Ising Lattice Gauge Theory with a Simple Matter Field

Ising Lattice Gauge Theory with a Simple Matter Field F. David Wandler 1 1 Department of Physics, University of Toronto, Toronto, Ontario, anada M5S 1A7. (Dated: December 8, 2018) I. INTRODUTION Quantum field theory is notorious for providing a very good description of particle physics interactions while at the same time being nearly impossible to solve exactly. To get around this hurdle, ingenious approximation methods have been developed. Lattice gauge theory is one such method. It is based on the idea of discretizing spacetime into a lattice and making that lattice euclidean by continuing to imaginary times. This transforms particle physics problems into statistical mechanics problems which are familiar and can be easily studied numerically. One particular connection is that different phases in the lattice description of a system corresponds to different force laws between particles. This idea will be developed more in this report. This report will study the simple model of Ising lattice gauge theory which is coupled to an Ising model in the place of a matter field. While this theory does not have a direct correspondence with any physical particle fields, it does provide qualitative insight into the nature of gauge fields coupled to matter fields. Gauge fields are responsible for the three fundamental forces described by the Standard Model of particle physics, so studying the behaviours of such fields can provide valuable insight into particle physics. It will be shown that the Ising lattice gauge theory with Ising model coupling has three phases, but that there exists a continuous connection between two of these phases. Section 2 covers how phases of a lattice gauge theory can correspond to different force laws in the particle physics description. Section 3 then introduces the model of interest, the Ising gauge theory with Ising model coupling. Section 4 proposes a phase diagram for this theory and justifies its main features. Section 5 concludes with a discussion of the results. II. PHASES AND FORE LAWS The goal of this section is to provide an intuitive picture of how the phases of Ising lattice gauge theory translate to force laws in the particle physics description. To begin, recall that Ising lattice gauge theory consists of a d-dimensional lattice with Ising spins located on the links of the lattice. Let e µ for µ = 1,..., d be the generators of the lattice (i.e. the difference between adjacent lattice vertices along the µ-th axis) and specify one generator as the time generator (i.e. e 1 corresponds to the forward time direction). Then define U µ (x) to be the spin (i.e. ±1) on the link between the lattice vertices at x and x + e µ. Then the Hamiltonian H 0 of Ising gauge theory is given by H 0 = K x,µν U µ (x)u ν (x + e µ )U µ (x + e ν )U ν (x) Where the sum is over all possible positions x and all pairs of directions µ and ν. This theory has a local Z 2 gauge symmetry, hence the name Ising gauge theory. This symmetry consists of transformations G(x) which flip the signs of the spins on all four links that touch the lattice vertex at x. For d > 2, this theory is known to have a phase transition [1]. The phases are distinguished by the behavior of gauge invariant correlation functions. These functions are the expectation values of closed loops of links. Denoting a closed loop as a set of links like {(x 1, µ 1 ), (x 2, µ 2 ),, (x n, µ n )} with x i+1 = x i + e µi and x 1 = x n + e µn, these correlation functions can be written as. Electronic address: f.wandler@mail.utoronto.ca

The phases are determined by how these correlation functions depend on the size of the loop. In the low K phase, for sufficiently large loops, the correlation function scales according to an area law e ca where c is some constant and A is the area contained in the loop. Similarly, in the high K phase, for sufficiently large loops, the correlation function scales according to a perimeter law: e bp where b is some constant and P is the perimeter of the loop. To see what these phases have to do with the force laws between particles, imagine coupling some static classical test charges to this potential. Note that the following argument is reproduced from Kogut [1] because of its importance to the topic at hand; it is not the original work of the author. This is done in particle physics by adding a term like e d d xj µ (x)u µ (x) to the action, where j µ (x) is the classical current density in the e µ direction at position x in spacetime and e is some charge characterizing the strength of this coupling. When considering two static charges a distance R apart over some time T, the term e d d xj µ (x)u µ (x) becomes e U µ(x) where the loop is now a rectangle with width R along a spatial direction and length T along the time direction. Recall that the Feynman integral for the particle physics description is given in terms of the action S by ( ) i D(U µ (x)) exp S Following the procedure for converting to the statistical mechanics picture the partition function with the charges included becomes ( Z = D(U µ (x)) exp 1 T (H 0 e ) U µ (x)) 2 From this it follows that exp e ) = Z Z 0 = e ln(z) ln(z0) where Z 0 is the partition function of the system without the charges present. The familiar formula[5] F = ln(z) relates this to the free energies. Therefore, the difference in the free energy of the theories is given by F F 0 = ln exp e ) Now the free energy is extensive so it makes sense to define the spatial potential energy by V (R) = 1 T (F F 0) Moreover, the expectation value can be evaluated as exp e ) = cosh(e) + sinh(e) Ignoring the constant terms to look at the qualitative behavior gives roughly V (R) = 1 T ln e Hence, in the low K phase over long times, V (R) 1 T A = 1 T (RT ) = R

