IA Scholar Query: Parameterized Computational Geometry via Decomposition Theorems.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgWed, 30 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Geometry and analysis of contact instantons and entanglement of Legendrian links I
https://scholar.archive.org/work/uebt2r7sovh6ppe4k37pjuon5m
The purposes of the present paper are two-fold. Firstly we further develop the interplay between the contact Hamiltonian geometry and the geometric analysis of Hamiltonian-perturbed contact instantons with the Legendrian boundary condition, which is initiated by the present author in . We introduce the class of tame contact manifolds (M,λ), which includes compact ones but not necessarily compact, and establish uniform a priori C^0-estimates for the contact instantons. Then we study the problem of estimating the Reeb-untangling energy of one Legendrian submanifold from another, and formulate a particularly designed parameterized moduli space for the study of the problem. We establish the Gromov-Floer-Hofer type convergence result for contact instantons of finite energy and construct its compactification of the moduli space, first by defining the correct energy and then by proving uniform a priori energy bounds in terms of the oscillation of the relevant contact Hamiltonian. Secondly, as an application of this geometry and analysis of contact instantons, we prove that the self Reeb-untangling energy of a compact Legendrian submanifold R in any tame contact manifold (M,λ) is greater than that of the period gap T_λ(M,R) of the Reeb chords of R. This is an optimal result in general.Yong-Geun Ohwork_uebt2r7sovh6ppe4k37pjuon5mWed, 30 Nov 2022 00:00:00 GMTGluing theories of contact instantons and of pseudoholomoprhic curves in SFT
https://scholar.archive.org/work/5dyvmd7oerg7xncznqt6gmtgki
We develop the gluing theory of contact instantons in the context of open strings and in the context of closed strings with vanishing charge, for example in the symplectization context. This is one of the key ingredients for the study of (virtually) smooth moduli space of (bordered) contact instantons needed for the construction of contact instanton Floer cohomology and more generally for the construction of Fukaya-type category of Legendrian submanifolds in contact manifold (M,ξ). As an application, we apply the gluing theorem to give the construction of (cylindrical) Legendrian contact instanton homology that enters in our solution to Sandon's question for the nondegenerate case. We also apply this gluing theory to that of moduli spaces of holomorphic buildings arising in Symplectic Field Theory (SFT) by canonically lifting the former to that of the latter.Yong-Geun Ohwork_5dyvmd7oerg7xncznqt6gmtgkiWed, 30 Nov 2022 00:00:00 GMTSampling from convex sets with a cold start using multiscale decompositions
https://scholar.archive.org/work/2hld6d4rm5cdbhgjzrtahyk5ey
Running a random walk in a convex body K⊆ℝ^n is a standard approach to sample approximately uniformly from the body. The requirement is that from a suitable initial distribution, the distribution of the walk comes close to the uniform distribution π_K on K after a number of steps polynomial in n and the aspect ratio R/r (i.e., when rB_2 ⊆ K ⊆ RB_2). Proofs of rapid mixing of such walks often require the probability density η_0 of the initial distribution with respect to π_K to be at most poly(n): this is called a "warm start". Achieving a warm start often requires non-trivial pre-processing before starting the random walk. This motivates proving rapid mixing from a "cold start", wherein η_0 can be as high as exp(poly(n)). Unlike warm starts, a cold start is usually trivial to achieve. However, a random walk need not mix rapidly from a cold start: an example being the well-known "ball walk". On the other hand, Lovász and Vempala proved that the "hit-and-run" random walk mixes rapidly from a cold start. For the related coordinate hit-and-run (CHR) walk, which has been found to be promising in computational experiments, rapid mixing from a warm start was proved only recently but the question of rapid mixing from a cold start remained open. We construct a family of random walks inspired by classical decompositions of subsets of ℝ^n into countably many axis-aligned dyadic cubes. We show that even with a cold start, the mixing times of these walks are bounded by a polynomial in n and the aspect ratio. Our main technical ingredient is an isoperimetric inequality for K for a metric that magnifies distances between points close to the boundary of K. As a corollary, we show that the CHR walk also mixes rapidly both from a cold start and from a point not too close to the boundary of K.Hariharan Narayanan, Amit Rajaraman, Piyush Srivastavawork_2hld6d4rm5cdbhgjzrtahyk5eyWed, 30 Nov 2022 00:00:00 GMTLow-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence
https://scholar.archive.org/work/4i3myuq5qncbrjsumyaxdkdzo4
In this paper, we consider the estimation of a low Tucker rank tensor from a number of noisy linear measurements. The general problem covers many specific examples arising from applications, including tensor regression, tensor completion, and tensor PCA/SVD. We consider an efficient Riemannian Gauss-Newton (RGN) method for low Tucker rank tensor estimation. Different from the generic (super)linear convergence guarantee of RGN in the literature, we prove the first local quadratic convergence guarantee of RGN for low-rank tensor estimation in the noisy setting under some regularity conditions and provide the corresponding estimation error upper bounds. A deterministic estimation error lower bound, which matches the upper bound, is provided that demonstrates the statistical optimality of RGN. The merit of RGN is illustrated through two machine learning applications: tensor regression and tensor SVD. Finally, we provide the simulation results to corroborate our theoretical findings.Yuetian Luo, Anru R. Zhangwork_4i3myuq5qncbrjsumyaxdkdzo4Tue, 29 Nov 2022 00:00:00 GMTPresymplectic BV-AKSZ formulation of Conformal Gravity
https://scholar.archive.org/work/iyv6xrqzbzfy3cdanf6xi5b66a
We elaborate on the presymplectic BV-AKSZ approach to local gauge theories and apply it to conformal gravity. More specifically, we identify a compatible presymplectic structure on the minimal model of the total BRST complex of this theory and show that together with the BRST differential it determines a full-scale BV formulation for a specific frame-like action which seems to be previously unknown. Remarkably, the underlying frame-like description requires no artificial off-shell constraints. Instead, the action becomes equivalent to the usual conformal gravity one, upon gauging away all the variables belonging to the kernel of the presymplectic structure. Finally, we show how the presymplectic BV-AKSZ approach extends to generic gauge theories.Ivan Dneprov, Maxim Grigorievwork_iyv6xrqzbzfy3cdanf6xi5b66aTue, 29 Nov 2022 00:00:00 GMTHierarchy of RG flows in 6d (1,0) orbi-instantons
https://scholar.archive.org/work/p3gq65h5vrds5bbqso4zqm3fyq
N M5-branes probing the intersection between the orbifold ℂ^2/Γ_ADE and an E_8 wall give rise to 6d (1,0) SCFTs known as ADE-type orbi-instantons. At fixed N and order of the orbifold, each element of Hom(Γ_ADE,E_8) defines a different SCFT. The SCFTs are connected by Higgs branch RG flows, which generically reduce the flavor symmetry of the UV fixed point. We determine the full hierarchy of these RG flows for type A, i.e. ℂ^2/ℤ_k, for any value of N and k. The hierarchy takes the form of an intricate Hasse diagram: each node represents an IR orbi-instanton (homomorphism), and each edge an allowed flow, compatibly with the 6d a-theorem. The partial order is defined via quiver subtraction of the 3d magnetic quivers associated with the 6d SCFTs, which is equivalent to performing a so-called Kraft-Procesi transition between homomorphisms.Marco Fazzi, Suvendu Giriwork_p3gq65h5vrds5bbqso4zqm3fyqTue, 29 Nov 2022 00:00:00 GMTRobust boundary integral equations for the solution of elastic scattering problems via Helmholtz decompositions
https://scholar.archive.org/work/rlgplxljgzcv7glwpjhcih5oma
