Low energy spectrum of SU(2) lattice gauge theory
Abstract
Prepotential formulation of gauge theories on honeycomb lattice yields local loop states, which are exact and orthonormal being free from any spurious loop degrees of freedom. We illustrate that, the dynamics of orthonormal loop states are exactly same in both the square and honeycomb lattices. We further extend this construction to arbitrary dimensions. Utilizing this result, we make a mean field ansatz for loop configurations for SU(2) lattice gauge theory in \(2+1\) dimension contributing to the low energy sector of the spectrum. Using variational analysis, we show that, this type of mean loop configurations has two distinct phases in the strong and weak coupling regime and shows a first order transition at \(g=1\). We also propose a reduced Hamiltonian to describe the dynamics of the theory within the mean field ansatz. We further work with the mean loop configuration obtained towards the weak coupling limit and analytically calculate the spectrum of the reduced Hamiltonian. The spectrum matches with that of the existing literature in this regime, establishing our ansatz to be a valid alternate one which is far more easier to handle for computation.
1 Introduction
Understanding the low energy behaviour of gauge theories is one of the most important problem of particle physics. Formulation of gauge theories on discrete lattice [1] has shown many major investigations in this direction over past few decades. Using Monte Carlo method, many important physical quantities can be computed numerically [2, 3]. However, understanding the vacuum as well as excited states in this sector is still open for investigation. Hence, there should always be attempts to make analytic approximations. This present work proposes such an approximation for SU(2) lattice gauge theories in \(2+1\) dimensions, which can as well be generalized to higher dimensions and higher gauge groups.
We work within the Hamiltonian formulation of lattice gauge theory [4] and use prepotential [5, 6, 7, 8, 9] framework, which gives an useful reformulation in terms of gauge invariant loops. Loop formulation of gauge theories is always desired by theorists [10, 11, 12, 13, 14, 15, 16, 17, 18] as one can get rid of spurious gauge degrees of freedom. However, working with loop does not guarantee to work with only physical degrees of freedom as the loop space itself is highly overcomplete [19, 20, 21]. Working in terms of gauge invariant loops in the weak coupling regime is again particularly difficult, as all possible loops of all shapes and sizes do contribute to the low energy spectrum of weakly coupled gauge field theories. In this scenario, the prepotential formulation [5, 6, 7, 8, 9] gives a great advantage over the standard Wilson loop approach as it is possible to extract only physically relevant loop degrees of freedom and study the dynamics of those.
In prepotenntial formulation of lattice gauge theory [5, 6, 7, 8, 9], one constructs gauge invariant loop variables, locally at each site. This particular feature is extremely useful for analytic calculations [22, 23, 24, 25, 26, 27] and also for the recent progress in quantum simulating lattice gauge theory [28, 29]. However, the original motivation for formulating prepotential approach was to make better understanding at the weak coupling regime of the theory. Towards this direction, a very important step is to construct the exact and orthonormal loop Hilbert space, containing only physical degrees of freedom. In this work, (also in another parallel recent work by Anishetty et al. [27]), we have proposed a general technique of constructing explicit orthonormal loop states for SU(2) lattice gauge theory in any arbitrary dimension. This is done even without going into the complicated ClebschGordon coeffiecients specific to SU(2) and hence is generalizable to SU(3) in a straightforward way.
The prepotential formulation on square lattice reveals that the physically relevant loops are the nonintersecting ones [22, 23]. We propose that, if one virtually splits each site of the lattice into two virtual ones following Fig. 8, the resulting lattice is an hexagonal one in two dimension. It has also been demonstrated that the dynamics of orthogonal loop states on square lattice is exactly equivalent to that of all possible loops on hexagonal one. The prepotential formulation on hexagonal lattice keeps all the important features of this particular formulation intact, such as local loop description by constructing intertwiners and Abelian weaving of those intertwiners leads to standard Wilson loops of the theory [5, 6, 7, 8, 9]. Moreover, the extra advantage on hexagonal lattice is, that the local loop space constructed at each site is exact and orthonormal. This is a tremendous advantage for the purpose of any computation be it analytic or numerical as one needs to work within a really small Hilbert space without bothering about complicated Mandelstam constraint. In [27], it has also been shown that there exists, more than one way to split each point, and the resulting lattices turn out to be of different types (i.e hexagon,octagon, square etc) as well as of different translational symmetries. It has also been argued in [27] that, dynamics on these virtual lattices are exactly equivalent to that of the original lattice by gauge fixing on the virtual links connecting splitted lattice sites. One can choose any splitting scheme as per the calculational convenience. For the purpose of present work we fix the splitting scheme given in Fig. 8 at each site, and get a hexagonal lattice to work with. We establish its equivalence with the original lattice by computing the dynamics explicitly for random loop configurations generated on both square and hexagonal lattice.
