Fuzzy Sphere Research Articles

Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization

The $3D$ Ising transition, the most celebrated and unsolved critical phenomenon in nature, has long been conjectured to have emergent conformal symmetry, similar to the case of the $2D$ Ising transition. Yet, the emergence of conformal invariance in the $3D$ Ising transition has rarely been explored directly, mainly due to unavoidable mathematical or conceptual obstructions. Here, we design an innovative way to study the quantum version of the $3D$ Ising phase transition on spherical geometry, using the ``fuzzy (non-commutative) sphere" regularization. We accurately calculate and analyze the energy spectra at the transition, and explicitly demonstrate the state-operator correspondence (i.e. radial quantization), a fingerprint of conformal field theory. In particular, we have identified 13 parity-even primary operators within a high accuracy and 2 parity-odd operators that were not known before. Our result directly elucidates the emergent conformal symmetry of the $3D$ Ising transition, a conjecture made by Polyakov half a century ago. More importantly, our approach opens a new avenue for studying $3D$ CFTs by making use of the state-operator correspondence and spherical geometry.

Chaotic dynamics of the mass deformed ABJM model

We explore the chaotic dynamics of the mass-deformed Aharony-Bergman-Jafferis-Maldacena model. To do so, we first perform a dimensional reduction of this model from $2+1$ to $0+1$ dimensions, considering that the fields are spatially uniform. Working in the 't Hooft limit and tracing over ansatz configurations involving fuzzy 2-spheres, which are described in terms of the Gomis-Rodriguez-Gomez-Van Raamsdonk-Verlinde matrices with collective time dependence, we obtain a family of reduced effective Lagrangians and demonstrate that they have chaotic dynamics by computing the associated Lyapunov exponents. In particular, we focus on how the largest Lyapunov exponent, $\lambda_L$, changes as a function of $E/N^2$. Depending on the structure of the effective potentials, we find either $\lambda_L \propto (E/N^2)^{1/3}$ or $\lambda_L \propto (E/N^2 - \gamma_N)^{1/3}$, where $\gamma_N(k, \mu)$ are constants determined in terms of the Chern-Simons coupling $k$, the mass $\mu$, and the matrix level $N$. Noting that the classical dynamics approximates the quantum theory only in the high-temperature regime, we investigate the temperature dependence of the largest Lyapunov exponents and give upper bounds on the temperature above which $\lambda_L$ values comply with the Maldacena-Shenker-Stanford bound, $ \lambda_L \leq 2 \pi T $, and below which it will eventually be not obeyed.

Qubitization strategies for bosonic field theories

Quantum simulations of bosonic field theories require a truncation in field space to map the theory onto finite quantum registers. Ideally, the truncated theory preserves the symmetries of the original model and has a critical point in the same universality class. In this paper, we explore two different truncations that preserve the symmetries of the 1+1-dimensional $O(3)$ non-linear $\sigma$-model - one that truncates the Hilbert space for the unit sphere by setting an angular momentum cutoff and a fuzzy sphere truncation inspired by non-commutative geometry. We compare the spectrum of the truncated theories in a finite box with the full theory. We use open boundary conditions, a novel method that improves on the correlation lengths accessible in our calculations. We provide evidence that the angular-momentum truncation fails to reproduce the $\sigma$-model and that the anti-ferromagnetic fuzzy model agrees with the full theory.

Sharpening the Conformal Symmetry of the 3D Ising Transition by Fuzzy Sphere Simulations

Uncovering conformal symmetry in the 3D Ising transition: State-operator correspondence from a fuzzy sphere regularization Authors: Wei Zhu, Chao Han, Emilie Huffman, Johannes S. Hofmann, and Yin-Chen He arXiv:2210.13482; DOI: 10.48550/arXiv.2210.13482 Recommended with a commentary by Ashvin Vishwanath, Harvard University |View Commentary (pdf)| This commentary may be cited as: DOI: 10.36471/JCCM_January_2023_02 https://doi.org/10.36471/JCCM_January_2023_02

Pinhole interference in three-dimensional fuzzy space

We investigate a quantum-to-classical transition which arises naturally within the fuzzy sphere formalism for three-dimensional non-commutative quantum mechanics. This transition may be understood as the mechanism of decoherence, but without requiring an additional external heat bath. We focus on treating a two-pinhole interference configuration within this formalism, as it provides an illustrative toy model for which this transition is readily observed and quantified. Specifically, we demonstrate a suppression of the quantum interference effects for objects passing through the pinholes with sufficiently-high energies or numbers of constituent particles.Our work extends a similar treatment of the double slit experiment, presented in Pittaway and Scholtz (2021), within the two-dimensional Moyal plane, only it addresses two key shortcomings that arise in that context. These are, firstly that the interference pattern in the Moyal plane lacks the expected reflection symmetry present in the pinhole setup, and secondly that the quantum-to-classical transition manifested in the Moyal plane occurs only at unrealistically high velocities and/or particle numbers. Both of these issues are solved in the fuzzy sphere framework.

