Showing papers on "Quantum geometry published in 2020"
TL;DR: In this paper, the quantum geometry of energy bands has been measured in a two-dimensional continuous medium, a high-finesse planar microcavity, together with the related anomalous Hall drift.
Abstract: Topological physics relies on the structure of the eigenstates of the Hamiltonians. The geometry of the eigenstates is encoded in the quantum geometric tensor1-comprising the Berry curvature2 (crucial for topological matter)3 and the quantum metric4, which defines the distance between the eigenstates. Knowledge of the quantum metric is essential for understanding many phenomena, such as superfluidity in flat bands5, orbital magnetic susceptibility6,7, the exciton Lamb shift8 and the non-adiabatic anomalous Hall effect6,9. However, the quantum geometry of energy bands has not been measured. Here we report the direct measurement of both the Berry curvature and the quantum metric in a two-dimensional continuous medium-a high-finesse planar microcavity10-together with the related anomalous Hall drift. The microcavity hosts strongly coupled exciton-photon modes (exciton polaritons) that are subject to photonic spin-orbit coupling11 from which Dirac cones emerge12, and to exciton Zeeman splitting, breaking time-reversal symmetry. The monopolar and half-skyrmion pseudospin textures are measured using polarization-resolved photoluminescence. The associated quantum geometry of the bands is extracted, enabling prediction of the anomalous Hall drift, which we measure independently using high-resolution spatially resolved epifluorescence. Our results unveil the intrinsic chirality of photonic modes, the cornerstone of topological photonics13-15. These results also experimentally validate the semiclassical description of wavepacket motion in geometrically non-trivial bands9,16. The use of exciton polaritons (interacting photons) opens up possibilities for future studies of quantum fluid physics in topological systems.
137 citations
TL;DR: In this article, the classical and quantum nature of edge modes and symmetries in gravity systems is investigated. But the main focus of this paper is to understand how different formulations of gravity provide non-trivial representations of different sectors of the corner symmetry algebra, and set the foundations of a new proposal for states of quantum geometry as representation states of this corner algebra.
Abstract: This is the first paper in a series devoted to understanding the classical and quantum nature of edge modes and symmetries in gravitational systems. The goal of this analysis is to: i) achieve a clear understanding of how different formulations of gravity provide non-trivial representations of different sectors of the corner symmetry algebra, and ii) set the foundations of a new proposal for states of quantum geometry as representation states of this corner symmetry algebra. In this first paper we explain how different formulations of gravity, in both metric and tetrad variables, share the same bulk symplectic structure but differ at the corner, and in turn lead to inequivalent representations of the corner symmetry algebra. This provides an organizing criterion for formulations of gravity depending on how big the physical symmetry group that is non-trivially represented at the corner is. This principle can be used as a “treasure map” revealing new clues and routes in the quest for quantum gravity. Building up on these results, we perform a detailed analysis of the corner pre-symplectic potential and symmetries of Einstein-Cartan-Holst gravity in [1], use this to provide a new look at the simplicity constraints in [2], and tackle the quantization in [3].
108 citations
TL;DR: In this paper, the classical and quantum nature of edge modes and symmetries in gravity systems has been investigated and a new proposal for states of quantum geometry as representation states of the corner symmetry algebra has been made.
Abstract: This is the first paper in a series devoted to understanding the classical and quantum nature of edge modes and symmetries in gravitational systems The goal of this analysis is to: i) achieve a clear understanding of how different formulations of gravity provide non-trivial representations of different sectors of the corner symmetry algebra, and ii) set the foundations of a new proposal for states of quantum geometry as representation states of this corner symmetry algebra In this first paper we explain how different formulations of gravity, in both metric and tetrad variables, share the same bulk symplectic structure but differ at the corner, and in turn lead to inequivalent representations of the corner symmetry algebra This provides an organizing criterion for formulations of gravity depending on how big the physical symmetry group that is non-trivially represented at the corner is This principle can be used as a "treasure map" revealing new clues and routes in the quest for quantum gravity Building up on these results, we perform a detailed analysis of the corner symplectic potential and symmetries of Einstein-Cartan-Holst gravity in [1], use this to provide a new look at the simplicity constraints in [2], and tackle the quantization in [3]
105 citations
TL;DR: This work performs the first experimental measurement of the complete quantum geometric tensor, and establishes coherent dynamical responses as a versatile probe for quantum geometry, and they pave the way for the detection of novel topological phenomena in solid state.