Within this framework, we propose a mean field configuration describing the low energy spectrum of the theory. It is true, that the mean field analysis cannot take us to the continuum limit, but is worth studying as it provides an excellent analytical tool to understand the vacuum structure of the for \(g<1\) and to study the low energy dynamics in this regime. In this work, we start from prepotential formulation of pure SU(2) gauge theory on square lattice. Then using the virtual point splitting technique of [27], we move to the virtual hexagonal lattice. On this particular lattice, we make a mean field ansatz that only an average loop configuration contribute to the low energy spectrum of SU(2) lattice gauge theory. We also show that this average loop configuration has two distinct phases at the weak and strong coupling regime of the theory and shows a first order phase transition at \(g=1\) denoting two distinct vacuum at two regimes of the theory. We also construct a reduced Hamiltonian, starting from the KogutSusskind Hamiltonian, which keeps the dynamics of the theory within the mean field ansatz. We further perform a variational calculation, to fix the mean field configuration at different values of coupling. This analysis confirms that the average fluxes flowing across each site shows a distinct jump as one moves from strong to weak coupling regime at \(g=1\). We are however interested in the the loop configuration at small values of g. This analysis reveals that large fluxes contribute to the low energy spectrum at weak coupling regime as opposed to the zero flux at strong coupling regime. For the original KogutSusskind Hamiltonian, largest contribution should come from large loops carrying large fluxes. Working with prepotentials makes us free from considering large loops at all, as all loops are now local [5, 6, 7, 8, 9, 22, 23, 27]. Hence, our ansatz of the vacuum in weak coupling regime, consists of only large fluxes flowing across the sites. We finally compute the lower lying spectrum of that reduced Hamiltonian at weak coupling regime and show that we get reasonably acceptable results within this approximation.
The organization of the paper is as follows: in Sect. 2 we briefly discuss the loop formulation of SU(2) gauge theory on hexagonal lattice and compare its dynamics with that of the square lattice. In Sect. 3, we illustrate the the origin of hexagonal lattice from square lattice by virtual point splitting in arbitrary dimension. In Sect. 4, we discuss the average loop configuration for prepotential formulation of SU(2) theory in \(2+1\) dimentions on virtual hexagonal lattice. In Sect. 5, we propose a reduced Hamiltonian and discuss its dynamics within the mean field ansatz and finally compute low energy spectrum within this ansatz. Finally we summarize and discuss future aspects of this study in Sect. 6.
