Showing papers on "Quantum geometry published in 2021"

A Short Review of Loop Quantum Gravity.

Abstract: An outstanding open issue in our quest for physics beyond Einstein is the unification of general relativity (GR) and quantum physics. Loop quantum gravity (LQG) is a leading approach toward this goal. At its heart is the central lesson of GR: Gravity is a manifestation of spacetime geometry. Thus, the approach emphasizes the quantum nature of geometry and focuses on its implications in extreme regimes -- near the big bang and inside black holes-- where Einstein's smooth continuum breaks down. We present a brief overview of the main ideas underlying LQG and highlight a few recent advances. This report is addressed to non-experts.

Topology and geometry under the nonlinear electromagnetic spotlight

Abstract: For many materials, a precise knowledge of their dispersion spectra is insufficient to predict their ordered phases and physical responses. Instead, these materials are classified by the geometrical and topological properties of their wavefunctions. A key challenge is to identify and implement experiments that probe or control these quantum properties. In this review, we describe recent progress in this direction, focusing on nonlinear electromagnetic responses that arise directly from quantum geometry and topology. We give an overview of the field by discussing new theoretical ideas, groundbreaking experiments, and the novel materials that drive them. We conclude by discussing how these techniques can be combined with new device architectures to uncover, probe, and ultimately control novel quantum phases with emergent topological and correlated properties.

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 (Bodendorfer et al 2019 Class Quantum Grav 36 195015), 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

Light-matter coupling and quantum geometry in moiré materials

Abstract: Quantum geometry has been identified as an important ingredient for the physics of quantum materials and especially of flat-band systems, such as moir\'e materials. On the other hand, the coupling between light and matter is of key importance across disciplines and especially for Floquet and cavity engineering of solids. Here we present fundamental relations between light-matter coupling and quantum geometry of Bloch wave functions, with a particular focus on flat-band and moir\'e materials, in which the quenching of the electronic kinetic energy could allow one to reach the limit of strong light-matter coupling more easily than in highly dispersive systems. We show that, despite the fact that flat bands have vanishing band velocities and curvatures, light couples to them via geometric contributions. Specifically, the intraband quantum metric allows diamagnetic coupling inside a flat band; the interband Berry connection governs dipole matrix elements between flat and dispersive bands. We illustrate these effects in two representative model systems: (i) a sawtooth quantum chain with a single flat band and (ii) a tight-binding model for twisted bilayer graphene. For (i) we highlight the importance of quantum geometry by demonstrating a nonvanishing diamagnetic light-matter coupling inside the flat band. For (ii) we explore the twist-angle dependence of various light-matter coupling matrix elements. Furthermore, at the magic angle corresponding to almost flat bands, we show a Floquet-topological gap opening under irradiation with circularly polarized light despite the nearly vanishing Fermi velocity. We discuss how these findings provide fundamental design principles and tools for light-matter-coupling-based control of emergent electronic properties in flat-band and moir\'e materials.

Capacity and Quantum Geometry of Parametrized Quantum Circuits

Abstract: Quantum geometric measures are presented to analyze parameterized quantum circuits, allowing one to identify improved quantum circuits and initialization techniques that enhance the performance of variational quantum algorithms.

Quantum Geometry and Flat Band Bose-Einstein Condensation.

Abstract: We study the properties of a weakly interacting Bose-Einstein condensate (BEC) in a flat band lattice system by using the multiband Bogoliubov theory and discover fundamental connections to the underlying quantum geometry. In a flat band, the speed of sound and the quantum depletion of the condensate are dictated by the quantum geometry, and a finite quantum distance between the condensed and other states guarantees stability of the BEC. Our results reveal that a suitable quantum geometry allows one to reach the strong quantum correlation regime even with weak interactions.

