Pub Date : 2023-10-31DOI: 10.1088/1751-8121/ad0885
Gernot Akemann, Sung-Soo Byun, Markus Ebke, Gregory Schehr
Abstract In this article, we compute and compare the statistics of the number of eigenvalues in a centred disc of radius $R$ in all three Ginibre ensembles. We determine the mean and variance as functions of $R$ in the vicinity of the origin, where the real and symplectic ensembles exhibit respectively an additional attraction to or repulsion from the real axis, leading to different results. In the large radius limit, all three ensembles coincide and display a universal bulk behaviour of $O(R^2)$ for the mean, and $O(R)$ for the variance. We present detailed conjectures for the bulk and edge scaling behaviours of the real Ginibre ensemble, having real and complex eigenvalues. For the symplectic ensemble we can go beyond the Gaussian case (corresponding to the Ginibre ensemble) and prove the universality of the full counting statistics both in the bulk and at the edge of the spectrum for rotationally invariant potentials, extending a recent work which considered the mean and the variance. This statistical behaviour coincides with the universality class of the complex Ginibre ensemble, which has been shown to be associated with the ground state of non-interacting fermions in a two-dimensional rotating harmonic trap. All our analytical results and conjectures are corroborated by numerical simulations.
{"title":"Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble","authors":"Gernot Akemann, Sung-Soo Byun, Markus Ebke, Gregory Schehr","doi":"10.1088/1751-8121/ad0885","DOIUrl":"https://doi.org/10.1088/1751-8121/ad0885","url":null,"abstract":"Abstract In this article, we compute and compare the statistics of the number of eigenvalues in a centred disc of radius $R$ in all three Ginibre ensembles. We determine the mean and variance as functions of $R$ in the vicinity of the origin, where the real and symplectic ensembles exhibit respectively an additional attraction to or repulsion from the real axis, leading to different results. In the large radius limit, all three ensembles coincide and display a universal bulk behaviour of $O(R^2)$ for the mean, and $O(R)$ for the variance. We present detailed conjectures for the bulk and edge scaling behaviours of the real Ginibre ensemble, having real and complex eigenvalues. For the symplectic ensemble we can go beyond the Gaussian case (corresponding to the Ginibre ensemble) and prove the universality of the full counting statistics both in the bulk and at the edge of the spectrum for rotationally invariant potentials, extending a recent work which considered the mean and the variance. This statistical behaviour coincides with the universality class of the complex Ginibre ensemble, which has been shown to be associated with the ground state of non-interacting fermions in a two-dimensional rotating harmonic trap. All our analytical results and conjectures are corroborated by numerical simulations.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"98 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135870901","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}
Pub Date : 2023-10-30DOI: 10.1088/1751-8121/ad0438
Apostol Vourdas
Abstract A set of n coherent states is introduced in a quantum system with d -dimensional Hilbert space H ( d ). It is shown that they resolve the identity, and also have a discrete isotropy property. A finite cyclic group acts on the set of these coherent states, and partitions it into orbits. A n -tuple representation of arbitrary states in H ( d ), analogous to the Bargmann representation, is defined. There are two other important properties of these coherent states which make them ‘ultra-quantum’. The first property is related to the Grothendieck formalism which studies the ‘edge’ of the Hilbert space and quantum formalisms. Roughly speaking the Grothendieck theorem considers a ‘classical’ quadratic form C that uses complex numbers in the unit disc, and a ‘quantum’ quadratic form Q that uses vectors in the unit ball of the Hilbert space. It shows that if C⩽1 , the corresponding Q might take values greater than 1, up to the complex Grothendieck constant kG . Q related to these coherent states is shown to take values in the ‘Grothendieck region’ (1,kG) , which is classically forbidden in the sense that C does not take values in it. The second property complements this, showing that these coheren