2 Loop formulation of SU(2) gauge theory on hexagonal lattice
Loop actions around the hexagonal plaquette, given in Fig. 6. The coefficients in the loop action are given in the last column
Sl no.  At vertices  Action on \(l_{12},l_{23},l_{31}\rangle \)  Coefficient 

1.  c and f  \({\mathscr {O}}^{++}_{12}l_{12},l_{23},l_{31}\rangle =C_1l_{12}+1\rangle \)  \(C_1=\sqrt{(\frac{l_{12}+1)(l_{12}+l_{23}+l_{31}+1)}{(l_{12}+l_{31}+2)(l_{12}+l_{23}+1)}}\) 
2.  c and f  \({\mathscr {O}}^{+}_{12}l_{12},l_{23},l_{31}\rangle =C_2l_{23}+1,l_{31}1\rangle \)  \(C_2=\sqrt{\frac{l_{31}(l_{23}+1)}{(l_{12}+l_{31}+2)(l_{12}+l_{23}+1)}}\) 
3.  c and f  \({\mathscr {O}}^{+}_{12}l_{12},l_{23},l_{31}\rangle =C_3l_{23}1,l_{31}+1\rangle \)  \(C_3=\sqrt{\frac{l_{23}(l_{31}+1)}{(l_{12}+l_{31}+1)(l_{12}+l_{23}+2)}}\) 
4.  c and f  \({\mathscr {O}}^{{}{}}_{12}l_{12},l_{23},l_{31}\rangle =C_4l_{12}1\rangle \)  \(C_4=\sqrt{\frac{l_{12}(l_{12}+l_{23}+l_{31}+1)}{(l_{12}+l_{31}+1)(l_{12}+l_{23}+2)}}\) 
5.  a and d  \({\mathscr {O}}^{++}_{31}l_{12},l_{23},l_{31}\rangle =C_5l_{31}+1\rangle \)  \(C_5=\sqrt{\frac{(l_{31}+1)(l_{12}+l_{23}+l_{31}+2)}{(l_{23}+l_{31}+1)(l_{12}+l_{31}+2)}}\) 
6.  a and d  \({\mathscr {O}}^{+}_{31}l_{12},l_{23},l_{31}\rangle =C_6l_{12}+1,l_{23}1\rangle \)  \(C_6=\sqrt{\frac{l_{23}(l_{12}+1)}{(l_{23}+l_{31}+2)(l_{12}+l_{31}+1)}}\) 
7.  a and d  \({\mathscr {O}}^{+}_{31}l_{12},l_{23},l_{31}\rangle =C_7l_{12}1,l_{23}+1\rangle \)  \(C_7=\sqrt{\frac{l_{12}(l_{23}+1)}{(l_{23}+l_{31}+1)(l_{12}+l_{31}+2)}}\) 
8.  a and d  \({\mathscr {O}}^{{}{}}_{31}l_{12},l_{23},l_{31}\rangle =C_8l_{31}1\rangle \)  \(C_8=\sqrt{\frac{l_{31}(l_{12}+l_{23}+l_{31}+1)}{(l_{23}+l_{31}+1)(l_{12}+l_{31}+2)}}\) 
9.  b and e  \({\mathscr {O}}^{++}_{23}l_{12},l_{23},l_{31}\rangle =C_9l_{23}+1\rangle \)  \(C_9=\sqrt{\frac{(l_{23}+1)(l_{12}+l_{23}+l_{31}+2)}{(l_{23}+l_{12}+1)(l_{23}+l_{31}+2)}}\) 
10.  b and e  \({\mathscr {O}}^{+}_{23}l_{12},l_{23},l_{31}\rangle =C_{10}l_{31}+1,l_{12}1\rangle \)  \(C_{10}=\sqrt{\frac{l_{31}(l_{12}+1)}{(l_{23}+l_{31}+2)(l_{23}+l_{12}+1)}}\) 
11.  b and e  \({\mathscr {O}}^{+}_{23}l_{12},l_{23},l_{31}\rangle =C_{11}l_{31}1,l_{12}+1\rangle \)  \(C_{11}=\sqrt{\frac{l_{12}(l_{31}+1)}{(l_{23}+l_{31}+1)(l_{23}+l_{12}+2)}}\) 
12.  b and e  \({\mathscr {O}}^{{}{}}_{23}l_{12},l_{23},l_{31}\rangle =C_{12}l_{23}1\rangle \)  \(C_{12}=\sqrt{\frac{l_{23}(l_{12}+l_{23}+l_{31}+1)}{(l_{23}+l_{12}+1)(l_{23}+l_{31}+2)}}\) 
Few sample results for the comparison between dynamics around square plaquette and hexagonal plaquette. The first and second and column denotes two loop states around a hexagonal plaquette between which there exist a non zero matrix element for plaquette term \(\text{ Tr }\, U_{plaquette}\) of (1). These configuration has been translated to a valid orthonormal loop configuration around a square plaquette using the prescription given in this section and the matrix element \(M_s\) is calculated following Appendix A. The same \(M_h\) is also calculated for hexagonal plaquette following (16). Results are shown upto 8 decimal places upto which these two matches exactly. Note, these are only few sample results from numerical simulation