Abstract: Geometry and topology are fundamental concepts, which underlie a wide range of fascinating physical phenomena such as topological states of matter and topological defects. In quantum mechanics, the geometry of quantum states is fully captured by the quantum geometric tensor. Using a qubit formed by an NV center in diamond, we perform the first experimental measurement of the complete quantum geometric tensor. Our approach builds on a strong connection between coherent Rabi oscillations upon parametric modulations and the quantum geometry of the underlying states. We then apply our method to a system of two interacting qubits, by exploiting the coupling between the NV center spin and a neighboring 13C nuclear spin. Our results establish coherent dynamical responses as a versatile probe for quantum geometry, and they pave the way for the detection of novel topological phenomena in solid state.
86 citations
TL;DR: In this article, the authors studied the low-frequency properties of the bulk photovoltaic effect in topological semimetals and provided a quantum geometric meaning of the second-order optical conductivity tensor.
Abstract: We study the low-frequency properties of the bulk photovoltaic effect in topological semimetals. The bulk photovoltaic effect is a nonlinear optical effect that generates DC photocurrents under uniform irradiation, allowed by noncentrosymmetry. It is a promising mechanism for a terahertz photodetection based on topological semimetals. Here, we systematically investigate the low-frequency behavior of the second-order optical conductivity in point-node semimetals. Through symmetry and power-counting analysis, we show that Dirac and Weyl points with tilted cones show the leading low-frequency divergence. In particular, we find new divergent behaviors of the conductivity of Dirac and Weyl points under circularly polarized light, where the conductivity scales as $\omega^{-2}$ and $\omega^{-1}$ near the gap-closing point in two and three dimensions, respectively. We provide a further perspective on the low-frequency bulk photovoltaic effect by revealing the complete quantum geometric meaning of the second-order optical conductivity tensor. The bulk photovoltaic effect has two origins, which are the transition of electron position and the transition of electron velocity during the optical excitation, and the resulting photocurrents are respectively called the shift current and the injection current. Based on an analysis of two-band models, we show that the injection current is controlled by the quantum metric and Berry curvature, whereas the shift current is governed by the Christoffel symbols near the gap-closing points in semimetals. Finally, for further demonstrations of our theory beyond simple two-band models, we perform first-principles calculations on magnetic Dirac semimetal MnGeO$_3$and Weyl semimetal PrGeAl. Our work brings out new insights into the structure of nonlinear optical responses as well as for the design of semimetal-based terahertz photodetectors.
46 citations
TL;DR: In this article, the interior structure, perturbations, and associated quasinormal modes of a quantum black hole model were investigated, and the violation of energy conditions near the transition surface was investigated.
Abstract: We investigate the interior structure, perturbations, and the associated quasinormal modes of a quantum black hole model recently proposed by Bodendorfer, Mele, and Munch (BMM). Within the framework of loop quantum gravity, the quantum parameters in the BMM model are introduced through polymerization, consequently replacing the Schwarzschild singularity with a spacelike transition surface. By treating the quantum geometry corrections as an `effective' matter contribution, we first prove the violation of energy conditions (in particular the null energy condition) near the transition surface and then investigate the required junction conditions on it. In addition, we study the quasinormal modes of massless scalar field perturbations, electromagnetic perturbations, and axial gravitational perturbations in this effective model. As expected, the quasinormal spectra deviate from their classical counterparts in the presence of quantum corrections. Interestingly, we find that the quasinormal frequencies of perturbations with different spins share the same qualitative tendency with respect to the change of the quantum parameters in this model.
