Spin Networks Research Articles

Projective Ponzano–Regge spin networks and their symmetries

We present a novel hierarchical construction of projective spin networks of the Ponzano-Regge type from an assembling of five quadrangles up to the combinatorial 4-simplex compatible with a geometrical realization in Euclidean 4-space. The key ingredients are the projective Desargues configuration and the incidence structure given by its space-dual, on the one hand, and the Biedenharn–Elliott identity for the 6j symbol of SU(2), on the other. The interplay between projective-combinatorial and algebraic features relies on the recoupling theory of angular momenta, an approach to discrete quantum gravity models carried out successfully over the last few decades. The role of Regge symmetry–an intriguing discrete symmetry of the 6j which goes beyond the standard tetrahedral symmetry of this symbol–will be also discussed in brief to highlight its role in providing a natural regularization of projective spin networks that somehow mimics the standard regularization through a q-deformation of SU(2).

Holographic spin networks from tensor network states

In the holographic correspondence of quantum gravity, a global onsite symmetry at the boundary generally translates to a local gauge symmetry in the bulk. We describe one way how the global boundary onsite symmetries can be gauged within the formalism of the multi-scale renormalization ansatz (MERA), in light of the ongoing discussion between tensor networks and holography. We describe how to "lift" the MERA representation of the ground state of a generic one dimensional (1D) local Hamiltonian, which has a global onsite symmetry, to a dual quantum state of a 2D "bulk" lattice on which the symmetry appears gauged. The 2D bulk state decomposes in terms of spin network states, which label a basis in the gauge-invariant sector of the bulk lattice. This decomposition is instrumental to obtain expectation values of gauge-invariant observables in the bulk, and also reveals that the bulk state is generally entangled between the gauge and the remaining ("gravitational") bulk degrees of freedom that are not fixed by the symmetry. We present numerical results for ground states of several 1D critical spin chains to illustrate that the bulk entanglement potentially depends on the central charge of the underlying conformal field theory. We also discuss the possibility of emergent topological order in the bulk using a simple example, and also of emergent symmetries in the non-gauge ("gravitational") sector in the bulk. More broadly, our holographic model translates the MERA, a tensor network state, to a superposition of spin network states, as they appear in lattice gauge theories in one higher dimension.

Intertwiner entanglement on spin networks

In the context of quantum gravity, we clarify entanglement calculations on spin networks: we distinguish the gauge-invariant entanglement between intertwiners located at the nodes and the entanglement between spin states located on the network's links. We compute explicitly these two notions of entanglement between neighboring nodes and show that they are always related to the typical $\ln(2j+1)$ term depending on the spin $j$ living on the link between them. This $\ln(2j+1)$ contribution comes from looking at non-gauge invariant states, thus we interpret it as gauge-breaking and unphysical. In particular, this confirms that pure spin network basis states do not carry any physical entanglement, so that true entanglement and correlations in loop quantum gravity comes from spin or intertwiner superpositions.

Entanglement entropy and correlations in loop quantum gravity

Black hole entropy is one of the few windows into the quantum aspects of gravitation, and its study over the years has highlighted the holographic nature of gravity. At the non-perturbative level in quantum gravity, promising explanations are being explored in terms of the entanglement entropy between regions of space. In the context of loop quantum gravity, this translates into an analysis of the correlations between the regions of the spin network states defining the quantum state of the geometry of space. In this paper, we explore a class of states, motivated by results in condensed matter physics, satisfying an area law for entanglement entropy and having non-trivial correlations. We highlight that entanglement comes from holonomy operators acting on loops crossing the boundary of the region.

Hamiltonian properties on a class of circulant interconnection networks

Classes of circulant graphs play an important role in modeling interconnection networks in parallel and distributed computing. They also find applications in modeling quantum spin networks supporting the perfect state transfer. It has been noticed that unitary Cayley graphs as a class of circulant graphs possess many good properties such as small diameter, mirror symmetry, recursive structure, regularity, etc. and therefore can serve as a model for efficient interconnection networks. In this paper we go a step further and analyze some other characteristics of unitary Cayley graphs important for the modeling of a good interconnection network. We show that all unitary Cayley graphs are hamiltonian. More precisely, every unitary Cayley graph is hamiltonian-laceable (up to one exception for X6) if it is bipartite, and hamiltonianconnected if it is not. We prove this by presenting an explicit construction of hamiltonian paths on Xnm using the hamiltonian paths on Xn and Xm for gcd(n,m) = 1. Moreover, we also prove that every unitary Cayley graph is bipancyclic and every nonbipartite unitary Cayley graph is pancyclic.

