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Exploiting analogies between the precessing quantum spin system and the charge-monopole system, we construct Bloch hyper-spheres with $\it{exact}$ spherical symmetries in arbitrary dimensions. Such Bloch hyper-spheres are realized as a collection of the orbits of a precessing quantum spin. The geometry of Bloch hyper-spheres is exactly equal to the quantum Nambu geometry of higher dimensional fuzzy spheres. The stabilizer group symmetry of the Bloch hyper-sphere necessarily introduces degenerate spin-coherent states, giving rise to the Wilczek-Zee geometric phase of non-Abelian monopoles associated with the hyper-sphere holonomy. The degenerate spin-coherent states induce matrix-valued quantum geometric tensors. While the minimal spin Bloch hyper-spheres exhibit similar properties in even and odd dimensions, their large spin counterparts differ qualitatively depending on the parity of the 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., Comment: 1+57 pages, 21 figures, 1 table, references added, typos corrected, to appear in NPB

Quantum matrix geometry is the underlying geometry of M(atrix) theory. Expanding upon the idea of level projection, we propose a quantum-oriented non-commutative 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 $\it{pure}$ quantum Nambu geometries: Their non-commutative 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., Comment: 1+51 pages, 17 figures, minor modifications, to appear in PRD

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., Comment: 12 pages, 9 figures, contribution to the proceedings for Workshop on Quantum Geometry, Field Theory and Gravity (dedicated to the memory of Professor John Madore), Corfu Summer Institute 2021 "School and Workshops on Elementary Particle Physics and Gravity" (CORFU2021) 29 August - 9 October 2021 Corfu, Greece

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 anti-monopole indices, $I_+$ and $I_-$. The energy levels of the $SO(5)$ Landau problem are grouped into $\text{Min}(I_+, I_-) +1$ sectors, each of which holds Landau levels. In the $n$-sector, $N$th 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 $SO(5)$ non-linear 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., Comment: 1+32 pages, 7 figures; minor corrections

The Bloch hyperboloid $H^2$ underlies the quantum geometry of the original $SO(2,1)$ squeezed states. In \cite{Hasebe-2019}, the author utilized a non-compact 2nd Hopf map and a Bloch four-hyperboloid $H^{2,2}$ to explore an $SO(2,3)$ extension of the squeezed states. In the present paper, we further pursue the idea to derive an $SO(4,1)$ version of squeezed vacuum based on the other Bloch four-hyperboloid $H^4$. We show that the obtained $SO(4,1)$ squeezed vacuum is a particular four-mode squeezed state not quite similar to the previous $SO(2,3)$ squeezed vacuum. In view of the Schwinger's formulation of angular momentum, the $SO(4,1)$ squeezed vacuum is interpreted as a superposition of an infinite number of maximally entangled spin-pairs of all integer spins. We clarify basic properties of the $SO(4,1)$ squeezed vacuum, such as von Neumann entropy of spin entanglement, spin correlations and uncertainty relations with emphasis on their distinctions to the original $SO(2,1)$ case., Comment: 1+37 pages, 8 figures, 1 table; explanations and figures modified; an abridged version was published in JPA

A curious correspondence has been known between Landau models and non-linear sigma models: Reinterpreting the base-manifolds of Landau models as field-manifolds, the Landau models are transformed to non-linear sigma models with same global and local symmetries. With the idea of the dimensional hierarchy of higher dimensional Landau models, we exploit this correspondence to present a systematic procedure for construction of non-linear sigma models in higher dimensions. We explicitly derive $O(2k+1)$ non-linear sigma models in $2k$ dimension based on the parent tensor gauge theories that originate from non-Abelian monopoles. The obtained non-linear sigma models turn out to be Skyrme-type non-linear sigma models with $O(2k)$ local symmetry. Through a dimensional reduction of Chern-Simons tensor field theories, we also derive Skyrme-type $O(2k)$ non-linear sigma models in $2k-1$ dimension, which realize the original and other Skyrme models as their special cases. As a unified description, we explore Skyrme-type $O(d+1)$ non-linear sigma models and clarify their basic properties, such as stability of soliton configurations, scale invariant solutions, and field configurations with higher winding number., Comment: 1+54 pages, 4 figures, 2 tables, discussions added, typos corrected, to appear in NPB

