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Matrix coordinates for I = 3. There are non-zero matrix elements in the shaded color regions. The blue, green, purple shaded regions are specified by ΔN = ±1, (Δn, ΔN ) = (±1, 0) and (Δs, Δn, ΔN ) = (±1, 0, 0), respectively. The red shaded regions correspond to ΔN = Δn = Δs = 0. The red-framed squares (with inner red and purple squares) denote the SO(4) Landau level subspaces. Obviously, the matrix geometry exhibits a nesting structure.
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The SO(5) Landau model is the mathematical platform of the 4D quantum Hall effect and provide a rare opportunity for a physical realization of the fuzzy four-sphere. We present an integrated analysis of the SO(5) Landau models and the associated matrix geometries through the Landau level projection. With the SO(5) monopole harmonics, we explicitly...
Contexts in source publication
Context 1
... (127a) indicates, there are two cases in which x m take finite values. The first case is (Δn, Δs) = (±1, 0) representing transition between two adjacent SO(4) Landau levels (two adjacent SO(4) lines in Fig.2) corresponding to the green shaded regions in Fig.4, while the second case ...
Context 2
... transition between the two adjacent sub-bands specified by s inside a SO(4) Landau level (two adjacent dots on an identical SO(4) line in Fig.2) corresponding to the small purple shaded regions in Fig.4. In the following, we focus on the second case, which in the language of the ...
Context 3
... each quasi-fuzzy four sphere does not possess the SO(5) covariance, since it is solely constructed by the SO(4) irreducible representations. There exist non-vanishing off-diagonal matrix elements between the adjacent SO(4) Landau levels (as represented by the green shaded rectangular blocks in Fig.4). Borrowing the string theory interpretation that the off-diagonal parts signify interactions between the fuzzy objects represented by the diagonal block matrices, one may say that the quasi-fuzzy four-spheres of the adjacent SO(4) Landau levels interact and conspire to maintain the SO(5) covariance of the nested fuzzy geometry. ...
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... constructing the generalized gamma matrices of large spin. This idea aligns with the recent developments of non-commutative geometry [53,54,55,56,57,57,58,59,60,61,62,63,64], especially from the quantum matrix geometry of the higher dimensional fuzzy spheres [64,61,59,55,54]. 4 We present a systematic construction of exactly spherical Bloch hyper-spheres and investigate their exotic properties. ...
... constructing the generalized gamma matrices of large spin. This idea aligns with the recent developments of non-commutative geometry [53,54,55,56,57,57,58,59,60,61,62,63,64], especially from the quantum matrix geometry of the higher dimensional fuzzy spheres [64,61,59,55,54]. 4 We present a systematic construction of exactly spherical Bloch hyper-spheres and investigate their exotic properties. We will see that higher dimensional Zeeman-Dirac models necessarily exhibit energy level degeneracies and realize the Wilczek-Zee connections of non-Abelian monopoles. ...
... For later convenience, we develop a geometric method for the present case. To orient the SO(5) spin coherent state to the direction x a , we introduce the SO(5) non-linear realization matrix [59,61]: ...
Leveraging analogies between precessing quantum spin systems and charge-monopole systems, we construct Bloch hyper-spheres with exact spherical symmetries in arbitrary dimensions. Such a Bloch hyper-sphere is realized as a collection of the orbits of precessing quantum spins, and its geometry mathematically aligns with the quantum Nambu geometry of a higher dimensional fuzzy sphere. Stabilizer group symmetry of the Bloch hyper-sphere necessarily introduces degenerate spin-coherent states and gives rise to Wilczek-Zee geometric phases of non-Abelian monopoles associated with the hyper-sphere holonomies. The degenerate spin-coherent states naturally induce matrix-valued quantum geometric tensors also. While the physical properties of Bloch hyper-spheres with minimal spin in even and odd dimensions are quite similar, their large spin counterparts differ qualitatively depending on the parity of dimensions. Exact correspondences between spin-coherent states and monopole harmonics in higher dimensions are established. We also investigate density matrices described by Bloch hyper-balls and elucidate their corresponding statistical and geometric properties such as von Neumann entropies and Bures quantum metrics.
