Large- reduction in QCD-like theories
with massive adjoint fermions
Tatsuo Azeyanagi111 E-mail address : email@example.com, Masanori Hanada222 E-mail address : firstname.lastname@example.org, Mithat Ünsal333 E-mail address : email@example.com and Ran Yacoby444 E-mail address : firstname.lastname@example.org
Department of Physics, Kyoto University,
Kyoto 606-8502, Japan
Department of Particle Physics and Astrophysics,
Weizmann Institute of Science,
Rehovot 76100, Israel
SLAC and Physics Department, Stanford University,
Stanford, California 94025/94305, USA
Large- QCD with heavy adjoint fermions emulates pure Yang-Mills theory at long distances. We study this theory on a four- and three-torus, and analytically argue the existence of a large-small volume equivalence. For any finite mass, the center-symmetry unbroken phase exists at sufficiently small volume and this phase can be used to study the large volume limit through the Eguchi-Kawai equivalence. A finite-temperature version of volume independence implies that thermodynamics on can be studied via a unitary matrix quantum mechanics on , by varying the temperature. To confirm this nonperturbatively, we numerically study both zero- and one-dimensional theories by using Monte-Carlo simulation. The order of finite- corrections turns out to be . We introduce various twisted versions of the reduced QCD which systematically suppress finite- corrections. Using a twisted model, we observe the confinement/deconfinement transition on a lattice. The result agrees with large volume simulations of Yang-Mills theory. We also comment that the twisted model can serve as a nonperturbative formulation of the noncommutative Yang-Mills theory.
- 1 Introduction
- 2 Review of original Eguchi-Kawai proposal
- 3 Eguchi-Kawai equivalence for massive QCD(Adj)
- 4 Lattice model and Monte-Carlo simulation
- 5 Stabilizing noncommutative Yang-Mills theory
- 6 Conclusions and Discussions
- A QCD(Adj) on continuum at finite- and one-loop analysis
Recently Yang-Mills (YM) theory with adjoint fermions, QCD(Adj), has attracted much interest. The main impetus behind this is a network of exact large- equivalences. The starting point is the large- orientifold equivalence, which states that the bosonic subsector of this theory is equivalent to the charge-conjugation even subsector of QCD with fermions in antisymmetric representation [QCD(AS)] , provided symmetries defining the neutral subsectors are not spontaneously broken . QCD(AS) reduces to the ordinary QCD with fundamental quarks when , and is a natural large- generalization thereof .111The phenomenology of QCD(AS) is examined in [4, 5, 6]. The second important link is an orbifold equivalence: when quarks are massless or light with respect to strong scale , QCD(Adj) compactified on a Euclidean four-torus exhibits volume independence, thanks to its unbroken prerequisite (center and translational) symmetries at any radius . Thus, through the Eguchi-Kawai (EK) equivalence , one can study large- QCD on by using a unitary matrix model on a single-site lattice. QCD(Adj) also provides new insights into gauge dynamics, especially on small . This theory exhibits new nonperturbative phenomena, most strikingly confinement due to magnetic bions, a new class of non-self-dual topological excitations [9, 10], distinct from monopoles and instantons.
The statement of the volume independence [7, 8, 11, 12] is as follows. Consider SU(N) gauge theories on , with or without fermions, toroidally compactified on a four manifold . For simplicity, we assume the matter fields are in adjoint representation and, hence, the theory has a global center symmetry, described most easily as gauge rotations aperiodic up to an element of the center group. The order parameters of this symmetry are Wilson lines wrapping distinct toroidal cycles. The observables singlet under the center transformation constitute the neutral sector. The volume independence states that dynamics of the neutral sector observables is independent of the size of the torus provided the center symmetry and translational invariance are not spontaneously broken. Among such observables are nonperturbative mass spectrum, free energy densities, and deconfinement transition temperature, just to count a few. This is clearly an extraordinarily well justified reason to study aspects of the small volume, large- gauge theories.
In fact, center symmetry is spontaneously broken in most examples. In the original Eguchi-Kawai model , which is a one-point reduction of Wilson’s bosonic lattice gauge theory, the breakdown can be shown by one-loop calculation around a diagonal background . The quenched [13, 14] and twisted  modifications of the Eguchi-Kawai model were proposed to preserve the symmetry, but after two decades, it is now understood that both modifications fail nontrivially due to nonperturbative effects , [17, 18, 19].222The failure of these modifications can be cured by introducing supersymmetry [19, 20]. Reference  proposed a concrete way to construct 4d super Yang-Mills theory by using the Eguchi-Kawai equivalence, which preserves 16 supersymmetries. Reference  studied the Eguchi-Kawai reduction in the strong coupling domain of super Yang-Mils by using AdS/CFT and D-branes. 333Recently, a new limiting procedure for the twisted Eguchi-Kawai model, which aims to prevent center breaking, has been proposed . Similarly, in any gauge theory compactified thermally on , or on a torus with at least one thermal boundary condition, center symmetry breaks spontaneously in the high temperature deconfined phase, and volume independence is only valid in the low temperature confined phase, above a critical volume .
