Phase structure of two-dimensional QED at zero temperature with flavor-dependent chemical potentials and the role of multidimensional theta functions
MMarch 12, 2018
Phase structure of two-dimensional QED at zero temperature with flavor-dependentchemical potentials and the role of multidimensional theta functions
Robert Lohmayer ∗ and Rajamani Narayanan † Department of Physics, Florida International University, Miami, FL 33199, USA. (Dated: March 12, 2018)We consider QED on a two-dimensional Euclidean torus with f flavors of massless fermions andflavor-dependent chemical potentials. The dependence of the partition function on the chemicalpotentials is reduced to a (2 f − f − f = 3and two, three, four or six phases can coexist for f = 4. We conjecture that the maximal numberof coexisting phases grows exponentially with increasing f . I. INTRODUCTION AND SUMMARY
QED in two dimensions is a useful toy model to gain an understanding of the theory at finite temperature andchemical potential [1–3]. In particular, the physics at zero temperature is interesting since one can study a systemthat can exist in several phases. The theory at zero temperature is governed by two degrees of freedom often referredto as the toron variables in a Hodge decomposition of the U(1) gauge field on a l × β torus where l is the circumferenceof the spatial circle and β is the inverse temperature. Integrating over the toron fields projects on to a state with netzero charge [4] and therefore there is no dependence on a flavor-independent chemical potential [5]. The dependenceon the isospin chemical potential for the two flavor case was studied in [6] and we extend this result to the case of f flavors in this paper. After integrating out the toron variables, the dependence on the ( f −
1) traceless chemicalpotential variables and the dimensionless temperature τ = lβ can be written in the form of a (2 f − f −
2) dimensional theta functionhas a non-trivial Riemann matrix and this is a consequence of the same gauge field (toron variables, in particular) thatcouples to all flavors. The resulting phase structure is quite intricate since it involves minimization of a quasi-periodicfunction over a set of integers. We will explicitly show:1.
Three flavors : The two-dimensional plane defined by the two traceless chemical potentials is filled by hexagonalcells (c.f. Fig. 4 in this paper) with the system having a specific value of the two traceless particle numbersin each cell and neighboring cells being separated by first-order phase transitions at zero temperature. Thevertices of the hexagon are shared by three cells and therefore two or three different phases can coexist at zerotemperature.2.
Four flavors : The three-dimensional space defined by the three traceless chemical potentials is filled by twotypes of cells (c.f. Fig. 8 in this paper). One of them can be viewed as a cube with the edges cut off. We thenstack many of these cells such that they join at the square faces. The remaining space is filled by the secondtype of cell. All edges of either one of the cells are shared by three cells but we have two types of vertices – onetype shared by four cells and another shared by six cells. At zero temperature, each cell can be identified by aunique value for the three different traceless particle numbers and neighboring cells are separated by first-orderphase transitions. Therefore, two, three, four or six phases can coexist at zero temperature.One can use the multidimensional theta function to study the phase structure when f > (cid:0) f (cid:98) f/ (cid:99) (cid:1) , increasing exponentially for large f .The organization of the paper is as follows. We derive the dependence of the partition function on the ( f − τ in section II. We briefly show the connection to the ∗ Electronic address: robert.lohmayer@fiu.edu † Electronic address: rajamani.narayanan@fiu.edu linear combinations that are invariant under uniform (flavor-independent) shifts a r X i v : . [ h e p - t h ] N ov two flavor case discussed in [6] and focus in detail on the three and four flavor cases in section III. We then concludethe paper with a discussion of some examples when f > II. THE PARTITION FUNCTION
