Spin Networks Research Articles

Transition Amplitudes in 3D Quantum Gravity: Boundaries and Holography in the Coloured Boulatov Model

We consider transition amplitudes in the coloured simplicial Boulatov model for three-dimensional Riemannian quantum gravity. First, we discuss aspects of the topology of coloured graphs with non-empty boundaries. Using a modification of the standard rooting procedure of coloured tensor models, we then write transition amplitudes systematically as topological expansions. We analyse the transition amplitudes for the simplest boundary topology, the 2-sphere, and prove that they factorize into a sum entirely given by the combinatorics of the boundary spin network state and that the leading order is given by graphs representing the closed 3-ball in the large N limit. This is the first step towards a more detailed study of the holographic nature of coloured Boulatov-type GFT models for topological field theories and quantum gravity.

Entanglement negativity on random spin networks

We investigate multipartite entanglement for quantum states of $3D$ space geometry, described via generalized random spin networks with fixed areas, in the context of background independent approaches to quantum gravity. We focus on entanglement negativity as a well defined witness of quantum correlations for mixed states, in our setting describing generic subregions of the boundary of a quantum $3D$ region of space. In particular, we consider a generic tripartition of the boundary of an open spin network state and we compute the typical R\'enyi negativity of two boundary subregions $A$ and $B$ immersed in the environment $C$, explicitly for a set of simple open random spin network states. We use the random character of the spin network to exploit replica and random average techniques to derive the typical R\'enyi negativty via a classical generalized Ising model correspondence generally used for random tensor networks in the large bond regime. For trivially correlated random spin network states, with only local entanglement between spins located on the network edges, we find that typical log negativity displays a holographic character, in agreement with the results for random tensor networks, in large spin limit. When nonlocal bulk entanglement between intertwiners at the vertices is considered the negativity increases, while at the same time the holographic scaling is generally perturbed by the bulk contribution.

Heat current magnification in classical and quantum spin networks.

We investigate heat current magnification (CM) due to asymmetry in the number of spins in two-branched classical as well as quantum spin systems that are kept between two heat baths at different temperatures. We study the classical Ising-like spin models using Q2R and Creutz cellular automaton dynamics. We show that just the difference in the number of spins is not enough and some other source of asymmetry like unequal spin-spin interaction strengths in the upper and lower branches is required for heat CM. We also provide a suitable physical motivation for CM along with ways to control and manipulate it. We then extend this study to a quantum system with modified Heisenberg XXZ interaction and preserved magnetization. Interestingly, in this case, just the asymmetry in the number of spins in the branches is enough to achieve heat CM. We observe that the onset of CM is accompanied by a dip in the total heat current flowing through the system. We then discuss how the observed CM characteristics can be attributed to the intersection of nondegenerate energy levels, population inversion, and atypical magnetization trends as a function of the asymmetry parameter in the Heisenberg XXZ Hamiltonian. Finally we use the concept of ergotropy to support our findings.

Spinor representation of the Hamiltonian constraint in 3D loop quantum gravity with a nonzero cosmological constant

We develop in a companion paper the kinematics of three-dimensional loop quantum gravity in Euclidean signature and with a negative cosmological constant, focusing in particular on the spinorial representation that is well known at zero cosmological constant. In this paper, we put this formalism to the test by quantizing the Hamiltonian constraint on the dual of a triangulation. The Hamiltonian constraints are obtained by projecting the flatness constraints onto spinors, as done in the flat case by the first author and Livine. Quantization then relies on $q$-deformed spinors. The quantum Hamiltonian constraint acts in the $q$-deformed spin network basis as difference equations on physical states, which are thus the Wheeler-DeWitt equations in this framework. Moreover, we study how physical states transform under Pachner moves of the canonical surface. We find that those transformations are in fact $q$ deformations of the transition amplitudes of the flat case as found by Noui and Perez. Our quantum Hamiltonian constraints, therefore, build a Turaev-Viro model at real $q$.

Analyzing and Unifying Robustness Measures for Excitation Transfer Control in Spin Networks

Recent achievements in quantum control have resulted in advanced techniques for designing controllers for applications in quantum communication, computing, and sensing. However, the susceptibility of such systems to noise and uncertainties necessitates robust controllers that perform effectively under these conditions to realize the full potential of quantum devices. The time-domain log-sensitivity and a recently introduced robustness infidelity measure (RIM) are two means to quantify controller robustness in quantum systems. The former can be found analytically, while the latter requires Monte-Carlo sampling. In this work, the correlation between the log-sensitivity and the RIM for evaluating the robustness of single excitation transfer fidelity in spin chains and rings in the presence of dephasing is investigated. We show that the expected differential sensitivity of the error agrees with the differential sensitivity of the RIM, where the expectation is over the error probability distribution. Statistical analysis also demonstrates that the log-sensitivity and the RIM are linked via the differential sensitivity, and that the differential sensitivity and RIM are highly concordant. This unification of two means (one analytic and one via sampling) to assess controller robustness in a variety of realistic scenarios provides a first step in unifying various tools to model and assess robustness of quantum controllers.

