Pub Date : 2024-08-01DOI: 10.1088/1751-8121/ad65a3
Ivan Dneprov, Maxim Grigoriev and Vyacheslav Gritzaenko
We elaborate on the recently proposed notion of a weak presymplectic gauge PDE. It is a -graded bundle over the space-time manifold, equipped with a degree 1 vector field and a compatible graded presymplectic structure. This geometrical data naturally defines a Lagrangian gauge field theory. Moreover, it encodes not only the Lagrangian of the theory but also its full-scale Batalin–Vilkovisky (BV) formulation. In particular, the respective field-antifield space arises as a symplectic quotient of the super-jet bundle of the initial fiber bundle. A remarkable property of this approach is that among the variety of presymplectic gauge PDEs encoding a given gauge theory we can pick a minimal one that usually turns out to be finite-dimensional, and unique in a certain sense. The approach can be considered as an extension of the familiar AKSZ construction to not necessarily topological and diffeomorphism-invariant theories. We present a variety of examples including p-forms, chiral Yang–Mills theory, Holst gravity, and conformal gravity. We also explain the explicit relation to the non-BV-BRST version of the formalism, which happens to be closely related to the covariant phase space and the multisymplectic approaches.
{"title":"Presymplectic minimal models of local gauge theories","authors":"Ivan Dneprov, Maxim Grigoriev and Vyacheslav Gritzaenko","doi":"10.1088/1751-8121/ad65a3","DOIUrl":"https://doi.org/10.1088/1751-8121/ad65a3","url":null,"abstract":"We elaborate on the recently proposed notion of a weak presymplectic gauge PDE. It is a -graded bundle over the space-time manifold, equipped with a degree 1 vector field and a compatible graded presymplectic structure. This geometrical data naturally defines a Lagrangian gauge field theory. Moreover, it encodes not only the Lagrangian of the theory but also its full-scale Batalin–Vilkovisky (BV) formulation. In particular, the respective field-antifield space arises as a symplectic quotient of the super-jet bundle of the initial fiber bundle. A remarkable property of this approach is that among the variety of presymplectic gauge PDEs encoding a given gauge theory we can pick a minimal one that usually turns out to be finite-dimensional, and unique in a certain sense. The approach can be considered as an extension of the familiar AKSZ construction to not necessarily topological and diffeomorphism-invariant theories. We present a variety of examples including p-forms, chiral Yang–Mills theory, Holst gravity, and conformal gravity. We also explain the explicit relation to the non-BV-BRST version of the formalism, which happens to be closely related to the covariant phase space and the multisymplectic approaches.","PeriodicalId":16763,"journal":{"name":"Journal of Physics A: Mathematical and Theoretical","volume":"22 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141883946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1088/1751-8121/ad6722
Katarzyna Siudzińska
Informationally overcomplete measurements find important applications in quantum tomography and quantum state estimation. The most popular are maximal sets of mutually unbiased bases, for which trace relations between measurement operators are well known. In this paper, we introduce a more general class of informationally overcomplete positive, operator-valued measure (POVMs) that are generated by equiangular tight frames of arbitrary rank. This class provides a generalization of equiangular measurements to non-projective POVMs, which include rescaled mutually unbiased measurements and bases. We provide a method of their construction, analyze their symmetry properties, and provide examples for highly symmetric cases. In particular, we find a wide class of generalized equiangular measurements that are conical two-designs, which allows us to derive the index of coincidence. Our results show benefits of considering a single informationally overcomplete measurement over informationally complete collections of POVMs.