Initial state  Final state  Square plaquette  Hexagonal plaquette 

\(2j_1^h,2j_2^h,2j_3^h,2j_4^h,2j_5^h,2j_6^h\rangle \)  \(\{2\bar{j}_1^h,2{\bar{j}}_2^h,2{\bar{j}}_3^h,2{\bar{j}}_4^h,2{\bar{j}}_5^h,2{\bar{j}}_6^h\}\)  \(M_s\)  \(M_h\) 
\( 8,15,13,9,5,11\rangle \)  \( 9,16,14,10,6,12\rangle \)  3.682168E−002  3.682168E−002 
\( 9,7,13,17,14,6\rangle \)  \( 10,6,14,18,15,7\rangle \)  1.097742E−002  1.097742E−002 
\( 9,16,12,14,14,7\rangle \)  \( 8,17,13,13,15,8\rangle \)  1.350154E−002  1.350154E−002 
\( 12,9,2,9,10,7\rangle \)  \( 11,8,3,10,9,8\rangle \)  8.383834E−002  8.383834E−002 
\( 7,10,6,10,10,16\rangle \)  \( 8,9,5,11,9,15\rangle \)  4.790649E−002  4.790649E−002 
\( 13,6,7,14,6,7\rangle \)  \( 12,5,6,13,5,8\rangle \)  1.420527E−002  1.420527E−002 
One important thing to note here is that, as per our convention of defining loop operators and loop states, some of the loop actions bring −ve sign in the coefficient as shown in Table 2. But this happens only for the mixed operators (i.e type \({\mathscr {O}}^{+}\)) which involves one creation and one annihilation operator. But when we consider a closed loop such as a plaquette, we see that these type of mixed vertices always appear in pairs. Moreover note that, for the full plaquette operator (or any closed loop), the mixed terms can only appear in pairs and hence, the plauette operators (or any closed loop operators) are always positive by our convention.
Further note that, the action of loop operators on any loop state consists of two parts, one contains a numerical coefficient or number operators and another is some shift operators for the linking numbers. The coefficient that appears in Table 1 are calculated follows from the convention that, the shift operators are always at right most position and coefficients (function of number operators) are at left.
Having set the action of the loop operators on arbitrary loop states, one can easily compute the matrix element of the magnetic part of the Hamiltonian (1) within orthonormal loop basis, characterized by \(l_{12}, l_{23},l_{31}(\text{ or, } n_1,n_2,n_3)\) basis. The magnetic Hamiltonian consists of \(2^6=64\) terms, each of which is a set of six local loop operator at each of the six vertices of the hexagon, the action of which on respective local loop states are computed following the Table 2.
2.1 Dynamics on hexagonal lattice vs. dynamics on square lattice
Having calculated numerical values of the matrix elements of the magnetic Hamiltonian for hexagonal plaquette, we will now compare those matrix elements with that for a square plaquette (Matrix elements of magnetic Hamiltonian on a square lattice is given in Appendix A).
