Showing papers on "Quantum geometry published in 2019"

Effective Quantum Extended Spacetime of Polymer Schwarzschild Black Hole

Abstract: The physical interpretation and eventual fate of gravitational singularities in a theory surpassing classical general relativity are puzzling questions that have generated a great deal of interest among various quantum gravity approaches. In the context of loop quantum gravity (LQG), one of the major candidates for a non-perturbative background-independent quantisation of general relativity, considerable effort has been devoted to construct effective models in which these questions can be studied. In these models, classical singularities are replaced by a ‘bounce’ induced by quantum geometry corrections. Undesirable features may arise however depending on the details of the model. In this paper, we focus on Schwarzschild black holes and propose a new effective quantum theory based on polymerisation of new canonical phase space variables inspired by those successful in loop quantum cosmology. The quantum corrected spacetime resulting from the solutions of the effective dynamics is characterised by infinitely many pairs of trapped and anti-trapped regions connected via a space-like transition surface replacing the central singularity. Quantum effects become relevant at a unique mass independent curvature scale, while they become negligible in the low curvature region near the horizon. The effective quantum metric describes also the exterior regions and asymptotically classical Schwarzschild geometry is recovered. We however find that physically acceptable solutions require us to select a certain subset of initial conditions, corresponding to a specific mass (de-)amplification after the bounce. We also sketch the corresponding quantum theory and explicitly compute the kernel of the Hamiltonian constraint operator.

Direct measurement of the quantum geometric tensor in a two-dimensional continuous medium

Abstract: Topological Physics relies on the specific structure of the eigenstates of Hamiltonians. Their geometry is encoded in the quantum geometric tensor containing both the celebrated Berry curvature, crucial for topological matter, and the quantum metric. The latter is at the heart of a growing number of physical phenomena such as superfluidity in flat bands, orbital magnetic susceptibility, exciton Lamb shift, and non-adiabatic corrections to the anomalous Hall effect. Here, we report the first direct measurement of both Berry curvature and quantum metric in a two-dimensional continuous medium. The studied platform is a planar microcavity of extremely high finesse, in the strong coupling regime. It hosts mixed exciton-photon modes (exciton-polaritons) subject to photonic spin-orbit-coupling which makes emerge Dirac cones and exciton Zeeman splitting breaking time-reversal symmetry. The monopolar and half-skyrmion pseudospin textures are measured by polarisation-resolved photoluminescence. The associated quantum geometry of the bands is straightforwardly extracted from these measurements. Our results unveil the intrinsic chirality of photonic modes which is at the basis of topological photonics. This technique can be extended to measure Bloch band geometries in artificial lattices. The use of exciton-polaritons (interacting photons) opens wide perspectives for future studies of quantum fluid physics in topological systems.

Semiclassical wave packet dynamics in nonuniform electric fields

Abstract: We study the semiclassical theory of wave packet dynamics in crystalline solids extended to include the effects of a nonuniform electric field. In particular, we derive a correction to the semiclassical equations of motion (EOMs) for the dynamics of the wave packet center that depends on the gradient of the electric field and on the quantum metric (also called the Fubini-Study, Bures, or Bloch metric) on the Brillouin zone. We show that the physical origin of this term is a contribution to the total energy of the wave packet that depends on its electric quadrupole moment and on the electric-field gradient. We also derive an equation relating the electric quadrupole moment of a sharply peaked wave packet to the quantum metric evaluated at the wave packet center in reciprocal space. Finally, we explore the physical consequences of this correction to the semiclassical EOMs. We show that in a metal with broken time-reversal and inversion symmetry, an electric-field gradient can generate a longitudinal current which is linear in the electric-field gradient, and which depends on the quantum metric at the Fermi surface. We then give two examples of concrete lattice models in which this effect occurs. Our results show that nonuniform electric fields can be used to probe the quantum geometry of the electronic bands in metals and open the door to further studies of the effects of nonuniform electric fields in solids.