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 derive matrix geometry of a four-sphere in any Landau level: In the lowest Landau level the matrix coordinates are given by the generalized $SO(5)$ gamma matrices of the fuzzy four-sphere satisfying the quantum Nambu algebra, while in higher Landau levels the matrix geometry becomes a nested fuzzy structure realizing a pure quantum geometry with no counterpart in classical geometry. The internal fuzzy geometry structure is discussed in the view of an $SO(4)$ Pauli-Schr\"odinger model and the $SO(4)$ Landau model, where we unveil a hidden singular gauge transformation between their background non-Abelian field configurations. Relativistic versions of the $SO(5)$ Landau model are also investigated and relationship to the Berezin-Toeplitz quantization is clarified. We finally discuss the matrix geometry of the Landau models in even higher dimensions., Comment: 1+52 papes, 8 figures, minor modifications

We explore the hyperbolic geometry of squeezed states in the perspective of the non-compact Hopf map. Based on analogies between squeeze operation and $Sp(2,\mathbb{R})$ hyperbolic rotation, two types of the squeeze operators, the (usual) Dirac- and the Schwinger-types, are introduced. We clarify the underlying hyperbolic geometry and $SO(2,1)$ representations of the squeezed states along the line of the 1st non-compact Hopf map. Following to the geometric hierarchy of the non-compact Hopf maps, we extend the $Sp(2; \mathbb{R})$ analysis to $Sp(4; \mathbb{R})$ --- the isometry of an split-signature four-hyperboloid. We explicitly construct the $Sp(4; \mathbb{R})$ squeeze operators in the Dirac- and Schwinger-types and investigate the physical meaning of the four-hyperboloid coordinates in the context of the Schwinger-type squeezed states. It is shown that the Schwinger-type $Sp(4;\mathbb{R})$ squeezed one-photon state is equal to an entangled superposition state of two $Sp(2;\mathbb{R})$ squeezed states and the corresponding concurrence has a clear geometric meaning. Taking advantage of the group theoretical formulation, basic properties of the $Sp(4;\mathbb{R})$ squeezed coherent states are also investigated. In particular, we show that the $Sp(4; \mathbb{R})$ squeezed vacuum naturally realizes a generalized squeezing in a 4D manner., Comment: 1+50 pages, 3 fugures, 1 table; minor corrections, an abridged version to appear in JPA

We develop an in-depth analysis of the $SO(4)$ Landau models on $S^3$ in the $SU(2)$ monopole background and their associated matrix geometry. The Schwinger and Dirac gauges for the $SU(2)$ monopole are introduced to provide a concrete coordinate representation of $SO(4)$ operators and wavefunctions. The gauge fixing enables us to demonstrate algebraic relations of the operators and the $SO(4)$ covariance of the eigenfunctions. With the spin connection of $S^3$, we construct an $SO(4)$ invariant Weyl-Landau operator and analyze its eigenvalue problem with explicit form of the eigenstates. The obtained results include the known formulae of the free Weyl operator eigenstates in the free field limit. Other eigenvalue problems of variant relativistic Landau models, such as massive Dirac-Landau and supersymmetric Landau models, are investigated too. With the developed $SO(4)$ technologies, we derive the three-dimensional matrix geometry in the Landau models. By applying the level projection method to the Landau models, we identify the matrix elements of the $S^3$ coordinates as the fuzzy three-sphere. For the non-relativistic model, it is shown that the fuzzy three-sphere geometry emerges in each of the Landau levels and only in the degenerate lowest energy sub-bands. We also point out that Dirac-Landau operator accommodates two fuzzy three-spheres in each Landau level and the mass term induces interaction between them., Comment: 1+59 pages, 8 figures, 1 table, minor corrections, published version