CDT Quantum Toroidal Spacetimes: An Overview

Abstract: Lattice formulations of gravity can be used to study non-perturbative aspects of quantum gravity. Causal Dynamical Triangulations (CDT) is a lattice model of gravity that has been used in this way. It has a built-in time foliation but is coordinate-independent in the spatial directions. The higher-order phase transitions observed in the model may be used to define a continuum limit of the lattice theory. Some aspects of the transitions are better studied when the topology of space is toroidal rather than spherical. In addition, a toroidal spatial topology allows us to understand more easily the nature of typical quantum fluctuations of the geometry. In particular, this topology makes it possible to use massless scalar fields that are solutions to Laplace’s equation with special boundary conditions as coordinates that capture the fractal structure of the quantum geometry. When such scalar fields are included as dynamical fields in the path integral, they can have a dramatic effect on the geometry.

Hybrid Loop Quantum Cosmology: An Overview

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.

Two-body problem in a multiband lattice and the role of quantum geometry

Abstract: We consider the two-body problem in a periodic potential, and study the bound-state dispersion of a spin-$\ensuremath{\uparrow}$ fermion that is interacting with a spin-$\ensuremath{\downarrow}$ fermion through a short-range attractive interaction Based on a variational approach, we obtain the exact solution of the dispersion in the form of a set of self-consistency equations, and apply it to tight-binding Hamiltonians with on-site interactions We pay special attention to the bipartite lattices with a two-point basis that exhibit time-reversal symmetry, and show that the lowest-energy bound states disperse quadratically with momentum, whose effective-mass tensor is partially controlled by the quantum metric tensor of the underlying Bloch states In particular, we apply our theory to the Mielke checkerboard lattice, and study the special role played by the interband processes in producing a finite effective mass for the bound states in a nonisolated flat band

Ground-state quantum geometry in superconductor–quantum dot chains

Abstract: Multiterminal Josephson junctions constitute engineered topological systems in arbitrary synthetic dimensions defined by the superconducting phases. Microwave spectroscopy enables the measurement of the quantum geometric tensor, a fundamental quantity describing both the quantum geometry and the topology of the emergent Andreev bound states in a unified manner. In this work we propose an experimentally feasible and scalable multiterminal setup of $N$ quantum dots connected to $N+1$ superconducting leads which allows us to deterministically study nontrivial topology in terms of the Chern number of the noninteracting ground state. An important result is that the nontrivial topology in a linear chain appears beyond a threshold value of the nonlocal proximity-induced pairing potential which represents the novel theoretical key ingredient of our proposal. Moreover, we generalize the microwave spectroscopy scheme to the multiband case and show that the elements of the quantum geometric tensor of the noninteracting ground state can be experimentally accessed from the measurable oscillator strengths at low temperature.

Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers

Abstract: Quantum geometry has emerged as a central and ubiquitous concept in quantum sciences, with direct consequences on quantum metrology and many-body quantum physics. In this context, two fundamental geometric quantities play complementary roles: the Fubini-Study metric, which introduces a notion of distance between quantum states defined over a parameter space, and the Berry curvature associated with Berry-phase effects and topological band structures. In fact, recent studies have revealed direct relations between these two important quantities, suggesting that topological properties can, in special cases, be deduced from the quantum metric. In this work, we establish general and exact relations between the quantum metric and the topological invariants of generic Dirac Hamiltonians. In particular, we demonstrate that topological indices (Chern numbers or winding numbers) are bounded by the quantum volume determined by the quantum metric. Our theoretical framework, which builds on the Clifford algebra of Dirac matrices, is applicable to topological insulators and semimetals of arbitrary spatial dimensions, with or without chiral symmetry. This work clarifies the role of the Fubini-Study metric in topological states of matter, suggesting unexplored topological responses and metrological applications in a broad class of quantum-engineered systems.