Kovtun, one of us (M.Ü.), and Yaffe, motivated by the quantitative differences between thermal and circle (nonthermal) compactifications, showed that if light or massless adjoint fermions endowed with periodic boundary conditions are added to Yang-Mills theory, then the center symmetry is stabilized at small volume dynamically . (Also see the discussion in Refs.[24, 25, 26].)
For heavy fermions, the infrared physics of QCD(Adj) on emulates the bosonic Yang-Mills theory. Since heavy fermions are also capable of restoring center symmetry at sufficiently small volume, this may provide an opportunity for a working Eguchi-Kawai reduction for an “almost” Yang-Mills theory. However, this is not straightforward. When one dimension is compactified, QCD(Adj) with massive fermions on small exhibits an intricate phase structure. This can be deduced from a one-loop effective action of the Wilson line [25, 27], simulations on an asymmetric torus mimicking , and studies on  which also mimics due to topological reasons, as explained in . In all these cases, the symmetry along is intact at large radius, and as one decreases the radius, it breaks down completely at some critical point and then gradually restores to various subgroups of the center symmetry. symmetry is restored fully only at , where is the compactification radius and is the fermion mass in continuum. This is troubling because the volume independence (strong coupling, non-Abelian confinement) domain is , whereas, for heavy fermions, , the first condition implies which is a volume dependent, weak-coupling Abelian confinement domain .
One might expect a similar pattern for a single-site lattice model based on this weak-coupling intuition. However, recent important work of Bringoltz and Sharpe shows that remains intact in a rather generous “funnel,” in the fermion mass, lattice coupling plane, covering the continuum limit of Yang-Mills theory  (also, see ), corresponding to the limit where the bare fermion mass is larger than cutoff scale. In particular, they observe a restoration at , where is bare lattice mass and is lattice spacing. Why and how the full center-symmetry restoration takes place at is the main theoretical problem that we wish to address in this work.
To set the notation, we first express the action of continuum QCD(Adj) on a four manifold:
where are Dirac fermions with mass . (Generalization to different values of masses is straightforward.) In general, we will consider this theory on four-torus , with sizes and , respectively. If we impose the antiperiodic boundary condition on fermions along the temporal direction, this action describes the finite-temperature system and corresponds to the inverse temperature. For periodic boundary conditions on fermions in all directions, we consider , a symmetric four-torus.
We show that, with periodic boundary conditions, symmetry is not broken at sufficiently small-, although it can be broken at some intermediate volume. (More precisely, symmetric and broken phases can coexist, while quantum tunneling between them is suppressed in the large- limit.) The argument, which will be quantified in Sec.3.1, is very simple – although the one-loop effective action suggests the existence of the attraction between Wilson line eigenvalues at small separation, the eigenvalues spread due to nonperturbative quantum fluctuations , and the one-loop calculation can be trusted only at large separations where it leads to repulsion. The estimates of nonperturbative fluctuations are outside the reach of one-loop perturbation theory and often overwhelm the implications of one-loop analysis. Because of nonperturbative effects, we find that the full center restoration takes place at , which is compatible with for .
Our main results are
Small-large volume equivalence: The QCD(Adj) with heavy fermions of mass on , or with radii , is equivalent to the theory on a small , with radii , due to unbroken center symmetry associated with the cycles on . This is a small-large volume equivalence with an intermediate center-symmetry broken phase, where volume independence is not valid. This is depicted in the following figure.
Finite-temperature equivalence: The theory on at finite temperature is equivalent to the one on provided the center symmetry associated with the cycles on is not spontaneously broken. The phase transition in the thermodynamic limit can be studied by using a large- reduced model on small by dialing . This form of equivalence is also useful for Hamiltonian formulation and extraction of the nonperturbative spectrum of the theory.
Twisted QCD: In compactified QCD(Adj), finite- corrections turn out to be order , as opposed to perturbative expectation on , which is order . There are two plausibly related explanations for this behavior, whose footprints can be seen in finite-volume perturbation theory. In a weak-coupling center-symmetric background, the volume is only enhanced by a factor of , and the effective volume is . Finite- corrections should scale as finite-volume corrections. The other is, in compact space, one cannot gauge away zero-momentum modes. Typically, there are order bosonic and fermionic zero modes, which may generate nonperturbative effects . Both problems can simultaneously be cured and corrections can systematically be improved by using twisted boundary conditions.444The relation between “effective” volume and is discussed in many places, for a review, see . The observations that twisted boundary conditions can be used i) to systematically reduce finite-volume corrections is given in Ref. and ii) to lift bosonic and fermionic zero modes is given in Ref.. We give analytic and numerical evidence suggesting that the two are indeed related. We refer the latter as TQCD(Adj) as per .