Fate of the Hoop Conjecture in Quantum Gravity.

We consider a closed region R of 3D quantum space described via SU(2) spin networks. Using the concentration of measure phenomenon we prove that, whenever the ratio between the boundary ∂R and the bulk edges of the graph overcomes a finite threshold, the state of the boundary is always thermal, with an entropy proportional to its area. The emergence of a thermal state of the boundary can be traced back to a large amount of entanglement between boundary and bulk degrees of freedom. Using the dual geometric interpretation provided by loop quantum gravity, we interpret such phenomenon as a pregeometric analogue of Thorne's "hoop conjecture," at the core of the formation of a horizon in general relativity.

Engineering the Eigenstates of Coupled Spin-1/2 Atoms on a Surface.

Quantum spin networks having engineered geometries and interactions are eagerly pursued for quantum simulation and access to emergent quantum phenomena such as spin liquids. Spin-1/2 centers are particularly desirable, because they readily manifest coherent quantum fluctuations. Here we introduce a controllable spin-1/2 architecture consisting of titanium atoms on a magnesium oxide surface. We tailor the spin interactions by atomic-precision positioning using a scanning tunneling microscope (STM) and subsequently perform electron spin resonance on individual atoms to drive transitions into and out of quantum eigenstates of the coupled-spin system. Interactions between the atoms are mapped over a range of distances extending from highly anisotropic dipole coupling to strong exchange coupling. The local magnetic field of the magnetic STM tip serves to precisely tune the superposition states of a pair of spins. The precise control of the spin-spin interactions and ability to probe the states of the coupled-spin network by addressing individual spins will enable the exploration of quantum many-body systems based on networks of spin-1/2 atoms on surfaces.

Selective addressing of solid-state spins at the nanoscale via magnetic resonance frequency encoding

The nitrogen vacancy centre in diamond is a leading platform for nanoscale sensing and imaging, as well as quantum information processing in the solid state. To date, individual control of two nitrogen vacancy electronic spins at the nanoscale has been demonstrated. However, a key challenge is to scale up such control to arrays of nitrogen vacancy spins. Here, we apply nanoscale magnetic resonance frequency encoding to realize site-selective addressing and coherent control of a four-site array of nitrogen vacancy spins. Sites in the array are separated by 100 nm, with each site containing multiple nitrogen vacancies separated by ~15 nm. Microcoils fabricated on the diamond chip provide electrically tuneable magnetic field gradients ~0.1 G/nm. Tailored application of gradient fields and resonant microwaves allow site-selective nitrogen vacancy spin manipulation and sensing applications, including Rabi oscillations, imaging, and nuclear magnetic resonance spectroscopy with nanoscale resolution. Microcoil-based magnetic resonance of solid-state spins provides a practical platform for quantum-assisted sensing, quantum information processing, and the study of nanoscale spin networks.

A molecular quantum spin network controlled by a single qubit.

Scalable quantum technologies require an unprecedented combination of precision and complexity for designing stable structures of well-controllable quantum systems on the nanoscale. It is a challenging task to find a suitable elementary building block, of which a quantum network can be comprised in a scalable way. We present the working principle of such a basic unit, engineered using molecular chemistry, whose collective control and readout are executed using a nitrogen vacancy (NV) center in diamond. The basic unit we investigate is a synthetic polyproline with electron spins localized on attached molecular side groups separated by a few nanometers. We demonstrate the collective readout and coherent manipulation of very few (≤ 6) of these S = 1/2 electronic spin systems and access their direct dipolar coupling tensor. Our results show that it is feasible to use spin-labeled peptides as a resource for a molecular qubit-based network, while at the same time providing simple optical readout of single quantum states through NV magnetometry. This work lays the foundation for building arbitrary quantum networks using well-established chemistry methods, which has many applications ranging from mapping distances in single molecules to quantum information processing.

