Local Degrees Of Freedom Research Articles

Hamiltonian constraints and unfree gauge symmetry

We study Hamiltonian form of unfree gauge symmetry where the gauge parameters have to obey differential equations. We consider the general case such that the Dirac-Bergmann algorithm does not necessarily terminate at secondary constraints, and tertiary and higher order constraints may arise. Given the involution relations for the first-class constraints of all generations, we provide explicit formulas for unfree gauge transformations in the Hamiltonian form, including the differential equations constraining gauge parameters. All the field theories with unfree gauge symmetry share the common feature: they admit sort of "global constants of motion" such that do not depend on the local degrees of freedom. The simplest example is the cosmological constant in the unimodular gravity. We consider these constants as modular parameters rather than conserved quantities. We provide a systematic way of identifying all the modular parameters. We demonstrate that the modular parameters contribute to the Hamiltonian constraints, while they are not explicitly involved in the action. The Hamiltonian analysis of the unfree gauge symmetry is precessed by a brief exposition for the Lagrangian analogue, including explicitly covariant formula for degrees of freedom number count. We also adjust the BFV-BRST Hamiltonian quantization method for the case of unfree gauge symmetry. The main distinction is in the content of the non-minimal sector and gauge fixing procedure. The general formalism is exemplified by traceless tensor fields of irreducible spin $s$ with the gauge symmetry parameters obeying transversality equations.

Dimer description of the SU(4) antiferromagnet on the triangular lattice

In systems with many local degrees of freedom, high-symmetry points in the phase diagram can provide an important starting point for the investigation of their properties throughout the phase diagram. In systems with both spin and orbital (or valley) degrees of freedom such a starting point gives rise to SU(4)-symmetric models. Here we consider SU(4)-symmetric "spin'' models, corresponding to Mott phases at half-filling, i.e. the six-dimensional representation of SU(4). This may be relevant to twisted multilayer graphene. In particular, we study the SU(4) antiferromagnetic "Heisenberg'' model on the triangular lattice, both in the classical limit and in the quantum regime. Carrying out a numerical study using the density matrix renormalization group (DMRG), we argue that the ground state is non-magnetic. We then derive a dimer expansion of the SU(4) spin model. An exact diagonalization (ED) study of the effective dimer model suggests that the ground state breaks translation invariance, forming a valence bond solid (VBS) with a 12-site unit cell. Finally, we consider the effect of SU(4)-symmetry breaking interactions due to Hund's coupling, and argue for a possible phase transition between a VBS and a magnetically ordered state.

On BF-type higher-spin actions in two dimensions

We propose a non-abelian higher-spin theory in two dimensions for an infinite multiplet of massive scalar fields and infinitely many topological higher-spin gauge fields together with their dilaton-like partners. The spectrum includes local degrees of freedom although the field equations take the form of flatness and covariant constancy conditions because fields take values in a suitable extension of the infinite-dimensional higher-spin algebra \U0001d525\U0001d530[λ]. The corresponding action functional is of BF-type and generalizes the known topological higher-spin Jackiw-Teitelboim gravity.

Efficient two-electron ansatz for benchmarking quantum chemistry on a quantum computer

Quantum chemistry provides key applications for near-term quantum computing, but these are greatly complicated by the presence of noise. In this work we present an efficient ansatz for the computation of two-electron atoms and molecules within a hybrid quantum-classical algorithm. The ansatz exploits the fundamental structure of the two-electron system, and treating the nonlocal and local degrees of freedom on the quantum and classical computers, respectively. Here the nonlocal degrees of freedom scale linearly with respect to basis-set size, giving a linear ansatz with only $\mathcal{O}(1)$ circuit preparations required for reduced state tomography. We implement this benchmark with error mitigation on two publicly available quantum computers, calculating accurate dissociation curves for 4- and 6- qubit calculations of ${\rm H}_\textrm{2}^{}$ and ${\rm H}_\textrm{3}^+$.