Our results have interesting spin-offs for noncommutative theories, phase transition in pure Yang-Mills theory and orientifold equivalence. Adding massive or massless adjoint fermions to the twisted Eguchi-Kawai model (TEK) model cures the global instability of the model [17, 19, 18]. Therefore, our formulation can be used to provide a nonperturbative definition of noncommutative bosonic Yang-Mills theory and noncommutative QCD(Adj). By using the reduced matrix model for TQCD(Adj) with very heavy adjoint fermions on lattices, we observe the confinement/deconfinement transition at bare coupling . Large volume simulations for pure Yang-Mills theory give similar results, in accordance with the finite-temperature version of equivalence (For example, YM theory simulated on lattices gives, in the extrapolated infinite volume limit,  (see also ). The small difference is due to effects, the existence of heavy fermions in our reduced model and the difference of numbers of sites along the temporal direction). We hope that the unitary matrix model can be used to gain insight into the nature of the deconfinement transition of the infinite volume theory.
Finally, combined with the large- orientifold equivalence between (T)QCD(Adj) and QCD(AS), thermal properties of the QCD(AS) can also be studied by using the large- reduced model for TQCD(Adj) at finite temperature [1, 2, 30].555Earlier work on deconfinement transition in large- QCD(Adj) and QCD(AS) in the weak-coupling limit of small showed that the critical temperatures agree , up to corrections. Our results for massive QCD(Adj) can be regarded as a first step towards this direction of research.
2 Review of original Eguchi-Kawai proposal
In this section, we give a brief review of the Eguchi-Kawai equivalence. Our aim is not to repeat the original works on the subject; rather we wish to point out two important ingredients which will be useful. These are perturbative and nonperturbative quantum fluctuations in the matrix model.
The EK model is the dimensional reduction of Wilson’s lattice gauge theory action down to a lattice, i.e., a one-site matrix model. The action of the reduced model is
where are unitary matrices. is the bare inverse ’t Hooft coupling constant. To probe continuum physics, should be chosen appropriately depending on the lattice spacing . This action has a global center symmetry
The reduced model (2) is equivalent to the translationally invariant subsector of the lattice theory with an arbitrary number of sites, provided center and translational symmetries are not broken. These are the necessary and sufficient symmetry realization conditions for the large- volume independence.
2.1 Perturbative fluctuations and one-loop potential
The realization of center symmetry can be determined by integrating out weakly coupled perturbative modes in the background of diagonal, commuting Wilson lines:
The resulting one-loop action reads (we parallel the discussion in the one-site matrix model and continuum theory on small )
where . Note that one-site matrix model and have the same symmetry properties, both invariant under .666The relation between the continuum one-loop effective action on and the one-loop effective action for one-site theory given in Eq. (5) is analogous of the one between the action of the XY model and its Villain form. It is also convenient to rewrite (5) by using the Wilson lines. This can be done by a Fourier transformation for the former and by a Poisson resummation for the latter:
and is the modified Bessel function of the second kind; . In the one-site model, Fourier coefficients are slightly more complicated, but the main result is the same as in continuum . The basic point is the negative definiteness of the integral in (7). Moreover, at a large-winding number, the integral can be evaluated analytically by localization, and converges to the continuum result quickly:
The one-loop action (5) has IR singularities whenever two (or more) eigenvalues coincide. This is also manifest in (6); both series are conditionally convergent at a large-winding number, and exhibit logarithmic divergence for coinciding eigenvalues. These issues are discussed throughly in the Appendix A. Physically, at these points, there are extra massless degrees of freedom which should not have been integrated out. In other words, the zeroth order assumption that one can expand the fluctuations around commuting saddles (4) is not always correct. As discussed in the Appendix A, the IR divergence can be regularized and meaningful results can be extracted from (5). The result is, of course, well-known; (5) generates an eigenvalue attraction, or in (6) the “masses” for Wilson lines are all negative and, consequently, eigenvalues clump. However, neither implies that all the eigenvalues are coincident. Because of nonperturbative effects, the eigenvalues spread. In order to estimate the size of the eigenvalue bunch, we may study the theory around one of its global minima, .
2.2 Nonperturbative fluctuations and size of eigenvalue bunch
Parametrizing , we observe that around the minimum of the one-loop potential, all the fluctuations are quartic (as opposed to being quadratic). The action expanded around one of the minima, the configuration gives
is the zero-dimensional ’t Hooft coupling. In the lattice, is, of course, a parameter that one can choose at will. However, to probe continuum physics, it needs to scale as by asymptotic freedom.
Since the Hermitian matrix model is given in terms of noncompact matrices, it is not a priori guaranteed that the theory actually exists quantum mechanically. Reference  shows that the partition function of the theory defined through (9) does not exist for SU(2), but exists for SU(N) for . In the next section, we will indeed see closely related Hermitian matrix models which do not exist for any . Despite this, the Hermitian matrix model defined through (9) is useful. It can be employed to determine the interaction between eigenvalues and to estimate the root-mean-square fluctuations of matrices. (We also refer to this as the size of the eigenvalue bunch.)