We demonstrate dimensional ladder of higher dimensional quantum Hall effects by exploiting quantum Hall effects on arbitrary odd dimensional spheres. Non-relativistic and relativistic Landau models are analyzed on $S^{2k-1}$ in the $SO(2k-1)$ monopole background. The total sub-band degeneracy of the odd dimensional lowest Landau level is shown to be equal to the winding number from the base-manifold $S^{2k-1}$ to the one-dimension higher $SO(2k)$ gauge group. Based on the chiral Hopf maps, we clarify the underlying quantum Nambu geometry for odd dimensional quantum Hall effect and the resulting quantum geometry is naturally embedded also in one-dimension higher quantum geometry. An origin of such dimensional ladder connecting even and odd dimensional quantum Hall effects is illuminated from a viewpoint of the spectral flow of Atiyah-Patodi-Singer index theorem in differential topology. We also present a BF topological field theory as an effective field theory in which membranes with different dimensions undergo non-trivial linking in odd dimensional space. Finally, an extended version of the dimensional hierarchy for higher dimensional quantum Hall liquids is proposed, and its relationship to quantum anomaly and D-brane physics is discussed., Comment: 1+49 pages, 8 figures, 4 tables, typos corrected

We present a relativistic formulation of the quantum Hall effect on Haldane sphere. An explicit form of the pseudopotential is derived for the relativistic quantum Hall effect with/without mass term. We clarify particular features of the relativistic quantum Hall states with the use of the exact diagonalization study of the pseudopotential Hamiltonian. Physical effects of the mass term to the relativistic quantum Hall states are investigated in detail. The mass term acts as an interpolating parameter between the relativistic and non-relativistic quantum Hall effects. It is pointed out that the mass term unevenly affects the many-body physics of the positive and negative Landau levels as a manifestation of the "parity anomaly". In particular, we explicitly demonstrate the instability of the Laughlin state of the positive first relativistic Landau level with the reduction of the charge gap., Comment: We have corrected the typographic errors in our article. There are three corrections: (1)2g -> 2j in Fig.3 (2)2g -> 2j in the caption of Fig.3 (3)2g=12 -> 2j = 64 in the caption of Fig.5

Non-commutative geometry naturally emerges in low energy physics of Landau models as a consequence of level projection. In this work, we proactively utilize the level projection as an effective tool to generate fuzzy geometry. The level projection is specifically applied to the relativistic Landau models. In the first half of the paper, a detail analysis of the relativistic Landau problems on a sphere is presented, where a concise expression of the Dirac-Landau operator eigenstates is obtained based on algebraic methods. We establish $SU(2)$ "gauge" transformation between the relativistic Landau model and the Pauli-Schr\"odinger non-relativistic quantum mechanics. After the $SU(2)$ transformation, the Dirac operator and the angular momentum operastors are found to satisfy the $SO(3,1)$ algebra. In the second half, the fuzzy geometries generated from the relativistic Landau levels are elucidated, where unique properties of the relativistic fuzzy geometries are clarified. We consider mass deformation of the relativistic Landau models and demonstrate its geometrical effects to fuzzy geometry. Super fuzzy geometry is also constructed from a supersymmetric quantum mechanics as the square of the Dirac-Landau operator. Finally, we apply the level projection method to real graphene system to generate valley fuzzy spheres., Comment: 1+56 pages, 13 figures, typos corrected, more explanations about the edth operators added, Appendix B and D expanded

We clarify relations between the higher dimensional quantum Hall effect and A-class topological insulator. In particular, we elucidate physical implications of the higher dimensional non-commutative geometry in the context of A-class topological insulator. This presentation is based on arXiv:1403.5066., Comment: 5 pages, 1 table; contribution to the proceedings of the Workshop on Noncommutative Field Theory and Gravity, Corfu, Greece, September 8-15, 2013, Fortschritte der Physik 2014

Chiral topological insulator (AIII-class) with Landau levels is constructed based on the Nambu 3-algebraic geometry. We clarify the geometric origin of the chiral symmetry of the AIII-class topological insulator in the context of non-commutative geometry of 4D quantum Hall effect. The many-body groundstate wavefunction is explicitly derived as a $(l,l,l-1)$ Laughlin-Halperin type wavefunction with unique $K$-matrix structure. Fundamental excitation is identified with anyonic string-like object with fractional charge ${1}/({1+2(l-1)^2})$. The Hall effect of the chiral topological insulators turns out be a color version of Hall effect, which exhibits a dual property of the Hall and spin-Hall effects., Comment: 6 pages, 1 figure, published version