This is a linear potential which generates a strong attractive force. This phase is called confinement because charges in this theory are so strongly attracted they tend to be confined in bound states. This resembles qualitatively the strong interaction of the Standard Model. For the high K phase overlong times, V (R) 1 (T + R) const. T This is a free potential, the charges feel no forces at long distances. For this reason, this is called the free phase. If the Z 2 gauge symmetry is upgraded to a continuous symmetry, this becomes a coulomb potential [1]. 3 III. OUPLING THE ISING LATTIE GAUGE THEORY TO A MATTER FIELD To couple a matter field to this Ising lattice gauge theory, imagine placing a regular Ising model of spins σ(x) on the vertices of the lattice, then use the gauge spins U µ (x) to determine the sign of the energy between the nearest neighbour spins, σ(x) and σ(x + e µ ). Following this construction gives the Hamiltonian H = K x,µν U µ (x)u ν (x + e µ )U µ (x + e ν )U ν (x) β x,µ σ(x)u µ (x)σ(x + e µ ). (1) The gauge symmetry still exists in this model if the generating transformations G(x) flip the signs of all the spins on the links touching the vertex at x as well as flipping the sign of σ(x). To see that these form a symmetry of the theory, consider picking just one vertex at x and applying G(x). Then for any UUUU term in the Hamiltonian, either zero or two signs have been flipped, leaving the term unchanged. Similarly, for the any σuσ term, if σ(x) is in the term, its sign in flipped and so is the sign of the U spin in the term, leaving the whole term unchanged. On the other hand, if σ(x) is not in the term, then none of the signs in the term are flipped, so the term is unchanged. Therefore, every G(x) leaves every term in the Hamiltonian unchanged, so they generate a local Z 2 gauge symmetry for the theory. IV. THE PHASE DIAGRAM Qualitatively, it is easy to see the three phases of this theory in the limiting cases of K and β. For example, in the case where the β = 0, this is just the usual Ising lattice gauge theory with a phase transition (assuming d > 2 [? ]) between a confining theory and a free theory. Therefore, it is expected that there will be free and confining phases in the theory. The third phase can be seen when K is taken to. This limit means that the energy cost of energy fluctuations in the U spins is infinitely large; therefore, the U µ (x) are effectively frozen into a given configuration. In this limit, one could simply pick a gauge and the system reduces to a regular Ising model in the σ spins. As Ising models in high enough dimensions exhibit a phase transition, it is expected that there will be a phase transition varying β in the K limit. This introduces another phase that is known as the Higgs phase in the region of high β and infinite K. The name Higgs phase comes from the fact that in this phase the coupled matter field picks up a nonzero magnetization which is analogous to the nonzero vacuum potential of the Higgs field in quantum field theory. Based on these intuitions, Fradkin and Shenker [2] proposed and justified the phase diagram given in Figure 1. Notice in particular that the transition along the β = 0 line extends into the region with nonzero β. Also, the phase transition along the K = line extends into the finite K region. Having these lines in the phase diagram means it makes sense to discuss the confinement, free, and Higgs phases in the context of the full theory, not just in certain limits. The dotted lines connecting these phase transitions is merely a guess at how the known solid lines extend. The shaded region is a particularly interesting feature of the diagram. Fradkin and Shenker were able to show that in this region no phase transitions can occur. This means that there is a continuous way to transition between the confining and Higgs phases. This is similar to the phase transition between the liquid and gas phases: there is a quantitative difference between the two phases but not a qualitative one. The rest of this section will focus on Fradkin and Shenker s justifications of these key features. First consider the phase transition between the confining and free phases. A renormalization type argument is used to show that this transition remains at small but nonzero β. To do this, they start with the Hamiltonian 1 with small positive β, then they replace the σuσ term with its free energy with respect to the σ spins. That is they define the effective Hamiltonian H eff = K ( )) U µ (x)u ν (x + e µ )U µ (x + e ν )U ν (x) ln exp β σ(x)u µ (x)σ(x + e µ x,µν x,µ x,σ(x) { 1,1}