Magnetic structure and spin dynamics of the quasi-two-dimensional antiferromagnet Zn-doped copper pyrovanadate

Magnetic properties of the antiferromagnet Zn$_{0.15}$Cu$_{1.85}$V$_2$O$_7$ (ZnCVO) have been thoroughly investigated on powder and single-crystal samples. The crystal structure determination using powder x-ray and neutron diffraction confirms that ZnCVO with Zn = 0.15 is isostructural with $\beta$-Cu$_{2}$V$_2$O$_7$ ($\beta$-CVO) with small deviation in the lattice parameters. Macroscopic magnetic properties measurements also confirm the similarity between the two compounds. The Cu$^{2+}$ spins were found to align along the crystallographic $c$-axis, antiparallel to their nearest neighbors connected by the leading exchange interaction $J_1$. Spin dynamics reveals a typical symmetric spin-wave dispersion with strong interactions in the $bc$-plane and weak interplane coupling. The exchange interaction analysis indicates that the spin network of ZnCVO is topologically consistent with the previous DFT prediction but the values of leading exchange interactions are contradictory. Furthermore, rather than the predicted 2D honeycomb structure, the spin network in ZnCVO could be better described by the anisotropic 2D spin network composing of $J_1$, $J_5$, and $J_6$ interactions, four bonds per one spin site, coupled by weak interplane interactions.

Helimagnets by disorder: Its role on the high-temperature magnetic spiral in the YBaCuFeO5 perovskite

Most of the spiral magnetoelectric multiferroics investigated in recent years are geometrically or exchange-frustrated magnets, where the presence of triangular or other frustrated spin networks produce low magnetic transition temperatures. This critically limits their potential uses. The exceptional stability of the spiral magnetic order (at ${T}_{S}$) in the layered structure of the ${\mathrm{YBaCuFeO}}_{5}$ double perovskite involves a nonconventional mechanism: spiral order by disorder. The model has been theoretically developed by Scaramucci et al. [Phys. Rev. Res. 2, 013273 (2020)] after the discovery of a huge impact of cation disorder on ${T}_{S}$ [M. Morin et al., Nat. Commun. 7, 13758 (2016)]. In this work the influence of disorder (and only disorder) on the magnetic phase diagram is studied on a quantitative basis extending the range of previous studies. We thoroughly investigate the impact of frustration due to $B$-site disorder (${n}_{d}$) on the magnetic spirals in the reference composition ${\mathrm{YBaCuFeO}}_{5}$. The interplay between disorder, stability, and the detailed features of the incommensurate spiral magnetic orders were systematic, quantitative, and methodically investigated in samples of identical composition, spanning a wide range of ${n}_{d}$ values. Three different regimes are distinguished in the ${\mathrm{YBaCuFeO}}_{5}$ phase diagram versus disorder. A triple point is found in ${\mathrm{YBaCuFeO}}_{5}$ driven by Fe/Cu disorder that sets limits to ${T}_{S}$ and the cycloidal component of the helicoidal order. These layered materials appear as a very efficient realization of the avenue ``spiral order by disorder'' to supply functional helimagnets at normal working temperatures.

Intertwiner entanglement excitation and holonomy operator

In the loop quantum gravity framework, spin network states carry entanglement between quantum excitations of the geometry at different space points. This intertwiner entanglement is gauge-invariant and comes from quantum superposition of spins and intertwiners. Bipartite entanglement can be interpreted as a witness of distance, while multipartite entanglement reflects the curvature of the quantum geometry. The present work investigates how the bipartite and multipartite intertwiner entanglement changes under the action of the holonomy operator, which is the basic building block of loop quantum gravity’s dynamics. We reveal the relation between entanglement excitation and the dispersion of the holonomy operator. This leads to a new interesting connection between bulk geometry and boundary observables via the dynamics of entanglement.