Towards higher-spin AdS2/CFT1 holography

We aim at formulating a higher-spin gravity theory around AdS2 relevant for holography. As a first step, we investigate its kinematics by identifying the low-dimensional cousins of the standard higher-dimensional structures in higher-spin gravity such as the singleton, the higher-spin symmetry algebra, the higher-rank gauge and matter fields, etc. In particular, the higher-spin algebra is given here by [λ] and parameterized by a real parameter λ. The singleton is defined to be a Verma module of the AdS2 isometry subalgebra so (2, 1) ⊂ [λ] with conformal weight Delta =frac{1pm lambda }{2}. On the one hand, the spectrum of local modes is determined by the Flato-Fronsdal theorem for the tensor product of two such singletons. It is given by an infinite tower of massive scalar fields in AdS2 with ascending masses expressed in terms of λ. On the other hand, the higher-spin fields arising through the gauging of [λ] algebra do not propagate local degrees of freedom. Our analysis of the spectrum suggests that AdS2 higher-spin gravity is a theory of an infinite collection of massive scalars with fine-tuned masses, interacting with infinitely many topological gauge fields. Finally, we discuss the holographic CFT1 duals of the kinematical structures identified in the bulk.

Uniqueness of Galilean conformal electrodynamics and its dynamical structure

We investigate the existence of action for both the electric and magnetic sectors of Galilean Electrodynamics using Helmholtz conditions. We prove the existence of unique action in magnetic limit with the addition of a scalar field in the system. The check also implies the non existence of action in the electric sector of Galilean electrodynamics. Dirac constraint analysis of the theory reveals that there are no local degrees of freedom in the system. Further, the theory enjoys a reduced but an infinite dimensional subalgebra of Galilean conformal symmetry algebra as global symmetries. The full Galilean conformal algebra however is realized as canonical symmetries on the phase space. The corresponding algebra of Hamilton functions acquire a state dependent central charge.

Non-Abelian gauge theories invariant under diffeomorphisms

We discuss diffeomorphism and gauge invariant theories in three dimensions motivated by the fact that some models of interest do not have a suitable action description yet. The construction is based on a canonical representation of symmetry generators and on building of the corresponding canonical action. We obtain a class of theories whose number of local degrees of freedom depends on the dimension of the gauge group and the number of the independent constraints. By choosing the latter, we focus on three special cases, starting with a theory with maximal local number of degrees of freedom and finishing with a theory with zero degrees of freedom (Chern-Simons).

Unitary Extension of Exotic Massive 3D Gravity from Bigravity.

We obtain a new 3D gravity model from two copies of parity-odd Einstein-Cartan theories. Using Hamiltonian analysis, we demonstrate that the only local degrees of freedom are two massive spin-2 modes. Unitarity of the model in anti-deSitter and Minkowski backgrounds can be satisfied for vast choices of the parameters without fine-tuning. The recent "exotic massive 3D gravity" model arises as a limiting case of the new model. We also show that there exist trajectories on the parameter space of the new model which cross the boundary between unitary and nonunitary regions. At the crossing point, one massive graviton decouples resulting in a unitary model with just one bulk degree of freedom but two positive central charges at odds with the usual expectation that the critical model has at least one vanishing central charge. Given the fact that a suitable nonrelativistic version of bigravity has been used as an effective theory for gapped spin-2 fractional quantum Hall states, our model may have interesting applications in condensed matter physics.

Boundaries and supercurrent multiplets in 3D Landau-Ginzburg models

Theories with 3D mathcal{N} = 2 bulk supersymmetry may preserve a 2D mathcal{N} = left(0, 2right) subalgebra when a boundary is introduced, possibly with localized degrees of freedom. We propose generalized supercurrent multiplets with bulk and boundary parts adapted to such setups. Using their structure, we comment on implications for the {overline{Q}}_{+} -cohomology. As an example, we apply the developed framework to Landau-Ginzburg models. In these models, we study the role of boundary degrees of freedom and matrix factorizations. We verify our results using quantization.

From topological to quantum entanglement

Entanglement is a special feature of the quantum world that reflects the existence of subtle, often non-local, correlations between local degrees of freedom. In topological theories such non-local correlations can be given a very intuitive interpretation: quantum entanglement of subsystems means that there are “strings” connecting them. More generally, an entangled state, or similarly, the density matrix of a mixed state, can be represented by cobordisms of topological spaces. Using a formal mathematical definition of TQFT we construct basic examples of entangled states and compute their von Neumann entropy.

The self-consistent matter coupling of a class of minimally modified gravity theories

The self-consistent matter coupling is found in a broad class of minimally modified gravity theories which was discovered recently. All constraints in the theories remain first class and thus a graviton has only 2 local degrees of freedom. The cosmological solution of one of the examples in this class, the so-called square root gravity, exhibits a singularity freeness at high energy limit. At low energy limit, the theory smoothly connects to GR. A general feature of the theories in this class, with the self-consistent matter coupling discovered in our current work, is the non-trivial interaction among different components of matter sector. We have also checked the Hamiltonian structure of a scalar QED coupling to the square root gravity in the same manner. All constraints in the theory are first class too and thus the local U(1) gauge symmetry in scalar QED is preserved.