4 FIG. 1: The phase diagram of the Ising lattice gauge theory coupled to an Ising model. Reprinted from Phase diagrams of lattice gauge theories with Higgs fields by Fradkin, E. and Shenker, S. Phys. Rev. D, 19(12). 3685. To leading order in β this is equivalent to H eff = K x,µν U µ (x)u ν (x + e µ )U µ (x + e ν )U ν (x) tanh(β) 4 x,µν U µ (x)u ν (x + e µ )U µ (x + e ν )U ν (x) Hence, the effect of adding a small β is to shift the constant K to K + tanh(β) 4 K + β 4. Therefore, if K c is the critical value of K in the β = 0 theory, in the small β theory a phase transition is expected at K = K c β 4. This justifies the line of phase transitions that intersects the β = 0 line in the phase diagram (Figure 1). The authors note that this argument also implies that the phase transition along the K = line extends to finite K by citing duality. Duality transformations transform the variables of the theory to produce a similar theory (potentially in a different number of dimensions) with different coefficients, and theories that are connected by such transformations are called dual. Fradkin and Shenker quote a result from Wegner [3] that the Ising lattice gauge theory with matter couplings are dual to Ising lattice gauge theories with matter couplings in an equal or greater number of dimensions with coefficients K and β. These transformations are such that K only depends on β, β only depends on K, and a small coefficient corresponds to a large dual coefficient. The argument above shows that there is a line of phase transitions given by K = K c (β ) 4 for small β which must correspond to a line of phase transitions β = F (K) for large K in the original model. The shaded region in Figure 1 is quite interesting. As mentioned above, Fradkin and Shenker proved that there can be no phase transitions in this region, continuously connecting the confining and Higgs phases. Notice that if any phase transition occurs, it must be signaled by some critical or singular behavior of the gauge invariant correlation functions; therefore, it is sufficient to prove that all such functions are analytic with respect to K and β in the shaded region. This is exactly what Fradkin and Shenker do in the appendix of [2]. General idea is to pick a nice gauge in which to calculate the correlation functions and then to break them down into a series of analytic functions f 1 (K, β) + f 2 (K, β) +. If this series of functions is bounded by a convergent geometric series(i.e. f n (K, β) α n for some number 1 < α < 1), then the series converges uniformly and the correlation function is analytic. They argue based on the forms of the functions f n that this condition is equivalent to the condition ( 4(e 2β + e 2β ) e 2K 1 p ) 1/p < (2) (e β + e β ) 4 For some integer p and some constant. The crux of the proof relies on the fact that p and are independent of K, β, and the overall correlation function. Therefore, the condition 2 defines a region in which no correlation function

can have singular behaviour. Notice that The right hand side of 2 vanishes as β at fixed K and as K 0 at fixed β, meaning that the condition holds for the regions of small K and/or large β. Moreover, in the limit of K, 2 becomes ( 4(e 2β + e 2β ) 1/p ) (e β + e β ) 4 < which still holds for finite value of β. This means that the region defined by 2 has the same qualitative shape as the shaded region in Figure 1. 5 V. DISUSSION The previous section showed that the phase diagram of the Lattice gauge theory coupled to a matter field is given by Figure 1. As discussed in Section 2, these phases correspond to different behaviours of the forces mediated by these particles. Of particular interest is the inclusion of the new Higgs phase. This phase has qualitative similarities to the Standard Model theory of the weak force, in which the W and Z gauge fields pick up a mass by being coupled to the Higgs field with a nonzero vacuum potential. As mentioned before, the magnetization of the matter field in the lattice model is analogous to this nonzero vacuum potential. Hence understanding this theory could give insight into the problems of particle physics. The other phases are also analogous to the other two Standard Model forces. For example, the confining phase is roughly analogous to how the gluon fields interacts with the quark fields in the Standard Model description of the strong force. The free phase allows for charges to roam freely which is somewhat similar to electrons coupled to the photon field, which are able to propagate freely unlike quarks. The analogies described here are vague and to use them properly one would need to upgrade the gauge symmetry from Z 2 to the continuous symmetries SU(n). Doing this would allow one to make useful statements about the Standard Model field. Fradkin and Shenker go on to do this later in their paper [2], but that is beyond the scope of this report. [1] Kogut, J. (1979). An introduction to lattice gauge theory and spin systems. Rev. Mod. Phys., 51(4), 659-713. DOI:10.1103/RevModPhys.51.659 [2] Fradkin, E. and Shenker, S. (1979). Phase diagrams of lattice gauge theories with Higgs fields. Phys. Rev. D, 19(12), 3682-2697. DOI:10.1103/PhysRevD.19.3682 [3] Wegner, F. (1971). Duality in generalized Ising models and phase transitions without local order parameters. J. Math. Phys., 12(10), 2259-2272. [4] Notice that here T has been set equal to 1. In lattice gauge theory T is related to the constant and therefore can be set to 1 in natural units. [5] For the rest of this paper, it will be assumed that d > 2 since this is necessary for any discussion of phases.