Mass and Horizon Dirac Observables in Effective Models of Quantum Black-to-White Hole Transition

Abstract: In the past years, black holes and the fate of their singularity have been heavily studied within loop quantum gravity. Effective spacetime descriptions incorporating quantum geometry corrections are provided by the so-called polymer models. Despite the technical differences, the main common feature shared by these models is that the classical singularity is resolved by a black-to-white hole transition. In a recent paper, we discussed the existence of two Dirac observables in the effective quantum theory respectively corresponding to the black and white hole mass. Physical requirements about the onset of quantum effects then fix the relation between these observables after the bounce, which in turn corresponds to a restriction on the admissible initial conditions for the model. In the present paper, we discuss in detail the role of such observables in black hole polymer models. First, we revisit previous models and analyse the existence of the Dirac observables there. Observables for the horizons or the masses are explicitly constructed. In the classical theory, only one Dirac observable has physical relevance. In the quantum theory, we find a relation between the existence of two physically relevant observables and the scaling behaviour of the polymerisation scales under fiducial cell rescaling. We present then a new model based on polymerisation of new variables which allows to overcome previous restrictions on initial conditions. Quantum effects cause a bound of a unique Kretschmann curvature scale, independently of the relation between the two masses.

Group field theory condensate cosmology: An appetizer

Abstract: This contribution is an appetizer to the relatively young and fast-evolving approach to quantum cosmology based on group field theory condensate states. We summarize the main assumptions and pillars of this approach which has revealed new perspectives on the long-standing question of how to recover the continuum from discrete geometric building blocks. Among others, we give a snapshot of recent work on isotropic cosmological solutions exhibiting an accelerated expansion, a bounce where anisotropies are shown to be under control, and inhomogeneities with an approximately scale-invariant power spectrum. Finally, we point to open issues in the condensate cosmology approach.

Effective Quantum Extended Spacetime of Polymer Schwarzschild Black Hole

Abstract: The physical interpretation and eventual fate of gravitational singularities in a theory surpassing classical general relativity are puzzling questions that have generated a great deal of interest among various quantum gravity approaches. In the context of loop quantum gravity (LQG), one of the major candidates for a non-perturbative background-independent quantisation of general relativity, considerable effort has been devoted to construct effective models in which these questions can be studied. In these models, classical singularities are replaced by a "bounce" induced by quantum geometry corrections. Undesirable features may arise however depending on the details of the model. In this paper, we focus on Schwarzschild black holes and propose a new effective quantum theory based on polymerisation of new canonical phase space variables inspired by those successful in loop quantum cosmology. The quantum corrected spacetime resulting from the solutions of the effective dynamics is characterised by infinitely many pairs of trapped and anti-trapped regions connected via a space-like transition surface replacing the central singularity. Quantum effects become relevant at a unique mass independent curvature scale, while they become negligible in the low curvature region near the horizon. The effective quantum metric describes also the exterior regions and asymptotically classical Schwarzschild geometry is recovered. We however find that physically acceptable solutions require us to select a certain subset of initial conditions, corresponding to a specific mass (de-)amplification after the bounce. We also sketch the corresponding quantum theory and explicitly compute the kernel of the Hamiltonian constraint operator.