{"title":"Ultra-quantum coherent states in a single finite quantum system","authors":"Apostol Vourdas","doi":"10.1088/1751-8121/ad0438","DOIUrl":"https://doi.org/10.1088/1751-8121/ad0438","url":null,"abstract":"Abstract A set of n coherent states is introduced in a quantum system with d -dimensional Hilbert space H ( d ). It is shown that they resolve the identity, and also have a discrete isotropy property. A finite cyclic group acts on the set of these coherent states, and partitions it into orbits. A n -tuple representation of arbitrary states in H ( d ), analogous to the Bargmann representation, is defined. There are two other important properties of these coherent states which make them ‘ultra-quantum’. The first property is related to the Grothendieck formalism which studies the ‘edge’ of the Hilbert space and quantum formalisms. Roughly speaking the Grothendieck theorem considers a ‘classical’ quadratic form <?CDATA ${mathfrak C}$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"fraktur\">C</mml:mi> </mml:mrow> </mml:mrow> </mml:math> that uses complex numbers in the unit disc, and a ‘quantum’ quadratic form <?CDATA ${mathfrak Q}$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"fraktur\">Q</mml:mi> </mml:mrow> </mml:mrow> </mml:math> that uses vectors in the unit ball of the Hilbert space. It shows that if <?CDATA ${mathfrak C}unicode{x2A7D} 1$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"fraktur\">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mtext>⩽</mml:mtext> <mml:mn>1</mml:mn> </mml:math> , the corresponding <?CDATA ${mathfrak Q}$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"fraktur\">Q</mml:mi> </mml:mrow> </mml:mrow> </mml:math> might take values greater than 1, up to the complex Grothendieck constant <?CDATA $k_mathrm{G}$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:msub> <mml:mi>k</mml:mi> <mml:mrow> <mml:mi mathvariant=\"normal\">G</mml:mi> </mml:mrow> </mml:msub> </mml:math> . <?CDATA ${mathfrak Q}$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"fraktur\">Q</mml:mi> </mml:mrow> </mml:mrow> </mml:math> related to these coherent states is shown to take values in the ‘Grothendieck region’ <?CDATA $(1,k_mathrm{G})$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mrow> <mml:mi mathvariant=\"normal\">G</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:math> , which is classically forbidden in the sense that <?CDATA ${mathfrak C}$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"fraktur\">C</mml:mi> </mml:mrow> </mml:mrow> </mml:math> does not take values in it. The second property complements this, showing that these coheren","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136019119","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}
Pub Date : 2023-10-30DOI: 10.1088/1751-8121/ad03cd
Ba Phi Nguyen, Kihong Kim
Abstract We present a numerical study of the transport and localization properties of excitations in one-dimensional lattices with diagonal disordered mosaic modulations. The model is characterized by the modulation period κ and the disorder strength W . We calculate the disorder averages ⟨T⟩ , 〈lnT〉 , and ⟨P⟩ , where T is the transmittance and P is the participation ratio, as a function of energy E and system size L , for different values of κ and W . For excitations at quasiresonance energies determined by κ , we find power-law scaling behaviors of the form ⟨T⟩∝L−γa , ⟨lnT⟩≈−γglnL , and ⟨P⟩∝Lβ , as L increases to a large value. In the strong disorder limit, the exponents are seen to saturate at the values γa∼0.5 , γg∼