We perform a detail study of higher dimensional quantum Hall effects and A-class topological insulators with emphasis on their relations to non-commutative geometry. There are two different formulations of non-commutative geometry for higher dimensional fuzzy spheres; the ordinary commutator formulation and quantum Nambu bracket formulation. Corresponding to these formulations, we introduce two kinds of monopole gauge fields; non-abelian gauge field and antisymmetric tensor gauge field, which respectively realize the non-commutative geometry of fuzzy sphere in the lowest Landau level. We establish connection between the two types of monopole gauge fields through Chern-Simons term, and derive explicit form of tensor monopole gauge fields with higher string-like singularity. The connection between two types of monopole is applied to generalize the concept of flux attachment in quantum Hall effect to A-class topological insulator. We propose tensor type Chern-Simons theory as the effective field theory for membranes in A-class topological insulators. Membranes turn out to be fractionally charged objects and the phase entanglement mediated by tensor gauge field transforms the membrane statistics to be anyonic. The index theorem supports the dimensional hierarchy of A-class topological insulator. Analogies to D-brane physics of string theory are discussed too., Comment: 1+56 pages, 4 figures, 4 tables, published version

Recent vigorous investigations of topological order have not only discovered new topological states of matter but also shed new light to "already known" topological states. One established example with topological order is the valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are disordered spin liquids with no spontaneous symmetry breaking but most typically manifest topological order known as hidden string order on 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy super-geometry in the construction of supersymmetric versions of VBS (SVBS) states, and give a pedagogical introduction of SVBS models and their properties [arXiv:0809.4885, 1105.3529, 1210.0299]. As concrete examples, we present detail analysis of supersymmetric versions of SU(2) and SO(5) VBS states, i.e. UOSp(N|2) and UOSp(N|4) SVBS states whose mathematics are closely related to fuzzy two- and four-superspheres. The SVBS states are physically interpreted as hole-doped VBS states with superconducting property that interpolate various VBS states depending on value of a hole-doping parameter. The parent Hamiltonians for SVBS states are explicitly constructed, and their gapped excitations are derived within the single-mode approximation on 1D SVBS chains. Prominent features of the SVBS chains are discussed in detail, such as a generalized string order parameter and entanglement spectra. It is realized that the entanglement spectra are at least doubly degenerate regardless of the parity of bulk (super)spins. Stability of topological phase with supersymmetry is discussed with emphasis on its relation to particular edge (super)spin states., Comment: Review article, 1+104 pages, 37 figures, published version

We present a detailed analysis of topological properties of the valence bond solid (VBS) states doped with fermionic holes. As concrete examples, we consider the supersymmetric extension of the SU(2)- and the SO(5) VBS states, dubbed UOSp(1|2) and UOSp(1|4) supersymmetric VBS states, respectively. Specifically, we investigate the string-order parameters and the entanglement spectra of these states to find that, even when the parent states (bosonic VBS states) do not support the string order, they recover it when holes are doped and the fermionic sector appears in the entanglement spectrum. These peculiar properties are discussed in light of the symmetry-protected topological order. To this end, we characterize a few typical classes of symmetry-protected topological orders in terms of supermatrix-product states (SMPS). From this, we see that the topological order in the bulk manifests itself in the transformation properties of the SMPS in question and thereby affects the structure of the entanglement spectrum. Then, we explicitly relate the existence of the string order and the structure of the entanglement spectrum to explain the recovery and the stabilization of the string order in the supersymmetric systems., Comment: 22 pages, 12 figures, supplementary material is available upon request, minor corrections

Fuzzy hyperboloids naturally emerge in the geometries of D-branes, twistor theory, and higher spin theories. In this work, we perform a systematic study of higher dimensional fuzzy hyperboloids (ultra-hyperboloids) based on non-compact Hopf maps. Two types of non-compact Hopf maps; split-type and hybrid-type, are introduced from the cousins of division algebras. We construct arbitrary even-dimensional fuzzy ultra-hyperboloids by applying the Schwinger operator formalism and indefinite Clifford algebras. It is shown that fuzzy hyperboloids, $H_F^{2p,2q}$, are represented by the coset, $H_F^{2p,2q}\simeq SO(2p,2q+1)/U(p,q)$, and exhibit two types of generalized dimensional hierarchy; hyperbolic-type (for $q\neq 0$) and hybrid-type (for $q=0$). Fuzzy hyperboloids can be expressed as fibre-bundle of fuzzy fibre over hyperbolic basemanifold. Such bundle structure of fuzzy hyperboloid gives rise to non-compact monopole gauge field. Physical realization of fuzzy hyperboloids is argued in the context of lowest Landau level physics., Comment: 1+59 pages, 2 tables, explanations added in Sec.2.2, references added