Scalar eigenvalues in 0d theory are related to the phases of the Wilson lines winding on temporal and spatial directions by
The ’t Hooft coupling has the dimension of and its value sets the typical mass scale of the 0d theory (9). In particular, typical fluctuation of the eigenvalues of the dynamical fields is set by this scale. Because of the generic noncommutativity of matrices , the relative positions of the eigenvalues make sense only when their separation is of order or larger than .
The one-loop effective action of the matrix model around the diagonal background can be calculated if all eigenvalue pairs are well-separated , corresponding to the case where all eigenvalues are weakly coupled. Integrating out massive “W bosons” (off-diagonal elements) by using the background field gauge yields
Clearly, this is nothing but Eq.(5) restricted to its Kaluza-Klein (KK) zero mode. Thus, as in the one-loop potential (5) of the full theory, the Hermitian model as well predicts eigenvalue attraction and also exhibits the same IR singularity whenever two eigenvalues coincide.
More importantly, the Hermitian matrix model (9) allows the determination of the typical size of eigenvalue fluctuations. In ’t Hooft’s large- limit, the root-mean-square fluctuations are
This result has various interesting implications.
The analysis of continuum and weak-coupling () domain of lattice gauge theory exhibits similar behavior. Let . Then, . At asymptotically small , the size of the eigenvalue bunch is much smaller than the size of the dual torus where eigenvalues are residing. In this domain, clearly, nonperturbative fluctuations cannot overwhelm broken center symmetry.
It is well-known that Eguchi-Kawai reduction holds in the strong coupling domain of lattice gauge theory, for . This is a lattice domain unrelated to continuum physics, whereas the above analysis is valid for sufficiently large . What happens to if we decrease to being of order few? Of course, this is unjustified, but demonstrates the trend of the root-mean-square fluctuations. Naive use of the shows that when is order one, the quantum fluctuations may be as large as the size of the dual torus. In this domain, eigenvalues feel the size of the compact space they live in and we expect the strong fluctuations to lead to center restoration. This is indeed the case.
In this paper, we will study Yang-Mills theory with massive adjoint fermions with mass . When , the discussion is the same as the original Eguchi-Kawai reduction. For , the perturbative-loop analysis indicates that only a finite subgroup of center symmetry would restore; however, such analysis does not take into account nonperturbative fluctuations. We propose that these fluctuations tend to uniformize the eigenvalue distribution rather quickly, even when , providing the resolution of the problem stated in the introduction.
3 Eguchi-Kawai equivalence for massive QCD(Adj)
In this section we explain why it is natural to expect that the Eguchi-Kawai reduction holds for the QCD(Adj) with massive adjoint fermions at sufficiently small volume. Our analysis is based on the one-loop potential for diagonal components of fields and estimates of nonperturbative quantum fluctuations. We start with the zero temperature case and then generalize to the finite temperature.
3.1 QCD(Adj) at zero temperature on at small
Let us consider the QCD(Adj) at zero temperature on a symmetric four-torus, with size . At sufficiently small , gauge coupling at the scale of the compactification is small and we may analytically compute the one-loop effective action on . The reason for doing this computation, apart from trying to determine the center-symmetry realization, is twofold. One is we would like to compare with the one-site theory, given in (LABEL:1loop-l). More importantly, we give evidence that some (not all) implications of one-loop action are, in full theory, overwhelmed by large nonperturbative quantum fluctuations, and therefore, incorrect.
The one-loop effective action, in the Wilson line background (4), induced by gauge and fermionic fluctuations with periodic boundary conditions can be written as
where , and is the fermion mass. Similar to original EK model (5), this expression has IR singularities whenever two (or more) eigenvalues coincide, which we discuss throughly in the Appendix A. The theory may have different saddles which are expressed in terms of noncommuting matrices. The classification of the saddles, using the techniques of Ref., of QCD(Adj) as a function of mass of the fermion is given in the Appendix A.
When we consider a regime where , we can split (14) into the zero and non-zero-momentum contribution. The eigenvalue dynamics is dominated by the interactions between nearby eigenvalues and the effect of high KK modes is negligible. Therefore, we will use the truncated Hermitian matrix model (15) to gain an understanding of the typical eigenvalue fluctuations. Strictly speaking, the truncation can be justified only when the center symmetry is completely broken and consequently there exists a clear separation of scales between the KK modes and zero modes. In a center-symmetric background, this is not the case. However, at large-, the states obtained by quantizing the theory on a center-symmetric vacuum fill the energy range uniformly. If we consider a finite but small range , there are still states (in perturbation theory) in this interval. We may therefore use the Hermitian model to probe the interaction of nearby eigenvalues, and their fluctuations. The eigenvalue dynamics of the full theory is mimicked rather accurately by the truncated Hermitian matrix model.
