Critical Dimension and Negative Specific Heat in One-dimensional Large-N Reduced Models

We investigate critical phenomena of the Yang-Mills (YM) type one-dimensional matrix model that is a large-N reduction (or dimensional reduction) of the D + 1 dimensional U(N) pure YM theory (bosonic BFSS model). This model shows a large-N phase transition at finite temperature, which is analogous to the confinement/deconfinement transition of the original YM theory. We study the matrix model at a three-loop calculation via the “principle of minimum sensitivity” and find that there is a critical dimension D = 35.5: At D ≤ 35, the transition is of first order, while it is of second order at D ≥ 36. Furthermore, we evaluate several observables in our method, and they nicely reproduce the existing Monte Carlo results. Through the gauge/gravity correspondence, the transition is expected to be related to a Gregory-Laflamme transition in gravity, and we argue that the existence of the critical dimension is qualitatively consistent with it. Besides, in the first order transition case, a stable phase having negative specific heat appears in the microcanonical ensemble, which is similar to Schwarzschild black holes. We study some properties of this phase.


I. INTRODUCTION
Critical phenomena in physics sometimes show interesting dependences on the numbers of the spatial dimensions. One remarkable example is the Gregory-Laflamme (GL) transition in the D + 1 dimensional gravity with a compact S 1 circle [1]. (See a review [2].) By changing the size of the S 1 from small to large, the stable configuration for a given energy changes from a uniform black string (UBS) to a localized black hole (LBH), and this transition is called the GL transition. A non-uniform black string (NUBS) may appear as an intermediate state in this transition. Surprisingly, the order of this phase transition does depend on D, and it is of first order at D ≤ 12, while is of second order at D ≥ 13 [3]. Hence, D = 12.5 can be regarded as a critical dimension of this transition. Curiously, if we fix the temperature instead of the energy, the critical dimension changes to D = 11.5 [4]. See Table  I. A similar critical dimension appears in the Rayleigh-Plateau (RP) instabilities in liquid too. If we consider a space time R D−1,1 × S 1 and set a liquid winding the S 1 with the same configuration as the UBS. Suppose that the volume of the liquid is fixed and the radius of the S 1 is increased. (Thus, the liquid is stretched along the S 1 .) Then, above a critical radius, this configuration becomes unstable due to the RP instability, and it tends to be non-uniform. The order of this transition depends on D similar to the GL transition, and it turned out that the critical dimension is D = 11.5 [5,6]. The connection between the GL and RP instabilities was also argued in [5]. * morita.takeshi(at)shizuoka.ac.jp † yoshida.hiroki. 16(at)shizuoka.ac.jp Critical dimension GL (fixed mass) 12.5 GL (fixed temperature) 11.5 RP 11.5 YM type matrix model (3-loop) 35.5 According to the gauge/gravity correspondence [7,8], the GL transition is expected to be qualitatively related to the confinement/deconfinement (CD) transition in the D+1 dimensional Yang-Mills (YM) type matrix quantum mechanics, whose action at finite temperature is given by [9][10][11][12][13][14][15], This model is a large-N reduction (or dimensional reduction) of the D + 1 dimensional U(N ) pure Yang-Mills (YM) theory to one dimension [16]. Here X I (I = 1, · · · , D) are the N × N Hermitian matrices that are the dimensional reductions of the spatial components of the original D + 1 dimensional gauge fields.
] is the covariant derivative and A t is the gauge field. g is the coupling constant, and we take the 't Hooft limit N → ∞ and g → 0 with a fixed 't Hooft coupling λ := g 2 N . Note that this model appears as low energy effective theories of D-branes and membranes in string theories in various situations, and is important in its own right [9,[17][18][19][20][21][22]. This model shows a large-N phase transition [23,24], which is an analog of the CD transition of the original YM theory [9,20,[25][26][27][28][29][30][31][32][33][34][35]. The order parameter of this transition is the Polyakov loop operators, If u n = 0, (∀n), it indicates a confinement, and, u n = 0, (∃n) shows a deconfinement. The relation between the CD transition and the GL transition can be intuitively understood as follows. The diagonal components of X I can be regarded as the positions of N particles (or D-branes). If we take the static diagonal gauge (A t ) ij = α i δ ij (i, j = 1, · · · , N ), α i also describe the positions of the particles. (Here the configuration space of the gauge field is regarded as a real space.) Particularly, the Polyakov loop (2) is invariant under the shift α i = α i +2π/β, and this space is actually an S 1 with the period 2π/β. At large-N , these particles may behave as a static fluid in the D + 1 dimension [36], and their distribution would be uniform, non-uniform or localized along the S 1 as schematically shown in FIG. 1. Now the connection to the GL transition in the gravity is clear. These configurations would correspond to a UBS, NUBS and LBH, respectively. Note that the temporal component α i of the gauge theory corresponds to the spatial S 1 direction in the gravity [37]. As we have mentioned, the UBS is stable when the size of the S 1 is small. Correspondingly, the uniform distribution in FIG 1 is stable at a small 2π/β, which means a low temperature. We can easily see that u n = 0 in the uniform distribution, and this is consistent with the confinement at low temperatures. (The localized distribution is characterized by u n = 0 for all n and the non-uniform distribution is characterized by u n = 0 for a finite number of n's. Thus, they are both deconfined.) Since the critical dimensions appear in the GL and RP transitions, the existence of a critical dimension in the CD transition of the matrix model is expected. Indeed, several evidences for this conjecture have been found [31]. For small D, Monte Carlo (MC) simulations show that the order of the CD transition up to D = 25 would be of first order [31,34]. On the other hand, at large-D, we can analyze the model analytically through the 1/D expansion, and find the second order CD transition [30]. Hence, a critical dimension would exist in the matrix model too. In this article, we analyze the matrix model by using so called "principle of minimum sensitivity" [38], and we will see that the critical dimension is D = 35.5 at a three-loop calculation.
In addition, in the first order transition case, a phase having negative specific heat arises in the microcanonical ensemble [24] [39]. Since some black holes such as Schwarzschild black holes and small black holes in AdS space-time [40] have negative specific heat too, the phases in the matrix model would be important to understand why these black holes have negative specific heat from the viewpoint of gauge theories [24]. Although, it is hard to explore such phases in MC calculations, we can easily access this phase in our method. We will derive several quantities in this phase near the critical temperature.