 Each site \((s\equiv a,b,c,d)\) on the square lattice has the following orthonormal angular momentum to have net angular momentum zero at site x:$$\begin{aligned} j_1^x,j_2^x,j_{{\bar{1}}}^x,j_{{\bar{2}}}^x,j_{12}^x=j_{{\bar{1}}{\bar{2}}}^s \end{aligned}$$
 Identify the flux around a square plaquette abcd as:$$\begin{aligned} j_1^a=j_{{\bar{1}}}^b\equiv j_1 ~&\quad j_2^b=j_{{\bar{2}}}^c\equiv j_2 \\ j_{{\bar{1}}}^c=j_{ 1}^d\equiv j_{{\bar{1}}} ~&\quad j_{\bar{2}}^d=j_{2}^a\equiv j_{{\bar{2}}} \end{aligned}$$
 At each site \((s\equiv a,b,c,d,e,f)\) of the hexagonal plaquette, the orthonormal states are characterized by \(l_{12}^x,l_{23}^x,l_{31}^x\) or equivalently byfollowing (9).^{3}$$\begin{aligned} n_1^x\equiv 2j_1^h_x, \quad n_2^x\equiv 2j_2^h_x, \quad n_3^x\equiv 2j_3^h_x \end{aligned}$$
 Identify the flux (marked with ‘h’) around the hexagonal plaquette ‘abcdef’ as:$$\begin{aligned} j_1^h= & {} \frac{n_1^a}{2}=\frac{n_1^f}{2},\quad j_2^h=\frac{n_3^a}{2}=\frac{n_3^b}{2}, \quad j_3^h=\frac{n_2^b}{2}=\frac{n_2^c}{2},\nonumber \\ j_4^h= & {} \frac{n_1^c}{2}=\frac{n_1^d}{2}, \quad j_5^h=\frac{n_3^d}{2}=\frac{n_3^e}{2}, \quad j_6^h=\frac{n_2^e}{2}=\frac{n_2^f}{2}.\nonumber \\ \end{aligned}$$(17)
 Now, the same flux around the hexagonal plaquette, can be identified as the dynamic flux around square plaquette ‘abcd’ in the following way:$$\begin{aligned} j_1^h= & {} j_2, \quad j_2^h=j_{12}^b, \quad j_3^h=j_1\nonumber \\ j_4^h= & {} j_4, \quad j_5^h=j_{12}^d, \quad j_6^h=j_{3} \end{aligned}$$(18)

Moreover, the external links of plaquettes ‘abcd’ and ‘abcdef’ can also be identified, but we are not writing them explicitly as we find them to remain unchanged in this particular plaquette dynamics.
We now compare the above calculated dynamics around an hexagonal plaquette with that of the square plaquette given in (A.6). For this purpose we identify the fluxes around the hexagonal plaquette with those around a square plaquette as given in Fig. 7. To compare the dynamics on square lattice and that on the hexagonal lattice, we simulate a random loop configuration on hexagonal lattice and compute the matrix element of the magnetic Hamiltonian as discussed above. Next we identify the same loop configuration on square lattice following prescription listed above, and compute the matrix element of Magnetic Hamiltonian on square lattice for this state following (A.6). In this comparison it is easy to observe that the nontrivial delta functions in (16) are exactly same as those arising in evaluating the 6j symbols in (A.6). More importantly, our numerical calculation using random loop configuration reveals that the numerical value of the nonzero matrix elements for each and every cases matches exactly (upto a sign) with each other for the calculations done on square lattice and hexagonal lattice. The discrepancy in sign arises as for the particular convention of defining the loop states on square plaquette, that we have chosen, each and every term becomes positive. We repeat this comparison for 1000 random loop configurations and find this exact matching for each and every case. For the purpose of illustration, we only quote a few sample results in Table 2. Hence, this numerical study proves that the dynamics of loop states on a square plaquette is identical to that on an hexagonal plaquette as long as one is interested only in orthonormal loop states, which are actually relevant for exact physical degrees of freedom.
3 Point Splitting and virtual hexagonal lattice
In the last section, we have established the equivalence in dynamics of orthonormal loop configurations in a square and hexagonal lattice. In this section, we prescribe a virtual point splitting technique, which translates any square lattice to its hexagonal counter part. As a result of this transition, we gain a theory formulated in terms of only explicitly orthonormal loop degrees of freedom at each site, and pay the price of an extra Abelian Gauss law constraint. This price is actually negligible as the square lattice already had the Abelian constraints to solve and the extra one in hexagonal lattice is on very same footing as those.
Now, prepotential formulation on this hexagonal lattice yields a local loop formulation of lattice gauge theory, exactly eqiuivalent to the original square lattice, but contains only orthonormal and physical loop degrees of freedom as this is free from complicated Mandelstam constraints. This makes the analysis on hexagonal lattice simpler for practical purpose of analytical as well as numerical computation.