Bubble networks: framed discrete geometry for quantum gravity

Abstract: In the context of canonical quantum gravity in 3 $$+$$ 1 dimensions, we introduce a new notion of bubble network that represents discrete 3d space geometries. These are natural extensions of twisted geometries, which represent the geometrical data underlying loop quantum geometry and are defined as networks of $$\mathrm {SU}(2)$$ holonomies. In addition to the $$\mathrm {SU}(2)$$ representations encoding the geometrical flux, the bubble network links carry a compatible $$\mathrm {SL}(2,{{\mathbb {R}}})$$ representation encoding the discretized frame field which composes the flux. In contrast with twisted geometries, this extra data allows to reconstruct the frame compatible with the flux unambiguously. At the classical level this data represents a network of 3d geometrical cells glued together. The $$\mathrm {SL}(2,{{\mathbb {R}}})$$ data contains information about the discretized 2d metrics of the interfaces between 3d cells and $$\mathrm {SL}(2,{{\mathbb {R}}})$$ local transformations are understood as the group of area-preserving diffeomorphisms. We further show that the natural gluing condition with respect to this extended group structure ensures that the intrinsic 2d geometry of a boundary surface is the same from the viewpoint of the two cells sharing it. At the quantum level this gluing corresponds to a maximal entanglement along the network edges. We emphasize that the nature of this extension of twisted geometries is compatible with the general analysis of gauge theories that predicts edge mode degrees of freedom at the interface of subsystems.

Probing localization and quantum geometry by spectroscopy

Abstract: This article introduces an efficient and universal detection method by which localization can be finely measured: the proposed protocol consists in shaking the system of interest and to monitor the resulting heating. This method opens an avenue for probing localization, but also quantum fluctuations and entanglement, in synthetic quantum matter.

Protected $SL(2,\mathbb{R})$ Symmetry in Quantum Cosmology

Abstract: The polymer quantization of cosmological backgrounds provides an alternative path to the original Wheeler-de Witt (WdW) quantum cosmology, based on a different representation the commutation relations of the canonical variables. This polymer representation allows to capture the lattice like structure of the quantum geometry and leads to a radically different quantum cosmology compared to the WdW construction. This new quantization scheme has attracted considerable attention due to the singularity resolution it allows in a wide class of symmetry reduced gravitational systems, in particular where the WdW scheme fails. However, as any canonical quantization scheme, ambiguities in the construction of the quantum theory, being regularization or factor-ordering ones, can drastically modify the resulting quantum dynamics. In this work, we propose a new criteria to restrict the quantization ambiguities in the simplest model of polymer quantum cosmology, for homogeneous and isotropic General Relativity minimally coupled to a massless scalar field. This new criteria is based on an underlying $\mathfrak{sl}(2,\mathbb{R})$ structure present in the phase space of this simple cosmological model. By preserving the symmetry of this cosmological system under this 1d conformal group, we derive a new regularization of the phase space. We perform both its polymer quantization and a quantization scheme directly providing a representation of the SL$(2,\mathbb{R})$ group action. The resulting quantum cosmology can be viewed as a lattice-like quantum mechanics with an SL$(2,\mathbb{R})$ invariance. This provides a new version of Loop Quantum Cosmology consistent with the conformal symmetry. This alternative construction opens new directions, among which a possible mapping with the conformal quantum mechanics as well as with recent matrix or tensor models constructions for quantum cosmological space-time.

Aspects of quantum gravity

Abstract: For more than 80 years theoretical physicists have been trying to develop a theory of quantum gravity which would successfully combine the tenets of Einstein's theory of general relativity (GR) together with those of quantum field theory. At the current stage, there are various competing responses to this challenge under construction. Attacking the problem of quantum gravity from the quantum geometry perspective, where space and spacetime are discrete, the focus of this thesis lies on the application of loop quantum gravity (LQG) and group field theory (GFT). We employ these two closely related nonperturbative approaches to two areas where quantum gravity effects are broadly expected to be relevant: black holes and quantum cosmology. Concerning black holes, apart from understanding their inner structure, a pressing issue is to give a microscopic explanation for the phenomenon of black hole entropy in terms of a discrete quantum geometry and relate it to the symmetries of the horizon. Black hole models in LQG are typically constructed via the isolated horizon boundary condition which gives rise to an effective description of the horizon geometry in terms SU(2) Chern-Simons theory. In this thesis we find a reinterpretation of the statistics of the horizon degrees of freedom as those of a system of non-Abelian anyons. As regards quantum cosmology, the challenge is to understand how the initial singularity problem of GR can be resolved by means of the discreteness of geometry and how a continuum spacetime can emerge from a large assembly of geometric building blocks. Most recent research in GFT aims at deriving the effective dynamics for condensate states directly from the microscopic GFT quantum dynamics and subsequently to extract a cosmological interpretation from them. In this thesis we elaborate on aspects of this approach and study phenomenological consequences in detail.