II. ANALYSIS VIA THE PRINCIPLE OF MINIMUM SENSITIVITY
To investigate the phase structure of the model (1), we employ the principle of minimum sensitivity [41]. Such a study was first done by Kabat and Lifschytz [25], but we use a different analysis in order to explore the details of the phase transition. We deform the model (1) as Here we have introduced the deformation parameter κ and M . If we take κ = 1, the M dependent terms are canceled, and this model goes back to the original model (1). We integrate out X I through the perturbative calculations with respect to κ, and derive the effective action of the Polyakov loop {u n }. The relevant terms at low temperatures, where all u n are small [20,24], are given by Here f 0 is a function of M while f i (i = 1, 2, 3, 4) are functions of M and T := 1/β [42]. The derivations and explicit expressions of f 0 and f i at three-loop order are shown in (A14) -(A18) in Appendix (B). (If we are interested in the two-loop results [43], we simply remove the terms proportional to κ 2 in these equations.) At this stage, we take κ = 1. Although the initial model (3) at κ = 1 is independent of the deformation parameter M , the obtained effective action does depend on M . Here, we fix M so that the M dependence of the effective action becomes a minimum. This prescription is so called "the principle of minimum sensitivity" [38]. Although the validity of such a prescription is generally not ensured, it works very well for many examples. We will compare our results with the existing studies in order to test our analysis.