... Nevertheless, this does not rule out the possibility of discovering new noncommutative geometries in higher dimensional systems. Following this idea, explorations of novel quantum matrix geometries have been conducted in various Landau models, such as relativistic models and supersymmetric models [53], odd dimensional models [54] and even dimensional models [55][56][57]. It is also worthwhile to mention that quantum matrix geometries associated with the Berezin-Toeplitz quantization have been intensively studied in recent years [58][59][60][61][62][63][64]. ...
... However, Berezin-Toeplitz quantization is primarily concerned with symplectic manifolds and is based on commutator formalism. The Kernel employed in the Berezin-Toeplitz quantization corresponds to the zero modes of the Dirac-Landau operator whose zero modes are essentially equivalent to the lowest Landau level eigenstates [53,55]. Therefore, the Berezin-Toeplitz quantization is thus closely related to the lowest Landau level matrix geometry and can be viewed as a special case of the present scheme. ...
... Using the SOð5Þ Landau model, we will derive the complete form of matrix coordinates in arbitrary Landau levels. This section also includes a review of Ref. [55]. ...
Quantum matrix geometry is the underlying geometry of M(atrix) theory. Expanding upon the idea of level projection, we propose a quantum-oriented noncommutative scheme for generating the matrix geometry of the coset space G/H. We employ this novel scheme to unveil unexplored matrix geometries by utilizing gauged quantum mechanics on higher dimensional spheres. The resultant matrix geometries manifest as pure quantum Nambu geometries: Their noncommutative structures elude capture through the conventional commutator formalism of Lie algebra, necessitating the introduction of the quantum Nambu algebra. This matrix geometry embodies a one-dimension-lower quantum internal geometry featuring nested fuzzy structures. While the continuum limit of this quantum geometry is represented by overlapping classical manifolds, their fuzzification cannot reproduce the original quantum geometry. We demonstrate how these quantum Nambu geometries give rise to novel solutions in Yang-Mills matrix models, exhibiting distinct physical properties from the known fuzzy sphere solutions.
... Moreover, it is also possible to consider non-compact manifolds and orbifolds [7]. Therefore, the Berezin-Toeplitz quantization of vector bundles might also be defined over more general manifolds than the closed Kähler case (for example the fuzzy S 4 [30][31][32][33]). We can also consider more challenging problems such as a quantization of odd-dimensional manifolds [34][35][36][37] or manifolds with boundaries. ...
We propose a matrix regularization of vector bundles over a general closed Kähler manifold. This matrix regularization is given as a natural generalization of the Berezin-Toeplitz quantization and gives a map from sections of a vector bundle to matrices. We examine the asymptotic behaviors of the map in the large-N limit. For vector bundles with algebraic structure, we derive a beautiful correspondence of the algebra of sections and the algebra of corresponding matrices in the large-N limit. We give two explicit examples for monopole bundles over a complex projective space CPn and a torus T2n.
... In the lowest Landau level ( = 0), the matrix coordinates (14) are obtained as [28] ...
... The decomposition rule (13) from (5) to (4) implies a nested structure in the corresponding matrix geometry [28]: Each oblique line of the (4) irreducible representations corresponds to a "fuzzy shell" in the matrix geometry side, and the ( + 1) sets of the (4) irreducible representations constitute the nested structure of the ( + 1) shells [ Fig.4]. ...
... We can find a similar hierarchical structure with respect to the monopole gauge fields [28] [ Fig.7]. One may wonder whether such a hierarchical relation is extended in even higher dimensions. ...