Quantum simulations of a qubit of space

Abstract: In loop quantum gravity approach to Planck scale physics, quantum geometry is represented by superposition of the so-called spin network states. In the recent literature, a class of spin networks promising from the perspective of quantum simulations of quantum gravitational systems has been studied. In this case, the spin network states are represented by graphs with four-valent nodes, and two dimensional intertwiner Hilbert spaces (qubits of space) attached to them. In this article, construction of quantum circuits for a general intertwiner qubit is presented. The obtained circuits are simulated on 5-qubit (Yorktown) and 15-qubit (Melbourne) IBM superconducting quantum computers, giving satisfactory fidelities. The circuits provide building blocks for quantum simulations of complex spin networks in the future. Furthermore, a class od maximally entangled states of spin networks is introduced. As an example of application, attempts to determine transition amplitudes for a monopole and a dipole spin networks with the use of superconducting quantum processor are made.

Capacity and quantum geometry of parametrized quantum circuits.

Abstract: To harness the potential of noisy intermediate-scale quantum devices, it is paramount to find the best type of circuits to run hybrid quantum-classical algorithms. Key candidates are parametrized quantum circuits that can be effectively implemented on current devices. Here, we evaluate the capacity and trainability of these circuits using the geometric structure of the parameter space via the effective quantum dimension, which reveals the expressive power of circuits in general as well as of particular initialization strategies. We assess the representation power of various popular circuit types and find striking differences depending on the type of entangling gates used. Particular circuits are characterized by scaling laws in their expressiveness. We identify a transition in the quantum geometry of the parameter space, which leads to a decay of the quantum natural gradient for deep circuits. For shallow circuits, the quantum natural gradient can be orders of magnitude larger in value compared to the regular gradient; however, both of them can suffer from vanishing gradients. By tuning a fixed set of circuit parameters to randomized ones, we find a region where the circuit is expressive, but does not suffer from barren plateaus, hinting at a good way to initialize circuits. Our results enhance the understanding of parametrized quantum circuits for improving variational quantum algorithms.

Holographic maps from quantum gravity states as tensor networks

Abstract: We define bulk/boundary maps corresponding to quantum gravity states in the tensorial group field theory formalism, for quantum geometric models sharing the same type of quantum states of loop quantum gravity. The maps are defined in terms of a partition of the quantum geometric data associated to a graph with open edges into bulk and boundary ones, in the spin representation. We determine the general condition on the entanglement structure of the state that makes the bulk/boundary map isometric (a necessary condition for holographic behaviour), and we analyse different types of quantum states, identifying those that define isometric bulk/boundary maps.

Excitations of a Bose-Einstein condensate and the quantum geometry of a flat band

Abstract: The quantum geometry of Bloch states fundamentally affects a wide range of physical phenomena. The quantum Hall effect, for example, is governed by the Chern number, and flat-band superconductivity by the distance between the Bloch states: the quantum metric. While understanding quantum geometry phenomena in the context of fermions is well established, less is known about the role of quantum geometry in bosonic systems where particles can undergo Bose-Einstein condensation (BEC). In conventional single-band or continuum systems, excitations of a weakly interacting BEC are determined by the condensate density and the interparticle interaction energy. In contrast to this, we discover here fundamental connections between the properties of a weakly interacting BEC and the underlying quantum geometry of a multiband lattice system. We show that, in the flat-band limit, the defining physical quantities of BEC, namely, the speed of sound and the quantum depletion, are dictated solely by the quantum geometry. We find that the speed of sound becomes proportional to the quantum metric of the condensed state. Furthermore, the quantum distance between the Bloch functions forces the quantum depletion and the quantum fluctuations of the density-density correlation to obtain finite values for infinitesimally small interactions. This is in striking contrast to dispersive bands where these quantities vanish with the interaction strength. Additionally, we show how in the flat-band limit the supercurrent is carried by the quantum fluctuations and is determined by the Berry connections of the Bloch states. Our results reveal how nontrivial quantum geometry allows reaching strong quantum correlation regime of condensed bosons even with weak interactions. This is highly relevant, for example, for polariton and photon BECs where interparticle interactions are inherently small. Our predictions can be experimentally tested with flat-band lattices already implemented in ultracold gases and various photonic platforms.