Protected $SL(2,\mathbb{R})$ Symmetry in Quantum Cosmology

Abstract: The polymer quantization of cosmological backgrounds provides an alternative path to the original Wheeler-de Witt (WdW) quantum cosmology, based on a different representation the commutation relations of the canonical variables. This polymer representation allows to capture the lattice like structure of the quantum geometry and leads to a radically different quantum cosmology compared to the WdW construction. This new quantization scheme has attracted considerable attention due to the singularity resolution it allows in a wide class of symmetry reduced gravitational systems, in particular where the WdW scheme fails. However, as any canonical quantization scheme, ambiguities in the construction of the quantum theory, being regularization or factor-ordering ones, can drastically modify the resulting quantum dynamics. In this work, we propose a new criteria to restrict the quantization ambiguities in the simplest model of polymer quantum cosmology, for homogeneous and isotropic General Relativity minimally coupled to a massless scalar field. This new criteria is based on an underlying SL(2,B) structure present in the phase space of this simple cosmological model. By preserving the symmetry of this cosmological system under this 1d conformal group, we derive a new regularization of the phase space. We perform both its polymer quantization and a quantization scheme directly providing a representation of the SL(2,B) group action. The resulting quantum cosmology can be viewed as a lattice-like quantum mechanics with an SL(2,B) invariance. This provides a new version of Loop Quantum Cosmology consistent with the conformal symmetry. This alternative construction opens new directions, among which a possible mapping with the conformal quantum mechanics as well as with recent matrix or tensor models constructions for quantum cosmological space-time.

Bouncing evolution in a model of loop quantum gravity

Abstract: To understand the dynamics of loop quantum gravity, the deparametrized model of gravity coupled to a scalar field is studied in a simple case, where the graph underlying the spin network basis is one loop based at a single vertex. The Hamiltonian operator ${\stackrel{^}{H}}_{v}$ is chosen to be graph-preserving, and the matrix elements of ${\stackrel{^}{H}}_{v}$ are explicitly worked out in a suitable basis. The nontrivial Euclidean part ${\stackrel{^}{H}}_{v}^{E}$ of ${\stackrel{^}{H}}_{v}$ is studied in details. It turns out that by choosing a specific symmetrization of ${\stackrel{^}{H}}_{v}^{E}$, the dynamics driven by the Hamiltonian give a picture of bouncing evolution. Our result in the model of full loop quantum gravity gives a significant echo of the well-known quantum bounce in the symmetry-reduced model of loop quantum cosmology, which indicates a closed relation between singularity resolution and quantum geometry.

Pre-inflationary dynamics of Starobinsky inflation and its generization in loop quantum Brans-Dicke cosmology

Abstract: Recently, the nonperturbative quantization scheme of loop quantum gravity has been extended to the Brans-Dicke theory and the corresponding loop quantum Brans-Dicke cosmology has been derived, which provides an essential platform to explore inflationary models in this framework. In this paper, we consider two inflation models, the Starobinsky and $\alpha$-attractor inflation whose cosmological predictions are in excellent agreement with Planck data, and study systematically their pre-inflationary dynamics as well as the slow-roll inflation. We show that for both models, the background evolution of a flat Friedmann-Lemaitre-Robertson-Walker universe in general can be divided into three different phases: the pre-inflationary quantum phase, quantum-to-classical transition, and the slow-roll inflation. The pre-inflationary dynamics are dominated by the quantum geometry effects of loop quantum Brans-Dicke cosmology and the corresponding Universe could be either initially expanding or contracting, depending on the initial velocity of inflaton field. It is shown that the detailed evolution of pre-inflationary quantum phase also depend on specific inflation models. After the pre-inflationary quantum phase, the universe gradually evolves into the slow-roll inflation with some of initial conditions for Starobinsky and $\alpha$-attractor potentials. In addition, to be consistent with observational data, we also find the restricted parameter space of initial conditions that could produce at least $60$ $e$-folds during the slow-roll inflation.