One of the most celebrated works of Professor Madore is the introduction of fuzzy sphere. I briefly review how the fuzzy two-sphere and its higher dimensional cousins are realized in the (spherical) Landau models in non-Abelian monopole backgrounds. For extracting quantum geometry from the Landau models, we evaluate the matrix elements of the coordinates of spheres in the lowest and higher Landau levels. For the lowest Landau level, the matrix geometry is identified as the geometry of fuzzy sphere. Meanwhile for the higher Landau levels, the obtained quantum geometry turns out to be a nested matrix geometry with no classical counterpart. There exists a hierarchical structure between the fuzzy geometries and the monopoles in different dimensions. That dimensional hierarchy signifies a Landau model counterpart of the dimensional ladder of quantum anomaly.
... In the lowest Landau level ( = 0), the matrix coordinates (14) are obtained as [28] ...
... The decomposition rule (13) from (5) to (4) implies a nested structure in the corresponding matrix geometry [28]: Each oblique line of the (4) irreducible representations corresponds to a "fuzzy shell" in the matrix geometry side, and the ( + 1) sets of the (4) irreducible representations constitute the nested structure of the ( + 1) shells [ Fig.4]. ...
... We can find a similar hierarchical structure with respect to the monopole gauge fields [28] [ Fig.7]. One may wonder whether such a hierarchical relation is extended in even higher dimensions. ...
... However, T 2n can be decomposed to the tensor product of T 2 s so that the results of [11] can also be applied to our case. Kähler case (for example the fuzzy S 4 [30][31][32][33]). We can also consider more challenging problems such as a quantization of odd-dimensional manifolds [34][35][36][37] or manifolds with boundaries. ...
We propose a matrix regularization of vector bundles over a general closed K\"ahler manifold. This matrix regularization is given as a natural generalization of the Berezin-Toeplitz quantization and gives a map from sections of a vector bundle to matrices. We examine the asymptotic behaviors of the map in the large-$N$ limit. For vector bundles with algebraic structure, we derive a beautiful correspondence of the algebra of sections and the algebra of corresponding matrices in the large-$N$ limit. We give two explicit examples for monopole bundles over a complex projective space $CP^n$ and a torus $T^{2n}$.
... Recent investigations have been extended to higherdimensional systems (n > 3), including the four-dimensional (4D) quantum Hall effect [15][16][17][18][19], Yang monopoles [20,21], two-dimensional (2D) Weyl surfaces in five-dimensional (5D) systems [22][23][24][25][26][27], and so on. The higher-dimensional systems are expected to possess properties that their lowerdimensional counterparts do not support. ...
Weyl semimetals are phases of matter with gapless electronic excitations that are protected by topology and symmetry. Their properties depend on the dimensions of the systems. It has been theoretically demonstrated that five-dimensional (5D) Weyl semimetals emerge as novel phases during the topological phase transition in analogy to the three-dimensional case. However, experimental observation of such a phenomenon remains a great challenge because the tunable 5D system is extremely hard to construct in real space. Here, we construct 5D electric circuit platforms in fully real space and experimentally observe topological phase transitions in five dimensions. Not only are Yang monopoles and linked Weyl surfaces observed experimentally, but various phase transitions in five dimensions are also proved, such as the phase transitions from a normal insulator to a Hopf link of twoWeyl surfaces and then to a 5D topological insulator. The demonstrated topological phase transitions in five dimensions leverage the concept of higher-dimensional Weyl physics to control electrical signals in the engineered circuits.
... Recent investigations have been extended to higherdimensional systems (n > 3), including the four-dimensional (4D) quantum Hall effect [15][16][17][18][19], Yang monopoles [20,21], two-dimensional (2D) Weyl surfaces in five-dimensional (5D) systems [22][23][24][25][26][27], and so on. The higher-dimensional systems are expected to possess properties that their lowerdimensional counterparts do not support. ...
Weyl semimetals are phases of matter with gapless electronic excitations that are protected by topology and symmetry. Their properties depend on the dimensions of the systems. It has been theoretically demonstrated that five-dimensional (5D) Weyl semimetals emerge as novel phases during the topological phase transition in analogy to the three-dimensional case. However, experimental observation of such a phenomenon remains a great challenge because the tunable 5D system is extremely hard to construct in real space. Here, we construct 5D electric circuit platforms in fully real space and experimentally observe topological phase transitions in five dimensions. Not only are Yang monopoles and linked Weyl surfaces observed experimentally, but various phase transitions in five dimensions are also proved, such as the phase transitions from a normal insulator to a Hopf link of two Weyl surfaces and then to a 5D topological insulator. The demonstrated topological phase transitions in five dimensions leverage the concept of higher-dimensional Weyl physics to control electrical signals in the engineered circuits.