The truncation of the KK modes in (1) yields the zero-dimensional Hermitian matrix model 777It should be understood that the Hermitian model (15) is used as auxiliary. In the strict limit, and for massless fermions, the partition function of the Hermitian model diverges. One-loop potential is unbounded from below as eigenvalue separations tend to infinity. At finite-, the target space of eigenvalues is bounded, cannot run-off to infinity and the partition function of the finite- theory exists.
where is the zero-dimensional ’t Hooft coupling, and is the strong scale of QCD(Adj).
The and are the two scales in the Hermitian matrix model (15). In particular, for the theory, which corresponds to the original EK model discussed in Sec.2, the root-mean-square fluctuations of matrices are set by this scale . With standard ’t Hooft scaling, this is in the large- limit. This is a crucial observation that will be important below. Because of the generic noncommutativity of matrices , the relative positions of the eigenvalues make sense only when their separation is of order or larger than .
The one-loop effective action of the matrix model around the diagonal background can be calculated if all eigenvalue pairs are well-separated , corresponding to the case where all eigenvalues are weakly coupled. Integrating out massive “W bosons” (off-diagonal elements) by using the background field gauge yields
Note that this is nothing but Eq.(14) restricted to its KK zero mode, as expected.
The effect of adjoint fermions, which is most manifest in (16), is to generate repulsion among eigenvalues. At , since the fermions do not contribute, they have no impact on . For finite fermions’ mass, the effect of fermions makes larger than that of the case. Although we will not prove this statement, it is easy to understand it on physical grounds. When fermion mass is zero, then the one-loop potential is unbounded from below only at large eigenvalue separation; hence . From now on, we assume . Somewhat conservatively, we use the smaller of the fluctuations in what follows.
Let us now discuss the realization of symmetry based on the effective action (16) for the Hermitian matrix model.
The or limits are same and are discussed in Sec.2. We repeat the main result for convenience. The one-loop action leads to the mutual attraction of eigenvalues at large eigenvalue separation and hence, eigenvalues must clump. This implies broken center symmetry. However at small eigenvalue separation, one-loop approximation is not valid; the background of the commuting Wilson line saddles breaks down. Because of nonperturbative quantum fluctuations, the eigenvalues do not collapse to a point; rather the eigenvalue clump has a finite extent of the order . Note that, although the size of the eigenvalue clump is suppressed with respect to at small and hence center symmetry is broken, it scales as in the large- limit. This will be crucial later.
3.1.2 (single Majorana)
When and , the theory is 4d pure super Yang-Mills theory. In this case, the one-loop effective action (16) vanishes and in fact, this is true to all loop orders due to supersymmetry. Taking nonperturbative fractional instanton effects into account, the center is unbroken on as discussed in . In supersymmetric theories with supersymmetry preserving boundary conditions, it is expected that there are no phase transitions as the volume is varied . At large-, the absence of phase transitions transmutes to exact volume independence  and unbroken center symmetry. However, it is also possible to construct a metastable center-broken sector , which becomes stable at large-.
When is fixed and nonzero, by taking , (16) is positive, and therefore, symmetry breaks down. If is fixed and , a center-symmetry preserving background exists for a finite range of . This aspect is discussed throughly in Ref.. This noncommutativity of limits requires care in drawing conclusions about this case.
3.1.3 : Uniformization of eigenvalue distribution
The effective action (16) predicts attraction at short distance , and repulsion at long distance . Then one may naively conclude that all eigenvalues clump and symmetry is broken. However, one should notice that this effective action is valid only at . When attractive force emerges at and hence is broken (left of Fig.1). When , however, it predicts only repulsion; eigenvalue fluctuation is of order and hence they cannot clump to small region where (16) predicts attraction (right of Fig.1).
At sufficiently small, but compactification radii , we can always guarantee that nonperturbative quantum fluctuations overwhelm fermion mass, i.e., . In this case, fermion mass is negligible and for the purpose of center-symmetry realization, the theory cannot be distinguished from the massless theory, for which center symmetry is unbroken. At such values of , since the target space of eigenvalues is compact four-torus with size , the repulsion implies that the eigenvalues will uniformly distribute over .
This is the sense in which lower dimensional nonperturbative quantum fluctuations help restoration of center symmetry at , as opposed to quantum field theory on where full center restoration requires .
As we will see in the Appendix A, the broken phase which consists of bunches of eigenvalues may exist when . However, the radius of each bunch is of order , which is smaller than the eigenvalue fluctuation in the symmetric phase , and hence even if the broken phase exists, tunneling to such a state is highly suppressed.
3.2 QCD(Adj) at finite temperature on asymmetric
We can generalize the argument of the previous subsection to the finite-temperature QCD(Adj) on an asymmetric four-torus . Circumferences of temporal and three spatial circles are taken to be and . On fermions, we impose antiperiodic and periodic boundary conditions along temporal and spatial circles, respectively.