40 citations
TL;DR: It is shown that there is a universal and simple relationship between the total Landau level spread of a flat band and the maximum Hilbert–Schmidt quantum distance, which can be verified in various candidate materials.
Abstract: Semiclassical quantization of electronic states under a magnetic field, as proposed by Onsager, describes not only the Landau level spectrum but also the geometric responses of metals under a magnetic field1-5. Even in graphene with relativistic energy dispersion, Onsager's rule correctly describes the π Berry phase, as well as the unusual Landau level spectrum of Dirac particles6,7. However, it is unclear whether this semiclassical idea is valid in dispersionless flat-band systems, in which an infinite number of degenerate semiclassical orbits are allowed. Here we show that the semiclassical quantization rule breaks down for a class of dispersionless flat bands called 'singular flat bands'8. The singular flat band has a band crossing with another dispersive band that is enforced by the band-flatness condition, and shows anomalous magnetic responses. The Landau levels of a singular flat band develop in the empty region in which no electronic states exist in the absence of a magnetic field, and exhibit an unusual 1/n dependence on the Landau level index n, which results in diverging orbital magnetic susceptibility. The total energy spread of the Landau levels of a singular flat band is determined by the quantum geometry of the relevant Bloch states, which is characterized by their Hilbert-Schmidt quantum distance. We show that there is a universal and simple relationship between the total Landau level spread of a flat band and the maximum Hilbert-Schmidt quantum distance, which can be verified in various candidate materials. The results indicate that the anomalous Landau level spectrum of flat bands is promising for the direct measurement of the quantum geometry of wavefunctions in condensed matter.
30 citations
TL;DR: In this paper, quantum geometric contributions to the BKT transition temperature in the presence of fluctuations beyond BCS theory were studied and a way to physically quantify the magnitude of these geometric terms is in terms of the ratio of the pairing onset temperature to BKT.
Abstract: We study quantum geometric contributions to the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature ${T}_{\text{BKT}}$ in the presence of fluctuations beyond BCS theory. Because quantum geometric effects become progressively more important with stronger pairing attraction, a full understanding of 2D multiorbital superconductivity requires the incorporation of preformed pairs. We find it is through the effective mass of these pairs that quantum geometry enters the theory and this suggests that the quantum geometric effects are present in the nonsuperconducting pseudogap phase as well. Increasing these geometric contributions tends to raise ${T}_{\text{BKT}}$, which then competes with fluctuation effects that generally depress it. We argue that a way to physically quantify the magnitude of these geometric terms is in terms of the ratio of the pairing onset temperature ${T}^{*}$ to ${T}_{\text{BKT}}$. Our paper calls attention to an experimental study demonstrating how both temperatures and, thus, their ratio may be currently accessible. They can be extracted from the same voltage-current measurements, which are generally used to establish BKT physics. We use these observations to provide rough preliminary estimates of the magnitude of the geometric contributions in, for example, magic angle twisted bilayer graphene.
30 citations
TL;DR: In this paper, the authors derived generalised uncertainty relations for orbital angular momentum and spin in the recently proposed smeared-space model of quantum geometry and obtained generalisations of the canonical so ( 3 ) and su ( 2 ) algebras.
Abstract: We derive generalised uncertainty relations (GURs) for orbital angular momentum and spin in the recently proposed 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 extended uncertainty principle (EUP), previously proposed in the quantum gravity literature, 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 so ( 3 ) and su ( 2 ) algebras. We find that, although SO ( 3 ) symmetry is preserved on three-dimensional slices of an enlarged phase space, corresponding to a superposition of background geometries, individual subcomponents of the generalised generators obey nontrivial subalgebras. These give rise to GURs for orbital angular momentum while leaving the canonical commutation relations intact except for a simple rescaling, ħ → ħ + β . The value of the new parameter, β ≃ ħ × 10 − 61 , is determined by the ratio of the dark energy density to the Planck density, and its existence is required by the presence of both minimum length and momentum uncertainties. 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 ∼ ħ Λ , where Λ is the cosmological constant, which is consistent with the existence of a finite cosmological horizon. In the smeared-space model, ħ and β 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 ± β / 2 . Finally, the modified spin algebra leads to GURs for spin measurements. The potential implications of these results for cosmology and high-energy physics, and for the description of spin and angular momentum in relativistic theories of quantum gravity, including dark energy, are briefly discussed.