Multi-fractal geometry of finite networks of spins: Nonequilibrium dynamics beyond thermalization and many-body-localization

Quantum spin networks overcome the challenges of traditional charge-based electronics by encoding information into spin degrees of freedom. Although beneficial for transmitting information with minimal losses when compared to their charge-based counterparts, the mathematical formalization of the information propagation in a spin(tronic) network is challenging due to its complicated scaling properties. In this paper, we propose a fractal geometric approach for unraveling the information-theoretic phenomena of spin chains and rings by abstracting them as weighted graphs, where the vertices correspond to single spin excitation states and the edges represent the information theoretic distance between pair of nodes. The weighted graph exhibits a complex self-similar structure. To quantify this complex behavior, we develop a new box-counting-inspired algorithm which assesses the mono-fractal versus multi-fractal properties of quantum spin networks. Mono- and multi-fractal properties are in the same spirit as, but different from, Eigenstate Thermalization Hypothesis (ETH) and Many-Body Localization (MBL), respectively. To demonstrate criticality in finite size systems, we define a thermodynamics inspired framework for describing information propagation and show evidence that some spin chains and rings exhibit an informational phase transition phenomenon, akin to the MBL transition.

Fock Representation of Gravitational Boundary Modes and the Discreteness of the Area Spectrum

In this article, we study the quantum theory of gravitational boundary modes on a null surface. These boundary modes are given by a spinor and a spinor-valued two-form, which enter the gravitational boundary term for self-dual gravity. Using a Fock representation, we quantise the boundary fields, and show that the area of a two-dimensional cross section turns into the difference of two number operators. The spectrum is discrete, and it agrees with the one known from loop quantum gravity with the correct dependence on the Barbero--Immirzi parameter. No discrete structures (such as spin network functions, or triangulations of space) are ever required---the entire derivation happens at the level of the continuum theory. In addition, the area spectrum is manifestly Lorentz invariant.

Pretty good quantum state transfer in symmetric spin networks via magnetic field

We study pretty good single-excitation quantum state transfer (i.e., state transfer that becomes arbitrarily close to perfect) between particles in symmetric spin networks, in the presence of an energy potential induced by a magnetic field. In particular, we show that if a network admits an involution that fixes at least one node or at least one link, then there exists a choice of potential on the nodes of the network for which we get pretty good state transfer between symmetric pairs of nodes. We show further that in many cases, the potential can be chosen so that it is only nonzero at the nodes between which we want pretty good state transfer. As a special case of this, we show that such a potential can be chosen on the endpoints of a spin chain to induce pretty good state transfer in chains of any length. This is in contrast to the result of Kempton et al. (Quantum Inf Comput 17(3):303–327, 2017), in which the authors show that there cannot be perfect state transfer in chains of length 4 or more, no matter what potential is chosen.

Spin statistics, partition functions and network entropy

Spin statistics, partition functions and network entropy

Quantum surface and intertwiner dynamics in loop quantum gravity

We introduce simple generic models of surface dynamics in loop quantum gravity (LQG). A quantum surface is defined as a set of elementary patches of area glued together. We provide it with an extra structure of locality (nearest neighbors), thought of as induced by the whole spin network state defining the 3d bulk geometry around the quantum surface. Here, we focus on classical surface dynamics, using a spinorial description of surface degrees of freedom. We introduce two classes of dynamics, to be thought as templates for future investigation of LQG dynamics with in mind the dynamics of quantum black holes. The first defines global dynamics of the closure defect of the surface, with two basic toy-models, either a dissipative dynamics relaxing towards the closure constraint or a Hamiltonian dynamics precessing the closure defect. The second class of dynamics describes the isolated regime, when both area and closure defect are conserved throughout the evolution. The surface dynamics is implemented through U(N) transformations and generalizes to a Bose-Hubbard Hamiltonian with a local quadratic potential interaction. We briefly discuss the implications of modeling the quantum black hole dynamics by a surface Bose-Hubbard model.

Magnetic Properties of the Distorted Kagomé Lattice Mn3(1,2,4-(O2C)3C6H3)2.

Kagomé lattice types have been of intense interest as idealized examples of extended frustrated spin systems. Here we demonstrate how the use of neutron diffraction and inelastic neutron scattering coupled with spin wave theory calculations can be used to elucidate the complex magnetic interactions of extended spin networks. We show that the magnetic properties of the coordination polymer Mn3(1,2,4-(O2C)3C6H3)2, a highly distorted kagomé lattice, have been erroneously characterized as a canted antiferromagnet in previous works. Our results demonstrate that, although the magnetic structure is ferrimagnetic, with a net magnetic moment, frustration persists in the system. We conclude by showing that the conventions of the Goodenough-Kanamori rules, which are often applied to similar magnetic exchange interactions, are not relevant in this case.