摘要本文对具有对角无序镶嵌调制的一维晶格中激发的输运和局域化性质进行了数值研究。该模型的特征是调制周期κ和无序强度W。我们计算⟨T⟩，< ln T >和⟨P⟩的无序平均值，其中T是透射率，P是参与率，作为能量E和系统大小L的函数，用于不同的κ和W值。对于由κ确定的准共振能量的激发，我们发现⟨T⟩∝L−γ a，⟨ln T⟩≈−γ g ln L，和⟨P⟩∝L β的幂律缩放行为，当L增加到一个大值时。在强无序极限下，指数在γ a ~ 0.5、γ g ~ 1和β ~ 0.3处饱和，与准共振能量值无关。这种行为与发生在所有其他能量的指数局域化行为相反。准共振能量下参与比谱的尖峰的出现为反常幂律局域化现象的存在提供了额外的证据。相应的特征态表现出多重分形行为和独特的节点结构。此外，我们研究了时间相关的波包动力学，并计算均方位移⟨m2 (t)⟩，空间概率分布，参与数和返回概率。当波包的初始动量满足准共振条件时，我们观察到波包的次扩散扩散，其特征为⟨m 2 (t)⟩∝t η，其中η总是小于1。我们还注意到在准共振能量处出现部分局域化，这可以通过参与数的饱和和长时间返回概率的非零值来表示。
{"title":"Transport and localization properties of excitations in one-dimensional lattices with diagonal disordered mosaic modulations","authors":"Ba Phi Nguyen, Kihong Kim","doi":"10.1088/1751-8121/ad03cd","DOIUrl":"https://doi.org/10.1088/1751-8121/ad03cd","url":null,"abstract":"Abstract We present a numerical study of the transport and localization properties of excitations in one-dimensional lattices with diagonal disordered mosaic modulations. The model is characterized by the modulation period κ and the disorder strength W . We calculate the disorder averages <?CDATA $langle Trangle$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo> <mml:mi>T</mml:mi> <mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo> </mml:math> , <?CDATA $langle ln Trangle$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mrow> <mml:mo>〈</mml:mo> <mml:mi>ln</mml:mi> <mml:mi>T</mml:mi> <mml:mo>〉</mml:mo> </mml:mrow> </mml:math> , and <?CDATA $langle Prangle$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo> <mml:mi>P</mml:mi> <mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo> </mml:math> , where T is the transmittance and P is the participation ratio, as a function of energy E and system size L , for different values of κ and W . For excitations at quasiresonance energies determined by κ , we find power-law scaling behaviors of the form <?CDATA $langle T rangle propto L^{-gamma_{a}}$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo> <mml:mi>T</mml:mi> <mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo> <mml:mo>∝</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msup> </mml:math> , <?CDATA $langle ln T rangle approx -gamma_g ln L$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo> <mml:mo form=\"prefix\">ln</mml:mo> <mml:mi>T</mml:mi> <mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo> <mml:mo>≈</mml:mo> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo form=\"prefix\">ln</mml:mo> <mml:mi>L</mml:mi> </mml:math> , and <?CDATA $langle P rangle propto L^{beta}$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo> <mml:mi>P</mml:mi> <mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo> <mml:mo>∝</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> </mml:msup> </mml:math> , as L increases to a large value. In the strong disorder limit, the exponents are seen to saturate at the values <?CDATA $gamma_a sim 0.5$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mo>∼</mml:mo> <mml:mn>0.5</mml:mn> </mml:math> , <?CDATA $gamma_g sim 1$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>∼</mml:","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"22 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136019105","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}
Pub Date : 2023-10-30DOI: 10.1088/1751-8121/ad0803
Martin Šípka, Michal Pavelka, Oğul Esen, M Grmela
Abstract In this paper, we present neural networks learning mechanical systems that are both symplectic
(for instance particle mechanics) and non-symplectic (for instance rotating rigid body). Mechanical
systems have Hamiltonian evolution, which consists of two building blocks: a Poisson bracket and an
energy functional. We feed a set of snapshots of a Hamiltonian system to our neural network models
which then find both the two building blocks. In particular, the models distinguish between symplectic
systems (with non-degenerate Poisson brackets) and non-symplectic systems (degenerate brackets).
In contrast with earlier works, our approach does not assume any further a priori information about
the dynamics except its Hamiltonianity, and it returns Poisson brackets that satisfy Jacobi identity.
Finally, the models indicate whether a system of equations is Hamiltonian or not.