Topological stability of the edge states is investigated for non-Hermitian systems. We examine two classes of non-Hermitian Hamiltonians supporting real bulk eigenenergies in weak non-Hermiticity: SU(1,1) and SO(3,2) Hamiltonians. As an SU(1,1) Hamiltonian, the tight-binding model on the honeycomb lattice with imaginary on-site potentials is examined. Edge states with ReE=0 and their topological stability are discussed by the winding number and the index theorem, based on the pseudo-anti-Hermiticity of the system. As a higher symmetric generalization of SU(1,1) Hamiltonians, we also consider SO(3,2) models. We investigate non-Hermitian generalization of the Luttinger Hamiltonian on the square lattice, and that of the Kane-Mele model on the honeycomb lattice, respectively. Using the generalized Kramers theorem for the time-reversal operator Theta with Theta^2=+1 [M. Sato et al., arXiv:1106.1806], we introduce a time-reversal invariant Chern number from which topological stability of gapless edge modes is argued., Comment: 29 pages, 19 figures, typos fixed

We argue supersymmetric generalizations of fuzzy two- and four-spheres based on the unitary-orthosymplectic algebras, $uosp(N|2)$ and $uosp(N|4)$, respectively. Supersymmetric version of Schwinger construction is applied to derive graded fully symmetric representation for fuzzy superspheres. As a classical counterpart of fuzzy superspheres, graded versions of 1st and 2nd Hopf maps are introduced, and their basic geometrical structures are studied. It is shown that fuzzy superspheres are represented as a "superposition" of fuzzy superspheres with lower supersymmetries. We also investigate algebraic structures of fuzzy two- and four-superspheres to identify $su(2|N)$ and $su(4|N)$ as their enhanced algebraic structures, respectively. Evaluation of correlation functions manifests such enhanced structure as quantum fluctuations of fuzzy supersphere., Comment: 56 pages, no figures, two tables, references added, minor corrections, to appear in NPB

For ordinary hermitian Hamiltonians, the states show the Kramers degeneracy when the system has a half-odd-integer spin and the time reversal operator obeys \Theta^2=-1, but no such a degeneracy exists when \Theta^2=+1. Here we point out that for non-hermitian systems, there exists a degeneracy similar to Kramers even when \Theta^2=+1. It is found that the new degeneracy follows from the mathematical structure of split-quaternion, instead of quaternion from which the Kramers degeneracy follows in the usual hermitian cases. Furthermore, we also show that particle/hole symmetry gives rise to a pair of states with opposite energies on the basis of the split quaternion in a class of non-hermitian Hamiltonians. As concrete examples, we examine in detail NxN Hamiltonians with N=2 and 4 which are non-hermitian generalizations of spin 1/2 Hamiltonian and quadrupole Hamiltonian of spin 3/2, respectively., Comment: 40 pages, 2 figures; typos fixed, references added

Supersymmetric valence bond solid models are extensions of the VBS model, a paradigmatic model of `solvable' gapped quantum antiferromagnets, to the case with doped fermionic holes. In this paper, we present a detailed analysis of physical properties of the models. For systematic studies, a supersymmetric version of the matrix product formalism is developed. On 1D chains, we exactly evaluate the hole-doping behavior of various physical quantities, such as the spin/charge excitation spectrum, superconducting order parameter. A generalized hidden order is proposed, and the corresponding string non-local order parameter is also calculated. The behavior of the string order parameter is discussed in the light of the entanglement spectrum., Comment: 20 pages, a problem with PdfLaTeX fixed. supplementary material is available upon request

This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to Clifford algebra is exactly analogous to generalization of fuzzy two-spheres to higher dimensional fuzzy spheres. Higher dimensional fuzzy spheres have an interesting hierarchical structure made of "compounds" of lower dimensional spheres. We give a physical interpretation for such particular structure of fuzzy spheres by utilizing Landau models in generic even dimensions. With Grassmann algebra, we also introduce a graded version of the Hopf map, and discuss its relation to fuzzy supersphere in context of supersymmetric Landau model., Comment: v2: note and references added; v3: references added