Matrix entanglement

In gauge/gravity duality, matrix degrees of freedom on the gauge theory side play important roles for the emergent geometry. In this paper, we discuss how the entanglement on the gravity side can be described as the entanglement between matrix degrees of freedom. Our approach, which we call ‘matrix entanglement’, is different from ‘target-space entanglement’ proposed and discussed recently by several groups. We consider several classes of quantum states to which our approach can play important roles. When applied to fuzzy sphere, matrix entanglement can be used to define the usual spatial entanglement in two-brane or five-brane world-volume theory nonperturbatively in a regularized setup. Another application is to a small black hole in AdS5×S5 that can evaporate without being attached to a heat bath, for which our approach suggests a gauge theory origin of the Page curve. The confined degrees of freedom in the partially-deconfined states play the important roles.

Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models

A fuzzy geometry is a certain type of spectral triple whose Dirac operator crucially turns out to be a finite matrix. This notion incorporates familiar examples like fuzzy spheres and fuzzy tori. In the framework of random noncommutative geometry, we use Barrett’s characterization of Dirac operators of fuzzy geometries in order to systematically compute the spectral action {S(D)= \operatorname{Tr} f(D)} for 2n -dimensional fuzzy geometries. In contrast to the original Chamseddine–Connes spectral action, we take a polynomial f with f(x)\to \infty as \vert x\vert \to\infty in order to obtain a well-defined path integral that can be stated as a random matrix model with action of the type {S(D)=N \cdot \operatorname{tr} F+\sum_i \operatorname{tr}A_i \cdot \operatorname{tr} B_i } , being F , \smash{A_i} and B_i noncommutative polynomials in \smash{2^{2n-1}} complex N\times N matrices that parametrize the Dirac operator D . For arbitrary signature—thus for any admissible KO-dimension—formulas for 2 -dimensional fuzzy geometries are given up to a sextic polynomial, and up to a quartic polynomial for 4 -dimensional ones, with focus on the octo-matrix models for Lorentzian and Riemannian signatures. The noncommutative polynomials F , \smash{A_i} and B_i are obtained via chord diagrams and satisfy: independence of N ; self-adjointness of the main polynomial F (modulo cyclic reordering of each monomial); also up to cyclicity, either self-adjointness or anti-self-adjointness of \smash{A_i} and B_i simultaneously, for fixed i . Collectively, this favors a free probabilistic perspective for the large- N limit we elaborate on.

Deformation quantization of the simplest Poisson orbifold

Whenever a given Poisson manifold is equipped with discrete symmetries the corresponding algebra of invariant functions or the algebra of functions twisted by the symmetry group can have new deformations, which are not captured by Kontsevich Formality. We consider the simplest example of this situation: R2 with the reflection symmetry Z2. The usual quantization leads to the Weyl algebra. While Weyl algebra is rigid, the algebra of even or twisted by Z2 functions has one more deformation, which was identified by Wigner and is related to Feigin's glλ and to fuzzy sphere. With the help of homological perturbation theory we obtain explicit formula for the deformed product, the first order of which can be extracted from Shoikhet–Tsygan–Kontsevich formality.

Massive Neutron Stars and White Dwarfs as Noncommutative Fuzzy Spheres

Over the last couple of decades, there have been direct and indirect evidences for massive compact objects than their conventional counterparts. A couple of such examples are super-Chandrasekhar white dwarfs and massive neutron stars. The observations of more than a dozen peculiar over-luminous type Ia supernovae predict their origins from super-Chandrasekhar white dwarf progenitors. On the other hand, recent gravitational wave detection and some pulsar observations provide arguments for massive neutron stars, lying in the famous mass-gap between lowest astrophysical black hole and conventional highest neutron star masses. We show that the idea of a squashed fuzzy sphere, which brings in noncommutative geometry, can self-consistently explain either of the massive objects as if they are actually fuzzy or squashed fuzzy spheres. Noncommutative geometry is a branch of quantum gravity. If the above proposal is correct, it will provide observational evidences for noncommutativity.