{"title":"Direct Poisson neural networks: learning non-symplectic mechanical systems","authors":"Martin Šípka, Michal Pavelka, Oğul Esen, M Grmela","doi":"10.1088/1751-8121/ad0803","DOIUrl":"https://doi.org/10.1088/1751-8121/ad0803","url":null,"abstract":"Abstract In this paper, we present neural networks learning mechanical systems that are both symplectic&#xD;(for instance particle mechanics) and non-symplectic (for instance rotating rigid body). Mechanical&#xD;systems have Hamiltonian evolution, which consists of two building blocks: a Poisson bracket and an&#xD;energy functional. We feed a set of snapshots of a Hamiltonian system to our neural network models&#xD;which then find both the two building blocks. In particular, the models distinguish between symplectic&#xD;systems (with non-degenerate Poisson brackets) and non-symplectic systems (degenerate brackets).&#xD;In contrast with earlier works, our approach does not assume any further a priori information about&#xD;the dynamics except its Hamiltonianity, and it returns Poisson brackets that satisfy Jacobi identity.&#xD;Finally, the models indicate whether a system of equations is Hamiltonian or not.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"11 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136019191","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}
Pub Date : 2023-10-30DOI: 10.1088/1751-8121/ad03a5
Tristan Lawrie, Sven Gnutzmann, Gregor K Tanner
Abstract In this work we present a three step procedure for generating a closed form expression of the Green’s function on both closed and open finite quantum graphs with general self-adjoint matching conditions. We first generalize and simplify the approach by Barra and Gaspard (2001 Phys. Rev. E 65 016205) and then discuss the validity of the explicit expressions. For compact graphs, we show that the explicit expression is equivalent to the spectral decomposition as a sum over poles at the discrete energy eigenvalues with residues that contain projector kernel onto the corresponding eigenstate. The derivation of the Green’s function is based on the scattering approach, in which stationary solutions are constructed by treating each vertex or subgraph as a scattering site described by a scattering matrix. The latter can then be given in a simple closed form from which the Green’s function is derived. The relevant scattering matrices contain inverse operators which are not well defined for wave numbers at which bound states in the continuum exists. It is shown that the singularities in the scattering matrix related to these bound states or perfect scars can be regularised. Green’s functions or scattering matrices can then be expressed as a sum of a regular and a singular part where the singular part contains the projection kernel onto the perfect scar.
本文给出了在具有一般自伴随匹配条件的封闭和开放有限量子图上生成格林函数的封闭形式表达式的三步过程。我们首先概括和简化了Barra和Gaspard(2001物理学)的方法。Rev. E 65 016205)，然后讨论显式表达式的有效性。对于紧致图，我们证明了显式表达式等价于谱分解为离散能量特征值的极点和，其残基包含投影核到相应的特征态上。格林函数的推导基于散射方法，其中通过将每个顶点或子图视为由散射矩阵描述的散射点来构造平稳解。后者可以用格林函数的简单封闭形式给出。相关的散射矩阵包含逆算符，这些逆算符对于连续体中存在束缚态的波数没有很好的定义。结果表明，与这些束缚态或完美伤痕相关的散射矩阵中的奇异点可以正则化。然后，格林函数或散射矩阵可以表示为正则部分和奇异部分的和，其中奇异部分包含到完美疤痕上的投影核。
{"title":"Closed form expressions for the Green’s function of a quantum graph – a scattering approach","authors":"Tristan Lawrie, Sven Gnutzmann, Gregor K Tanner","doi":"10.1088/1751-8121/ad03a5","DOIUrl":"https://doi.org/10.1088/1751-8121/ad03a5","url":null,"abstract":"Abstract In this work we present a three step procedure for generating a closed form expression of the Green’s function on both closed and open finite quantum graphs with general self-adjoint matching conditions. We first generalize and simplify the approach by Barra and Gaspard (2001 Phys. Rev. E 65 016205) and then discuss the validity of the explicit expressions. For compact graphs, we show that the explicit expression is equivalent to the spectral decomposition as a sum over poles at the discrete energy eigenvalues with residues that contain projector kernel onto the corresponding eigenstate. The derivation of the Green’s function is based on the scattering approach, in which stationary solutions are constructed by treating each vertex or subgraph as a scattering site described by a scattering matrix. The latter can then be given in a simple closed form from which the Green’s function is derived. The relevant scattering matrices contain inverse operators which are not well defined for wave numbers at which bound states in the continuum exists. It is shown that the singularities in the scattering matrix related to these bound states or perfect scars can be regularised. Green’s functions or scattering matrices can then be expressed as a sum of a regular and a singular part where the singular part contains the projection kernel onto the perfect scar.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"73 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136019237","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}
Pub Date : 2023-10-30DOI: 10.1088/1751-8121/ad04a6
Xinyu Mu, Shulin Lyu
Abstract We study the Hankel determinant generated by a Gaussian weight with Fisher–Hartwig singularities of root type at t j , j=1,⋯,N . It characterizes a type of average characteristic polynomial of matrices from Gaussian unitary ensembles. We derive the ladder operators satisfied by the associated monic orthogonal polynomials and three compatibility conditions. By using them and introducing 2 N auxiliary quantities {Rn,j,rn,j,j=1,⋯,N} , we build a series of difference equations. Furthermore, we prove that {Rn,j,rn,j} satisfy Riccati equations. From them we deduce a system of second order PDEs satisfied by {Rn,j,j=1,⋯,N} , which reduces to a Painlevé IV equation for N = 1. We also show that the logarithmic derivative of the Hankel determinant satisfies the generalized σ -form of a Painlevé IV equation.