We develop a noncompact version of the Hopf maps based on the split algebras. The split algebras consist of three species: split-complex numbers, split quaternions, and split octonions. They correspond to three noncompact Hopf maps that represent topological maps between hyperboloids in different dimensions with hyperboloid bundle. We realize such noncompact Hopf maps in two ways: one is to utilize the split-imaginary unit, and the other is to utilize the ordinary imaginary unit. Topological structures of the hyperboloid bundles are explored, and the canonical connections are naturally regarded as noncompact gauge field of monopoles., Comment: 40 pages, no figures, + 8 pages, discussions and references added, published version

Introducing a non-compact version of the Hopf map, we demonstrate remarkable close relations between quantum Hall effect and twistor theory. We first construct quantum Hall effect on a hyperboloid based on the noncompact 2nd Hopf map of split-quaternions. We analyze a hyperbolic one-particle mechanics, and explore many-body problem, where a many-body groundstate wavefunction and membrane-like excitations are derived explicitly. In the lowest Landau level, the symmetry is enhanced from $SO(3, 2)$ to the $SU(2, 2)$ conformal symmetry. We point out that the quantum Hall effect naturally realizes the philosophy of twistor theory. In particular, the emergence mechanism of fuzzy space-time is discussed somehow in detail., Comment: 5 pages, 1 table, no figures, references added, published version

We construct a supersymmetric quantum Hall liquid with a deformed supersymmetry. One parameter is introduced in the supersymmetric Laughlin wavefunction to realize the original Laughlin wavefunction and the Moore-Read wavefunction in two extremal limits of the parameter. The introduced parameter corresponds to the coherence factor in the BCS theory. It is pointed out that the parameter-dependent supersymmetric Laughlin wavefunction enjoys a deformed supersymmetry. Based on the deformed supersymmetry, we construct a pseudo-potential Hamiltonian whose groundstate is exactly the parameter-dependent supersymmetric Laughlin wavefunction. Though the SUSY pseudo-potential Hamiltonian is parameter-dependent and non-Hermitian, its eigenvalues are parameter-independent and real., Comment: 14 pages, contribution to the proceedings of the Group 27 conference, Yerevan, Armenia, August 13-19, 2008, published version

In this work we investigate the supersymmetric version of the valence bond solid (SVBS) state. In one dimension, the SVBS states continuously interpolate between the valence bond states for integer and half-integer spin chains, and they generally describe superconducting valence bond liquid states. Spin and superconducting correlation functions can be computed exactly for these states, and their correlation lengths are equal at the supersymmetric point. In higher dimensions, the wave function for the SVBS states can describe resonating valence bond states (RVB). The SVBS states for the spin models are shown to be precisely analogous to the bosonic Pfaffian states of the quantum Hall effect. We also give microscopic Hamiltonians for which the SVBS state is the exact ground state., Comment: 21 pages, 8 figures, 1 table, minor corrections

Developing a non-compact version of the SUSY Hopf map, we formulate the quantum Hall effect on a super-hyperboloid. Based on $OSp(1|2)$ group theoretical methods, we first analyze the one-particle Landau problem, and successively explore the many-body problem where Laughlin wavefunction, hard-core pseudo-potential Hamiltonian and topological excitations are derived. It is also shown that the fuzzy super-hyperboloid emerges in the lowest Landau level., Comment: 14 pages, two columns, no figures, published version, typos corrected

Supertwistor theory is geometrically constructed based on the SUSY Hopf map. We derive a new incidence relation for the geometrical supertwistor theory. The present supertwistor exhibits remarkable properties: Minkowski space need not be complexified to introduce spin degrees of freedom, and even number SUSY is {automatically} incorporated by the geometrical set-up. We also develop a theory for massless free particle in Minkowski superspace, which physically corresponds to the geometrical supertwistor theory. The spin degrees of freedom are originated from fermionic momenta as well as fermionic coordinates. The geometrical supertwistor is quantized to reproduce same physical contents as in the original supertwistor theory. Relationships to superspin formalism and SUSY quantum Hall effect are also discussed., Comment: 9 pages, no figures, two columns