On the Geometry of No-Boundary Instantons in Loop Quantum Cosmology

Abstract: We study the geometry of Euclidean instantons in loop quantum cosmology (LQC) such as those relevant for the no-boundary proposal. Confining ourselves to the simplest case of a cosmological constant in minisuperspace cosmologies, we analyze solutions of the semiclassical (Euclidean) path integral in LQC. We find that the geometry of LQC instantons have the peculiar feature of an infinite tail which distinguishes them from Einstein gravity. Moreover, due to quantum-geometry corrections, the small-a behaviour of these instantons seem to naturally favor a closing-off of the geometry in a regular fashion, as was originally proposed for the no-boundary wavefunction.

Generic absence of strong singularities and geodesic completeness in modified loop quantum cosmologies

Abstract: Different regularizations of the Hamiltonian constraint in loop quantum cosmology yield modified loop quantum cosmologies, namely mLQC-I and mLQC-II, which lead to qualitatively different Planck scale physics. We perform a comprehensive analysis of resolution of various singularities in these modified loop cosmologies using effective spacetime description and compare with earlier results in standard loop quantum cosmology. We show that the volume remains non-zero and finite in finite time evolution for all considered loop cosmological models. Interestingly, even though expansion scalar and energy density are bounded due to quantum geometry, curvature invariants can still potentially diverge due to pressure singularities at a finite volume. These divergences are shown to be harmless since geodesic evolution does not break down and no strong singularities are present in the effective spacetimes of loop cosmologies. Using a phenomenological matter model, various types of exotic strong and weak singularities, including big rip, sudden, big freeze and type-IV singularities, are studied. We show that as in standard loop quantum cosmology, big rip and big freeze singularities are resolved in mLQC-I and mLQC-II, but quantum geometric effects do not resolve sudden and type-IV singularities.

Generalised uncertainty relations for angular momentum and spin in quantum geometry

Abstract: We derive generalised uncertainty relations (GURs) for angular momentum and spin in the smeared-space model of quantum geometry. The model implements a minimum length and a minimum linear momentum, and recovers both the generalised uncertainty principle (GUP) and the extended uncertainty principle (EUP) within a single formalism. In this paper, we investigate the consequences of these results for particles with extrinsic and intrinsic angular momentum, and obtain generalisations of the canonical ${\rm so(3)}$ and ${\rm su(2)}$ algebras. We find that, although ${\rm SO(3)}$ symmetry is preserved on three-dimensional slices of an enlarged phase space, individual subcomponents of the generalised generators obey nontrivial subalgebras. These give rise to GURs for angular momentum while leaving the canonical commutation relations intact except for a simple rescaling, $\hbar \rightarrow \hbar + \beta$. The value of the new parameter, $\beta \simeq \hbar \times 10^{-61}$, is determined by the ratio of the dark energy density to the Planck density. Here, we assume the former to be of the order of the Planck length and the latter to be of the order of the de Sitter momentum $\sim \hbar\sqrt{\Lambda}$, where $\Lambda$ is the cosmological constant. In the smeared-space model, $\hbar$ and $\beta$ are interpreted as the quantisation scales for matter and geometry, respectively, and a quantum state vector is associated with the spatial background. We show that this also gives rise to a rescaled Lie algebra for generalised spin operators, together with associated subalgebras that are analogous to those for orbital angular momentum. Remarkably, consistency of the algebraic structure requires the quantum state associated with a flat background to be fermionic, with spin eigenvalues $\pm \beta/2$. Finally, the modified spin algebra leads to GURs for spin measurements.