Locality from the Spectrum

Essential to the description of a quantum system are its local degrees of freedom, which enable the interpretation of subsystems and dynamics in the Hilbert space. While a choice of local tensor factorization of the Hilbert space is often implicit in the writing of a Hamiltonian or Lagrangian, the identification of local tensor factors is not intrinsic to the Hilbert space itself. Instead, the only basis-invariant data of a Hamiltonian is its spectrum, which does not manifestly determine the local structure. This ambiguity is highlighted by the existence of dualities, in which the same energy spectrum may describe two systems with very different local degrees of freedom. We argue that in fact, the energy spectrum alone almost always encodes a unique description of local degrees of freedom when such a description exists, allowing one to explicitly identify local subsystems and how they interact. As a consequence, we can almost always write a Hamiltonian in its local presentation given only its spectrum. In special cases, multiple dual local descriptions can be extracted from a given spectrum, but generically the local description is unique.

Binding complexity and multiparty entanglement

We introduce “binding complexity”, a new notion of circuit complexity which quantifies the difficulty of distributing entanglement among multiple parties, each consisting of many local degrees of freedom. We define binding complexity of a given state as the minimal number of quantum gates that must act between parties to prepare it. To illustrate the new notion we compute it in a toy model for a scalar field theory, using certain multiparty entangled states which are analogous to configurations that are known in AdS/CFT to correspond to multiboundary wormholes. Pursuing this analogy, we show that our states can be prepared by the Euclidean path integral in (0 + 1)-dimensional quantum mechanics on graphs with wormhole-like structure. We compute the binding complexity of our states by adapting the Euler-Arnold approach to Nielsen’s geometrization of gate counting, and find a scaling with entropy that resembles a result for the interior volume of holographic multiboundary wormholes. We also compute the binding complexity of general coherent states in perturbation theory, and show that for “double-trace deformations” of the Hamiltonian the effects resemble expansion of a wormhole interior in holographic theories.

Chip-on-tip endoscope incorporating a soft robotic pneumatic bending microactuator.

In the ever advancing field of minimally invasive surgery, flexible instruments with local degrees of freedom are needed to navigate through the intricate topologies of the human body. Although cable or concentric tube driven solutions have proven their merits in this field, they are inadequate for realizing small bending radii and suffer from friction, which is detrimental when automation is envisioned. Soft robotic actuators with locally actuated degrees of freedom are foreseen to fill in this void, where elastic inflatable actuators are very promising due to their S3-principle, being Small, Soft and Safe. This paper reports on the characterization of a chip-on-tip endoscope, consisting out of a soft robotic pneumatic bending microactuator equipped with a 1.1 × 1.1mm2 CMOS camera. As such, the total diameter of the endoscope measures 1.66mm. To show the feasibility of using this system in a surgical environment, a preliminary test on an eye mock-up is conducted.

Minimally modified theories of gravity: a playground for testing the uniqueness of general relativity

In a recent paper [1], it was introduced a new class of gravitational theories with two local degrees of freedom. The existence of these theories apparently challenges the distinctive role of general relativity as the unique non-linear theory of massless spin-2 particles. Here we perform a comprehensive analysis of these theories with the aim of (i) understanding whether or not these are actually equivalent to general relativity, and (ii) finding the root of the variance in case these are not. We have found that a broad set of seemingly different theories actually pass all the possible tests of equivalence to general relativity (in vacuum) that we were able to devise, including the analysis of scattering amplitudes using on-shell techniques. These results are complemented with the observation that the only examples which are manifestly not equivalent to general relativity either do not contain gravitons in their spectrum, or are not guaranteed to include only two local degrees of freedom once radiative corrections are taken into account. Coupling to matter is also considered: we show that coupling these theories to matter in a consistent way is not as straightforward as one could expect. Minimal coupling, as well as the most straightforward non-minimal couplings, cannot be used. Therefore, before being able to address any issues in the presence of matter, it would be necessary to find a consistent (and in any case rather peculiar) coupling scheme.