Completely analogous to the discussion of Sec.3.1, at small , the eigenvalue dynamics of the full theory is mimicked rather accurately by the truncated Hermitian matrix quantum mechanics with action:888We repeat the same cautionary note as in Section 3.1. The classical theory written in Eq.(17) does not exist as a quantum theory. In the strict limit, and for massless fermions (or fermions with a finite mass), the theory does not have a ground state. In this paper, we study finite-, limit of (1) which is a well-defined quantum theory. The Hermitian model (17) will be useful as auxiliary to infer some general lessons about the finite- case.
The ’t Hooft coupling has the dimension of and its value sets the typical mass scale of the theory. In particular, typical fluctuation of the eigenvalues of the dynamical fields is given by this scale. The mapping between the scalar eigenvalues and the phases of the Wilson loops is the same as in (11), but now only running over spatial directions. Since the direction is compact, unlike its decompactification limit , the Wilson line along that direction cannot be gauged away. We parametrize , or .
The one-loop effective action of the matrix quantum mechanics around the static diagonal background in the background field gauge can be computed in the weak-coupling domain,
where and represent diagonal components of the gauge field and three scalars , respectively. By subtracting a constant factor, (LABEL:MQM1) can be rewritten as  (for reference, we quote both periodic and antiperiodic boundary conditions along circle)
From the expression of , one might naively conclude that eigenvalues and coincide, because the one-loop action is negative infinity at that point. However, the analysis is valid only when condition (19) is satisfied; thus drawing reliable conclusions requires care.
The limit of (21) is both intuitive and insightful. Since it is the limit of arbitrarily low temperatures, the fermionic boundary conditions should not matter, and indeed, this is transparent from (21); the hyperbolic cosine increases unboundedly, and the trigonometric cosine is bounded, hence the one-loop expression is dominated by the former.999Also note that at limit, can be gauged away. Hence, it should not appear in one-loop potential. The ground-state energy of the system at one-loop order (or one-loop potential) can be deduced from the limit and the result is
This is an intuitive and simple result101010 We could have guessed this result by physical reasoning as follows: In the weak-coupling domain where is small and one-loop analysis is reliable, the Lagrangian (17) reduces to a collection of bosonic and fermionic harmonic oscillators: (25) The eigenvalue differences of background matrices are identified with the frequencies of harmonic oscillators, and . In the chiral limit, . In gauge quantum mechanics, there are many massive bosonic fluctuations and many massive fermionic fluctuations. Hence, the background dependence of the ground-state energy of the system is (26) which is given in (24). and various remarks are in order regarding the chiral limit on .
For , is unbounded from below and eigenvalues mutually repel each other. The classical minima corresponding to the space of commuting triples are unstable against perturbative quantum fluctuations. This means, at the limit, the Hermitian matrix quantum mechanics (17) does not have a ground state.
At finite , since the target space of eigenvalues is compact three-torus with size , the repulsion implies that eigenvalues will uniformly distribute over . This implies unbroken center symmetry in the limit and the theory obeys volume independence.
For , is bounded from below. The minimum is at . However, in this domain, one-loop analysis is not reliable, and there are quantum fluctuations of order . At , , the ground-state energy is zero to all orders in perturbation theory due to supersymmetry.
In the following, we study in more detail the phase structure of the four-dimensional theory by using (21) and estimates of the nonperturbative quantum fluctuations. We first explain (bosonic) and (or one Majorana fermion) cases. We then study the case with , our main interest in this paper.
The one-loop action (21) generates an attraction between eigenvalues at long distance and the center symmetry is broken. More precisely, there are saddles related to each other by center conjugations. The tunneling between these saddles is suppressed in the large- limit and the theory is in a center-broken phase. However the eigenvalues do not collapse to a single point due to the nonperturbative quantum fluctuations; the clump has a finite size of order which is in the large- limit.
3.2.2 (single Majorana)
The supersymmetric theory ( and ) has been studied extensively, because its maximally supersymmetric cousin has a dual description as the D0-brane system. In this case the one-loop potential falls off exponentially,
and the ground-state energy is . This follows from the cancellation of the attractions and repulsions between eigenvalues due to supersymmetry. Consequently, there exists a space of flat directions. Therefore, once eigenvalues are well separated, they will propagate freely like the gas of D0-branes to all order in perturbation theory.
In order to study nonperturbative aspects, it is more useful in this case to recall that the theory is formulated on , and to discuss the system in Hamiltonian formulation. As asserted above, to all orders in perturbation theory, the theory has a moduli space, and all possible realizations of center symmetry are possible. However, non-perturbatively, this is not the case. Quantization of the zero-momentum bosonic modes gives rise to a discrete spectrum and a gap.