4 The Hamiltonian and Average loop configurations
On hexagonal lattice, the hexagonal plaquettes are surrounded by six links, out of which four are links of the original square lattice along directions \( 1~ \& ~2\), and remaining two are virtual links, resulted from point splitting along direction 3. The electric fields are defined at each end of the links of the original square lattice. Hence even for hexagonal lattice, only the electric fields for links along directions \( 1~ \& ~2\) contribute to the Hamiltonian given in (20). In prepotential formulation, this electric part of Hamiltonian \(H_E\) counts the fluxes which are actually related to number of prepotentials sitting at each end of the links of the original lattice [5, 6, 7, 8, 9].
The magnetic part of the Hamiltonian \(H_{B}\) is anyway more complicated to analyse. Clearly, at weak coupling regime, this part contributes most, and hence it is essential to simplify it as much as possible to make analytic calculations feasible. The part however is altered from the square lattice, as it must contain trace of the product of link operators along the full hexagonal plaquette to make the smallest closed loop.
The magnetic Hamiltonian on hexagonal lattice contains \(2^6\) different gauge invariant plaquette terms for prepotential formulation of SU(2) gauge theory on hexagonal lattice. Pictorially, these plaquette terms contain \(n_d\) number of dotted link and \(6n_d\) number of solid links around a plaquette, for \(n_d=1,2,..,6\). These solid and dashed line comes in different combination yielding all of the 64 plaquette terms as illustrated in Fig. 11.
Let us now, concentrate on the loop configurations of the system and proceed towards a mean field ansatz (i.e average loop configuration) for low energy limit of the theory.
4.1 Mean Field Ansatz
In this subsection, we make an ansatz for the vacuum loop configuration of the SU(2) lattice gauge theory. Strong coupling vacuum of the system is wellknown and consists of 0 flux state. Whereas, in the naive continuum limit, as \(g\rightarrow 0\), all the loop configurations contribute to the low energy spectrum. However, the maximum contribution is expected to come from large loops carrying large fluxes. In prepotential formulation, the size of the loop is not relevant, as all the loops has been made local. Abelian weaving along the links give rise to the standard Wilson loops.
List of coefficients under the mean field ansatz
5 The reduced Hamiltonian and its spectrum
5.1 The SubHamiltonian
(1) This term together with its Hermitian conjugate pair (2 terms) creates or annihilates flux around a full plaquette. (2) This term along with its rotationally symmetric (6 terms) and hermitian conjugate pairs (6 more terms) increases or decreases the length of the Wilson loops by 5 units. (3) Same as (2), total 12 terms, changes length by 4 units. (4), (5), (6) and the rotationally symmetric 9 terms ( 9 more hermitian conjugate terms) merge (separate) two loops and construct one (two) bigger (smaller) loop(s) of their combined length \(+()\) 4 units of length. (7) and its hermitian conjugate term (total 2 terms) are rotationally symmetric and merges three loops to construct e bigger loop of their combined length. (8), (9) and their rotationally symmetric 6 terms for each constitute a hermitian conjugate set of 12 terms, each of which merges two loops and construct a bigger one of the same length. Finally (10) and its 6 rotationally symmetric terms are hermitian conjugate set and changes shape of a loop without changing its length.
