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Ising Spin Network States for Loop Quantum Gravity by Feller and Livine

This week I have been studying a great paper by Feller and Livine on Ising Spin Network States for Loop Quantum Gravity. In the context of loop quantum gravity, quantum states of geometry are defined as
spin networks. These are graphs decorated with spin and intertwiners, which represent quantized excitations of areas and volumes of the space geometry. In this paper the authors develop the condensed matter point of view on extracting the physical and geometrical information out of spin network states: they
introduce  Ising spin network states, both in 2d on a square lattice and in 3d on a hexagonal lattice, whose correlations map onto the usual Ising model in statistical physics. They construct these states from the basic holonomy operators of loop gravity and derive a set of local Hamiltonian constraints which entirely characterize the states. By studying their phase diagram distance can be reconstructed from the correlations in the various phases.

The line of research pursued in this paper is at the interface between condensed matter and quantum information and quantum gravity on the other: the aim is to understand how the distance can be recovered from correlation and entanglement between sub-systems of the quantum gravity state. The core of the investigation is the correlations and entanglement entropy on spin network states. Correlations and especially entropy are of special importance
for the understanding of black holes dynamics. Understanding the microscopic origin of black holes entropy is one the major test of any attempt to quantify gravity and entanglement between the horizon and its environment degrees of freedom appears crucial

A spin network state is defined on a graph, dressed
with spins on the edge and intertwiners at the vertices.
A spin on an edge e is a half-integer je 2 N=2 giving an
irreducible representation of SU(2) while an intertwiner
at a vertex v is an invariant tensor, or singlet state, between
the representations living on the edges attached
to that vertex. Spins and intertwiners respectively carry
the basic quanta of area and volume. The authors build the spin
network states based on three clear simplifications:

  1.  Use a fixed graph, discarding graph superposition and graph changing dynamics and  work with a fixed regular
  2. Freeze all the spins on all the graph edges. Fix to smallest possible value, ½, which correspond to the most basic excitation of geometry in loop quantum gravity, thus representing a quantum geometry directly at the Planck scale.
  3. Restricted  to 4-valent vertices, which represent the basic quanta of volume in loop quantum gravity, dual to quantum tetrahedra.

These simplifications provide us with the perfect setting to map spin network states, describing the Planck scale quantum geometry, to qubit-based condensed matter models. Such models have been extensively studied in statistical physics and much is known on their phase diagrams and correlation functions, and we hope to be able to import these results to the context of loop quantum gravity. One of the most useful model is the Ising model whose relevance goes from modeling binary mixture to the magnetism of matter. We thus naturally propose to construct and investigate Ising spin
network states.

The paper reviews the definition of spin network and analyzes the structure of 4-valent intertwiners between spins ½ leading to the
effective two-state systems used to define the Ising spin network states. Different equivalent definitions are given in terms of the high and low temperature expansions of the Ising model. The loop representation of the spin network is then obtained and studied as well as the associated density which gives information about
parallel transport in the classical limit. Section III introduces

It then  introduces a set of local Hamiltonian constraints for which
the Ising state is a unique solution and elaborates on their usefulness for understanding the coarse-graining of
spin network  and the dynamic of loop quantum gravity.

After this, it discusses the phase diagram of the Ising
states and their continuum limit as well as the distance
from correlation point of view.


Ising Spin Network State

Spin network basis states  define the basic excitations of the quantum geometry and they are provided with a natural interpretation in terms of discrete geometry with the spins giving the quanta of area and the intertwiners giving the quanta of volume.


These 4-valent vertices will be organized along a regular lattice. The 3d diamond lattice and the 2d square lattice are considered . Looking initially at the 2d square the square lattice: in this setting, the space of 4-valent interwiners between four spins ½ is two dimensional – it can be decomposed into spin 0 and spin 1 states by combining the spins by pairs, as


Different such decompositions exist and are shown as  a graphical representation below. There are three such decompositions, depending on which spins are paired together, the  s, t and u channels.


The spin 0 and 1 states in the s channel  can be explicitly written in terms of the up and down states of the four spins:


Those two states form a basis of the intertwiner space.  There transformation matrices between this basis and the two other channels:


Let’s look at the  intertwiner basis  defined in terms of the square volume operator U of loop quantum gravity. Since the spins, and the area quanta, are fixed, the only freedom left in the spin network states are the volume quanta defined by the intertwiners. This will provide the geometrical interpretation of our spin network states as
excitations of volumes located at each lattice node. For a 4-valent vertex, this operator is defined as:



where J are the spin operators acting on the i link.

The volume itself can then obtained by taking the square root of the absolute value of U . Geometrically, 4-valent intertwiners are interpreted as representing quantum tetrahedron, which becomes the building block of the quantum geometry in loop quantum gravity  and spinfoam models. U takes the following form in the s channel basis:


The smallest  possible value of a chunk of space is the square volume ±√3/4 in Plank units.

The two oriented volume states of û,  | u ↑,↓〉 , can be considered as
the two levels of an effective qubit. Let’s now define a pure spin network state which maps its quantum fluctuations on the thermal fluctuations of a given classical statistical model such as the Ising model by


This state represents a particular configuration of the spin network and the full state is a quantum superposition of them all. Defined as such, the state is unnormalized but its norm is easily computed using the Ising partition function ZIsing:


The intertwiner states living at each vertex are now entangled and carry non-trivial correlations. More precisely this state exhibits Ising correlations between two vertex i, j:


Those correlations are between two volume operators at different vertex which are in fact components of the 2- point function of the gravitational field. So understanding how those correlations can behave in a non-trivial way is a first step toward understanding the behavior of the full 2-point gravity correlations and for instance recover the inverse square law of the propagator.

The generalization to 3d is straightforward. Keeping the requirement that the lattice be 4-valent the natural
regular lattice is the diamond lattice



Using the usual geometrical interpretation of loop quantum gravity, this lattice can be seen as dual to a triangulation of the 3d space in terms of tetrahedra dual to each vertex. This can be seen as an extension of the more used cubic lattice better suited to loop quantum gravity. The Ising spin network state and the whole
set of results which followed are then identical :

  • The wave function


  • The Hamiltonian constraints and their algebra  are the same.


In 3d, the Ising model also exhibits a phase transition

Information about the 2-points correlation functionssuch as long distance behavior at the phase transition or near it can be obtained using methods of quantum field theory. In d dimensions we have


where K(r) are modified Bessel functions and ξ is the
correlation length. For the three-dimensional case, we
have the simple and exact expression



In this paper, the authors have introduced a class of spin network
states for loop quantum gravity on 4-valent graph. Such 4-valent graph allows for a natural geometrical interpretation in terms of quantum tetrahedra glued together into a 3d triangulation of space, but it also allows them to be map the degrees of freedom of those states to effective qubits. Then we can define spin network states corresponding to known statistical spin models, such as the Ising model, so that the correlations living on the spin network are exactly the same as those models.
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