Witten studied the gauge quantum mechanics for finite- within Born-Oppenheimer approximation and showed that the bosonic ground-state wave function (ignoring fermionic zero modes which are not crucial in what follows) is constant
and the excited states have an energy gap . The center symmetry and volume independence was not discussed in Ref.; however, its results have natural implications which apply to our discussion. In particular, we can introduce an eigenvalue distribution function measuring the density of eigenvalues. The density is everywhere non-negative and obeys . Since the ground-state wave function spreads uniformly over the perturbative flat directions,
and the center is unbroken. Unlike the discussion in Sec.3.2.1, it is also unbroken at large-. Nonperturbatively, we have a unique saddle singlet under center conjugations, and this is the main difference with respect to purely bosonic theory which has saddles.
When the fermion is massive, there is a subtlety due to order of limits of small mass vs. small- analogous to the discussion in Sec. 3.1.2, with similar conclusions.
First consider the case with . From the one-loop effective action (21), it is apparent that repulsive force coming from fermions dominates at long distance. In the limit where is shrunk to zero size, the resulting Hermitian quantum matrix model is not well defined; the one-loop potential is unbounded from below for large eigenvalue separations. For finite , the eigenvalues can no longer run off to infinity; instead they spread over the dual uniformly. We expect the center-symmetric phase to continue upon compactification, down to so long as . In this domain, the center symmetry is intact, and in the large- limit, volume independence must hold.
In the high temperature limit, , fermions decouple due to thermal mass, eigenvalues clump, and a metastable bound state of eigenvalues exists. Let us assume , where and . Then the second term in the right hand side of (21) is negligible, due to the large thermal mass of fermions, and the effective action becomes that of the zero-dimensional bosonic matrix model,
where we used the identification . This potential produces attraction between eigenvalues. This model is studied extensively, and taking into account nonperturbative effects, a bound state of eigenvalues exists and it satisfies conditions . In this domain, center symmetry is completely broken.
Generalization to nonzero is straightforward and follows from the discussion of Sec.3.1.3 on . At sufficiently small volume such that , the effect of mass is small, and hence attraction in one-loop action is overwhelmed by nonperturbative quantum fluctuations. In this domain, the center symmetry is intact.
4 Lattice model and Monte-Carlo simulation
In Sec.3.1, we explained on continuum and why the Eguchi-Kawai reduction holds for the QCD(Adj) at zero and finite temperature, based on perturbative-loop analysis, supplemented crucially with the estimates of nonperturbative quantum fluctuations. Below, we study the unitary matrix model and one-dimensional lattice model by using Monte Carlo simulation.
It is hard to implement the thermal model on a computer since the temporal direction is not reduced. Moreover, in order to describe phenomena typical to the finite-temperature system, we need to take the effective spatial volume111111 The space size in its ordinary sense is fixed. By developing perturbation theory around a center symmetric background, say for simplicity with only one-dimension compact, we observe that both the lowest states and the spacing between the states is suppressed by a factor of and is given by as opposed to the usual (center-broken) KK-spectrum where the level spacing is . In other words, to all orders in perturbation theory, the effective space size is enhanced into . (See the discussion in  for QCD(Adj) and related discussions in a review  for the quenched Eguchi-Kawai model (QEK) and the twisted Eguchi-Kawai model (TEK).) This is the perturbative essence of volume independence, and in the sense of neutral sector observables, decompactification limit can be reached at while keeping fixed. sufficiently large compared to . As for a lattice model with an isotropic lattice spacing, roughly speaking, we need to take the spatial lattice size to be twice larger than the temporal one , . Since the is related to the matrix size for the large- reduced models in the sense described in footnote 11, we need to take large enough to satisfy this condition. When we use the original Eguchi-Kawai reduction for the spatial directions, finite- correction behaves as rather than , as we will show numerically. This fact indicates that the effective lattice size scales as , and it is not practical for numerical simulations. Therefore, to gain more effective spatial volume, we impose a twisted boundary condition on the spatial directions. For the latter, the effective volume is more enhanced, and volume independent domain can be reached more quickly. We call the QCD(Adj) with the twisted boundary condition as TQCD(Adj). The introduction of TQCD(Adj), which is algorithmically more convenient, is our main improvement over the one-site model of Ref..
To explain the efficiency of the TQCD(Adj), we start with the zero temperature case, that is, the single-site model with periodic boundary condition along the temporal direction on fermions. Then we apply the twist to the finite-temperature case, that is, the one-dimensional lattice model with an antiperiodic boundary condition on fermions along the temporal direction, and then see the Eguchi-Kawai reduction holds in this case. Especially we show this reduced model at finite-temperature describes the confinement/deconfinement transition.
4.1 Single-site theories
4.1.1 QCD(Adj) on lattice
The single-site matrix model can be obtained from a four-dimensional lattice gauge theory by reducing the number of lattice sites to one in all directions . The action is
where corresponds to the link variable in the four-dimensional theory. The inverse ’t Hooft coupling constant should be chosen appropriately depending on the lattice spacing . The fermionic part is obtained as dimensional reduction of the Wilson-Dirac fermion term
The hopping parameter can be expressed as
where is the bare mass.