Let us now make the following observations:
 1.Plaquette consisting of alternate solid and dashed line. There are only two options for this, which constitutes the rotationally symmetric and Hermitian operator. We denote this operator by$$\begin{aligned} H_{\text{ pmpmpm }}+\text{ rotation } \end{aligned}$$(30)
 2.Plaquette consisting of three consecutive solid and dashed lines. There are six options for this, which constitutes the rotationally symmetric and Hermitian operator. We denote this operator by$$\begin{aligned} H_{\text{ pppmmm }}+\text{ rotations } \end{aligned}$$(31)
 3.Plaquette consisting of two consecutive solid line, two consecutive dashed line and then a single solid and dashed line along with their Hermitian conjugate plaquette terms. We denote this type of operator byEach of these two types of plaquette terms has 6 rotationally symmetric contribution. Hence this particular type of rotational symmetric Hermitian operator contains total of 12 individual plaquette operators.$$\begin{aligned} H_{\text{ ppmmpm }}+H_{\text{ mmppmp }}+\text{ rotations } \end{aligned}$$(32)
The smallest eigenvalue of the Hamiltonian matrix reaches its minima for these values of L, M, at the particular g mentioned in the first column
g  \(L_{\lambda _{min}}\)  \(M_{\lambda _{min}}\) 

10  0  0 
1  0  0 
0.1  10  3 
0.01  374  32 
0.001  11,620  320 

For some fixed g, we numerically calculate (using Mathematica) the value of L and M, for which lowest eigenvalue of the Hamiltonian matrix reaches a minima. In the strong coupling regime, i.e for \(g\ge 1\), that minima is always at \(L=M=0\). However, for smaller and smaller values of g, the minima is at larger and larger values of \( L \& M\), as listed in Table 4. This calculation establishes the naive analysis done in Sect. 4 of this paper which shows the existence of two different mean field phases of the system in strong and weak coupling regimes. Note that, we are interested in the weak coupling regime of the theory as the continuum limit lies there. Up to this point of this work, we have not used any assumption for weak coupling limit, except taking a mean field ansatz. We now fix the mean field configuration in a way, such that we are in the weak coupling regime of the theory. From the variational study discussed above, we see, that the ground state energy shows a perfect first order phase transition at coupling \(g=1\), above which the vacuum is the strong coupling vacuum, which in the mean field ansatz gives \(L=0,M=0\rangle \). However, as \(g\rightarrow 0\), the lowest energy mean field state turns out to be a state comprising of large average flux at each site, implying the weak coupling vacuum to be consisted of loops carrying large fluxes, throughout the lattice. As \(g\rightarrow 0\), \(L>>M>>0\). The coefficients, listed in Table 3 also changes to particular limiting values as \(g\rightarrow 0\).

We now consider a particlular value of g, in the weak coupling regime, and the corresponding mean field configuration. For each configuration, we exactly diagonalize the Hamiltonian matrix, and calculate the eigenvalues. We list the spacings between the lowest one and first excited one as \(\varDelta \lambda _1\) and similarly between the first and second excited one as \(\varDelta \lambda _2\). We list these two gaps and their ratios at different values of the coupling constant \(g\rightarrow 0\) in Table 5. It is very much clear from Table 5, that the gaps are scaling as \(\sim g^2\) and the ratio of lowest two energy gaps converges to the numerical value of 1. Note that, the absolute value of masses are subjected to be renormalized. However, the ratio of the two consecutive massgaps are always physical. In our study we have shown it to converge to numerical value of 1, for \(g\rightarrow 0\) for arbitrarily large lattices, as all the computation was done locally at each site.
The gaps between three consecutive energy levels, and their ratios are listed for different values of coupling g in weak coupling regime
g  \(\varDelta \lambda _1\)  \(\varDelta \lambda _2\)  \(\frac{\varDelta \lambda _1}{\varDelta \lambda _2}\) 

0.1  0.00998711  0.00916295  1.08994 
0.01  0.0000999971  0.0000989114  1.01098 
0.001  0.00000099896  0.000000998378  1.00058 
6 Summary and future directions
In this work we have proposed and justified an effective mean field description for the low energy spectrum of SU(2) lattice gauge theory in \(2+1\) dimension and have analytically calculated the spectrum at the weak coupling regime. Starting from prepotential formulation on the spatial 2D square lattice, we perform virtual splitting of each lattice site into two and end up with a virtual hexagonal lattice. On this hexagonal lattice, all of the local loop states in prepotential formulations constitutes an exact and orthonormal loop basis, with no further Mandelstam constraints. We have proposed a mean value ansatz for the loop configurations throughout the lattice contributing to the low energy spectrum of the theory. We have shown that such average loop configurations have two distinct phases at the strong and weak coupling regime. Next, we have chosen a reduced Hamiltonian, from the full KogutSusskind Hamiltonian, which keeps the dynamics of the loops confined into our ansatz. Variational study shows that this reduced system with mean valueloop configuration shows a clear jump between the weak and strong coupling vacuum. As we are interested to explore weak coupling regime of the theory, we choose the relevant average loop configuration in that regime and calculate the spectrum for the reduced Hamiltonian we choose. In this spectrum we find \(\varDelta E\sim g^2\) which is the expected weak coupling behaviour for mass gap of the theory. The spacings of the spectrum obtained in this work is as well consistent with the available literature at weak coupling regime of \(2+1\) dimensional SU(2) lattice gauge theory. We have discussed in detail, how the point splitting lattices are constructed in higher dimensions, which can as well be exploited to extend this work beyond \(2+1\) dimensions in a straight forward way.