... Not only for its elegant mathematical structure, the SUð2Þ monopole found its physical applications in the SOð5Þ Landau model and four-dimensional (4D) quantum Hall effect [4], which, from a modern point of view, is the first theoretical model of a topological insulator in higher dimension. The underlying geometry of the system is the nested quantum Nambu geometry that does not have any counterpart in classical geometry [5], which renders the system to be quite unique also in view of the noncommutative geometry [6,7]. Tensor-type Chern-Simons theories are proposed as effective field theories [6,7] that naturally induce generalized fractional statistics of extended objects [8][9][10]. ...
... Sandwiching the SOð5Þ angular momentum operators with the SOð5Þ monopole harmonics, we can in principle derive the SOð5Þ matrix generators of arbitrary representations. In this section, we review Yang's work with a modern notation [5] and derive a general matrix form of the SOð5Þ generators. ...
... The symbols, j and k, denote the bi-spin indices of the SOð4Þ ≃ SUð2Þ ⊗ SUð2Þ group, while n ¼ j þ k− p−q 2 ð¼ 0; 1; 2; …; qÞ and s ¼ j − kð¼ − p−q 2 ; − p−q þ 1; − p−q 2 þ 2; …; p−q 2 Þ indicate the Landau level index and the chirality parameter in the SOð4Þ Landau model [5]. The notations, ðj; kÞ 4 and ½n; s, are both useful according to context, and we hereafter utilize them interchangeably: ...
We investigate the SO(5) Landau problem in the SO(4) monopole gauge field background by applying the techniques of the non-linear realization of quantum field theory. The SO(4) monopole carries two topological invariants, the second Chern number and a generalized Euler number, specified by the SU(2) monopole and antimonopole indices, I+ and I−. The energy levels of the SO(5) Landau problem are grouped into Min(I+,I−)+1 sectors, each of which holds Landau levels. In the n-sectors, Nth Landau level eigenstates constitute the SO(5) irreducible representation with (p,q)5=(N+I++I−−n,N+n)5 whose function form is obtained from the SO(5) nonlinear realization matrix. In the n=0 sector, the emergent quantum geometry of the lowest Landau level is identified as the fuzzy four-sphere with radius being proportional to the difference between I+ and I−. The Laughlin-like wavefunction is constructed by imposing the SO(5) lowest Landau level projection to the many-body wavefunction made of the Slater determinant. We also analyze the relativistic version of the SO(5) Landau model to demonstrate the Atiyah-Singer index theorem in the SO(4) gauge field configuration.
... Recently, the topological properties of higherdimensional systems (n > 3) have drawn con-siderable attention, including the 4D quantum Hall effect (14)(15)(16)(17)(18)(19), Yang monopoles (20)(21)(22), and 2D Weyl surfaces in 5D systems (23)(24)(25)(26)(27). Such systems are expected to possess properties that their lower-dimensional counterparts do not support, that is, those associated with a nonzero second Chern number (C 2 ). ...
Topology in higher dimensions
In condensed-matter systems, the band structure of a material has often been equated with functionality. However, consideration of the topology of the band structure now provides a route to develop a functionality that goes far beyond the expected properties of the materials. Using electromagnetic metamaterials as building blocks, Ma et al . realized a five-dimensional generalization of a topological Weyl semimetal. Along with the three real momentum dimensions, these included two bi-anisotropy material parameters as synthetic dimensions to demonstrate both linked Weyl surfaces and Yang monopoles. The metamaterial platform provides a powerful route to explore the exotic physics associated with higher-order topological phenomena. —ISO