This action has a center symmetry
If this symmetry is not broken, then the model is equivalent to the translationally invariant subsector of lattice theory with an arbitrary number of sites, including an infinite lattice limit.
Although detailed analytic evaluation of one-loop effective potential depends on the choice of lattice fermions,121212Here, we use Dirac-Wilson fermion with Wilson parameter ,  uses overlap fermions, and also discusses naive fermions. intuitively, the absence of the center-symmetry breaking phase follows closely the discussion on continuum . The discussion on can easily be generalized to the lattice. The role of the compactification scale is replaced by the lattice spacing in the one-site model. The main lesson that we learn is that the center symmetry on the lattice model is in fact much more robust than the center on continuum .
The one-loop action for the one-site theory in the classical background of commuting Wilson lines where is given by
The first term is induced by gauge fluctuations and leads to eigenvalue attraction . Geometrically,
is the separation between two eigenvalues of the Wilson line in the direction. is the spectrum of massive gauge fluctuations ( bosons), familiar from the usual D-brane pictures as the spectrum of open strings ending on branes, where eigenvalues of Wilson line are identified with branes. The second term, proportional to , is induced by fermionic fluctuations, and is equal to where is the spectrum of fermions.
Eq.(LABEL:1loop-l) may be rewritten in a form mimicking the continuum expression on given in (14):
where we have subtracted a holonomy-independent constant term. This expression can be Poisson resummed and be written in terms of Wilson lines as
as discussed in the Appendix A. The only difference with respect to the continuum expression on is that
where is an enhancement factor of a one-site model over continuum , for both bosonic and fermionic contribution. In general, due to peculiarities of the dispersion relation of the one-site model, the center stability is further enhanced on the one-site model with respect to continuum . Asymptotically, for , as expected on physical grounds, by just inspecting the dispersion relations.
The fermionic contribution to is numerically sizeable and has interesting implications. For example, on , start with . Following the discussion of the Appendix A, the singly-winding Wilson lines do get stabilized (in perturbation theory) at , whereas for the one-site model, the same phenomena take place at . This is due to the fact that in the domain of heavy fermion bare mass, the last term in (37), which may roughly be viewed as an “effective mass” , is suppressed with respect to the bare mass, . For theory, the fermions’ contribution dominates, leading to eigenvalue repulsion, and unbroken center symmetry for the theory defined on a lattice, or EK reduction of QCD(Adj) to a single-site lattice.
If one-loop perturbation theory was the whole description, this would be the transition to a restored phase. However, as explained above, many phases can coexist and it is beyond perturbation to determine which phase is chosen in the end. Interestingly, the point where the center symmetry (partially) restores in simulation is very close to the prediction by perturbation theory . This is approximately the value of where Ref. observed the full center restoration, not just . In Fig. 4, we can also see the restoration around the same value. Although we could not see partial breaking clearly by measuring the observable employed in Ref., the discrepancy between in large- start and the one in small- start may indicate the existence of a partial breaking phase.
4.1.2 TQCD(Adj) at zero temperature
In compactified QCD(Adj), as we will see explicitly from simulations, the finite- corrections turn out to be order , as opposed to the perturbative expectation on . As explained in the results section of Sec.1.1, there are two plausibly related explanations for this behavior. One is related to the discussion of effective volume in the reduced model. In the reduced model, serves the role of an emergent spacetime volume, at least in a perturbative description in finite volume around a center-symmetric configuration. Finite- corrections should scale as finite-volume corrections. However, what is not always clear is the factor via which volume enhancement takes place , 131313Of course, for QEK-like configuration, and for TEK-like configurations, , see, for example, . However, these deductions are in perturbation theory around particular backgrounds, and the determination of is likely non-perturbative. and may in fact be determined nonperturbatively.
In compact space, one cannot gauge away zero-momentum modes, and these modes are crucial in studying perturbation theory in finite volume. In perturbation theory, the spectrum of the theory relies on the background for the Wilson lines. If theory has massless adjoint fermions, there will also be fermionic zero modes in the spectrum. Typically, there are order light or massless bosonic and order fermionic zero modes, which may generate nonperturbative effects .
Both problems can simultaneously be solved and corrections can systematically be improved by using boundary conditions which cannot be obeyed by either bosonic or fermionic (if there are any) zero modes. This can be done by using the twisted boundary conditions of ’t Hooft . This idea is, of course, not new, and is used by Witten in Ref. to lift the zero modes in super Yang-Mills theory in the context of supersymmetric theories on , and by Gonzalez-Arroyo and Okawa  in the context of large- reduced models.
Our main observation can be summarized by using the following pedagogical exercise. (The generalization to the theories that we use in simulations is straightforward. The prescription given below works equally well on lattice and continuum.) Let denote either a unitary gauge field or an adjoint fermion field. We impose the following generalized boundary conditions on fields