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Eguchi-Kawai reduction with one flavor of adjoint Moebius fermion

We study the single site lattice gauge theory of SU(N) coupled to one Dirac flavor of fermion in the adjoint representation. We utilize M\"obius fermions for this study, and accelerate the calculation with graphics processing units (GPUs). Our Monte Carlo simulations indicate that for sufficiently large inverse 't Hooft coupling b = 1/g^2 N, and for N \leq 10 the distribution of traced Polyakov loops has"fingers"that extend from the origin. However, in the massless case the distribution of eigenvalues of the untraced Polyakov loop becomes uniform at large N, indicating preservation of center symmetry in the thermodynamic limit. By contrast, for a large mass and large b, the distribution is highly nonuniform in the same limit, indicating spontaneous center symmetry breaking. These conclusions are confirmed by comparing to the quenched case, as well as by examining another observable based on the average value of the modulus of the traced Polyakov loop. The result of this investigation is that with massless adjoint fermions center symmetry is stabilized and the Eguchi-Kawai reduction should be successful; this is in agreement with most other studies.


I. INTRODUCTION
Eguchi-Kawai reduction [1] is an attractive idea in which a gauge theory becomes spacetime volume independent for a large number of colors N in a U (N ) or SU (N ) gauge theory. Originally this was shown by demonstrating that the Schwinger-Dyson equations (loop equations) for suitably defined Wilson loops are equivalent in two lattice Yang-Mills theories at large N : a model with an infinite number of sites in all four spacetime dimensions and a model with a single site. However, after the appearance of [1] it was rapidly shown that there is a phase transition at which the center symmetry (U (1) d for gauge group U (N ) or Z d N for SU (N ), where d is the number of spacetime dimensions) is broken [2,3], invalidating the reduction in the continuum limit. Various solutions to this problem have been suggested over the years, but the one that will occupy us here is the one that by adding fermions in the adjoint representation the center symmetry may remain unbroken for all values of the coupling [4].
If this is successful, some important things can be learned about quantum gauge field theories in large or infinite spacetime volume by studying a much simpler system. The large volume theory (e.g., K d sites in d dimensions) and the small volume theory (e.g., one site in d dimensions) have a parent-daughter relationship under an orbifold by a discrete translation group Z d K [4]. The basic observables in the parent theory are single trace operators which are averaged over all sites of the lattice, whereas in the daughter theory one has simply as the corresponding operator. Connected "correlation functions" of this class of operators are equal to each other in the large N limit: In the single site lattice case that we consider, L = a, the lattice spacing. For M = 2 these are susceptibilities in the large volume theory, and so we could learn about critical exponents by studying a single site lattice. However, correlations between operators on different timeslices are not accessible due to the sum over Euclidean time t, and so the usual spectral studies could not be performed by this approach.
In this article we examine the proposal of "center stabilization" (i.e., the absence of center symmetry breaking) by adjoint representation fermion flavors nonperturbatively by Monte Carlo simulations, restricting ourselves to the case of a single flavor of Dirac fermion in the adjoint representation (two Majorana flavors). We will show that for finite N there is a center symmetry breaking phase transition at finite gauge coupling. This is to be contrasted with various previous findings. The continuum one-loop effective potential in the N → ∞ limit indicates that the eigenvalues of the (untraced) Polyakov loop are uniformly distributed, and hence center symmetry is unbroken [4]. The single site lattice study of [5], which used Wilson fermions, found that for a certain range of masses center symmetry was unbroken. However, for the largest value of the inverse 't Hooft coupling b = 1/g 2 N that they studied, the findings are somewhat ambiguous for their smallest value of N . They are also using a different discretization, which may have important effects on irrelevant operators which change the symmetry at a given choice of m, b, N . See the related discussion in [6]. We will demonstrate that the critical b at which center symmetry breaks increases with N , so that for the larger values of N that they studied it may be possible to miss the phase transition if b is not taken large enough. For this reason, we find that it is quite important to first study modest values of N where the phase transition is easily accessible. In [7] the distribution of was studied as an indicator whether or not center symmetry is broken. It should be approximately 1/2 if the theory is symmetric, since the Polyakov loop Tr U µ will be distributed close to zero. In Fig. 1 of [7] it can be seen that P µ = 1/2 to a very good approximation for the b and N values that they have studied in the one Dirac flavor theory. However, it is important to note that they use the Wilson kernel and a value of the Wilson mass (what we call m 5 below) that has much larger magnitude than the one in our study, which may have a significant impact on the critical value of b where center symmetry breaks. The lack of symmetry breaking found in [7] is consistent with the findings of the lattice perturbation theory calculation of [8] for b = 1 (see Fig. 4 of that paper). By contrast, the analysis of Ref. [9], which is a sort of semiclassical approach that only keeps the eigenvalues of the link matrices and throws away the rest of the gauge field information, indicates that center symmetry is broken at large enough b.