{"title":"Informationally overcomplete measurements from generalized equiangular tight frames","authors":"Katarzyna Siudzińska","doi":"10.1088/1751-8121/ad6722","DOIUrl":"https://doi.org/10.1088/1751-8121/ad6722","url":null,"abstract":"Informationally overcomplete measurements find important applications in quantum tomography and quantum state estimation. The most popular are maximal sets of mutually unbiased bases, for which trace relations between measurement operators are well known. In this paper, we introduce a more general class of informationally overcomplete positive, operator-valued measure (POVMs) that are generated by equiangular tight frames of arbitrary rank. This class provides a generalization of equiangular measurements to non-projective POVMs, which include rescaled mutually unbiased measurements and bases. We provide a method of their construction, analyze their symmetry properties, and provide examples for highly symmetric cases. In particular, we find a wide class of generalized equiangular measurements that are conical two-designs, which allows us to derive the index of coincidence. Our results show benefits of considering a single informationally overcomplete measurement over informationally complete collections of POVMs.","PeriodicalId":16763,"journal":{"name":"Journal of Physics A: Mathematical and Theoretical","volume":"88 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1088/1751-8121/ad6721
Adrian Koenigstein, Marc Winstel
This work shows that the known phase boundary between the phase with chiral symmetry and the phase of spatially inhomogeneous chiral symmetry breaking in the phase diagram of the (1+1)-dimensional chiral Gross–Neveu (GN) model can be detected from the bosonic two-point function alone and thereby confirms and extends previous results (Schön and Thies 2000 At The Frontier of Particle Physics: Handbook of QCD, Boris Ioffe Festschrift vol 3 (World Scentific) ch 33, pp 1945–2032; Boehmer et al 2008 Phys. Rev. D 78 065043; Boehmer and Thies 2009 Phys. Rev. D 80 125038; Thies 2018 Phys. Rev. D 98 096019; Thies 2022 Phys. Rev. D 105 116003). The analysis is referred to as the stability analysis of the symmetric phase and does not require knowledge about spatial modulations of condensates. We perform this analysis in the infinite-N limit at nonzero temperature and nonzero quark and chiral chemical potentials also inside the inhomogeneous phase. Thereby we observe an interesting relation between the bosonic 1-particle irreducible two-point vertex function of the chiral GN model and the spinodal line of the GN model.
{"title":"Revisiting the spatially inhomogeneous condensates in the (1+1) -dimensional chiral Gross–Neveu model via the bosonic two-point function in the infinite-N limit","authors":"Adrian Koenigstein, Marc Winstel","doi":"10.1088/1751-8121/ad6721","DOIUrl":"https://doi.org/10.1088/1751-8121/ad6721","url":null,"abstract":"This work shows that the known phase boundary between the phase with chiral symmetry and the phase of spatially inhomogeneous chiral symmetry breaking in the phase diagram of the <inline-formula>\u0000<tex-math><?CDATA $(1 + 1)$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math>\u0000<inline-graphic xlink:href=\"aad6721ieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>-dimensional chiral Gross–Neveu (GN) model can be detected from the bosonic two-point function alone and thereby confirms and extends previous results (Schön and Thies 2000 <italic toggle=\"yes\">At The Frontier of Particle Physics: Handbook of QCD, Boris Ioffe Festschrift</italic> vol 3 (World Scentific) ch 33, pp 1945–2032; Boehmer <italic toggle=\"yes\">et al</italic> 2008 <italic toggle=\"yes\">Phys. Rev.</italic> D <bold>78</bold> 065043; Boehmer and Thies 2009 <italic toggle=\"yes\">Phys. Rev.</italic> D <bold>80</bold> 125038; Thies 2018 <italic toggle=\"yes\">Phys. Rev.</italic> D <bold>98</bold> 096019; Thies 2022 <italic toggle=\"yes\">Phys. Rev.</italic> D <bold>105</bold> 116003). The analysis is referred to as the stability analysis of the symmetric phase and does not require knowledge about spatial modulations of condensates. We perform this analysis in the infinite-<italic toggle=\"yes\">N</italic> limit at nonzero temperature and nonzero quark and chiral chemical potentials also inside the inhomogeneous phase. Thereby we observe an interesting relation between the bosonic 1-particle irreducible two-point vertex function of the chiral GN model and the spinodal line of the GN model.","PeriodicalId":16763,"journal":{"name":"Journal of Physics A: Mathematical and Theoretical","volume":"145 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1088/1751-8121/ad6651