Experimental estimation of the quantum Fisher information from randomized measurements

Abstract: The quantum Fisher information (QFI) represents a fundamental concept in quantum physics. It quantifies the metrological potential of quantum states in quantum parameter estimation measurements, and is intrinsically related to quantum geometry and multipartite entanglement of many-body systems. Using a nitrogen-vacancy center spin in diamond, we experimentally demonstrate a randomized-measurement method to extract the QFI of the qubit, for both pure and mixed states. We then apply this scheme to a 4-qubit state, using a superconducting quantum computer, and show that it provides access to the sub-QFI, which sets a lower bound on the QFI for general mixed states. We numerically study the scaling of statistical error, considering $N$-qubit states, to illustrate the advantage of our randomized-measurement approach in estimating the QFI and multipartite entanglement. Our results highlight the general applicability of our method to different quantum platforms, including solid-state spin systems, superconducting quantum computers, and trapped ions.

Ground-0 Axioms vs. First Principles and Second Law: From the Geometry of Light and Logic of Photon to Mind-Light-Matter Unity-AI&QI

Abstract: Without the geometry of light and logic of photon, observer-observability forms a paradox in modern science, truth-equilibrium finds no unification, and mind-light-matter unity is unreachable in spacetime. Subsequently, quantum mechanics has been shrouded with mysteries preventing itself from reaching definable causality for a general purpose analytical quantum computing paradigm. Ground-0 Axioms are introduced as an equilibrium-based, dynamic, bipolar set-theoretic unification of the first principles of science and the second law of thermodynamics. Related literatures are critically reviewed to justify the self-evident nature of Ground-0 Axioms. A historical misinterpretation by the founding fathers of quantum mechanics is identified and corrected. That disproves spacetime geometries (including but not limited to Euclidean and Hilbert spaces) as the geometries of light and truth-based logics (including but not limited to bra-ket quantum logic) as the logics of photon. Backed with logically definable causality and Dirac 3-polarizer experiment, bipolar quantum geometry (BQG) and bipolar dynamic logic (BDL) are identified as the geometry of light and the logic of photon, respectively, and wave-particle complementarity is shown less fundamental than bipolar complementarity. As a result, Ground-0 Axioms lead to a geometrical and logical illumination of the quantum and classical worlds as well as the physical and mental worlds. With logical resolutions to the EPR and Schrodinger's cat paradoxes, an analytical quantum computing paradigm named quantum intelligence (QI) is introduced. It is shown that QI makes mind-light-matter unity and quantum-digital compatibility logically reachable for quantum-neuro-fuzzy AI-machinery with groundbreaking applications. It is contended that Ground-0 Axioms open a new era of science and philosophy—the era of mind-light-matter unity in which human-level white-box AI&QI is logically prompted to join Einstein's grand unification to foster major scientific advances.

Quantum Gravity and Riemannian Geometry on the Fuzzy Sphere

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 $$ \frac{1}{2}(\mathrm{Tr}(g^2)-\frac{1}{2}\mathrm{Tr}(g)^2)/\det (g)$$ . 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.

Topological features without a lattice in Rashba spin-orbit coupled atoms.

Abstract: Topological order can be found in a wide range of physical systems, from crystalline solids, photonic meta-materials and even atmospheric waves to optomechanic, acoustic and atomic systems. Topological systems are a robust foundation for creating quantized channels for transporting electrical current, light, and atmospheric disturbances. These topological effects are quantified in terms of integer-valued ‘invariants’, such as the Chern number, applicable to the quantum Hall effect, or the $${{\mathbb{Z}}}_{2}$$ invariant suitable for topological insulators. Here, we report the engineering of Rashba spin-orbit coupling for a cold atomic gas giving non-trivial topology, without the underlying crystalline structure that conventionally yields integer Chern numbers. We validated our procedure by spectroscopically measuring both branches of the Rashba dispersion relation which touch at a single Dirac point. We then measured the quantum geometry underlying the dispersion relation using matter-wave interferometry to implement a form of quantum state tomography, giving a Berry’s phase with magnitude π. This implies that opening a gap at the Dirac point would give two dispersions (bands) each with half-integer Chern number, potentially implying new forms of topological transport. Here, the authors study topology in spin-orbit coupled 87Rb atoms by using time domain spectroscopy and quantum state tomography. They measure full quantum state to extract the Berry phase of the system and show signatures of a half-integer Chern index.