We review the recent developments of the SUSY quantum Hall effect [hep-th/0409230, hep-th/0411137, hep-th/0503162, hep-th/0606007, arXiv:0705.4527]. We introduce a SUSY formulation of the quantum Hall effect on supermanifolds. On each of supersphere and superplane, we investigate SUSY Landau problem and explicitly construct SUSY extensions of Laughlin wavefunction and topological excitations. The non-anti-commutative geometry naturally emerges in the lowest Landau level and brings particular physics to the SUSY quantum Hall effect. It is shown that SUSY provides a unified picture of the original Laughlin and Moore-Read states. Based on the charge-flux duality, we also develop a Chern-Simons effective field theory for the SUSY quantum Hall effect., Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

Based on the recently proposed SUSY quantum Hall effect, we show that Laughlin and Moore-Read states are related by a hidden SUSY transformation. Regarding the SUSY Laughlin wavefunction as a master wavefunction, Laughlin and Moore-Read states appear as two extreme limits of component wavefunctions. Realizations of topological excitations on Laughlin and Moore-Read states are also discussed in the SUSY formalism. We develop a streographically projected formulation of the SUSY quantum Hall effect. With appropriate interpretation of Grassmann odd coordinates, we illustrate striking analogies between SUSY quantum Hall effect and superfluidity., Comment: 5 pages, 1 figure, typos fixed

We develop a supersymmetric extension of Chern-Simons theory and Chern-Simons-Landau-Ginzburg theory for supersymmetric quantum Hall liquid. Supersymmetric counterparts of topological and gauge structures peculiar to the Chern-Simons theory are inspected in the supersymmetric Chern-Simons theory. We also explore an effective field theoretical description for the supersymmetric quantum Hall liquid. The key observation is the the charge-flux duality. Based on the duality, we derive a dual supersymmetric Chern-Simons-Landau-Ginzburg theory, and discuss physical properties of the topological excitations in supersymmetric quantum Hall liquid., Comment: 12 pages, no figures, published version in PRD

Supersymmetric quantum Hall liquids are constructed on a noncommutative superplane. We explore a supersymmetric formalism of the Landau problem. In the lowest Landau level, there appear spin-less bosonic states and spin-1/2 down fermionic states, which exhibit a super-chiral property. It is shown the Laughlin wavefunction and topological excitations have their superpartners. Similarities between supersymmetric quantum Hall systems and bilayer quantum Hall systems are discussed., Comment: 11 pages, 3 figures, 1 table, minor corrections, published in Phys.Rev.D

Supersymmetric quantum Hall liquids are constructed on a supersphere in a supermonopole background. We derive a supersymmetric generalization of the Laughlin wavefunction, which is a ground state of a hard-core $OSp(1|2)$ invariant Hamiltonian. We also present excited topological objects, which are fractionally charged deficits made by super Hall currents. Several relations between quantum Hall systems and their supersymmetric extensions are discussed., Comment: Typos corrected, 5 pages, to be published in PRL

It is well-known that coordinates of a charged particle in a monopole background become noncommutative. In this paper, we study the motion of a charged particle moving on a supersphere in the presence of a supermonopole. We construct a supermonopole by using a supersymmetric extension of the first Hopf map. We investigate algebras of angular momentum operators and supersymmetry generators. It is shown that coordinates of the particle are described by fuzzy supersphere in the lowest Landau level. We find that there exist two kinds of degenerate wavefunctions due to the supersymmetry. Ground state wavefunctions are given by the Hopf spinor and we discuss their several properties., Comment: 22 pages

We construct higher dimensional quantum Hall systems based on fuzzy spheres. It is shown that fuzzy spheres are realized as spheres in colored monopole backgrounds. The space noncommutativity is related to higher spins which is originated from the internal structure of fuzzy spheres. In $2k$-dimensional quantum Hall systems, Laughlin-like wave function supports fractionally charged excitations, $q=m^{-{1/2}k(k+1)}$ (m is odd). Topological objects are ($2k-2$)-branes whose statistics are determined by the linking number related to the general Hopf map. Higher dimensional quantum Hall systems exhibit a dimensional hierarchy, where lower dimensional branes condense to make higher dimensional incompressible liquid., Comment: 4 pages, 1 figure