Sanjib Sabhapandit and Satya N Majumdar
We study a system of N noninteracting particles on a line in the presence of a harmonic trap , where the trap center z(t) undergoes a stochastic modulation that remains bounded in time. We show that this stochastic modulation drives the system into a nonequilibrium stationary state, where the joint distribution of the positions of the particles is not factorizable. This indicates strong correlations between the positions of the particles that are not inbuilt, but rather get generated by the dynamics itself. Moreover, we show that the stationary joint distribution can be fully characterized and has a special conditionally independent and identically distributed structure. This special structure allows us to compute several observables analytically even in such a strongly correlated system, for an arbitrary drive z(t) that remains bounded in time. These observables include the average density profile, the correlations between particle positions, the order and gap statistics, as well as the full counting statistics. We then apply our general results to two specific examples where (i) z(t) represents a dichotomous telegraphic noise, and (ii) z(t) represents an Ornstein–Uhlenbeck process. Our analytical predictions are verified in numerical simulations, finding excellent agreement.
我们研究了一个存在谐波陷阱的直线上 N 个非相互作用粒子系统，在这个系统中，陷阱中心 z(t) 会发生随机调制，而这种调制在时间上是有界的。我们的研究表明，这种随机调制促使系统进入非平衡静止状态，在这种状态下，粒子位置的联合分布是不可因式分解的。这表明粒子位置之间存在很强的相关性，而这种相关性并不是内在的，而是由动力学本身产生的。此外，我们还证明了静态联合分布可以被完全表征，并且具有特殊的条件独立同分布结构。这种特殊结构使我们甚至可以在这样一个强相关系统中，针对时间上保持有界的任意驱动 z(t)，分析计算多个观测值。这些观测值包括平均密度曲线、粒子位置之间的相关性、阶次和间隙统计量以及完整的计数统计量。然后，我们将我们的一般结果应用于两个具体的例子：(i) z(t) 代表二分电报噪声，(ii) z(t) 代表奥恩斯坦-乌伦贝克过程。我们的分析预测在数值模拟中得到了验证，结果非常吻合。
{"title":"Noninteracting particles in a harmonic trap with a stochastically driven center","authors":"Sanjib Sabhapandit and Satya N Majumdar","doi":"10.1088/1751-8121/ad6651","DOIUrl":"https://doi.org/10.1088/1751-8121/ad6651","url":null,"abstract":"We study a system of N noninteracting particles on a line in the presence of a harmonic trap , where the trap center z(t) undergoes a stochastic modulation that remains bounded in time. We show that this stochastic modulation drives the system into a nonequilibrium stationary state, where the joint distribution of the positions of the particles is not factorizable. This indicates strong correlations between the positions of the particles that are not inbuilt, but rather get generated by the dynamics itself. Moreover, we show that the stationary joint distribution can be fully characterized and has a special conditionally independent and identically distributed structure. This special structure allows us to compute several observables analytically even in such a strongly correlated system, for an arbitrary drive z(t) that remains bounded in time. These observables include the average density profile, the correlations between particle positions, the order and gap statistics, as well as the full counting statistics. We then apply our general results to two specific examples where (i) z(t) represents a dichotomous telegraphic noise, and (ii) z(t) represents an Ornstein–Uhlenbeck process. Our analytical predictions are verified in numerical simulations, finding excellent agreement.","PeriodicalId":16763,"journal":{"name":"Journal of Physics A: Mathematical and Theoretical","volume":"12 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141883947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1088/1751-8121/ad67bb
Michael (Misha) Chertkov
The paper reflects on the future role of artificial intelligence (AI) in scientific research, with a special focus on turbulence studies, and examines the evolution of AI, particularly through Diffusion Models rooted in non-equilibrium statistical mechanics. It underscores the significant impact of AI on advancing reduced, Lagrangian models of turbulence through innovative use of Deep Neural Networks. Additionally, the paper reviews various other AI applications in turbulence research and outlines potential challenges and opportunities in the concurrent advancement of AI and statistical hydrodynamics. This discussion sets the stage for a future where AI and turbulence research are intricately intertwined, leading to more profound insights and advancements in both fields.