26 citations
TL;DR: In this paper, the interior structure, perturbations, and associated quasinormal modes of a quantum black hole model were investigated, and the violation of energy conditions near the transition surface was investigated.
Abstract: We investigate the interior structure, perturbations, and the associated quasinormal modes of a quantum black hole model recently proposed by Bodendorfer, Mele, and Munch (BMM). Within the framework of loop quantum gravity, the quantum parameters in the BMM model are introduced through polymerization, consequently replacing the Schwarzschild singularity with a spacelike transition surface. By treating the quantum geometry corrections as an `effective' matter contribution, we first prove the violation of energy conditions (in particular the null energy condition) near the transition surface and then investigate the required junction conditions on it. In addition, we study the quasinormal modes of massless scalar field perturbations, electromagnetic perturbations, and axial gravitational perturbations in this effective model. As expected, the quasinormal spectra deviate from their classical counterparts in the presence of quantum corrections. Interestingly, we find that the quasinormal frequencies of perturbations with different spins share the same qualitative tendency with respect to the change of the quantum parameters in this model.
26 citations
TL;DR: In this paper, the authors theoretically study stability of moir\'e flatband ferromagnetism against collective excitations, with a focus on the effects of Bloch band quantum geometry.
Abstract: Several moir\'e systems created by various twisted bilayers have manifested magnetism under flatband conditions leading to enhanced interaction effects. We theoretically study stability of moir\'e flatband ferromagnetism against collective excitations, with a focus on the effects of Bloch band quantum geometry. The spin magnon spectrum is calculated using different approaches, including Bethe-Salpeter equation, single mode approximation, and an analytical theory. One of our main results is an analytical expression for the spin stiffness in terms of the Coulomb interaction potential, the Berry curvatures, and the quantum metric tensor, where the last two quantities characterize the quantum geometry of moir\'e bands. This analytical theory shows that Berry curvatures play an important role in stiffening the spin magnons. Furthermore, we construct an effective field theory for the magnetization fluctuations and show explicitly that skyrmion excitations bind an integer number of electrons that is proportional to the Bloch band Chern number and the skyrmion winding number.
TL;DR: In this article, a variational wave function for matrix quantum mechanics with multiple bosonic and fermionic matrices is provided, and the ground state (and also long-lived metastable states) of SU(N) matrix quantum physics with three bosonic matrices, and also its supersymmetric mini-BMN extension, are studied as a function of coupling and N. The variational quantum Monte Carlo method is implemented with deep generative flows to search for gauge-invariant low-energy states.
Abstract: We employ machine learning techniques to provide accurate variational wave functions for matrix quantum mechanics, with multiple bosonic and fermionic matrices. The variational quantum Monte Carlo method 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 SU(N) matrix quantum mechanics with three bosonic matrices, and also 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 wave function. 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.
TL;DR: In this paper, the authors extend the standard treatments in two ways: by using a class of thermal condensates, the coherent thermal states, which encode statistical fluctuations in quantum geometry; and by introducing a suitable class of smearing functions as nonsingular, well-behaved generalizations for relational clock frames in group field theory.