我们研究了在t j, j = 1，⋯N处由具有根型Fisher-Hartwig奇点的高斯权值生成的Hankel行列式。刻画了高斯酉系综中矩阵的一类平均特征多项式。导出了相应的单正交多项式和三个相容条件所满足的阶梯算子。利用它们并引入2n辅助量{R N, j, R N, j, j = 1，⋯，N}，我们建立了一系列差分方程。进一步证明了{R n, j, R n, j}满足Riccati方程。从它们我们推导出一个二阶偏微分方程系统，满足于{R n, j, j = 1，⋯，n}，它简化为n = 1时的painlevev方程。我们还证明了汉克尔行列式的对数导数满足painlevev方程的广义σ -形式。
{"title":"Hankel determinants for a Gaussian weight with Fisher-Hartwig singularities and generalized Painlevé IV equation","authors":"Xinyu Mu, Shulin Lyu","doi":"10.1088/1751-8121/ad04a6","DOIUrl":"https://doi.org/10.1088/1751-8121/ad04a6","url":null,"abstract":"Abstract We study the Hankel determinant generated by a Gaussian weight with Fisher–Hartwig singularities of root type at t j , <?CDATA $j = 1,cdots ,N$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> </mml:math> . It characterizes a type of average characteristic polynomial of matrices from Gaussian unitary ensembles. We derive the ladder operators satisfied by the associated monic orthogonal polynomials and three compatibility conditions. By using them and introducing 2 N auxiliary quantities <?CDATA ${R_{n,j}, r_{n,j}, j = 1,cdots,N}$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:math> , we build a series of difference equations. Furthermore, we prove that <?CDATA ${R_{n,j}, r_{n,j}}$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:math> satisfy Riccati equations. From them we deduce a system of second order PDEs satisfied by <?CDATA ${R_{n,j}, j = 1,cdots,N}$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:math> , which reduces to a Painlevé IV equation for N = 1. We also show that the logarithmic derivative of the Hankel determinant satisfies the generalized σ -form of a Painlevé IV equation.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"69 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018875","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}
Pub Date : 2023-10-30DOI: 10.1088/1751-8121/acfcf5
Filippus Stefanus Roux
Abstract A Wigner functional approach is used to derive an evolution equation for a photonic state propagating through a Kerr medium. The resulting evolution equation incorporates all the spatiotemporal degrees of freedom together with the photon-number degrees of freedom and thus allows thorough analyses of the effects of experimental parameters in physical quantum information systems. We then use the evolution equation to consider four-wave mixing as a spontaneous process and finally we impose some approximations to obtain an expression for the optical field due to self-phase modulation.