II. FORMULATION
The action consists of a gauge part and a fermion part: S = S g + S f , where we use the Wilson plaquette gauge action and a Mobius Dirac operator. The latter is five dimensional, like domain wall fermions, with the four ordinary spacetime dimensions reduced to a single site. For the purposes of the Monte Carlo simulation, we only need the fermion measure, and for this the manipulations of [10] are essential for reducing this to the determinant of a version of the four-dimensional overlap operatorone involving the Möbius kernel rather than the more conventional Wilson kernel. Explicitly, we compute the determinant of the overlap operator where for Möbius fermions the polar approximation to the sign function is used, The Hermitian operator H 5 is γ 5 times the Möbius kernel, It involves a Möbius transformation of the Wilson kernel D W , which in turn depends on the bare "mass" m 5 . This "mass" is also known as the "domain wall height" and it does not correspond to a physical mass, but is rather a parameter of the regulator. In (2.1) the parameter m is the fermion mass in the sense that the partially conserved axial current mass will vanish as m → 0, although of course it still requires a multiplicative renormalization in order to obtain a physical quantity. In our studies we have taken based on findings in [10], and we have considered various values of L s . For us increasing L s does not increase our cost significantly, so we take it to be large enough to have a negligible residual mass. For N = 8 and L s = 64, the residual mass is m res = O(10 −8 ), which is quite acceptable, so we use this value in all that follows.

III. SIMULATION
We begin by thermalizing the gauge fields over a period of 2500 to 5000 Monte Carlo iterations, and measurements are subsequently taken every 10 iterations, accumulating a total of 2500 to 5000 samples. Randomization and rethermalization is performed every 100 samples in order to overcome the free energy barriers between the N ground states in the broken phase. Various steps of numerical linear algebra are required: we compute the inverse in (2.3) and diagonalize this operator to compute the sign function in (2.2). After rotating back to the original basis, we form the operator (2.1) and compute its determinant by an LU decomposition. For all of these manipulations we accelerate the calculations with NVIDIA C2075 graphics processing units (GPUs), making use of the MAGMA library [11]. Our simulation proceeds along the lines of the method used in [5]. We update the gauge fields using the Metropolis algorithm, with S f = − ln det D Möbius for the fermion effective action and the Wilson plaquette gauge action. In doing this the new fields are chosen by U µ → V µ U µ with V µ a random SU (N ) matrix. The random matrix is generated using standard SU (2) subgroup methods of Cabibbo and Marinari [12]. In order to keep acceptance rates to a reasonable level (50% to 80%), we find that V µ must not be too far from a unit matrix. We use a Gaussian distribution about the unit matrix, and adjust the variance as a function of b, since the acceptance rate is a sensitive function of this parameter. Further, only one subgroup is used per update, though all four gauge matrices are updated each iteration. Autocorrelations are monitored by observing how well the Polyakov loops sample the full range of possible values. In particular, in the cases of broken center symmetry, we check that all N vacua appear in the distribution, which requires an adequate tunneling between minima of the free energy. Unitarity of the gauge matrices is monitored, due to the possibility of accumulated roundoff error. In practice we never found an instance of unitarity being violated beyond our tolerance of 1.0 × 10 −10 .

IV. CENTER SYMMETRY PHASE DIAGRAM
We have explored the fate of the Z N center symmetry as a function of N , b and m. To do this we have examined the Polyakov loop in each direction, which for a single site lattice is just P µ = Tr U µ . When the distribution of P µ form a lump centered at zero, the phase is center symmetric (Fig. 1). When the distribution forms N islands away from zero, the center symmetry is broken (Fig. 2). We have studied N = 3 to 10 and have found the critical value of b above which center symmetry is broken in each case. The results for m = 0.0001 are shown in Fig. 3 and it can be seen that b c increases with N . We also find that increasing m decreases b c . The simulations become more difficult as b and N increase. In the case of N = 10, and only in this case, this gave rise to slightly ambiguous results for the largest value of b. For instance we found that b = 0.3, N = 10 was clearly broken, whereas the case of b = 0.4, N = 10 can be classified as "almost certainly broken" because it suffered from significant autocorrelation and only appeared to explore two of the ten ground states, giving rise to two "islands" or "legs" far away from the origin.