Abstract: The condensate cosmology program of group field theory has produced several interesting results. The key idea is in the suggestion that a macroscopic homogeneous spacetime can be approximated by a dynamical condensate phase of the underlying microscopic system of an arbitrarily large number of candidate quanta of geometry. In this work, we extend the standard treatments in two ways: by using a class of thermal condensates, the coherent thermal states, which encode statistical fluctuations in quantum geometry; and by introducing a suitable class of smearing functions as nonsingular, well-behaved generalizations for relational clock frames in group field theory. In particular, we investigate an effective relational cosmological dynamics for homogeneous and isotropic spacetimes, extracted from a class of free group field theory models, and subsequently investigate aspects of its late and early times evolution. We find the correct classical limit of Friedmann equations at late times, with a bounce and an accelerated expansion at early times. Specifically, we find additional correction terms in the evolution equations corresponding to the statistical contribution of the new thermal condensates in general, and a higher upper bound on the number of $e$-folds, even without including any interactions.
TL;DR: In this article, the Ricci curvature was investigated in full, four-dimensional quantum gravity, formulated nonperturbatively in terms of Causal Dynamical Triangulations (CDT).
Abstract: We investigate the quantum Ricci curvature, which was introduced in earlier work, in full, four-dimensional quantum gravity, formulated nonperturbatively in terms of Causal Dynamical Triangulations (CDT). A key finding of the CDT approach is the emergence of a universe of de Sitter-type, as evidenced by the successful matching of Monte Carlo measurements of the quantum dynamics of the global scale factor with a semiclassical minisuperspace model. An important question is whether the quantum universe exhibits semiclassicality also with regard to its more local geometric properties. Using the new quantum curvature observable, we examine whether the (quasi-)local properties of the quantum geometry resemble those of a constantly curved space. We find evidence that on sufficiently large scales the curvature behaviour is compatible with that of a four-sphere, thus strengthening the interpretation of the dynamically generated quantum universe in terms of a de Sitter space.
TL;DR: In this article, the Ricci curvature was investigated in full, four-dimensional quantum gravity, formulated nonperturbatively in terms of Causal Dynamical Triangulations (CDT).
Abstract: We investigate the quantum Ricci curvature, which was introduced in earlier work, in full, four-dimensional quantum gravity, formulated nonperturbatively in terms of Causal Dynamical Triangulations (CDT). A key finding of the CDT approach is the emergence of a universe of de Sitter-type, as evidenced by the successful matching of Monte Carlo measurements of the quantum dynamics of the global scale factor with a semiclassical minisuperspace model. An important question is whether the quantum universe exhibits semiclassicality also with regard to its more local geometric properties. Using the new quantum curvature observable, we examine whether the (quasi-)local properties of the quantum geometry resemble those of a constantly curved space. We find evidence that on sufficiently large scales the curvature behaviour is compatible with that of a four-sphere, thus strengthening the interpretation of the dynamically generated quantum universe in terms of a de Sitter space.
TL;DR: In this paper, the SO ( 5 ) Landau model and the associated matrix geometries are analyzed at the Landau level projection. But the authors do not consider the matrix geometry of a 4D fuzzy four-sphere.
TL;DR: In this article, the Yetter lattice model and its continuum counterpart, the BFCG theory, generalize BF theory to 2-gauge groups and provide an exactly solvable topologically-flat version of 4D general relativity.
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 two-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 two-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 two-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.
TL;DR: In this paper, two strategies for constructing general geometric operators in all dimensional loop quantum gravity are proposed, which mainly come from the two different regularization methods for the basic building blocks of the spatial geometry.
Abstract: Two strategies for constructing general geometric operators in all dimensional loop quantum gravity are proposed. The different constructions mainly come from the two different regularization methods for the basic building blocks of the spatial geometry. The first regularization method is a generalization of the regularization of the length operator in standard ($1+3$)-dimensional loop quantum gravity, while the second method is a natural extension of those for standard ($\mathrm{D}\ensuremath{-}1$)-area and usual D-volume operators. Two versions of general geometric operators to measure arbitrary $m$-areas are constructed, and their properties are discussed and compared. They serve as valuable candidates to study the quantum geometry in arbitrary dimensions.