Helmholtz decompositions of the elastic fields open up new avenues for the solution of linear elastic scattering problems via boundary integral equations (BIE) [Dong, Lai, Li, Mathematics of Computation,2021]. The main appeal of this approach is that the ensuing systems of BIE feature only integral operators associated with the Helmholtz equation. However, these BIE involve non standard boundary integral operators that do not result after the application of either the Dirichlet or the Neumann trace to Helmholtz single and double layer potentials. Rather, the Helmholtz decomposition approach leads to BIE formulations of elastic scattering problems with Neumann boundary conditions that involve boundary traces of the Hessians of Helmholtz layer potential. As a consequence, the classical combined field approach applied in the framework of the Helmholtz decompositions leads to BIE formulations which, although robust, are not of the second kind. Following the regularizing methodology introduced in [Boubendir, Dominguez, Levadoux, Turc, SIAM Journal on Applied Mathematics 2015] we design and analyze novel robust Helmholtz decomposition BIE for the solution of elastic scattering that are of the second kind in the case of smooth scatterers in two dimensions. We present a variety of numerical results based on Nystrom discretizations that illustrate the good performance of the second kind regularized formulations in connections to iterative solvers.V. Dominguez, C. Turcwork_rlgplxljgzcv7glwpjhcih5omaTue, 29 Nov 2022 00:00:00 GMTNonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective
https://scholar.archive.org/work/m2tum4ngx5fv3ob2fa4laxzblq
We study a general matrix optimization problem with a fixed-rank positive semidefinite (PSD) constraint. We perform the Burer-Monteiro factorization and consider a particular Riemannian quotient geometry in a search space that has a total space equipped with the Euclidean metric. When the original objective f satisfies standard restricted strong convexity and smoothness properties, we characterize the global landscape of the factorized objective under the Riemannian quotient geometry. We show the entire search space can be divided into three regions: (R1) the region near the target parameter of interest, where the factorized objective is geodesically strongly convex and smooth; (R2) the region containing neighborhoods of all strict saddle points; (R3) the remaining regions, where the factorized objective has a large gradient. To our best knowledge, this is the first global landscape analysis of the Burer-Monteiro factorized objective under the Riemannian quotient geometry. Our results provide a fully geometric explanation for the superior performance of vanilla gradient descent under the Burer-Monteiro factorization. When f satisfies a weaker restricted strict convexity property, we show there exists a neighborhood near local minimizers such that the factorized objective is geodesically convex. To prove our results we provide a comprehensive landscape analysis of a matrix factorization problem with a least squares objective, which serves as a critical bridge. Our conclusions are also based on a result of independent interest stating that the geodesic ball centered at Y with a radius 1/3 of the least singular value of Y is a geodesically convex set under the Riemannian quotient geometry, which as a corollary, also implies a quantitative bound of the convexity radius in the Bures-Wasserstein space. The convexity radius obtained is sharp up to constants.Yuetian Luo, Nicolas Garcia Trilloswork_m2tum4ngx5fv3ob2fa4laxzblqTue, 29 Nov 2022 00:00:00 GMTThe Poisson boundary of hyperbolic groups without moment conditions
https://scholar.archive.org/work/ucfn4qjtavfntpgineijdocn7m
We prove that the Poisson boundary of a random walk with finite entropy on a non-elementary hyperbolic group can be identified with its hyperbolic boundary, without assuming any moment condition on the measure. We also extend our method to groups with an action by isometries on a hyperbolic metric space containing a WPD element; this applies to a large class of non-hyperbolic groups such as relatively hyperbolic groups, mapping class groups, and groups acting on CAT(0) spaces.Kunal Chawla, Behrang Forghani, Joshua Frisch, Giulio Tiozzowork_ucfn4qjtavfntpgineijdocn7mTue, 29 Nov 2022 00:00:00 GMTTopological Matter and Fractional Entangled Geometry
https://scholar.archive.org/work/ijlpsyes2jbe7nryynyciyaaea
