). An example with ${cal\u0000Q}(theta)in (1,k_G)$ is given, which is related to phenomena where\u0000classically isolated by high potentials regions of space, communicate through\u0000quantum tunnelling. Other examples show that `ultra-quantumness' according to\u0000the Grothendieck formalism (${cal Q}(theta)in (1,k_G)$), is different from\u0000quantumness according to other criteria (like quantum interference or the\u0000uncertainty principle).","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sebastian Bahamonde, Jorge Gigante Valcarcel, José M. M. Senovilla
We present the algebraic classification of the gravitational field in four-dimensional general metric-affine geometries, thus extending the current results of the literature in the particular framework of Weyl-Cartan geometry by the presence of the traceless nonmetricity tensor. This quantity switches on four of the eleven fundamental parts of the irreducible representation of the curvature tensor under the pseudo-orthogonal group, in such a way that three of them present similar algebraic types as the ones obtained in Weyl-Cartan geometry, whereas the remaining one includes thirty independent components and gives rise to a new algebraic classification. The latter is derived by means of its principal null directions and their levels of alignment, obtaining a total number of sixteen main algebraic types, which can be split into many subtypes. As an immediate application, we determine the algebraic types of the broadest family of static and spherically symmetric black hole solutions with spin, dilation and shear charges in Metric-Affine Gravity.
{"title":"Algebraic classification of the gravitational field in general metric-affine geometries","authors":"Sebastian Bahamonde, Jorge Gigante Valcarcel, José M. M. Senovilla","doi":"arxiv-2409.07153","DOIUrl":"https://doi.org/arxiv-2409.07153","url":null,"abstract":"We present the algebraic classification of the gravitational field in\u0000four-dimensional general metric-affine geometries, thus extending the current\u0000results of the literature in the particular framework of Weyl-Cartan geometry\u0000by the presence of the traceless nonmetricity tensor. This quantity switches on\u0000four of the eleven fundamental parts of the irreducible representation of the\u0000curvature tensor under the pseudo-orthogonal group, in such a way that three of\u0000them present similar algebraic types as the ones obtained in Weyl-Cartan\u0000geometry, whereas the remaining one includes thirty independent components and\u0000gives rise to a new algebraic classification. The latter is derived by means of\u0000its principal null directions and their levels of alignment, obtaining a total\u0000number of sixteen main algebraic types, which can be split into many subtypes.\u0000As an immediate application, we determine the algebraic types of the broadest\u0000family of static and spherically symmetric black hole solutions with spin,\u0000dilation and shear charges in Metric-Affine Gravity.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We utilize three complementary approaches to pinpoint the exact form of scattering amplitudes in Schwarzschild spacetime. First, we solve the Regge-Wheeler equation perturbatively in the small-frequency regime. We use the obtained solutions to determine the monodromy in the near-spatial infinity region, which leads to a specific partial differential equation on the elements of the scattering matrix. As a result, it can be written in terms of the elements of the infinitesimal generator of the monodromy transformation and an integration constant. This constant is further related to the Nekrasov-Shatashvili free energy through the resummation of infinitely many instantons. The quasinormal mode frequencies are also studied in the small-frequency approximation.
{"title":"Black hole scattering amplitudes via analytic small-frequency expansion and monodromy","authors":"Gleb Aminov, Paolo Arnaudo","doi":"arxiv-2409.06681","DOIUrl":"https://doi.org/arxiv-2409.06681","url":null,"abstract":"We utilize three complementary approaches to pinpoint the exact form of\u0000scattering amplitudes in Schwarzschild spacetime. First, we solve the\u0000Regge-Wheeler equation perturbatively in the small-frequency regime. We use the\u0000obtained solutions to determine the monodromy in the near-spatial infinity\u0000region, which leads to a specific partial differential equation on the elements\u0000of the scattering matrix. As a result, it can be written in terms of the\u0000elements of the infinitesimal generator of the monodromy transformation and an\u0000integration constant. This constant is further related to the\u0000Nekrasov-Shatashvili free energy through the resummation of infinitely many\u0000instantons. The quasinormal mode frequencies are also studied in the\u0000small-frequency approximation.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present thought experiments to measure the Arnowitt-Deser-Misner and Bondi-Sachs energy of isolated systems in general relativity. The expression of the Bondi-Sachs energy used in the protocol is likely to have other applications. In particular, it is well-suited to to be promoted to an operator in non-perturbative loop quantum gravity.
{"title":"The Operational Meaning of Total Energy of Isolated Systems in General Relativity","authors":"Abhay Ashtekar, Simone Speziale","doi":"arxiv-2409.06698","DOIUrl":"https://doi.org/arxiv-2409.06698","url":null,"abstract":"We present thought experiments to measure the Arnowitt-Deser-Misner and\u0000Bondi-Sachs energy of isolated systems in general relativity. The expression of\u0000the Bondi-Sachs energy used in the protocol is likely to have other\u0000applications. In particular, it is well-suited to to be promoted to an operator\u0000in non-perturbative loop quantum gravity.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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