A. Low Temperature and Confinement
To explore the phase structure, we start from considering the low temperature regime. At low temperatures, we observe f 1 , f 2 , f 3 > 0 from (A15) ∼ (A17). Then, the stable configuration in the effective action (4) is given by u 1 = u 2 = 0, and it is in the confinement phase. Thus, we can approximate S eff = N 2 βf 0 , and M at low temperatures is fixed so that the M dependence of f 0 is minimized, hence where M 0 denotes the value of M that minimizes |∂ M f 0 |. In the two-loop effective action, f 0 has a single extremum ∂ M f 0 = 0 via (A14), and it gives M 0 as In the three-loop effective action, f 0 does not have any extremum. However, it has an inflection point ∂ 2 M f 0 = 0, which minimizes (5), and we obtain In order to test whether these results are reliable, we evaluate the free energy F := S eff /β = N 2 f 0 (M 0 ) and compare them with the MC results at low temperatures [44]. By using (A14), we obtain F in the confinement phase as These results are shown in Fig. 2 and Table II, and both the two-and three-loop analyses show good agreement [45]. Furthermore, in Appendix D, we also compare our results (8) at large-D with the 1/D expansion [30], which would provide reliable results there, and again find good agreement. Thus, we expect that our analysis via the principle of minimum sensitivity appropriately works in our model (1).   [31].

B. Confinement/Deconfinement Transition
As temperature increases, f 1 (M 0 , T ) becomes negative, and u 1 and u 2 may obtain non-zero vevs, indicating a deconfinement. This is the CD transition in our model. Near the critical temperature, u 1 and u 2 would be small and we can perturbatively treat them in the effective action (4). Correspondingly, M can be expanded as Here, in the two-loop theory, M 0 is given by (6) and M i (i = 1, · · · , 4) are fixed through the condition ∂ M S eff = 0 in (4). In the three-loop theory, M 0 is given by (7) and Then, by substituting (9) into the effective action (4) and using the small {u n } expansion, we obtain Herē where f i are evaluated at M = M 0 . Finally, by integrating out u 2 , we reach a Landau-Ginzburg type effective action for u 1 , Although the explicit formulas for a(T ) and b(T ) are complicated and we omit to show them, it is straightforward to obtain them from (A14) -(A18) by using Mathematica. Now we can easily see the phase structure [20,24,26]. If a > 0, u 1 = 0 is (meta-)stable and the system may be confined. If a < 0, u 1 = 0 is unstable and u 1 has to develop a non-zero vev, and it is deconfinement. Thus, we can derive the critical temperature T c by solving a(T c ) = 0. Numerical solutions of this equation are shown in FIG. 2 and Table III.
In order to determine the order of the transition, we expand a(T ) = −c(T − T c ) + · · · (c := ∂a/∂T > 0) near T = T c , and obtain the classical solution of u 1 in (12) as Therefore, if b(T c ) is positive, it indicates a non-trivial solution in T ≥ T c , which implies a continuous second order phase transition. If b(T c ) is negative, an unstable solution exists in T ≤ T c , and a first order phase transition occurs at a temperature, which is slightly below T c . We define this transition temperature as T 0 . See FIG. 3.

C. Critical Dimension
We plot b(T c ) with respect to D in FIG. 4. At two-loop order, b is always negative and it predicts the first order In the second order phase transition case (the right panel), the stable u1 = 0 solution (13) appears in T ≥ Tc. Therefore, through (13), the signature of b at T = Tc determines the order of the phase transition. Note that there is another transition point TGWW, at which a third order transition between the non-uniform distribution and the localized one in FIG. 1 occurs [ 9,20,26,30]. This transition is so called the Gross-Witten-Wadia (GWW) transition [46,47], and is important in the context of the resolution of the naked singularities in the gravity [26,48].      (12) and (13) through the ordinary thermodynamical relations, Here C is negative because b < 0. We plot C in FIG. 5 (right). See also Table IV. At three-loop, as D approaches to the critical dimension D = 35.5, C diverges, since b → 0.