Here, we review our progress on a geometrical approach of quantum physics and topological crystals starting from nature, electrodynamics of planets and linking with Dirac magnetic monopoles and gauge fields. The Bloch sphere of a quantum spin-1/2 particle can also acquire an integer topological charge in the presence of a radial magnetic field. We show that the global topological properties are revealed from the poles of the surface allowing a correspondence between smooth fields, metric and quantum distance. The information is transported from each pole to the equatorial plane on a thin Dirac string. We develop the theory, "the quantum topometry" in space and time, and present applications on transport from a Newtonian approach, on a quantized photo-electric effect from circular dichroism of light towards topological band structures of crystals. The occurrence of robust edge modes related to the topological lattice models are revealed analytically when deforming the sphere or ellipse onto a cylinder. The topological properties of the quantum Hall effect, the quantum anomalous Hall effect and the quantum spin Hall effect on the honeycomb lattice can be measured locally in the Brillouin zone from the light-matter coupling. The formalism allows us to include interaction effects from the momentum space. Interactions may also result in fractional entangled geometry within the curved space. We develop a relation between entangled wavefunction in quantum mechanics, coherent superposition of geometries, a way to one-half topological numbers and Majorana fermions. We show realizations in topological matter. We present a relation between axion electrodynamics, topological insulators on a surface of a cube and the two-spheres' model via the meron.Karyn Le Hurwork_ijlpsyes2jbe7nryynyciyaaeaTue, 29 Nov 2022 00:00:00 GMTBilliards and Teichmüller curves
https://scholar.archive.org/work/7o5ocb25bjg3li63utodongkbe
A Teichmüller curve V ⊂ M g V \subset \mathcal {M}_g is an isometrically immersed algebraic curve in the moduli space of Riemann surfaces. These rare, extremal objects are related to billiards in polygons, Hodge theory, algebraic geometry and surface topology. This paper presents the six known families of primitive Teichmüller curves that have been discovered over the past 30 years, and a selection of open problems.Curtis McMullenwork_7o5ocb25bjg3li63utodongkbeMon, 28 Nov 2022 00:00:00 GMTA reduced basis super-localized orthogonal decomposition for reaction-convection-diffusion problems
https://scholar.archive.org/work/5wiyunxupbg3vmdleacjfnepoq
This paper presents a method for the numerical treatment of reaction-convection-diffusion problems with parameter-dependent coefficients that are arbitrary rough and possibly varying at a very fine scale. The presented technique combines the reduced basis (RB) framework with the recently proposed super-localized orthogonal decomposition (SLOD). More specifically, the RB is used for accelerating the typically costly SLOD basis computation, while the SLOD is employed for an efficient compression of the problem's solution operator requiring coarse solves only. The combined advantages of both methods allow one to tackle the challenges arising from parametric heterogeneous coefficients. Given a value of the parameter vector, the method outputs a corresponding compressed solution operator which can be used to efficiently treat multiple, possibly non-affine, right-hand sides at the same time, requiring only one coarse solve per right-hand side.Francesca Bonizzoni and Moritz Hauck and Daniel Peterseimwork_5wiyunxupbg3vmdleacjfnepoqMon, 28 Nov 2022 00:00:00 GMTTautological relations and integrable systems
https://scholar.archive.org/work/szgkqx4vmffw7p7k7v4pczeeey
We present a family of conjectural relations in the tautological cohomology of the moduli spaces of stable algebraic curves of genus g with n marked points. A large part of these relations has a surprisingly simple form: the tautological classes involved in the relations are given by stable graphs that are trees and that are decorated only by powers of the psi-classes at half-edges. We show that the proposed conjectural relations imply certain fundamental properties of the Dubrovin-Zhang (DZ) and the double ramification (DR) hierarchies associated to F-cohomological field theories. Our relations naturally extend a similar system of conjectural relations, which were proposed in an earlier work of the first author together with Guéré and Rossi and which are responsible for the normal Miura equivalence of the DZ and the DR hierarchy associated to an arbitrary cohomological field theory. Finally, we prove all the above mentioned relations in the case n=1 and arbitrary g using a variation of the method from a paper by Liu and Pandharipande, this can be of independent interest. In particular, this proves the main conjecture from our previous joined work together with Hernández Iglesias. We also prove all the above mentioned relations in the case g=0 and arbitrary n.Alexandr Buryak, Sergey Shadrinwork_szgkqx4vmffw7p7k7v4pczeeeyMon, 28 Nov 2022 00:00:00 GMTStatistical Shape Analysis of Shape Graphs with Applications to Retinal Blood-Vessel Networks