Signatures of spin-liquid state in a 3D frustrated lattice compound KSrFe2(PO4)3 with S = 5/2

A quantum spin-liquid is a spin disordered state of matter in which spins are strongly correlated and highly entangled with low-energy excitations. It has been often found in two-dimensional S = ½, highly frustrated spin networks but rarely observed in three-dimensional (3D) frustrated quantum magnets. Here, KSrFe2(PO4)3, forming a complicated 3D frustrated lattice with a spin moment S = 5/2, is investigated by thermodynamic, neutron diffraction measurements and electronic structure calculations. Despite the relatively sizable Curie–Weiss temperature θCW = −70 K, a conventional magnetic long-range order is confirmed to be absent down to 0.19 K. The magnetic heat capacity data follow the power-law behavior at the lowest temperature region, supporting gapless excitations in a 3D spin-liquid state. Strong geometrical spin frustration responsible for the spin-liquid feature is understood as originating from the almost comparable five competing nearest-neighbor antiferromagnetic exchange interactions, which form the complicated 3D frustrated spin network. All these results suggest that the compound KSrFe2(PO4)3, representing a unique 3D spin frustrated network, could be a rare example of forming a gapless spin-liquid state even with a large spin moment of S = 5/2.

Fundamentals of fractional revival in graphs

We develop a general spectral framework to analyze quantum fractional revival in quantum spin networks. In particular, we determine when the adjacency algebra of a graph contains a matrix of a block diagonal form required for fractional revival, and introduce generalizations of the notions of cospectral and strongly cospectral vertices to arbitrary subsets of vertices. We give several constructions of graphs admitting fractional revival. This work resolves two open questions of Chan et al. (2019) [6].

Minimal model for the Bekenstein-Hawking entropy

In this work we derive the Bekenstein-Hawking entropy formula, $S=\frac{A}{4{l}_{p}^{2}}$, from the following minimal assumptions: (i) there is a minimum area, ${A}_{\mathrm{min}}$, proportional to ${l}_{p}^{2}$; (ii) the event horizon area, $A$, is tessellated by $N=A/{A}_{\mathrm{min}}$ distinguishable units; and (iii) the internal structure of these units is that of an infinite tower of internal levels. Although our results are model independent, this internal structure can be realized as the excitations of more fundamental entities such as, for instance, strings or loop quantum gravity spin networks. Even more, once the microstates of the black hole are taken to be singlets formed within the infinite tower of states describing the whole event horizon, the correction term $\ensuremath{-}\frac{3}{2}\mathrm{log}A$ emerges from our model. Finally, some comments regarding the applicability of the present model to extremal black holes, as well as possible relationships with spectral geometry and other approaches are pointed out. Our results are independent of the dimension of the black hole and whether it is rotating or not.

Unitary Design of Quantum Spin Networks for Robust Routing, Entanglement Generation, and Phase Sensing

AbstractSpin chains can be used to describe a wide range of platforms for quantum computation and quantum information. They enable the understanding, demonstration, and modeling of numerous useful phenomena, such as high fidelity transfer of quantum states, creation and distribution of entanglement, and creation of resources for measurement‐based quantum processing. In this paper, a more complex spin system, a 2D spin network (SN) engineered by applying suitable unitaries to two uncoupled spin chains, is studied. Considering only the single‐excitation subspace of the SN, it is demonstrated that the system can be operated as a router, directing information through the SN. It is also shown that it can serve to generate maximally entangled states between two sites. Furthermore, it is illustrated that this SN system can be used as a sensor device able to determine an unknown phase applied to a system spin. A detailed modeling investigation of the effects of static disorder in the system shows that this system is robust against different types of disorder.

Classical dynamics for loop gravity: The two-vertex model

The study of toy models in loop quantum gravity (LQG), defined as truncations of the full theory, is relevant to both the development of the LQG phenomenology, in cosmology and astrophysics, and the progress towards the resolution of the open issues of the theory, in particular the implementation of the dynamics. Here, we study the dynamics of spin network states of quantum geometry defined on the family of graphs consisting in 2 vertices linked by an arbitrary number of edges, or 2-vertex model in short. A symmetry reduced sector of this model -- to isotropic and homogeneous geometries -- was successfully studied in the past, where interesting cosmological insights were found. We now study the evolution of the classical trajectories for this system in the general case, for arbitrary number of edges with random initial configurations. We use the spinorial formalism and its clear interpretation of spin networks in terms of discrete twisted geometries, with the quantum 3d space made of superpositions of polyhedra glued together by faces of equal area. Remarkably, oscillatory and divergent regimes are found with a universal dependence on the coupling constants of the Hamiltonian and independent of the initial spinors or the number of edges. Furthermore, we explore the evolution of the associated polyhedra as well as their volumes and areas.

Quantum geometric maps and their properties

Quantum geometric maps, which relate SU(2) spin networks and Lorentz covariant projected spin networks, are an important ingredient of spin foam models (and tensorial group field theories) for four-dimensional quantum gravity. We give a general definition of such maps, that encompasses all current spin foam models, and we investigate their properties at such general level. We then specialize the definition to see how the precise implementation of simplicity constraints affects features of the quantum geometric maps in specific models.