String modes, propagators and loops on fuzzy spaces

We present a systematic organization of functions and operators on the fuzzy 2-sphere in terms of string modes, which are optimally localized in position and momentum space. This allows to separate the semi-classical and the deep quantum regime of non-commutative quantum field theory, and exhibits its nonlocal nature. This organization greatly simplifies the computation of loop contributions, avoiding oscillatory integrals and providing the effective action directly in position space. UV/IR mixing is understood as nonlocality arising from long string modes in the loops. The method is suited for any quantized symplectic space.

The AdS𝜃2/CFT1 correspondence and noncommutative geometry III: Phase structure of the noncommutative AdS𝜃2 × 𝕊N2

The near-horizon noncommutative geometry of black holes, given by [Formula: see text], is discussed and the phase structure of the corresponding Yang–Mills matrix models is presented. The dominant phase transition as the system cools down, i.e. as the gauge coupling constant is decreased is an emergent geometry transition between a geometric noncommutative [Formula: see text] phase (discrete spectrum) and a Yang–Mills matrix phase (continuous spectrum) with no background geometrical structure. We also find a possibility for topology change transitions in which space or time directions grow or decay as the temperature is varied. Indeed, the noncommutative near-horizon geometry [Formula: see text] can evaporate only partially to a fuzzy sphere [Formula: see text] (emergence of time) or to a noncommutative anti-de Sitter space–time [Formula: see text] (topology change).

Eigenvalue-flipping algorithm for matrix Monte Carlo

Many physical systems can be described in terms of matrix models that we often cannot solve analytically. Fortunately, they can be studied numerically in a straightforward way. Many commonly used algorithms follow the Monte Carlo method, which is efficient for small matrix sizes but cannot guarantee ergodicity when working with large ones. In this paper, we propose an improvement of the algorithm that, for a large class of matrix models, allows to tunnel between various vacua in a proficient way, where sign change of eigenvalues is proposed externally. We test the method on two models: the pure potential matrix model and the scalar field theory on the fuzzy sphere.

The Podleś Spheres Converge to the Sphere

We prove that the Podles spheres $S_q^2$ converge in quantum Gromov-Hausdorff distance to the classical 2-sphere as the deformation parameter $q$ tends to 1. Moreover, we construct a $q$-deformed analogue of the fuzzy spheres, and prove that they converge to $S_q^2$ as their linear dimension tends to infinity, thus providing a quantum counterpart to a classical result of Rieffel.

Spectral triple with real structure on fuzzy sphere

In this paper, we have illustrated the construction of a real structure on a fuzzy sphere S*2 in its spin-1/2 representation. Considering the SU(2) covariant Dirac and chirality operator on S*2 given by U. C. Watamura and Watamura [Commun. Math. Phys. 183, 365–382 (1997) and Commun. Math. Phys. 212, 395–413 (2000)], we have shown that the real structure is consistent with other spectral data for KO dimension-4 fulfilling the zero order condition, where we find it necessary to enlarge the symmetry group from SO(3) to the full orthogonal group O(3). However, the first order condition is violated, thus paving the way to construct a toy model for an SU(2) gauge theory to capture some features of physics beyond the standard model following Chamseddine et al. (J. High Energy Phys. 2013, 132).

Geometric Dirac operator on the fuzzy sphere

We construct a Connes spectral triple or ‘Dirac operator’ on the non-reduced fuzzy sphere mathbb {C}_lambda [S^2] as realised using quantum Riemannian geometry with a central quantum metric g of Euclidean signature and its associated quantum Levi-Civita connection. The Dirac operator is characterised uniquely up to unitary equivalence within our quantum Riemannian geometric setting and an assumption that the spinor bundle is trivial and rank 2 with a central basis. The spectral triple has KO dimension 3 and in the case of the round metric, essentially, recovers a previous proposal motivated by rotational symmetry.

The fuzzy sphere morphology is responsible for the increase in light scattering during the shrinkage of thermoresponsive microgels.