V. CONCLUSIONS
We have found that on the single site lattice with Ginsparg-Wilson-type fermions (in our case Möbius), center symmetry is spontaneously broken for modest val- ues of N . By contrast in the continuum large N limit the one loop effective potential for the eigenvalues of the (untraced) Polyakov loop shows that they repel and are uniformly distributed for any N f = 0 [4]. A different result has been obtained recently using an approach which truncates the link matrices to diagonal matrices only consisting of their eigenvalues, finding that in the the large N limit, N f = 1 has spontaneously broken center symmetry [9]. Our nonperturbative lattice results differ from Ref. [7] which found that to a good approximation (1.3) was 1/2, corresponding to Polyakov loops near zero on most configurations, and hence unbroken center symmetry. Similarly, the older study [5] which used Wilson fermions found unbroken center symmetry for a wide range of fermion masses. We have found in our study that the critical b where center symmetry breaks increases with the number of colors N . The study [7] used b in the range 0.32 ≤ b ≤ 0.70 and N = 18 and 25. It is quite possible that for these large values of N the critical b is larger than 0.70, in which case [7] simply would have been performed in the symmetric phase. It is also important that the critical b most likely increases as |m 5 | is increased, based on the results in Fig. 5 of [8]. Since we perform our calculations with more modest values of |m 5 | and N , we observe symmetry breaking at smaller b. In the study [5], the (b, N ) pairs of (0.3, 10), (0.5, 15) and (1.0, 13) were found to be center symmetric for an appropriate choice of fermion mass. However, as mentioned above, the pair (1.0, 10) was somewhat ambiguous, based on inspection of their Polyakov loop distribution (see for instance the upper left hand panel of Fig. 19 of [5] which seems to show some structure away from the origin). One possible interpretation is that clear symmetry breaking for N = 10 only occurs for some b c > 1.0 and in the case of N = 13 an even larger value of b c is required. In future work we intend to test this conjecture. We remark that b c may be larger for Wilson fermions than for the Möbius fermions we have used. It is also possible that the two discretizations simply have different symmetry properties of the ground state at large b, though this is difficult to reconcile with the apparent universality of the b → ∞ limit.
Also difficult to reconcile are the nonperturbative findings of [8,13], where the theory of a single Majorana fermion ("half a Dirac flavor") in the adjoint representation was studied on a single site lattice. For N = 11 and b = 7 they found in [8] that the quantity (1.3) was approximately 1/2 for the right range of the Wilson kernel mass (what we are calling m 5 ). In [13] they found that for N = 11, 15 and 18 the quantity (1.3) was approximately 1/2 for b = 5, provided the fermion mass m in lattice units satisfied m < 0.1. According to the analysis of [9], this theory should have broken center symmetry. One possible way out is that the trace of a single power of the Polyakov loop may not be the right observable in this "half flavor" theory. It would be quite interesting to examine the eigenvalue spectrum of the untraced Polyakov loop in this case. It may also be the case that b c depends sensitively on the Wilson kernel mass m 5 and that b c becomes quite large for the value of m 5 that was used in [13].
Since we have found that b c increases with N , it may be that in the N → ∞ limit b c → ∞ as well. Then the claim of [4] would be substantiated, since it is an infinite N hypothesis. To further explore this possibility, we will in future work more extensively study b c versus N for a wider range of N and will perform a scaling analysis in order to make the infinite N extrapolation. We have found that the Metropolis algorithm used in this paper in inadequate for this purpose, as for larger N and b acceptance rates cannot be kept large enough without giving up on reasonable autocorrelation times-essentially the simulation becomes frozen about the initial configuration. To overcome this we will transition to the hybrid Monte Carlo algorithm where acceptance rates can be kept close to unity while still sampling configuration space in a representative way.
The gauge contribution to the effective potential V eff tends to order (nonzero Polyakov loop) while the fermion contribution to V eff tends to disorder (zero Polyakov loop), as emphasized in [14]. But for fixed b, why would the winner of this competition depend on N ? Naively, for adjoint fermions, they enter at the same order in N as the gauge fields. So it must be that the subleading corrections in the 1/N 2 expansion are different for fermions and gauge bosons, and these lead to a b c = b c (N ). However, if corrections to b c (N ) follows a 1/N 2 expansion, then the only way for b c (N ) to increase monotonically with N is if b c (N ) ∼ N p , p > 0 at leading order. Then we conclude that b c (∞) = ∞ and center symmetry is unbroken in the N → ∞ limit.