Building blocks of topological quantum chemistry: Elementary band representations

The link between chemical orbitals described by local degrees of freedom and band theory, which is defined in momentum space, was proposed by Zak several decades ago for spinless systems with and without time-reversal in his theory of "elementary" band representations. In Nature 547, 298-305 (2017), we introduced the generalization of this theory to the experimentally relevant situation of spin-orbit coupled systems with time-reversal symmetry and proved that all bands that do not transform as band representations are topological. Here, we give the full details of this construction. We prove that elementary band representations are either connected as bands in the Brillouin zone and are described by localized Wannier orbitals respecting the symmetries of the lattice (including time-reversal when applicable), or, if disconnected, describe topological insulators. We then show how to generate a band representation from a particular Wyckoff position and determine which Wyckoff positions generate elementary band representations for all space groups. This theory applies to spinful and spinless systems, in all dimensions, with and without time reversal. We introduce a homotopic notion of equivalence and show that it results in a finer classification of topological phases than approaches based only on the symmetry of wavefunctions at special points in the Brillouin zone. Utilizing a mapping of the band connectivity into a graph theory problem, which we introduced in Nature 547, 298-305 (2017), we show in companion papers which Wyckoff positions can generate disconnected elementary band representations, furnishing a natural avenue for a systematic materials search.

Local Degree of Freedom of Clutter for Reduced-Dimension Space-Time Adaptive Processing with MIMO Radar

Degree of freedom (DOF) of clutter in the reduced-dimension (RD) domain, which is called local DOF (LDOF), is of great importance for RD MIMO-STAP (space-time adaptive processing for multiple-input multiple-output radar) algorithms. In this paper, the LDOF equivalence of different RD MIMO-STAP algorithms are firstly proved, and then a generalized LDOF estimation rule under different conditions is developed to estimate the clutter LDOF for MIMO radar effectively. The accuracy of the proposed rule is verified, and how to design RD MIMO-STAP processors under the guidance of the proposed rule is presented through numerical simulations.

The Morley-Type Virtual Element for Plate Bending Problems

We propose a simple nonconforming virtual element for plate bending problems, which has few local degrees of freedom and provides the optimal convergence in $$H^2$$ -norm. Moreover, we prove the optimal error estimates in $$H^1$$ - and $$L^2$$ -norm. The nonconforming virtual element is constructed for any order of accuracy, but not $$C^0$$ -continuous. It is worth mentioning that, for the lowest-order case on triangular meshes the simplified nonconforming virtual element coincides with the well-known Morley element, so it can be taken as the extension of the Morley element to polygonal meshes. Finally, we verify the optimal convergence in $$H^2$$ -norm for the nonconforming virtual element by some numerical tests.

Physics-Based Quadratic Deformation Using Elastic Weighting.

This paper presents a spatial reduction framework for simulating nonlinear deformable objects interactively. This reduced model is built using a small number of overlapping quadratic domains as we notice that incorporating high-order degrees of freedom (DOFs) is important for the simulation quality. Departing from existing multi-domain methods in graphics, our method interprets deformed shapes as blended quadratic transformations from nearby domains. Doing so avoids expensive safeguards against the domain coupling and improves the numerical robustness under large deformations. We present an algorithm that efficiently computes weight functions for reduced DOFs in a physics-aware manner. Inspired by the well-known multi-weight enveloping technique, our framework also allows subspace tweaking based on a few representative deformation poses. Such elastic weighting mechanism significantly extends the expressivity of the reduced model with light-weight computational efforts. Our simulator is versatile and can be well interfaced with many existing techniques. It also supports local DOF adaption to incorporate novel deformations (i.e., induced by the collision). The proposed algorithm complements state-of-the-art model reduction and domain decomposition methods by seeking for good trade-offs among animation quality, numerical robustness, pre-computation complexity, and simulation efficiency from an alternative perspective.

Yang–Baxter maps, discrete integrable equations and quantum groups

For every quantized Lie algebra there exists a map from the tensor square of the algebra to itself, which by construction satisfies the set-theoretic Yang–Baxter equation. This map allows one to define an integrable discrete quantum evolution system on quadrilateral lattices, where local degrees of freedom (dynamical variables) take values in a tensor power of the quantized Lie algebra. The corresponding equations of motion admit the zero curvature representation. The commuting Integrals of Motion are defined in the standard way via the Quantum Inverse Problem Method, utilizing Baxter's famous commuting transfer matrix approach. All elements of the above construction have a meaningful quasi-classical limit. As a result one obtains an integrable discrete Hamiltonian evolution system, where the local equation of motion are determined by a classical Yang–Baxter map and the action functional is determined by the quasi-classical asymptotics of the universal R-matrix of the underlying quantum algebra. In this paper we present detailed considerations of the above scheme on the example of the algebra Uq(sl(2)) leading to discrete Liouville equations, however the approach is rather general and can be applied to any quantized Lie algebra.