{"title":"Four-wave mixing in all degrees of freedom","authors":"Filippus Stefanus Roux","doi":"10.1088/1751-8121/acfcf5","DOIUrl":"https://doi.org/10.1088/1751-8121/acfcf5","url":null,"abstract":"Abstract A Wigner functional approach is used to derive an evolution equation for a photonic state propagating through a Kerr medium. The resulting evolution equation incorporates all the spatiotemporal degrees of freedom together with the photon-number degrees of freedom and thus allows thorough analyses of the effects of experimental parameters in physical quantum information systems. We then use the evolution equation to consider four-wave mixing as a spontaneous process and finally we impose some approximations to obtain an expression for the optical field due to self-phase modulation.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"58 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018673","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}
Pub Date : 2023-10-30DOI: 10.1088/1751-8121/ad043b
L K Eraso-Hernandez, Alejandro P Riascos
Abstract In this paper, we study the synchronization of identical Kuramoto phase oscillators under cumulative stochastic damage to the edges of networks. We analyze the capacity of coupled oscillators to reach a coherent state from initial random phases. The process of synchronization is a global function performed by a system that gradually changes when the damage weakens individual connections of the network. We explore diverse structures characterized by different topologies. Among these are deterministic networks as a wheel or the lattice formed by the movements of the knight on a chess board, and random networks generated with the Erdős–Rényi and Barabási–Albert algorithms. In addition, we study the synchronization times of 109 non-isomorphic graphs with six nodes. The synchronization times and other introduced quantities are sensitive to the impact of damage, allowing us to measure the reduction of the capacity of synchronization and classify the effect of damage in the systems under study. This approach is general and paves the way for the exploration of the effect of damage accumulation in diverse dynamical processes in complex systems.
{"title":"Influence of cumulative damage on synchronizationof Kuramoto oscillators on networks","authors":"L K Eraso-Hernandez, Alejandro P Riascos","doi":"10.1088/1751-8121/ad043b","DOIUrl":"https://doi.org/10.1088/1751-8121/ad043b","url":null,"abstract":"Abstract In this paper, we study the synchronization of identical Kuramoto phase oscillators under cumulative stochastic damage to the edges of networks. We analyze the capacity of coupled oscillators to reach a coherent state from initial random phases. The process of synchronization is a global function performed by a system that gradually changes when the damage weakens individual connections of the network. We explore diverse structures characterized by different topologies. Among these are deterministic networks as a wheel or the lattice formed by the movements of the knight on a chess board, and random networks generated with the Erdős–Rényi and Barabási–Albert algorithms. In addition, we study the synchronization times of 109 non-isomorphic graphs with six nodes. The synchronization times and other introduced quantities are sensitive to the impact of damage, allowing us to measure the reduction of the capacity of synchronization and classify the effect of damage in the systems under study. This approach is general and paves the way for the exploration of the effect of damage accumulation in diverse dynamical processes in complex systems.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"53 S2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018987","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}
Pub Date : 2023-10-27DOI: 10.1088/1751-8121/ad07c6
María José Benac, Pedro Massey, Noelia Belén Rios, Mariano Ruiz
Abstract Let $mathcal{D}(d)$ denote the convex set of density matrices of size $d$ and let $rho,,sigmainmathcal{D}(d)$ be such that $rhonotprec sigma$. Consider the majorization flows $mathcal{L}(sigma)={mu inmathcal{D}(d) : muprec sigma}$ and $mathcal{U}(rho)={nuinmathcal{D}(d) : rhoprec nu}$, where $prec$ stands for the majorization pre-order relation. We endow $mathcal{L}(sigma)$ and $mathcal{U}(rho)$ with the metric induced by the spectral norm. Let $N(cdot)$ be a strictly convex unitarily invariant norm and let $mu_0in mathcal{L}(sigma)$ and $nu_0inmathcal{U}(rho)$ be local minimizers of the distance functions 
$Phi_N(mu)=N(rho-mu)$, for $muinmathcal{L}(sigma)$ and $Psi_N(nu)=N(sigma-nu)$, for $nuinmathcal{U}(rho)$. In this work we show that, for every unitarily invariant norm $tilde N(cdot)$ we have that 
$$
tilde N(rho-mu_0)leq tilde N(rho-mu), , , muinmathcal{L}(sigma)peso{and} 
tilde N(sigma-nu_0)leq tilde N(sigma-nu), , , nuinmathcal{U}(rho),.
$$ That is, $mu_0$ and $nu_0$ are global minimizers of the distances to the corresponding majorization flows, with respect to every unitarily invariant norm. We describe the (unique) spectral structure (eigenvalues) of $mu_0$ and $nu_0$ in terms of a simple finite step algorithm; we also describe the geometrical structure (eigenvectors) of $mu_0$ and $nu_0$ in terms of the geometrical structure of $sigma$ and $rho$, respectively. We include a discussion of the physical and computational implications of our results. We also compare our results to some recent related results in the context of quantum information theory.