Quantum geometry from higher gauge theory.

Abstract: Higher gauge theories play a prominent role in the construction of 4d topological invariants and have been long ago proposed as a tool for 4d quantum gravity. The Yetter lattice model and its continuum counterpart, the BFCG theory, generalize BF theory to 2-gauge groups and -- when specialized to 4d and the Poincare 2-group -- they provide an exactly solvable topologically-flat version of 4d general relativity. The 2-Poincare Yetter model was conjectured to be equivalent to a state sum model of quantum flat spacetime developed by Baratin and Freidel after work by Korepanov (KBF model). This conjecture was motivated by the origin of the KBF model in the theory of 2-representations of the Poincare 2-group. Its proof, however, has remained elusive due to the lack of a generalized Peter-Weyl theorem for 2-groups. In this work we prove this conjecture. Our proof avoids the Peter-Weyl theorem and rather leverages the geometrical content of the Yetter model. Key for the proof is the introduction of a kinematical boundary Hilbert space on which 1- and 2-Lorentz invariance is imposed. Geometrically this allows the identification of (quantum) tetrad variables and of the associated (quantum) Levi-Civita connection. States in this Hilbert space are labelled by quantum numbers that match the 2-group representation labels. Our results open exciting opportunities for the construction of new representations of quantum geometries. Compared to loop quantum gravity, the higher gauge theory framework provides a quantum representation of the ADM-Regge initial data, including an identification of the intrinsic and extrinsic curvature. Furthermore, it leads to a version of the diffeomorphism and Hamiltonian constraints that acts on the vertices of the discretization, thus providing a prospect for a quantum realization of the hypersurface deformation algebra in 4d.

Unitarity and information in quantum gravity: a simple example

Abstract: In approaches to quantum gravity, where smooth spacetime is an emergent approximation of a discrete Planckian fundamental structure, any effective smooth field theoretical description would miss part of the fundamental degrees of freedom and thus break unitarity. This is applicable also to trivial gravitational field (low energy) idealizations realized by the use of the Minkowski background geometry which, as any other spacetime geometry, corresponds, in the fundamental description, to infinitely many different and closely degenerate discrete microstates. The existence of such microstates provides a large reservoir for information to be coded at the end of black hole evaporation and thus opens the way to a natural resolution of the black hole evaporation information puzzle. In this paper we show that these expectations can be made precise in a simple quantum gravity model for cosmology motivated by loop quantum gravity. Concretely, even when the model is fundamentally unitary, when microscopic degrees of freedom irrelevant to low-energy cosmological observers are suitably ignored, pure states in the effective description evolve into mixed states due to decoherence with the Planckian microscopic structure. Moreover, in the relevant physical regime these hidden degrees freedom do not carry any `energy' and thus realize in a fully quantum gravitational context the idea (emphasized before by Unruh and Wald) that decoherence can take place without dissipation, now in a concrete gravitational model strongly motivated by quantum gravity. All this strengthen the perspective of a quite conservative and natural resolution of the black hole evaporation puzzle where information is not destroyed but simply degraded (made unavailable to low energy observers) into correlations with the microscopic structure of the quantum geometry at the Planck scale.