Quantum (matrix) geometry and quasi-coherent states

Abstract: A general framework is described which associates geometrical structures to any set of $D$ finite-dimensional hermitian matrices $X^a, \ a=1,...,D$. This framework generalizes and systematizes the well-known examples of fuzzy spaces, and allows to extract the underlying classical space without requiring the limit of large matrices or representation theory. The approach is based on the previously introduced concept of quasi-coherent states. In particular, a concept of quantum Kahler geometry arises naturally, which includes the well-known quantized coadjoint orbits such as the fuzzy sphere $S^2_N$ and fuzzy $\mathbb{C} P^n_N$. A quantization map for quantum Kahler geometries is established. Some examples of quantum geometries which are not Kahler are identified, including the minimal fuzzy torus.

Quantum K-theory of quiver varieties and many-body systems

Abstract: We define quantum equivariant K-theory of Nakajima quiver varieties. We discuss type A in detail as well as its connections with quantum XXZ spin chains and trigonometric Ruijsenaars-Schneider models. Finally we study a limit which produces a K-theoretic version of results of Givental and Kim, connecting quantum geometry of flag varieties and Toda lattice.

Riemannian Geometry of Resonant Optical Responses.

Abstract: Geometry of quantum states has proved to be a useful concept for understanding responses of electronic systems to static electromagnetic fields. However, it has been challenging to relate quantum geometry with resonant optical responses. The main obstacle is that optical transitions are properties of a pair of states, while existing geometrical properties are defined for a single state. Therefore, concrete geometric understanding of optical responses has been limited to two-level systems, where one of two states determines the Hilbert space completely. Here, we construct a general theory of Riemannian geometry for resonant optical processes, by identifying transition dipole moment matrix elements as tangent vectors. This theory applies to arbitrarily high-order responses, suggesting that optical responses can be generally thought of as manifestations of the Riemannian geometry of quantum states. We use our theory to show that third-order photovoltaic Hall effects are related to the Riemann curvature tensor and demonstrate an experimentally accessible regime where they dominate the response.

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 q-bits 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 of 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 strengthens 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.

A-type Quiver Varieties and ADHM Moduli Spaces

Abstract: We study quantum geometry of Nakajima quiver varieties of two different types—framed A-type quivers and ADHM quivers. While these spaces look completely different we find a surprising connection between equivariant K-theories thereof with a nontrivial match between their equivariant parameters. In particular, we demonstrate that quantum equivariant K-theory of $$A_n$$ quiver varieties in a certain $$n\rightarrow \infty $$ limit reproduces equivariant K-theory of the Hilbert scheme of points on $$\mathbb {C}^2$$ . We analyze the correspondence from the point of view of enumerative geometry, representation theory and integrable systems. We also propose a conjecture which relates spectra of quantum multiplication operators in K-theory of the ADHM moduli spaces with the solution of the elliptic Ruijsenaars–Schneider model.

Superconductivity in monolayer FeSe enhanced by quantum geometry

Abstract: We formulate the superfluid weight in unconventional superconductors with $\bm k$-dependent Cooper pair potentials based on the geometric properties of Bloch electrons. We apply the formula to a model of the monolayer FeSe obtained by the first-principles calculation. Our numerical calculations point to a significant enhancement of the Berezinskii-Kosterlitz-Thouless transition temperature due to the geometric contribution to the superfluid weight, which is not included in the Fermi liquid theory. The $\bm k$-dependence of the gap function also stabilizes the superconducting state. Our results reveal that the geometric properties of Bloch electrons play an essential role in superconducting materials and pave the way for clarifying hidden aspects of superconductivity from the viewpoint of quantum geometry.