TL;DR: In this paper, the authors used the non-extensive entropy of Tsallis to study the properties of the Schwarzschild BH and found that the results are strikingly identical to those obtained by Hawking and Page in anti-de Sitter space within the original of the Boltzmann entropy formula.
Abstract: In this letter, using the non-extensive entropy of Tsallis, we study some properties of the Schwarzschild black holes (BHs), based on the loop quantum gravity (LQG), some novel characteristics and results of the Schwarzschild BH can be obtained in Mejrhit and Ennadifi (Phys Lett B 794:45–49, 2019). Here we find that these findings are strikingly identical to ones obtained by Hawking and Page in anti-de Sitter space within the original of the Boltzmann entropy formula. By using the semi-classical estimate analysis on the energy at this minimum $$M_{min}$$
, an approximate relationship between the q and $$\gamma $$
parameters of BHs can be found, (
$$q\approx \frac{\sqrt{3}\gamma }{\pi \ln 2}+1$$
), which is remarkable approaching to q-parameters of cosmic ray spectra and quarks coalescing to hadrons in high energy.
Posted Content•
TL;DR: In this paper, the authors studied the quantum geometry of the fuzzy sphere defined as the angular momentum algebra and showed that for each metric there is a unique quantum Levi-Civita connection with constant coefficients, with scalar curvature.
Abstract: We study the quantum geometry of the fuzzy sphere defined as the angular momentum algebra $[x_i,x_j]=2\imath\lambda_p \epsilon_{ijk}x_k$ modulo setting $\sum_i x_i^2$ to a constant, using a recently introduced 3D rotationally invariant differential structure. Metrics are given by symmetric $3 \times 3$ matrices $g$ and we show that for each metric there is a unique quantum Levi-Civita connection with constant coefficients, with scalar curvature $ \frac{1}{2}({\rm Tr}(g^2)-\frac{1}{2}{\rm Tr}(g)^2)/\det(g)$. As an application, we construct Euclidean quantum gravity on the fuzzy unit sphere. We also calculate the charge 1 monopole for the 3D differential structure.
TL;DR: In this article, the authors present a gauge-invariant treatment of singularity resolution using loop quantum gravity techniques with respect to local SU(2) transformations, which reveals many novel features of quantum geometry which were till now hidden in models based on non-gauge-inariant discretizations.
Abstract: We present a gauge-invariant treatment of singularity resolution using loop quantum gravity techniques with respect to local SU(2) transformations. 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 resolve the big bang singularity replacing it with a non-singular bounce when spacetime curvature reaches Planckian value. The bounce is found to be generically asymmetric in the sense that pre-bounce and post-bounce branches are not mirrored to each other and effective constants, such as Newton's constant, are rescaled across the bounce. Furthermore, in the vicinity of the bounce, minimally coupled matter behaves as non-minimally coupled. These ramifications of quantum geometry open a rich avenue for potential phenomenological signatures.
TL;DR: In this article, two inequivalent loop quantizations are investigated, one based on holonomies of the Ashtekar-Barbero connection using closed loops, and another based on the connection operator.
Abstract: Difference in pressure during expansion and contraction stages of the cosmic evolution can result in a hysteresislike phenomenon in nonsingular cyclic models sourced with a massive scalar field. We discuss this phenomenon for spatially closed isotropic spacetimes in loop quantum cosmology (LQC) for a quadratic and a coshlike potential, with and without a negative cosmological constant using an effective spacetime description of the underlying quantum geometry. Two inequivalent loop quantizations are investigated---one based on holonomies of the Ashtekar-Barbero connection using closed loops, and another based on the connection operator. Due to the underlying quantum geometric effects, both of the models avoid classical singularities, but unlike the holonomy based quantization, the connection based quantization results in two quantum bounces. In spite of the differences in nonsingular effective dynamics of both of the models, the phenomena of hysteresis is found to be robust for the ${\ensuremath{\phi}}^{2}$ potential. Quasiperiodic beats exist for the coshlike potential, irrespective of the nature of the classical recollapse whether by the spatial curvature, or a negative cosmological constant. An interplay of the negative cosmological constant and the spatial curvature in the presence of potentials results in rich features such as islands of cluster of bounces separated by an accelerated expansion and a universe which either undergoes a steplike expansion with multiple turnarounds or quasiperiodic beats depending on a ``tuning'' of the steepness parameter of the potential.