https://scholar.archive.org/work/sbrte3ncajg65lzvbo2xihs27q
This paper provides theoretical and computational developments in statistical shape analysis of shape graphs, and demonstrates them using analysis of complex data from retinal blood-vessel (RBV) networks. The shape graphs are represented by a set of nodes and edges (planar articulated curves) connecting some of these nodes. The goals are to utilize shapes of edges and connectivities and locations of nodes to: (1) characterize full shapes, (2) quantify shape differences, and (3) model statistical variability. We develop a mathematical representation, elastic Riemannian shape metrics, and associated tools for such statistical analysis. Specifically, we derive tools for shape graph registration, geodesics, summaries, and shape modeling. Geodesics are convenient for visualizing optimal deformations, and PCA helps in dimension reduction and statistical modeling. One key challenge here is comparisons of shape graphs with vastly different complexities (in number of nodes and edges). This paper introduces a novel multi-scale representation of shape graphs to handle this challenge. Using the notions of (1) "effective resistance" to cluster nodes and (2) elastic shape averaging of edge curves, one can reduce shape graph complexity while maintaining overall structures. This way, we can compare shape graphs by bringing them to similar complexity. We demonstrate these ideas on Retinal Blood Vessel (RBV) networks taken from the STARE and DRIVE databases.Aditi Basu Bal, Xiaoyang Guo, Tom Needham, Anuj Srivastavawork_sbrte3ncajg65lzvbo2xihs27qMon, 28 Nov 2022 00:00:00 GMTPrecision Studies of QCD in the Low Energy Domain of the EIC
https://scholar.archive.org/work/q2sym423sfbkbagov27ikefimm
The manuscript focuses on the high impact science of the EIC with objective to identify a portion of the science program for QCD precision studies that requires or greatly benefits from high luminosity and low center-of-mass energies. The science topics include (1) Generalized Parton Distributions, 3D imagining and mechanical properties of the nucleon (2) mass and spin of the nucleon (3) Momentum dependence of the nucleon in semi-inclusive deep inelastic scattering (4) Exotic meson spectroscopy (5) Science highlights of nuclei (6) Precision studies of Lattice QCD in the EIC era (7) Science of far-forward particle detection (8) Radiative effects and corrections (9) Artificial Intelligence (10) EIC interaction regions for high impact science program with discovery potential. This paper documents the scientific basis for supporting such a program and helps to define the path toward the realization of the second EIC interaction region.V. Burkert, L. Elouadrhiri, A. Afanasev, J. Arrington, M. Contalbrigo, W. Cosyn, A. Deshpande, D. Glazier, X. Ji, S. Liuti, Y. Oh, D. Richards, T. Satogata, A. Vossenwork_q2sym423sfbkbagov27ikefimmMon, 28 Nov 2022 00:00:00 GMTIdentifying good directions to escape the NTK regime and efficiently learn low-degree plus sparse polynomials
https://scholar.archive.org/work/enbx7evtmzaepcqkzhcqsqwx3m
A recent goal in the theory of deep learning is to identify how neural networks can escape the "lazy training," or Neural Tangent Kernel (NTK) regime, where the network is coupled with its first order Taylor expansion at initialization. While the NTK is minimax optimal for learning dense polynomials (Ghorbani et al, 2021), it cannot learn features, and hence has poor sample complexity for learning many classes of functions including sparse polynomials. Recent works have thus aimed to identify settings where gradient based algorithms provably generalize better than the NTK. One such example is the "QuadNTK" approach of Bai and Lee (2020), which analyzes the second-order term in the Taylor expansion. Bai and Lee (2020) show