{"title":"Mixing artificial and natural intelligence: from statistical mechanics to AI and back to turbulence","authors":"Michael (Misha) Chertkov","doi":"10.1088/1751-8121/ad67bb","DOIUrl":"https://doi.org/10.1088/1751-8121/ad67bb","url":null,"abstract":"The paper reflects on the future role of artificial intelligence (AI) in scientific research, with a special focus on turbulence studies, and examines the evolution of AI, particularly through Diffusion Models rooted in non-equilibrium statistical mechanics. It underscores the significant impact of AI on advancing reduced, Lagrangian models of turbulence through innovative use of Deep Neural Networks. Additionally, the paper reviews various other AI applications in turbulence research and outlines potential challenges and opportunities in the concurrent advancement of AI and statistical hydrodynamics. This discussion sets the stage for a future where AI and turbulence research are intricately intertwined, leading to more profound insights and advancements in both fields.","PeriodicalId":16763,"journal":{"name":"Journal of Physics A: Mathematical and Theoretical","volume":"187 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141883949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1088/1751-8121/ad6380
Ofek Lauber Bonomo, Itamar Shitrit and Shlomi Reuveni
‘With persistence, a drop of water hollows out the stone’ goes the ancient Greek proverb. Yet, canonical percolation models do not account for interactions between a moving tracer and its environment. Recently, we have introduced the Sokoban model, which differs from this convention by allowing a tracer to push single obstacles that block its path. To test how this newfound ability affects percolation, we hereby consider a Bethe lattice on which obstacles are scattered randomly and ask for the probability that the Sokoban percolates through this lattice, i.e. escapes to infinity. We present an exact solution to this problem and determine the escape probability as a function of obstacle density. Similar to regular percolation, we show that the escape probability undergoes a second-order phase transition. We exactly determine the critical obstacle density at which this transition occurs and show that it is higher than that of a tracer without obstacle-pushing abilities. Our findings assert that pushing facilitates percolation on the Bethe lattice, as intuitively expected. This result, however, sharply contrasts with our previous findings on the 2D square lattice, where the Sokoban cannot escape even at obstacle densities well below the regular percolation threshold. This indicates that the presence of a regular percolation transition does not guarantee a percolation transition for a pushy tracer. The stark contrast between the Bethe and 2D lattices also highlights the significant impact of network topology on the effects of obstacle pushing and underscores the necessity for a more comprehensive understanding of percolation phenomena in systems with tracer-media interactions.