Abstract: The asymptotic safety program builds on a high-energy completion of gravity based on the Reuter fixed point, a non-trivial fixed point of the gravitational renormalization group flow. At this fixed point the canonical mass-dimension of coupling constants is balanced by anomalous dimensions induced by quantum fluctuations such that the theory enjoys quantum scale invariance in the ultraviolet. The crucial role played by the quantum fluctuations suggests that the geometry associated with the fixed point exhibits non-manifold like properties. In this work, we continue the characterization of this geometry employing the composite operator formalism based on the effective average action. Explicitly, we give a relation between the anomalous dimensions of geometric operators on a background d-sphere and the stability matrix encoding the linearized renormalization group flow in the vicinity of the fixed point. The eigenvalue spectrum of the stability matrix is analyzed in detail and we identify a perturbative regime where the spectral properties are governed by canonical power counting. Our results recover the feature that quantum gravity fluctuations turn the (classically marginal) R^2-operator into a relevant one. Moreover, we find strong indications that higher-order curvature terms present in the two-point function play a crucial role in guaranteeing the predictive power of the Reuter fixed point.
TL;DR: In this article, the role of the principle of equivalence in quantum gravity effects in the local inertial frames was investigated from a specific perspective and it was shown that at mesoscopic scales close to, but somewhat larger than, Planck length one could describe quantum spacetime and matter in terms of an effective geometry.
Abstract: Principle of equivalence makes effects of classical gravity vanish in local inertial frames. What role does the principle of equivalence play as regards quantum gravitational effects in the local inertial frames? I address this question here from a specific perspective. At mesoscopic scales close to, but somewhat larger than, Planck length one could describe quantum spacetime and matter in terms of an effective geometry. The key feature of such an effective quantum geometry is the existence of a zero-point-length. When we proceed from quantum geometry to quantum matter, the zero-point-length will introduce corrections in the propagator for matter fields in a specific manner. On the other hand, one cannot ignore the self gravity of matter fields at the mesoscopic scales and this will also modify the form of the propagator. Consistency demands that, these two modifications—coming from two different directions—are the same. I show that this non-trivial demand is actually satisfied. Surprisingly, the principle of equivalence, operating at sub-Planck scales, ensures this consistency in a subtle manner.
TL;DR: The notion of superconnection was introduced by Quillen in 1985 and used in gauge-Higgs field theory in the 1990's to represent the finite quantum geometry of one generation of fermions in the Standard Model of particle physics as discussed by the authors.
Posted Content•
TL;DR: In this article, a generalisation of statistical equilibrium, specifically Gibbs states, suitable for background independent systems is investigated. But the authors focus on an information theoretic characterisation based on the maximum entropy principle, which is a challenging open issue due to the absence of absolute notions of time and energy.
Abstract: Statistical equilibrium configurations are important in the physics of macroscopic systems with a large number of constituent degrees of freedom. They are expected to be crucial also in discrete quantum gravity, where dynamical spacetime should emerge from the collective physics of the underlying quantum gravitational degrees of freedom. However, defining statistical equilibrium in a background independent system is a challenging open issue, mainly due to the absence of absolute notions of time and energy. This is especially so in non-perturbative quantum gravity frameworks that are devoid of usual space and time structures. In this thesis, we investigate aspects of a generalisation of statistical equilibrium, specifically Gibbs states, suitable for background independent systems. We emphasise on an information theoretic characterisation based on the maximum entropy principle. Subsequently, we explore the resultant generalised Gibbs states in a discrete quantum gravitational system composed of many candidate quanta of geometry, utilising their field theoretic formulation of group field theory and various many-body techniques. We construct several concrete examples of quantum gravitational generalised Gibbs states. We further develop inequivalent thermal representations based on entangled, two-mode squeezed, thermofield double vacua, induced by a class of generalised Gibbs states. In these representations, we define a class of thermal condensates which encode statistical fluctuations in a given observable, e.g. volume of the quantum geometry. We apply these states in the condensate cosmology programme of group field theory to study a relational effective cosmological dynamics extracted from a class of free models, for homogeneous and isotropic spacetimes. We find the correct classical limit of Friedmann equations at late times, with a bounce and accelerated expansion at early times.