that the second-order term can learn sparse polynomials efficiently; however, it sacrifices the ability to learn general dense polynomials. In this paper, we analyze how gradient descent on a two-layer neural network can escape the NTK regime by utilizing a spectral characterization of the NTK (Montanari and Zhong, 2020) and building on the QuadNTK approach. We first expand upon the spectral analysis to identify "good" directions in parameter space in which we can move without harming generalization. Next, we show that a wide two-layer neural network can jointly use the NTK and QuadNTK to fit target functions consisting of a dense low-degree term and a sparse high-degree term -- something neither the NTK nor the QuadNTK can do on their own. Finally, we construct a regularizer which encourages our parameter vector to move in the "good" directions, and show that gradient descent on the regularized loss will converge to a global minimizer, which also has low test error. This yields an end to end convergence and generalization guarantee with provable sample complexity improvement over both the NTK and QuadNTK on their own.Eshaan Nichani, Yu Bai, Jason D. Leework_enbx7evtmzaepcqkzhcqsqwx3mSun, 27 Nov 2022 00:00:00 GMTThe de Sitter group and its representations: a window on the notion of de Sitterian elementary systems
https://scholar.archive.org/work/ls6z4rds2rdebjxnfqqi3ejchq
We review the construction of ("free") elementary systems in de Sitter (dS) spacetime, in the Wigner sense, as associated with unitary irreducible representations (UIR's) of the dS (relativity) group. This study emphasizes the conceptual issues arising in the formulation of such systems and discusses known results in a mathematically rigorous way. Particular attention is paid to: "smooth" transition from classical to quantum theory; physical content under vanishing curvature, from the point of view of a local ("tangent") Minkowskian observer; and thermal interpretation (on the quantum level), in the sense of the Gibbons-Hawking temperature. We review three decompositions of the dS group physically relevant for the description of dS spacetime and classical phase spaces of elementary systems living on it. We review the construction of (projective) dS UIR's issued from these group decompositions. (Projective) Hilbert spaces carrying the UIR's (in some restricted sense) identify quantum ("one-particle") states spaces of dS elementary systems. Adopting a well-established Fock procedure, based on the Wightman-Gärding axioms and on analyticity requirements in the complexified Riemannian manifold, we proceed with a consistent quantum field theory (QFT) formulation of elementary systems in dS spacetime. This dS QFT formulation closely parallels the corresponding Minkowskian one, while the usual spectral condition is replaced by a certain geometric Kubo-Martin-Schwinger (KMS) condition equivalent to a precise thermal manifestation of the associated "vacuum" states. We end our study by reviewing a consistent and univocal definition of mass in dS relativity. This definition, presented in terms of invariant parameters characterizing the dS UIR's, accurately gives sense to terms like "massive" and "massless" fields in dS relativity according to their Minkowskian counterparts, yielded by the group contraction procedures.Mohammad Enayati, Jean-Pierre Gazeau, Hamed Pejhan, Anzhong Wangwork_ls6z4rds2rdebjxnfqqi3ejchqSun, 27 Nov 2022 00:00:00 GMTInformation Geometry of Dynamics on Graphs and Hypergraphs
https://scholar.archive.org/work/zzobax2mnngiljvk6zetumgap4
We introduce a new information-geometric structure of dynamics on discrete objects such as graphs and hypergraphs. The setup consists of two dually flat structures built on the vertex and edge spaces, respectively. The former is the conventional duality between density and potential, e.g., the probability density and its logarithmic form induced by a convex thermodynamic function. The latter is the duality between flux and force induced by a convex and symmetric dissipation function, which drives the dynamics on the manifold. These two are connected topologically by the homological algebraic relation induced by the underlying discrete objects. The generalized gradient flow in this doubly dual flat structure is an extension of the gradient flows on Riemannian