{"title":"Sokoban percolation on the Bethe lattice","authors":"Ofek Lauber Bonomo, Itamar Shitrit and Shlomi Reuveni","doi":"10.1088/1751-8121/ad6380","DOIUrl":"https://doi.org/10.1088/1751-8121/ad6380","url":null,"abstract":"‘With persistence, a drop of water hollows out the stone’ goes the ancient Greek proverb. Yet, canonical percolation models do not account for interactions between a moving tracer and its environment. Recently, we have introduced the Sokoban model, which differs from this convention by allowing a tracer to push single obstacles that block its path. To test how this newfound ability affects percolation, we hereby consider a Bethe lattice on which obstacles are scattered randomly and ask for the probability that the Sokoban percolates through this lattice, i.e. escapes to infinity. We present an exact solution to this problem and determine the escape probability as a function of obstacle density. Similar to regular percolation, we show that the escape probability undergoes a second-order phase transition. We exactly determine the critical obstacle density at which this transition occurs and show that it is higher than that of a tracer without obstacle-pushing abilities. Our findings assert that pushing facilitates percolation on the Bethe lattice, as intuitively expected. This result, however, sharply contrasts with our previous findings on the 2D square lattice, where the Sokoban cannot escape even at obstacle densities well below the regular percolation threshold. This indicates that the presence of a regular percolation transition does not guarantee a percolation transition for a pushy tracer. The stark contrast between the Bethe and 2D lattices also highlights the significant impact of network topology on the effects of obstacle pushing and underscores the necessity for a more comprehensive understanding of percolation phenomena in systems with tracer-media interactions.","PeriodicalId":16763,"journal":{"name":"Journal of Physics A: Mathematical and Theoretical","volume":"38 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141887155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1088/1751-8121/ad6654
Arnoldo Guerra IV, Narciso Román-Roy
We define canonical lifts of vector fields to the multisymplectic multimomentum bundles of De Donder–Weyl Hamiltonian (first-order) field theories and to the appropriate premultisymplectic embedded constraint submanifolds on which singular field theories are studied. These new canonical lifts are used to study the so-called natural Noether symmetries present in both regular and singular Hamiltonian field theories along with their associated conserved quantities obtained from Noether’s theorem. The Klein–Gordon field, the Polyakov bosonic string, and Einstein–Cartan gravity in 3+1 dimensions are analyzed in depth as applications of these concepts; as a peripheral result obtained in the analysis of the bosonic string, we provide a new geometrical interpretation of the well-known Virasoro constraint.
我们定义了向量场到 De Donder-Weyl 哈密顿（一阶）场论的多折射多动量束以及奇异场论所研究的适当前多折射嵌入约束子曼形上的典型提升。这些新的典型提升用于研究规则和奇异哈密顿场论中存在的所谓自然诺特对称性，以及从诺特定理中获得的相关守恒量。作为这些概念的应用，我们深入分析了 3+1 维中的克莱因-戈登场、波利亚科夫玻色弦和爱因斯坦-卡尔坦引力；作为玻色弦分析中获得的一个外围结果，我们为著名的维拉索罗约束提供了一种新的几何解释。
{"title":"Canonical lifts in multisymplectic De Donder–Weyl Hamiltonian field theories","authors":"Arnoldo Guerra IV, Narciso Román-Roy","doi":"10.1088/1751-8121/ad6654","DOIUrl":"https://doi.org/10.1088/1751-8121/ad6654","url":null,"abstract":"We define canonical lifts of vector fields to the multisymplectic multimomentum bundles of De Donder–Weyl Hamiltonian (first-order) field theories and to the appropriate premultisymplectic embedded constraint submanifolds on which singular field theories are studied. These new canonical lifts are used to study the so-called <italic toggle=\"yes\">natural Noether symmetries</italic> present in both regular and singular Hamiltonian field theories along with their associated conserved quantities obtained from Noether’s theorem. The <italic toggle=\"yes\">Klein–Gordon field</italic>, the <italic toggle=\"yes\">Polyakov bosonic string</italic>, and <italic toggle=\"yes\">Einstein–Cartan gravity in 3+1 dimensions</italic> are analyzed in depth as applications of these concepts; as a peripheral result obtained in the analysis of the bosonic string, we provide a new geometrical interpretation of the well-known <italic toggle=\"yes\">Virasoro constraint</italic>.","PeriodicalId":16763,"journal":{"name":"Journal of Physics A: Mathematical and Theoretical","volume":"96 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-31DOI: 10.1088/1751-8121/ad65a4
Wojciech Domitrz, Marcin Zubilewicz
This paper focuses on local curvature invariants associated with bi-Lagrangian structures. We establish several geometric conditions that determine when the canonical connection is flat, building on our previous findings regarding divergence-free webs (Domitrz and Zubilewicz 2023 Anal. Math. Phys.13 4). Addressing questions raised by Tabachnikov (1993 Differ. Geom. Appl.3 265–84), we provide complete solutions to two problems: the existence of flat bi-Lagrangian structures within the space of rays induced by a pair of hypersurfaces, and the existence of flat bi-Lagrangian structures induced by tangents to Lagrangian curves in the symplectic plane.