TL;DR: In this paper, the effective dynamics of the loop quantized Janis-Newman-Winicour (JNW) spacetime was investigated, which is an extension of the Schwarzschild spacetime with an extra minimally coupled massless scalar field.
Abstract: Inspired by the recent proposal for the quantum effective dynamics of the Schwarzschild spacetime given in \cite{AOS1}, we investigate the effective dynamics of the loop quantized Janis-Newman-Winicour (JNW) spacetime which is an extension of the Schwarzschild spacetime with an extra minimally coupled massless scalar field. Two parameters are introduced in order to regularize the Hamiltonian constraint in the quantum effective dynamics. These two parameters are assumed to be Dirac observables when the effective dynamics is solved. By carefully choosing appropriate conditions for these two parameters, we completely determine them, and the resulted new effective description of the JNW spacetime leads to a well behaved quantum dynamics which on one hand resolves the classical singularities, and on the other hand, agrees with the classical dynamics in the low curvature region.
TL;DR: In this article, the authors consider the background-independent quantization of a general scale-invariant theory of gravity with matter, which supports a conserved Weyl current recently suggested as a natural flow of time.
TL;DR: Loop Quantum Gravity as discussed by the authors is a nonperturbative and background independent program for the quantization of General Relativity, which has been applied successfully to the study of cosmological spacetimes, both to test the principles and techniques of the theory and to discuss its physical consequences.
Abstract: Loop Quantum Gravity is a nonperturbative and background independent program for the quantization of General Relativity. Its underlying formalism has been applied successfully to the study of cosmological spacetimes, both to test the principles and techniques of the theory and to discuss its physical consequences. These applications have opened a new area of research known as Loop Quantum Cosmology. The hybrid approach addresses the quantization of cosmological systems that include fields. This proposal combines the description of a finite number of degrees of freedom using Loop Quantum Cosmology, typically corresponding to a homogeneous background, and a Fock quantization of the field content of the model. In this review we first present a summary of the foundations of homogeneous Loop Quantum Cosmology and we then revisit the hybrid quantization approach, applying it to the study of Gowdy spacetimes with linearly polarized gravitational waves on toroidal spatial sections, and to the analysis of cosmological perturbations in preinflationary and inflationary stages of the Universe. The main challenge is to extract predictions about quantum geometry effects that eventually might be confronted with cosmological observations. This is the first extensive review of the hybrid approach in the literature on Loop Quantum Cosmology.
TL;DR: In this article, the authors introduced the framework of gauge-invariant variables to describe fractional quantum Hall (FQH) states, and proved that the wave function can always be represented by a unique holomorphic multivariable complex function.
Abstract: Herein, we introduce the framework of gauge-invariant variables to describe fractional quantum Hall (FQH) states, and prove that the wave function can always be represented by a unique holomorphic multivariable complex function. As a special case, within the lowest Landau level, this function reduces to the well-known holomorphic coordinate representation of wave functions in the symmetric gauge. Using this framework, we derive an analytic guiding center Schr\"odinger's equation governing FQH states; it has a novel structure. We show how the electronic interaction is parametrized by generalized pseudopotentials, which depend on the Landau level occupancy pattern; they reduce to the Haldane pseudopotentials when only one Landau level is considered. Our formulation is apt for incorporating a new combination of techniques, from symmetric functions, Galois theory and complex analysis, to accurately predict the physics of FQH states using first principles.