III. DISCUSSIONS
We have shown that the critical dimension of the matrix model (1) is D = 35.5 at three-loop. The existence of a critical dimension has been predicted through the MC [31] and the 1/D expansion [30], and our result is consistent with them. Besides, the strong similarity between the GL, RP and the CD in the matrix model (1) are sharpened. This similarity may arise because the matrix model may describe a kind of fluid as depicted in FIG. 1. (The obtained critical dimension is different from the gravity, but it would not be a problem because we cannot expect any quantitative agreement in this correspondence [9,12].) However, our analysis relies on the perturbative calculation and the principle of the minimum sensitivity, and D = 35.5 is not conclusive. We need the higher order loop calculations to ensure it. At large-D, these corrections may make our results closer to those of the 1/D expansion [30]. (See Appendix D for the results at large-D in our analysis.) Also, there are several varieties of the principle of the minimum sensitivity [49], and we need to check whether our results depend on these schemes.
Another remaining problem is understanding the properties of the first order phase transition at T 0 in D ≤ 35. Above T 0 , the stable configuration would be a nonuniform distribution or a localized one depending on D [50]. If the stable configuration is a non-uniform distribution, another phase transition to a localized distribution must occur at a higher temperature. Indeed, these transitions have been found in the GL and RP transitions [2,6,51,52]. Besides, they would be important for a deeper understanding of the negative specific heat phase in the microcanonical ensemble.
In order to investigate them, we need to evaluate the effective action at finite {u n }, and thus we cannot use the expansion (9). Besides, we need to calculate higher order couplings of the Polyakov loops such as |u 1 | 6 in the effective action (4). We leave this problem for future work.  In this Appendix, we derive the effective action (4) at three-loop order. Starting from the deformed action (3), we compute the effective action of the Polyakov loop {u n } by integrating out X I through the standard perturbative calculation with respect to κ. It will lead to the expansion, The analysis mainly follows that of the massive BFSS model [20]. In order to compute this expansion, we use the propagator of X I in the static diagonal gauge (A t ) ij = α i δ ij [30], x n u i −n u j n . (A2) Here x := e −βM and ||t|| denotes ||t + nβ|| = t for 0 ≤ t < β. u i n := e iβnαi , which satisfies N i=1 u i n = N u n , where u n is the n-th Polyakov loop defined in (2).
Through the one-loop integral, we obtain At two-loop, we obtain Here (A5) has been computed via the planar diagram depicted in FIG. 6, and · · · denotes the irrelevant terms at low temperatures. On the other hand, (A6) can be generated from the one-loop result (A3). In order to compute the three-loop corrections, we need to evaluate Here the last three terms are from the three diagrams depicted in FIG. 6, and we obtain In addition, we need to compute and The three-loop correction S 3-loop in (A1) is given as the sum of (A7), (A11) and (A12).
By substituting these results to (A1), we can read off the effective action at three-loop order, Here We have used x = e −βM . If we are interested in the two-loop effective action, we should simply ignore O(κ 2 ) terms in this result.

