## Yang-Mills theory and tensor networks

In my two previous posts I discussed a quantum-information inspired approach to the study of quantum field theory.

So far everything I’ve talked about applies only to standard bosonic quantum fields. However, there are, apart from fermionic quantum fields, another type of quantum field which requires a more careful approach, namely, gauge fields.

I’m not going to get into the details of classical gauge fields, and their quantisation. Rather, in keeping with the modern conception of field theory as an effective theory, I’m simply going to define quantum gauge fields in the presence of a regulator, in this case the lattice, and then discuss their continuum limit, quantum Yang-Mills theory, separately.

Thus, from now on, a quantum (lattice) gauge theory is given as follows. Consider a regular lattice in ${D}$ spatial dimensions (the integer lattice ${\mathbb{Z}^D}$ will do just fine). Pick a group ${G}$ which for us is the lie group ${SU(2)}$ (although any compact topological group will do just fine). Now, on each edge ${e}$ or link of our lattice we put a copy of a quantum system ${C}$ which is defined to be a quantum particle moving on the group ${G}$. That is, ${C}$ has a hilbert space ${L^2(G)}$ with (improper) basis given by ${|g\rangle}$, for all ${g\in G}$. Thus the hilbert space for our lattice is simply

$\displaystyle \mathcal{H} = \bigotimes_{e} L^2(G), \ \ \ \ \ (1)$

where tensor product is over all the edges of the lattice. In our case the hilbert space for a link, ${L^2(G)}$, is infinite dimensional. However, this isn’t really a serious complication as it is only slightly more involved than a simple harmonic oscillator.

What makes a gauge theory a gauge theory is the presence of an enormous local symmetry. What this means is that there is a group,

$\displaystyle LG = \prod_{v} G, \ \ \ \ \ (2)$

where the product is over the vertices of the lattice, which acts on hilbert space and that all physical states lie in the sector of ${\mathcal{H}}$ invariant under the action of ${LG}$.

To describe the action of ${LG}$ we introduce the unitary operators

$\displaystyle L_h |g\rangle = |hg\rangle, \ \ \ \ \ (3)$

and

$\displaystyle R_h |g\rangle = |gh^{-1}\rangle, \ \ \ \ \ (4)$

on ${L^2(G)}$. These are simply the “multiply by ${h}$ on the left” (respectively, right) operators. Using these two unitary operators we can build our represention of ${LG}$ on ${\mathcal{H}}$ as follows. First decorate each link in the lattice with an arrow and then define the action of the subgroup ${LG_v \cong G}$ of elements of ${LG}$ which are not equal to the identity except on the factor corresponding to vertex ${v}$:

$\displaystyle h\mapsto \bigotimes_{e_- = v} L_h \otimes \bigotimes_{e_+ = v} R_h, \ \ \ \ \ (5)$

where ${e_-}$ (respectively, ${e_+}$) denotes the vertex at the source (respectively, target) of the edge ${e}$. This is just a unitary operator acting nontrivially only on the edges touching the vertex ${v}$. The action of ${l = (l_{v_1}, l_{v_2}, \ldots) \in LG}$ is then the product over all the vertices of these unitary operators, one for each ${l_v}$. (All of these unitaries commute.) The subspace ${\mathcal{H}_L}$ of ${\mathcal{H}}$ of states invariant under this action is called the gauge invariant sector.

A microscopic hamiltonian giving the dynamics of quantum Yang-Mills theory is the Kogut-Susskind model ${H(g_H)}$, which is a lattice hamiltonian defined on ${\mathcal{H}}$ but which is symmetric under the action of ${LG}$, so its ground state, and excited states, live in ${\mathcal{H}_L}$. I won’t write it out here except to say it is a combination of two terms, a “kinetic energy term” KE, diagonal in the momentum or fourier basis of ${L^2(G)}$, and a “potential energy term” PE, diagonal in the improper position basis. The Kogut-Susskind model depends on one parameter ${g_H}$ and looks quite a lot like a transverse Ising model (actually, it is a rotor model). Indeed, much of the physics of the transverse Ising model has an analogous counterpart in the KG model.

There is a competition between the KE and PE terms and when the coupling constant ${g_H = 0}$ the KG model is diagonal in the position basis and when ${g_H = \infty}$ the KG model is diagonal in the conjugate momentum basis. To actually get the quantum Yang-Mills gauge theory we need to tune the KG model to its quantum phase transition and apply the continuum limit procedure mentioned in the previous post. It is now widely believed that this transition occurs at ${g_H = 0}$ — this is called asymptotic freedom. What makes this model especially tricky to work with is that we need to approach the phase transition from the “strong-coupling” limit ${g_H=\infty}$ rendering many perturbative approaches inefficient or insufficient.

The structure of the gauge-invariant sector where the physical states live is far from trivial. Because we know the eigenstates of the KG model are gauge invariant much effort has been devoted at obtaining a manifestly gauge-invariant description of states in this sector. There are, roughly speaking, three approaches: (a) the spin-network basis of Baez which spends the local gauge symmetry by explicitly building the projector onto the gauge-invariant sector as a product of commuting projections and diagonalising these projectors separately; (b) the KKN (after Karabali-Kim-Nair) or Bars corner variable basis which works in the position basis and identifies a complete set of gauge invariant quantities diagonal in the position basis and writes all wavefunctions in terms of these variables; and (c) fixing a gauge by spending the gauge freedom. (The currently popular quantum link model approach — see, e.g., this paper today — works by truncating the infinite-dimensional projections used in the construction of the spin-network basis to some finite-dimensional subspace.)

Unfortunately none of the aforementioned approaches really exposes a compact and physically intuitive representation of the ground state of the KG model as it approaches its quantum phase transition. This is because they are either good at diagonalising the PE terms (Bars corner variables or gauge fixing) or the KE term (spin-network variables). What we’d really like is a compromise between these approaches.

Approaching quantum phase transitions is a business that tensor networks are rather good at. In particular, the multiscale entanglement renormalisation ansatz (MERA) is a natural candidate for such a situation: it gives a very natural interpolation between two fixed points of the RG. What I found was that when we exploit this idea in the context of lattice gauge theory we obtain a description which interpolates between a spin-network type description to a Bars corner-variable type description.

For the past three years I have devoted much of my energy into developing this idea; I’ve been searching for a good ansatz for the quantum phase transition of the KG model which could be used as input for the continuum limit procedure. This goal has taken me on a rather circuitous journey, via many dead ends, culminating in a ground-state ansatz compactly expressed as a gauge-invariant tensor MERA. The fundamental tool I’ve used in this construction is something I’ve called the quantum parallel transport operation, which is a quantum gate that can be used to construct arbitrary gauge-invariant tensor networks in a way that is not tied to spin-network or corner variables.

Today I’d like to announce my second open-science project aimed at exploring this idea. This project is hosted on github in the repository Lattice-gauge-theory-and-tensor-networks.

This project will run under the same conditions of the continuous limits project, please feel free to fork it!