本文重点研究与双拉格朗日结构相关的局部曲率不变式。我们在之前关于无发散网的发现（Domitrz and Zubilewicz 2023 Anal.）针对塔巴奇尼科夫（1993 Differ. Geom. Appl.3，265-84）提出的问题，我们提供了两个问题的完整解决方案：一对超曲面诱导的射线空间内平面双拉格朗日结构的存在性，以及交错平面内拉格朗日曲线切线诱导的平面双拉格朗日结构的存在性。
{"title":"Bi-Lagrangian structures and the space of rays","authors":"Wojciech Domitrz, Marcin Zubilewicz","doi":"10.1088/1751-8121/ad65a4","DOIUrl":"https://doi.org/10.1088/1751-8121/ad65a4","url":null,"abstract":"This paper focuses on local curvature invariants associated with bi-Lagrangian structures. We establish several geometric conditions that determine when the canonical connection is flat, building on our previous findings regarding divergence-free webs (Domitrz and Zubilewicz 2023 <italic toggle=\"yes\">Anal. Math. Phys.</italic>\u0000<bold>13</bold> 4). Addressing questions raised by Tabachnikov (1993 <italic toggle=\"yes\">Differ. Geom. Appl.</italic>\u0000<bold>3</bold> 265–84), we provide complete solutions to two problems: the existence of flat bi-Lagrangian structures within the space of rays induced by a pair of hypersurfaces, and the existence of flat bi-Lagrangian structures induced by tangents to Lagrangian curves in the symplectic plane.","PeriodicalId":16763,"journal":{"name":"Journal of Physics A: Mathematical and Theoretical","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-30DOI: 10.1088/1751-8121/ad653f
Pieter W Claeys, Austen Lamacraft, Jamie Vicary
Dual-unitary brickwork circuits are an exactly-solvable model for many-body chaotic quantum systems, based on 2-site gates which are unitary in both the time and space directions. Prosen has recently described an alternative model called dual-unitary interactions round-a-face, which we here call clockwork, based on 2-controlled 1-site unitaries composed in a non-brickwork structure, yet with many of the same attractive global properties. We present a 2-categorical framework that simultaneously generalizes these two existing models, and use it to show that brickwork and clockwork circuits can interact richly, yielding new types of generalized heterogeneous circuits. We show that these interactions are governed by quantum combinatorial data, which we precisely characterize. These generalized circuits remain exactly-solvable and we show that they retain the attractive features of the original models such as single-site correlation functions vanishing everywhere except on the causal light-cone. Our framework allows us to directly extend the notion of solvable initial states to these biunitary circuits, and we show these circuits demonstrate maximal entanglement growth and exact thermalization after finitely many time steps.