Finite geometric toy model of spacetime as an error correcting code

Abstract: A finite geometric model of space-time (which we call the bulk) is shown to emerge as a set of error correcting codes. The bulk is encoding a set of messages located in a blow up of the Gibbons-Hoffman-Wootters (GHW) discrete phase space for $n$-qubits (which we call the boundary). Our error correcting code is a geometric subspace code known from network coding, and the correspondence map is the finite geometric analogue of the Pl\"ucker map well-known from twistor theory. The $n=2$ case of the bulk-boundary correspondence is precisely the twistor correspondence where the boundary is playing the role of the twistor space and the bulk is a finite geometric version of compactified Minkowski space-time. For $n\ensuremath{\ge}3$ the bulk is identified with the finite geometric version of the Brody-Hughston quantum space-time. For special regions on both sides of the correspondence we associate certain collections of qubit observables. On the boundary side this association gives rise to the well-known GHW quantum net structure. In this picture the messages are complete sets of commuting observables associated to Lagrangian subspaces giving a partition of the boundary. Incomplete subsets of observables corresponding to subspaces of the Lagrangian ones are regarded as corrupted messages. Such a partition of the boundary is represented on the bulk side as a special collection of space-time points. For a particular message residing in the boundary, the set of possible errors is described by the fine details of the light-cone structure of its representative space-time point in the bulk. The geometric arrangement of representative space-time points, playing the role of the variety of codewords, encapsulates an algebraic algorithm for recovery from errors on the boundary side.

Spectral dimension on spatial hypersurfaces in causal set quantum gravity

Abstract: An important probe of quantum geometry is its spectral dimension, defined via a spatial diffusion process. In this work we study the spectral dimension of a ``spatial hypersurface'' in a manifoldlike causal set using the induced spatial distance function. In previous work, the diffusion was taken on the full causal set, where the nearest neighbours are unbounded in number. The resulting super-diffusion leads to an increase in the spectral dimension at short diffusion times, in contrast to other approaches to quantum gravity. In the current work, by using a temporal localisation in the causal set, the number of nearest spatial neighbours is rendered finite. Using numerical simulations of causal sets obtained from $d=3$ Minkowski spacetime, we find that for a flat spatial hypersurface, the spectral dimension agrees with the Hausdorff dimension at intermediate scales, but shows clear indications of dimensional reduction at small scales, i.e., in the ultraviolet. The latter is a direct consequence of ``discrete asymptotic silence'' at small scales in causal sets.

Manifest non-locality in quantum mechanics, quantum field theory and quantum gravity

Abstract: We summarize our recent work on the foundational aspects of string theory as a quantum theory of gravity. We emphasize the hidden quantum geometry (modular spacetime) behind the generic representat...

Deep Quantum Geometry of Matrices

Abstract: We employ machine learning techniques to provide accurate variational wavefunctions for matrix quantum mechanics, with multiple bosonic and fermionic matrices. Variational quantum Monte Carlo is implemented with deep generative flows to search for gauge invariant low energy states. The ground state, and also long-lived metastable states, of an $\mathrm{SU}(N)$ matrix quantum mechanics with three bosonic matrices, as well as its supersymmetric `mini-BMN' extension, are studied as a function of coupling and $N$. Known semiclassical fuzzy sphere states are recovered, and the collapse of these geometries in more strongly quantum regimes is probed using the variational wavefunction. We then describe a factorization of the quantum mechanical Hilbert space that corresponds to a spatial partition of the emergent geometry. Under this partition, the fuzzy sphere states show a boundary-law entanglement entropy in the large $N$ limit.

Quantum geometry of Boolean algebras and de Morgan duality

Abstract: We take a fresh look at the geometrization of logic using the recently developed tools of `quantum Riemannian geometry' applied in the digital case over the field $\Bbb F_2=\{0,1\}$, extending de Morgan duality to this context of differential forms and connections. The 1-forms correspond to graphs and the exterior derivative of a subset amounts to the arrows that cross between the set and its complement. The line graph $0-1-2$ has a non-flat but Ricci flat quantum Riemannian geometry. The previously known four quantum geometries on the triangle graph, of which one is curved, are revisited in terms of left-invariant differentials, as are the quantum geometries on the dual Hopf algebra, the group algebra of $\Bbb Z_3$. For the square, we find a moduli of four quantum Riemannian geometries, all flat, while for an $n$-gon with $n>4$ we find a unique one, again flat. We also propose an extension of de Morgan duality to general algebras and differentials over $\Bbb F_2$.