Appendix B: The Details of the Derivation of the Critical Dimension
To show the details of the derivations of the critical dimension in Sec. II C, we analyze the effective action (A13) and discuss how we determine the phase structure. We will mainly show the analysis at two-loop, since the three-loop analysis is almost parallel. (Recall that we remove O(κ 2 ) terms in (A13) when we consider the twoloop effective theory.) We set κ = 1 in (A13) hereafter to use the principle of the minimum sensitivity.
First, we consider a low temperature regime. There, x = e −βM would be small, and f 1 and f 3 would be positive. Then, to make the effective action (A13) small, u 1 = u 2 = 0 would be favored. Thus, the effective action (A13) would become S eff = βN 2 f 0 (M ).
Here, we need to determine M . As we have discussed in (5), we fix M such that the M dependence of the effective action is minimized. From (A14), we find that at ∂ M f 0 at two-loop becomes 0 and is minimized. Then we obtain the free energy at low temperatures as This result is shown in FIG. 2 and Table II. We find good agreement with the MC results even at two-loop order. Next, in order to investigate the phase transition, we compute M i andf i defined in (9) and (11). However, since ∂ M f 0 = 0 at M = M 0 , we obtainf i = f i for i = 1, 3, 4 and we need to evaluate only M 1 andf 2 . By substituting the expansion (9) into the equation ∂ M S eff = 0, we find The explicit formulas for these equations are rather messy, and we omit showing them, but one can obtain them easily by using Mathematica. Now, we are ready to discuss the critical phenomena. As we have argued below (12), the critical temperature can be found through This equation can be solved numerically and the result is summarized in FIG. 2 and Table III. Again our results seem to be consistent with the MC results. Finally, we determine the order of the phase transition. Through the discussions around (13), it is determined by the signature of b defined in (12) at the critical temperature. We numerically see that it is always negative as shown in FIG. 4 and indicates the first order phase transition for any D at two-loop order. (As we will shown in (D9), we can confirm it analytically, if D is large.) So far, we have shown the two-loop results. Now we will move on to the three-loop case. The three-loop calculation is almost parallel to the two-loop analysis. One significant difference is that the minimum of |∂ M f 0 | in (5) is not zero. Hence we need to find the minimum via ∂ 2 M f 0 = 0, and obtain The rest of the calculations are straightforward. We obtain the free energy in the confinement phase as We can also compute specific heat C from the free energy through the ordinary thermodynamical relation, and we obtain where we have used (14) and (11) and (12).) The specific heat is very small in the confinement phase due to the large-N volume independence, which strongly suppresses temperature dependence of physical quantities [16,53]. It becomes positive in the second order phase transition case, since b > 0, while it becomes negative in the first order transition case (b < 0). Phases with negative specific heat are unphysical in usual thermodynamical systems. However, in our case, it becomes physical in the microcanonical ensemble [24]. (See [54][55][56] for some discussions on phases with negative specific heat.) Also, the specific heat (C4) at the critical temperature tends to diverge as  Table IV.

Appendix D: Large-D Limit
At large-D, the 1/D expansion [30] would be reliable. Hence, it would be valuable to evaluate our results at large-D and compare them with the 1/D expansion [30].
In the large-D expansion, we obtain the following quantities: These are from (4.27), (4.33) with (4.25) and (4.30) in [30], respectively. Besides, we evaluate b(T c ) in the effective action (12), which fixes the order of the transition, as where we have used (4.29) and (4.30) in [30]. This is always positive and the 1/D expansion predicts the second order phase transition at large-D. Through (C4), we obtain the specific heat at T = T c as C/N 2 | T =Tc = 3 2 D log D We will compare these quantities with our results at large-D. First, we evaluate our two-loop results at large-D. At two-loop, we can solve (B4) at large-D and obtain the critical temperature analytically. Then, we obtain Thus, b is negative, and it does not agree with the 1/D expansion (D4). On the other hand, the leading order terms of F , R 2 and β c in our results are precisely coincident with those of the 1/D expansion, although the 1/D corrections differ. Since the results of the 1/D expansion [30] would be reliable at large-D, these quantities at two-loop order are accidentally very good at large-D.
Next, we consider the three-loop results. Different from the two-loop case, we cannot solve T c in the threeloop case analytically even at large-D. From (B6) and (C3), we obtain Thus, they do not agree with (D1) and (D2) in the 1/D expansion. However, these are numerically not bad. For F , if we compare the coefficients of the leading terms of (D1) and (D11), we obtain 3/8 = 0.375 and 1412/160(120) 2/3 = 0.363... and the error is 3% only. Similarly, for R 2 , we have 1/2 = 0.5 and 148/60(120) 1/3 = 0.500092..., and they are very close. Hence, we presume that the convergence of the principle of the minimum sensitivity at large-D would be good in our model.