Fermion coupling to loop quantum gravity: Canonical formulation

In the model of a fermion field coupled to loop quantum gravity, we consider the Gauss and the Hamiltonian constraints. According to the explicit solutions to the Gauss constraint, the fermion spins and the gravitational spin networks intertwine with each other so that the fermion spins contribute to the volume of the spin network vertices. For the Hamiltonian constraint, the regularization and quantization procedures are presented in detail. By introducing an adapted vertex Hilbert space to remove the regulator, we propose a diffeomorphism covariant graph-changing Hamiltonian constraint operator of the fermion field. This operator shows how fermions move in the loop quantum gravity spacetime and simultaneously influences the background quantum geometry. Moreover, as an innovation of our work, introducing the vertex Hilbert space also fixes issues so that a densely defined symmetric Hamiltonian constraint operator can be obtained.

Holographic entanglement in spin network states: A focused review

In the long-standing quest to reconcile gravity with quantum mechanics, profound connections have been unveiled between concepts traditionally pertaining to a quantum information theory, such as entanglement, and constitutive features of gravity, like holography. Developing and promoting these connections from the conceptual to the operational level unlock access to a powerful set of tools which can be pivotal toward the formulation of a consistent theory of quantum gravity. Here, we review recent progress on the role and applications of quantum informational methods, in particular tensor networks, for quantum gravity models. We focus on spin network states dual to finite regions of space, represented as entanglement graphs in the group field theory approach to quantum gravity, and illustrate how techniques from random tensor networks can be exploited to investigate their holographic properties. In particular, spin network states can be interpreted as maps from bulk to boundary, whose holographic behavior increases with the inhomogeneity of their geometric data (up to becoming proper quantum channels). The entanglement entropy of boundary states, which are obtained by feeding such maps with suitable bulk states, is then proved to follow a bulk area law with corrections due to the entanglement of the bulk state. We further review how exceeding a certain threshold of bulk entanglement leads to the emergence of a black hole-like region, revealing intriguing perspectives for quantum cosmology.

Nonbistable rectified linear unit-based gain-dissipative Ising spin network with stochastic resonance effect

The gain-dissipative Ising machine (GIM), comprising a nonlinear transfer function and spin interaction feedback, is one of the most promising quantum analog annealers. However, previous GIM constructions have been limited to only bistable nonlinear transfer functions with low noise intensity condition. To provide a new insight and broaden the potential design of GIMs, we construct a nonbistable rectified linear unit-based gain-dissipative Ising spin network (RISN). In contrast to the traditional GIM, spin dynamics analysis in the uncoupled state proves that RISN is nonbistable. Furthermore, it is demonstrated that the proposed RISN can effectively solve combinatorial optimization problems with a high noise level, which can be regarded as a stochastic resonance phenomenon. The proposed configuration, which does not comply with known conclusions, can become a stepping-stone for new GIM designs.

SU(N) toric code and non-Abelian anyons

We construct SU($N$) toric code model describing the dynamics of SU($N$) electric and magnetic fluxes on a two dimensional torus. We show that the model has $N^2$ topologically distinct ground states $|\psi_0\rangle_{({\mathsf p},{\mathsf q})}$ which are loop states characterized by $Z_N \otimes Z_N$ centre charges $({\mathsf p},{\mathsf q} =0,1,2,\cdots, N-1)$. We explicitly construct them in terms of coherent superpositions of all possible spin network states on torus with Wigner coefficients as their amplitudes. All excited quasiparticle states with SU($N$) electric charges and magnetic fluxes are constructed. We show that the braiding statistics of these SU(N) electric, magnetic quasiparticles or nonabelian anyons is encoded in the Wigner rotation matrices.

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

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

A Scalable CMOS Ising Computer Featuring Sparse and Reconfigurable Spin Interconnects for Solving Combinatorial Optimization Problems

No existing algorithms can find exact solutions to the combinatorial optimization problems (COPs) classified as non-deterministic polynomial-time (NP) hard problems. Alternatively, Ising computer based on the Ising model and annealing process has recently drawn significant attention. The Ising computers can find approximate solutions to the NP-hard COPs by observing the convergence of dynamic spin states. However, they have encountered challenges in mapping the optimization problems to the inflexible Ising computers with fixed spin interconnects. In this article, we propose a scalable CMOS Ising computer with sparse and reconfigurable spin interconnects for arbitrary mapping of spin networks with minimal overhead. Without a mapping algorithm, the proposed Ising computer provides a method for directly mapping COPs to the reconfigurable hardware. A 65-nm CMOS Ising test chip with 252 spins is fabricated and used for solving COPs, including max-cut problems.