Thermoresponsive microgels undergo a volume phase transition from a swollen state under good solvent conditions to a collapsed state under poor solvent conditions. The most prominent examples of such responsive systems are based on poly-(N-isopropylacrylamide). When cross-linked with N,N'-methylenebisacrylamide, such microgels typically possess a fuzzy-spherelike morphology with a higher cross-linked core and a loosely cross-linked fuzzy shell. Despite the efforts devoted to understanding the internal structure of microgels and their kinetics during collapse/swelling, the origins of the accompanying changes in light scattering intensity have barely been addressed. In this work, we study core-shell microgels that contain small gold nanoparticle cores with microgel shells of different thicknesses and cross-linker densities. All microgels are small enough to fulfill the Rayleigh-Debye-Gans criterion at all stages of swelling. Due to the high X-ray contrast of the gold cores, we can use absolute intensity small-angle X-ray scattering to determine the number density in the dilute dispersions. This allows us to extract polymer volume fractions of the microgels at different stages of swelling from form factor analysis of small-angle neutron scattering data. We match our findings to results from temperature-dependent absorbance measurements. The increase in absorbance during the shrinkage of the microgels is related to the transition from fuzzy spheres to hard sphere-like scattering objects with a rather homogeneous density profile. We provide a first attempt to model experimental spectra using finite difference time domain simulations that take into account the structural changes during the volume phase transition. Our findings significantly contribute to the understanding of the optical properties of thermoresponsive microgels. Further, we provide polymer volume fractions and microgel refractive indices as a function of the swelling state.

A noncommutative spacetime realization of quantum black holes, Regge trajectories and holography

It is shown that the radial spectrum associated with a fuzzy sphere in a noncommutative phase space characterized by the Yang algebra, leads exactly to a Regge-like spectrum GMl2=l=1,2,3,…, for all positive values of l, and which is consistent with the extremal quantum Kerr black hole solution that occurs when the outer and inner horizon radius coincide r+=r−=GM. The condition GMl2=l is tantamount to the mass-angular momentum relation Ml2=lMp2 implying a mass-squared quantization in multiples of the Planck-mass-squared. Another important feature (also pointed out by Tanaka) is the holographic nature of these results that are based in recasting the Yang algebra associated with an 8D noncommuting phase space, involving xμ,pν,μ,ν=0,1,2,3, in terms of the undeformed realizations of the Lorentz algebra generators JAB corresponding to a 6D-spacetime, and associated to a 12D-phase-space with coordinates XA,PA;A=0,1,2,…,5. We finalize with a discussion of the noncommuting 3D isotropic and Born oscillators. Finding solutions to these oscillators merit investigation because they introduce explicit dynamics to the quantum black holes. We hope that the findings in this work, relating the Regge-like spectrum l=GM2 and the quantized area of black hole horizons in Planck bits, via the Yang algebra in Noncommutative phase spaces, will help us elucidate some of the impending issues pertaining the black hole information paradox and the role that string theory and quantum information will play in its resolution.

Batalin\u2013Vilkovisky quantization of fuzzy field theories

We apply the modern Batalin–Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite-dimensional analogues of the field theories proposed recently through the notion of ‘braided L_infty -algebras’. The techniques are illustrated by computing perturbative correlation functions for scalar and Chern–Simons theories on the fuzzy 2-sphere, as well as for braided scalar field theories on the fuzzy 2-torus.

Momentum-space entanglement in scalar field theory on fuzzy spheres

Quantum field theory defined on a noncommutative space is a useful toy model of quantum gravity and is known to have several intriguing properties, such as nonlocality and UV/IR mixing. They suggest novel types of correlation among the degrees of freedom of different energy scales. In this paper, we investigate such correlations by the use of entanglement entropy in the momentum space. We explicitly evaluate the entanglement entropy of scalar field theory on a fuzzy sphere and find that it exhibits different behaviors from that on the usual continuous sphere. We argue that these differences would originate in different characteristics; non-planar contributions and matrix regularizations. It is also found that the mutual information between the low and the high momentum modes shows different scaling behaviors when the effect of a cutoff becomes important.

Two approaches to quantum gravity and M-(atrix) theory at large number of dimensions

A Gaussian approximation to the bosonic part of M-(atrix) theory with mass deformation is considered at large values of the dimension [Formula: see text]. From the perspective of the gauge/gravity duality this action reproduces with great accuracy the stringy Hagedorn phase transition from a confinement (black string) phase to a deconfinement (black hole) phase whereas from the perspective of the matrix/geometry approach this action only captures a remnant of the geometric Yang–Mills-to-fuzzy-sphere phase where the fuzzy sphere solution is only manifested as a three-cut configuration termed the “baby fuzzy sphere” configuration. The Yang–Mills phase retains most of its characteristics with two exceptions: (i) the uniform distribution inside a solid ball suffers a crossover at very small values of the gauge coupling constant to a Wigner’s semicircle law, and (ii) the uniform distribution at small values of the temperatures is nonexistent.