manifolds, which include Markov jump processes and nonlinear chemical reaction dynamics as well as the natural gradient and mirror descent. The information-geometric projections on this doubly dual flat structure lead to the information-geometric generalizations of Helmholtz-Hodge-Kodaira decomposition and Otto structure in L^2 Wasserstein geometry. The structure can be extended to non-gradient nonequilibrium flow, from which we also obtain the induced dually flat structure on cycle spaces. This abstract but general framework can extend the applicability of information geometry to various problems of linear and nonlinear dynamics.Tetsuya J. Kobayashi, Dimitri Loutchko, Atsushi Kamimura, Shuhei Horiguchi, Yuki Sughiyamawork_zzobax2mnngiljvk6zetumgap4Sat, 26 Nov 2022 00:00:00 GMTRobust fast direct integral equation solver for three-dimensional quasi-periodic scattering problems with a large number of layers
https://scholar.archive.org/work/uuureauuwbbzvlgt4rfd6i6rre
A boundary integral equation method for the 3-D Helmholtz equation in multilayered media with many quasi-periodic layers is presented. Compared with conventional quasi-periodic Green's function method, the new method is robust at all scattering parameters. A periodizing scheme is used to decompose the solution into near- and far-field contributions. The near-field contribution uses the free-space Green's function in an integral equation on the interface in the unit cell and its immediate eight neighbors; the far-field contribution uses proxy point sources that enclose the unit cell. A specialized high-order quadrature is developed to discretize the underlying surface integral operators to keep the number of unknowns per layer small. We achieve overall linear computational complexity in the number of layers by reducing the linear system into block tridiagonal form and then solving the system directly via block LU decomposition. The new solver is capable of handling a 100-interface structure with 961.3k unknowns to 10^-5 accuracy in less than 2 hours on a desktop workstation.Bowei Wu, Min Hyung Chowork_uuureauuwbbzvlgt4rfd6i6rreSat, 26 Nov 2022 00:00:00 GMTCriticality and entanglement in non-unitary quantum circuits and tensor networks of non-interacting fermions
https://scholar.archive.org/work/o2uuo7bvkjbxnoyiwvef6mkzn4
Models for non-unitary quantum dynamics, such as quantum circuits that include projective measurements, have been shown to exhibit rich quantum critical behavior. There are many complementary perspectives on this behavior. For example, there is a known correspondence between d-dimensional local non-unitary quantum circuits and tensor networks on a D=(d+1)-dimensional lattice. Here, we show that in the case of systems of non-interacting fermions, there is furthermore a full correspondence between non-unitary circuits in d spatial dimensions and unitary non-interacting fermion problems with static Hermitian Hamiltonians in D=(d+1) spatial dimensions. This provides a powerful new perspective for understanding entanglement phases and critical behavior exhibited by non-interacting circuits. Classifying the symmetries of the corresponding non-interacting Hamiltonian, we show that a large class of random circuits, including the most generic circuits with randomness in space and time, are in correspondence with Hamiltonians with static spatial disorder in the ten Altland-Zirnbauer symmetry classes. We find the criticality that is known to occur in all of these classes to be the origin of the critical entanglement properties of the corresponding random non-unitary circuit. To exemplify this, we numerically study the quantum states at the boundary of Haar-random Gaussian fermionic tensor networks of dimension D=2 and D=3. We show that the most general such tensor network ensemble corresponds to a unitary problem of non-interacting fermions with static disorder in Altland-Zirnbauer symmetry class DIII, which for both D=2 and D=3 is known to exhibit a stable critical metallic phase. Tensor networks and corresponding random non-unitary circuits in the other nine Altland-Zirnbauer symmetry classes can be obtained from the DIII case by implementing Clifford algebra extensions for classifying spaces.Chao-Ming Jian, Bela Bauer, Anna Keselman, Andreas W. W. Ludwigwork_o2uuo7bvkjbxnoyiwvef6mkzn4Fri, 25 Nov 2022 00:00:00 GMT