Exploring alternatives to the Hamiltonian calculation of the Ashtekar-Olmedo-Singh black hole solution

Abstract: In this article, we reexamine the derivation of the dynamical equations of the Ashtekar-Olmedo-Singh black hole model in order to determine whether it is possible to construct a Hamiltonian formalism where the parameters that regulate the introduction of quantum geometry effects are treated as true constants of motion. After arguing that these parameters should capture contributions from two distinct sectors of the phase space that had been considered independent in previous analyses in the literature, we proceed to obtain the corresponding equations of motion and analyze the consequences of this more general choice. We restrict our discussion exclusively to these dynamical issues. We also investigate whether the proposed procedure can be reconciled with the results of Ashtekar, Olmedo, and Singh, at least in some appropriate limit.

Quantum geometric exciton drift velocity

Abstract: In many situations, excitons---bound particle-hole pairs above an insulating ground state---carry an electric dipole moment, allowing them to be manipulated via coupling to an electric field. For two-dimensional systems, we demonstrate that this property of an exciton is uniquely determined by the quantum geometry of its eigenstates, and demonstrate its intimate connection with a quantity which we call the quantum geometric dipole. We demonstrate that this quantity arises naturally in the semiclassical equations of motion of an exciton in an electric field, adding a term additional to the anomalous velocity coming from Berry's curvature. In a uniform electric field, this contributes a drift velocity to the exciton akin to that expected for excitons in crossed electric and magnetic fields, even in the absence of a real magnetic field. We compute the quantities relevant to semiclassical exciton dynamics for several interesting examples of bilayer systems with weak interlayer tunneling and Fermi energy in a gap, where the exciton may be sensibly described as a two-body problem. These quantities include the exciton dispersion, its quantum geometric dipole, and its Berry's curvature. For a simple example of two gapped-graphene layers in a vanishing magnetic field, we demonstrate that there is a nonvanishing quantum geometric dipole when the layers are different, e.g., have different gaps, but vanishes when the layers are identical. We further analyze examples in the presence of magnetic fields, allowing us to examine cases involving graphene, in which a gap is opened by Landau level splitting. Heterostructures involving transition metal dichalcogenides materials are also considered. In each case, the quantum geometric dipole and Berry's curvatures play out in different ways. In some cases, the lowest energy exciton state is found to reside at finite momentum, with interesting possible consequences for Bose condensation in these systems. Additionally, we find situations in which the quantum geometric dipole increases monotonically with exciton momentum, suggesting that the quantum geometry can be exploited to produce photocurrents from initially bound excitons with electric fields, without the need to overcome an effective barrier via tunneling or thermal excitation. We speculate on further possible effects of the semiclassical dynamics in geometries where the constituent layers are subject to the same or different electric fields.

Properties of dynamical fractal geometries in the model of causal dynamical triangulations

Abstract: We investigate the geometry of a quantum universe with the topology of the four-torus. The study of noncontractible geodesic loops reveals that a typical quantum geometry consists of a small semiclassical toroidal bulk part, dressed with many outgrowths, which contain most of the four-volume and which have almost spherical topologies, but nevertheless are quite fractal.

Superconductivity-induced spectral weight transfer due to quantum geometry

Abstract: Optical spectral weight transfer associated with the onset of superconductivity at high-energy scales compared with the superconducting gap has been observed in several systems such as high-${T}_{c}$ cuprates. While there are still debates on the origin of this phenomenon, a consensus is that it is due to strong correlation effects beyond the BCS theory. Here, we show that there is another route to a nonzero spectral weight transfer based on the quantum geometry of the conduction band in multiband systems. We discuss applying this idea to the cuprates and twisted multilayer graphene.