The closure constraint for the hyperbolic tetrahedron as a Bianchi identity

The closure constraint is a central piece of the mathematics of loop quantum gravity. It encodes the gauge invariance of the spin network states of quantum geometry and provides them with a geometrical interpretation: each decorated vertex of a spin network is dual to a quantized polyhedron in $\mathbb{R}^{3}$. For instance, a 4-valent vertex is interpreted as a tetrahedron determined by the four normal vectors of its faces. We develop a framework where the closure constraint is re-interpreted as a Bianchi identity, with the normals defined as holonomies around the polyhedron faces of a connection (constructed from the spinning geometry interpretation of twisted geometries). This allows us to define closure constraints for hyperbolic tetrahedra (living in the 3-hyperboloid of unit future-oriented spacelike vectors in $\mathbb{R}^{3,1}$) in terms of normals living all in $SU(2)$ or in $SB(2,\mathbb{C})$. The latter fits perfectly with the classical phase space developed for $q$-deformed loop quantum gravity supposed to account for a non-vanishing cosmological constant $\Lambda>0$. This is the first step towards interpreting $q$-deformed twisted geometries as actual discrete hyperbolic triangulations.

Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence configurations

The basic ingredients of the quantum theory of orbital and spin angular momentum (vector coefficients, 3nj symbols) encounter continuing relevance in wide areas beyond the traditional ones (molecular, atomic and nuclear spectroscopies and dynamics). This paper offers insight on the connection at the most elementary of levels with the diagrammatic approaches to projective geometry. In particular here we exhibit how the Fano, Desargues and related incidence configurations emerge in the Racah and in the Biedenharn-Elliott identities, corresponding respectively to the hexagonal and pentagonal relationships that provide the basis for the construction of 3nj symbols and of spin networks. It is shown that the treatment, although mostly confined to the quadrangulation of the real projective plane, permits however the introduction of networks involving seven and ten spins, and preludes to developments towards computational and asymptotic approaches for quantum and semi-classical applications to spectroscopy and dynamics.

Guest Editorial – Special Issue on Quantum Circuits

Guest Editorial – Special Issue on Quantum Circuits

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

We apply the recently suggested strategy to lift state spaces and operators for (2 + 1)-dimensional topological quantum field theories to state spaces and operators for a (3 + 1)-dimensional TQFT with defects. We start from the (2 + 1)-dimensional TuraevViro theory and obtain a state space, consistent with the state space expected from the Crane-Yetter model with line defects.This work has important applications for quantum gravity as well as the theory of topological phases in (3 + 1) dimensions. It provides a self-dual quantum geometry realization based on a vacuum state peaked on a homogeneously curved geometry. The state spaces and operators we construct here provide also an improved version of the Walker-Wang model, and simplify its analysis considerably.We in particular show that the fusion bases of the (2 + 1)-dimensional theory lead to a rich set of bases for the (3 + 1)-dimensional theory. This includes a quantum deformed spin network basis, which in a loop quantum gravity context diagonalizes spatial geometry operators. We also obtain a dual curvature basis, that diagonalizes the Walker-Wang Hamiltonian.Furthermore, the construction presented here can be generalized to provide state spaces for the recently introduced dichromatic four-dimensional manifold invariants.

Ab initio computations of zero-field splitting parameters and effective exchange integrals for single-molecule magnets (Mn12- and Mn11Cr-acetate clusters)

In this study, the zero-field splitting parameters (D, E) and effective exchange integrals (J) for a Mn12/Mn11Cr mixed crystal were evaluated using ab initio molecular orbital (MO) computations based on the hybrid density functional theory (DFT) methods for Mn11Cr∗-acetate cluster (Mn12 and Mn11Cr mixing). In this study, Mn12-acetate, Mn11Cr-acetate, and Mn12−nCrn-acetate clusters were focused on. The zero-field splitting (ZFS) parameters are obtained by UB3LYP methods, i.e. −0.339cm−1 (Mn12-acetate) and −0.351cm−1 (Mn11Cr-acetate). The energy barriers were estimated to be 53.7K and 50.4K, which showed good agreement with previously reported experimental values of 68.8K and 56.8K, respectively. The origin of such anisotropy is assumed to be the eight MnIII ions with d4-electrons in their outer ring subunits, where elongated conformation along the axial direction becomes very important and leads to negative D values. The spin networks in these clusters were discussed by using the effective exchange integral sets. The Js value, as estimated by the UB3LYP method revealed that the J2, J3, and J3′ interactions played important roles in describing the parallel spin ordering in cubane and ring subunits as well as the antiferromagnetic coupling between two subunits. The other interactions, i.e. J1, J1′, J4, and J4′ somewhat disturbed the spin ordering in the Mn12-acetate cluster. The substitution of MnIV by CrIII did not change the spin network. However, the total S value changed after the substitution.