{"title":"From dual-unitary to biunitary: a 2-categorical model for exactly-solvable many-body quantum dynamics","authors":"Pieter W Claeys, Austen Lamacraft, Jamie Vicary","doi":"10.1088/1751-8121/ad653f","DOIUrl":"https://doi.org/10.1088/1751-8121/ad653f","url":null,"abstract":"Dual-unitary brickwork circuits are an exactly-solvable model for many-body chaotic quantum systems, based on 2-site gates which are unitary in both the time and space directions. Prosen has recently described an alternative model called <italic toggle=\"yes\">dual-unitary interactions round-a-face</italic>, which we here call <italic toggle=\"yes\">clockwork</italic>, based on 2-controlled 1-site unitaries composed in a non-brickwork structure, yet with many of the same attractive global properties. We present a 2-categorical framework that simultaneously generalizes these two existing models, and use it to show that brickwork and clockwork circuits can interact richly, yielding new types of generalized heterogeneous circuits. We show that these interactions are governed by quantum combinatorial data, which we precisely characterize. These generalized circuits remain exactly-solvable and we show that they retain the attractive features of the original models such as single-site correlation functions vanishing everywhere except on the causal light-cone. Our framework allows us to directly extend the notion of solvable initial states to these biunitary circuits, and we show these circuits demonstrate maximal entanglement growth and exact thermalization after finitely many time steps.","PeriodicalId":16763,"journal":{"name":"Journal of Physics A: Mathematical and Theoretical","volume":"74 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-30DOI: 10.1088/1751-8121/ad65a5
Philippe Di Francesco, Hieu Trung Vu
We study the T-system of type A∞, also known as the octahedron recurrence/equation, viewed as a 2+1-dimensional discrete evolution equation. Generalizing earlier work on arctic curves for the Aztec Diamond obtained from solutions of the octahedron recurrence with ‘flat’ initial data, we consider initial data along parallel ‘slanted’ planes perpendicular to an arbitrary admissible direction (r,s,t)∈Z+3. The corresponding solutions of the T-system are interpreted as partition functions of dimer models on some suitable ‘pinecone’ graphs introduced by Bousquet–Mélou, Propp, and West in 2009. The T-system formulation and some exact solutions in uniform or periodic cases allow us to explore the thermodynamic limit of the corresponding dimer models and to derive exact arctic curves separating the various phases of the system. This direct approach bypasses the standard general theory of dimers using the Kasteleyn matrix approach and uses instead the theory of Analytic Combinatorics in Several Variables, by focusing on a linear system obeyed by the dimer density generating function.
{"title":"Arctic curves of the T-system with slanted initial data","authors":"Philippe Di Francesco, Hieu Trung Vu","doi":"10.1088/1751-8121/ad65a5","DOIUrl":"https://doi.org/10.1088/1751-8121/ad65a5","url":null,"abstract":"We study the <italic toggle=\"yes\">T</italic>-system of type <inline-formula>\u0000<tex-math><?CDATA $A_infty$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:msub></mml:mrow></mml:math>\u0000<inline-graphic xlink:href=\"aad65a5ieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>, also known as the octahedron recurrence/equation, viewed as a <inline-formula>\u0000<tex-math><?CDATA $2+1$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>\u0000<inline-graphic xlink:href=\"aad65a5ieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>-dimensional discrete evolution equation. Generalizing earlier work on arctic curves for the Aztec Diamond obtained from solutions of the octahedron recurrence with ‘flat’ initial data, we consider initial data along parallel ‘slanted’ planes perpendicular to an arbitrary admissible direction <inline-formula>\u0000<tex-math><?CDATA $(r,s,t)in {mathbb{Z}}_+^3$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∈</mml:mo><mml:msubsup><mml:mrow><mml:mrow><mml:mi mathvariant=\"double-struck\">Z</mml:mi></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mn>3</mml:mn></mml:msubsup></mml:mrow></mml:math>\u0000<inline-graphic xlink:href=\"aad65a5ieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>. The corresponding solutions of the <italic toggle=\"yes\">T</italic>-system are interpreted as partition functions of dimer models on some suitable ‘pinecone’ graphs introduced by Bousquet–Mélou, Propp, and West in 2009. The <italic toggle=\"yes\">T</italic>-system formulation and some exact solutions in uniform or periodic cases allow us to explore the thermodynamic limit of the corresponding dimer models and to derive exact arctic curves separating the various phases of the system. This direct approach bypasses the standard general theory of dimers using the Kasteleyn matrix approach and uses instead the theory of Analytic Combinatorics in Several Variables, by focusing on a linear system obeyed by the dimer density generating function.","PeriodicalId":16763,"journal":{"name":"Journal of Physics A: Mathematical and Theoretical","volume":"12 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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