$\operatorname{SL}(2,\mathbb{C})$ Chern-Simons theory, flat connections, and four-dimensional quantum geometry

Abstract: The present paper analyze SL(2,C) Chern-Simons theory on a class of graph complement 3- manifolds, and its relation with classical and quantum geometries on 4-dimensional manifolds. In classical theory, we explain the correspondence between a class of SL(2,C) flat connections on 3-manifold and the Lorentzian simplicial geometries in 4 dimensions. The class of flat connections on the graph complement 3-manifold is specified by a certain boundary condition. The corresponding simplicial 4-dimensional geome- tries are made by constant curvature 4-simplices. The quantization of 4d simplicial geometry can be carried out via the quantization of flat connection on 3-manifold in Chern-Simons theory. In quantum SL(2,C) Chern-Simons theory, a basis of physical wave functions is the class of (holomorphic) 3d block, defined by analytically continued Chern-Simons path integral over Lefschetz thimbles. Here we propose that the (holomorphic) 3d block with the proper boundary condition imposed gives the quantization of simplicial 4- dimensional geometry. Interestingly in the semiclassical asymptotic expansion of (holomorphic) 3d block, the leading contribution gives the classical action of simplicial Einstein-Hilbert gravity in 4 dimensions, i.e. Lorentzian 4d Regge action on constant curvature 4-simplices with a cosmological constant. Such a result suggests a relation between SL(2,C) Chern-Simons theory on a class of 3-manifolds and simplicial quantum gravity on 4-dimensional manifolds. This paper presents the details for the results reported in (1).

Self-duality of the 6j-symbol and Fisher zeros for the Tetrahedron

Abstract: The relation between the 2d Ising partition function and spin network evaluations, reflecting a bulk-boundary duality between the 2d Ising model and 3d quantum gravity, promises an exchange of results and methods between statistical physics and quantum geometry. We apply this relation to the case of the tetrahedral graph. First, we find that the high/low temperature duality of the 2d Ising model translates into a new self-duality formula for Wigner's 6j-symbol from the theory of spin recoupling. Second, we focus on the duality between the large spin asymptotics of the 6j-symbol and Fisher zeros. Using the Ponzano-Regge formula for the asymptotics for the 6j-symbol at large spins in terms of the tetrahedron geometry, we obtain a geometric formula for the zeros of the (inhomogeneous) Ising partition function in terms of triangle angles and dihedral angles in the tetrahedron. While it is well-known that the 2d intrinsic geometry can be used to parametrize the critical point of the Ising model, e.g. on isoradial graphs, it is the first time to our knowledge that the extrinsic geometry is found to also be relevant.This outlines a method towards a more general geometric parametrization of the Fisher zeros for the 2d Ising model on arbitrary graphs.

Gauge invariant bounce from quantum geometry

Abstract: We present a gauge-invariant treatment of singularity resolution using quantum gravity techniques. Our analysis reveals many novel features of quantum geometry which were till now hidden in models based on non-gauge-invariant discretizations. Quantum geometric effects result in a non-singular bounce which is generically asymmetric with effective constants getting rescaled beyond the singularity, and minimally coupled matter behaving as non-minimally coupled. These ramifications of quantum geometry open a rich avenue for potential phenomenological signatures.

A localization marker from many-body quantum geometry measurements

Abstract: The spatial localization of quantum states plays a central role in condensed-matter phenomena, ranging from many-body localization to topological matter. Building on the dissipation-fluctuation theorem, we propose that the localization properties of a quantum-engineered system can be probed by spectroscopy, namely, by measuring its excitation rate upon a periodic drive. We apply this method to various examples that are of direct experimental relevance in ultracold atomic gases, including Anderson localization, topological edge modes, and interacting particles in a harmonic trap. Moreover, inspired by a relation between quantum fluctuations and the quantum metric, we describe how our scheme can be generalized in view of extracting the full quantum-geometric tensor of many-body systems. Our approach opens an avenue for probing localization, as well as quantum fluctuations, geometry and entanglement, in synthetic quantum matter.