{"title":"Local minimizers of the distances to the majorization flows","authors":"María José Benac, Pedro Massey, Noelia Belén Rios, Mariano Ruiz","doi":"10.1088/1751-8121/ad07c6","DOIUrl":"https://doi.org/10.1088/1751-8121/ad07c6","url":null,"abstract":"Abstract Let $mathcal{D}(d)$ denote the convex set of density matrices of size $d$ and let $rho,,sigmainmathcal{D}(d)$ be such that $rhonotprec sigma$. Consider the majorization flows $mathcal{L}(sigma)={mu inmathcal{D}(d) : muprec sigma}$ and $mathcal{U}(rho)={nuinmathcal{D}(d) : rhoprec nu}$, where $prec$ stands for the majorization pre-order relation. We endow $mathcal{L}(sigma)$ and $mathcal{U}(rho)$ with the metric induced by the spectral norm. Let $N(cdot)$ be a strictly convex unitarily invariant norm and let $mu_0in mathcal{L}(sigma)$ and $nu_0inmathcal{U}(rho)$ be local minimizers of the distance functions &#xD;$Phi_N(mu)=N(rho-mu)$, for $muinmathcal{L}(sigma)$ and $Psi_N(nu)=N(sigma-nu)$, for $nuinmathcal{U}(rho)$. In this work we show that, for every unitarily invariant norm $tilde N(cdot)$ we have that &#xD;$$&#xD;tilde N(rho-mu_0)leq tilde N(rho-mu), , , muinmathcal{L}(sigma)peso{and} &#xD;tilde N(sigma-nu_0)leq tilde N(sigma-nu), , , nuinmathcal{U}(rho),.&#xD;$$ That is, $mu_0$ and $nu_0$ are global minimizers of the distances to the corresponding majorization flows, with respect to every unitarily invariant norm. We describe the (unique) spectral structure (eigenvalues) of $mu_0$ and $nu_0$ in terms of a simple finite step algorithm; we also describe the geometrical structure (eigenvectors) of $mu_0$ and $nu_0$ in terms of the geometrical structure of $sigma$ and $rho$, respectively. We include a discussion of the physical and computational implications of our results. We also compare our results to some recent related results in the context of quantum information theory.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"14 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136317243","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}
Pub Date : 2023-10-27DOI: 10.1088/1751-8121/ad07c7
Hai Zhang, Kai Wu, Delong Wang
We study the relationship between integrable systems with a position-dependent mass (PDM) and complex holomorphic functions and the potential applications of the latter to deduce the former. For a prescribed mass term the associated complex function is derived. The complex function and related plane transformation are used to generate the PDM systems of three integrable Hénon–Heiles systems and a Holt system as well. We also figure out a holomorphic function, which ensures separability of the corresponding PDM systems in the polar-like coordinates. The holomorphic function together with Jacobi method have yielded a variety of generalized separable systems. At last we put forward an example of a family of separable systems to show that not all PDM systems can be deduced through some holomorphic function.
{"title":"Two-dimensional integrable systems with position-dependent mass via complex holomorphic functions","authors":"Hai Zhang, Kai Wu, Delong Wang","doi":"10.1088/1751-8121/ad07c7","DOIUrl":"https://doi.org/10.1088/1751-8121/ad07c7","url":null,"abstract":"We study the relationship between integrable systems with a position-dependent mass (PDM) and complex holomorphic functions and the potential applications of the latter to deduce the former. For a prescribed mass term the associated complex function is derived. The complex function and related plane transformation are used to generate the PDM systems of three integrable Hénon–Heiles systems and a Holt system as well. We also figure out a holomorphic function, which ensures separability of the corresponding PDM systems in the polar-like coordinates. The holomorphic function together with Jacobi method have yielded a variety of generalized separable systems. At last we put forward an example of a family of separable systems to show that not all PDM systems can be deduced through some holomorphic function.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234986","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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