In a recent and parallel work [27], the point splitting lattice is constructed and utilized to analytically study the weak coupling limit of SU(2) lattice gauge theory in \(2+1\) dimension as well. In that work, they have used the path integral representation of the phase space to analytically compute the dispersion relation at the lowest order in weak coupling perturbation expansion.
However, the particular study demonstrated in this paper shows that the physical results at the weak coupling regime of SU(2) gauge theory can be extracted from a much simpler mean field approximation made within prepotential formulation of the theory. Being completely gauge invariant, and formulated only in terms of relevant physical degrees of freedom, this technique is suited for both analytic calculations and numerical simulations. From analytic perspective, this study gives a clear notion of the weak coupling vacuum for pure gauge theory and its dynamics. From numerical perspective, this particular formulation is most suited for quantum MonteCarlo simulation of Hamiltonian lattice gauge theory using a complete gauge invariant basis characterized by only integers. Till date, this aspect has not been studied extensively, but worth investigating in near future. That study will lead to explore some of the very important physics such as calculation of the entanglement entropy of lattice gauge theory. Last but not the least, there is a tremendous progress going on, in the recently developed research interests for quantum simulating gauge theories using both analog [28, 29, 35, 36, 37, 38] and digital quantum simulators [39, 40, 41, 42, 43]. The prepotential formulation has already been explored to propose quantum simulator for gauge theories [28, 29] and is a promising framework to define a whole new prepotential paradigm in quantum simulating QCD [43]. This present work, shows the way to construct quantum simuator to simulate the loop dynamics of nonAbelian lattice gauge theory beyond strong coupling limit. The work in this direction is in progress and will be reported shortly.
Footnotes
 1.
On hexagonal lattice, i=1,2,3 as denoted in Fig. 1, although the physical dimension of lattice is only two.
 2.
Fixing the loop quantum numbers \(n_1,n_2,n_3\) at all the even sites throughout the lattice, automatically fixes the configurations at all of the odd sites. A valid loop configuration is obtained if triangle inequality is valid at each and every sites. However, for that case generating linking numbers throughout the lattice and picking valid configurations when Abelian Gauss law (21) is satisfied is another option. It requires a detail study to find out which one is the most efficient one. Here, as we are only interested in dynamics around a chosen plaquette, we do not bother to generate loop configurations throughout the lattice.
 3.
Note that, the hexagonal lattice contains alternate odd (b,d,f) and even (a,c,e) sites. The links emerge emerge in the direction 1, 2, 3 from even sites and in \({\bar{1}}, {\bar{2}},{\bar{3}}\) from odd sites. However, for most of the purposes, we will not differentiate even and odd sites in general and will consider links to emerge from all sites in direction 1, 2, 3.
 4.
\(\varDelta E=0.2637g^2\).
 5.
\(\varDelta E\approx 2g^2\).
 6.
\(\varDelta E\approx 2.2g^2\).
 7.
\(\varDelta E\approx 2.1 g^2\).
Notes
Acknowledgements
We would like to thank Ramesh Anishetty for numerous discussions throughout the entire project as well as for his valuable suggestions on the manuscript. We would also like to thank Nathan Goldman and Center for Nonlinear Phenomena and Complex Systems, University libre de Bruxelles for hospitality during a substantial part of the project.
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