Consider f -flavored massless QED on a finite torus with spatial length l and dimensionless temperature τ . Allflavors have the same gauge coupling el where e is dimensionless. Let µ t = (cid:0) µ µ · · · µ f (cid:1) (1)be the flavor-dependent chemical potential vector. The partition function is [3, 6] Z ( µ , τ, e ) = Z b ( τ, e ) Z t ( µ , τ ) , (2)where the bosonic part is given by Z b ( τ, e ) = 1 η f ( iτ ) ∞ (cid:89) (cid:48) k ,k = −∞ (cid:114)(cid:0) k + τ k (cid:1) (cid:16) k + τ (cid:104) k + fe π (cid:105)(cid:17) (3)(with k = k = 0 excluded from the product and η ( iτ ) being the Dedekind eta function) and the toronic part reads Z t ( µ , τ ) = (cid:90) − dh (cid:90) − dh f (cid:89) i =1 g ( h , h , τ, µ i ) ,g ( h , h , τ, µ ) = ∞ (cid:88) n,m = −∞ exp (cid:20) − πτ (cid:20)(cid:16) n + h − i µτ (cid:17) + (cid:16) m + h − i µτ (cid:17) (cid:21) + 2 πih ( n − m ) (cid:21) . (4)We will only consider ourselves with the physics at zero temperature and therefore focus on the toronic part andperform the integration over the toronic variables, h and h . A. Multidimensional theta function
Statement
The toronic part of the partition function has a representation in the form of a (2 f − Z t ( µ , τ ) = 1 √ τ f ∞ (cid:88) n = −∞ exp (cid:20) − πτ (cid:18) n t T t + iτ s t (cid:19) (cid:18) ¯Ω ¯Ω (cid:19) (cid:18) T n + iτ s (cid:19)(cid:21) (5)where n is a (2 f − f − × (2 f −
2) transformation matrix T is T = · · · · · · · · · · · · − − · · · − f ; T − = · · · · · · · · · · · · f f · · · f f . (6)The ( f − × ( f −
1) matrix ¯Ω is¯Ω = − f − f · · · − f − f − f · · · − f ... ... . . . ... − f − f · · · − f ; ¯Ω − = · · ·
11 2 · · · · · · ;¯Ω = R · · · · · · · · · · · · f R t , R ij = √ j ( j +1) i ≤ j < ( f − − j √ j ( j +1) i = j + 1 ≤ ( f − i > j + 1 ≤ ( f − √ f − j = ( f − ∀ i . (7)The dependence on the chemical potentials comes from s t = (cid:0) ¯ µ ¯ µ · · · ¯ µ f − ¯ µ − ¯ µ · · · − ¯ µ f (cid:1) (8)where we have separated the chemical potentials into a flavor-independent component and ( f −
1) traceless componentsusing ¯ µ ¯ µ ...¯ µ f = M µ , M = · · · − · · ·
01 0 − · · · · · · − . (9) Proof:
Consider the sum a t a = f (cid:88) i =1 a i a i . (10)Noting that N = · · · − ( f −
1) 1 · · ·
11 1 − ( f − · · · · · · − ( f − , N M = f, (11)it follows that a t a = 1 f f (cid:88) i =1 b i ¯ a i with ¯ a = M a , b = N a . (12)Explicitly, b = ¯ a and b i = ¯ a − f a i = f ¯ a i − f (cid:88) j =2 ¯ a j , i = 2 , . . . , f , (13)where we have used the relation f a = f (cid:88) i =1 ¯ a i . (14)Therefore, a t a = 1 f ¯ a t N ¯ a = 1 f ¯ a − f f (cid:88) i,j =2 ¯ a i ¯ a j + f (cid:88) i =2 ¯ a i . (15)Setting ¯ n = M n , ¯ m = M m , ¯ µ = M µ (16)in (4) and using the relation (14) to rewrite ¯ n and ¯ m , we obtain Z t ( µ , τ ) = ∞ (cid:88) n ,m , { ¯ n i , ¯ m i } = −∞ (cid:90) − dh (cid:90) − dh exp (cid:34) πih (cid:32) f ( n − m ) − f (cid:88) i =2 (¯ n i − ¯ m i ) (cid:33)(cid:35) ×× exp (cid:34) − πτ (cid:32) f (cid:40) f n − f (cid:88) i =2 ¯ n i + f h − i ¯ µ τ (cid:41) + 1 f (cid:40) f m − f (cid:88) i =2 ¯ m i + f h − i ¯ µ τ (cid:41) − f f (cid:88) i,j =2 (cid:110)(cid:16) ¯ n i − i ¯ µ i τ (cid:17) (cid:16) ¯ n j − i ¯ µ j τ (cid:17) + (cid:16) ¯ m i − i ¯ µ i τ (cid:17) (cid:16) ¯ m j − i ¯ µ j τ (cid:17)(cid:111) + f (cid:88) i =2 (cid:26)(cid:16) ¯ n i − i ¯ µ i τ (cid:17) + (cid:16) ¯ m i − i ¯ µ i τ (cid:17) (cid:27)(cid:33)(cid:35) (17)where n , m , ¯ n i and ¯ m i , i = 2 , . . . , f , are the new set of summation variables. The integral over h results in Z t ( µ , τ ) = ∞ (cid:88) (cid:48) n , { ¯ n i , ¯ m i } = −∞ (cid:90) − dh exp (cid:34) − πτ (cid:32) f (cid:40) f n − f (cid:88) i =2 ¯ n i + f h − i ¯ µ τ (cid:41) − f f (cid:88) i,j =2 (cid:110)(cid:16) ¯ n i − i ¯ µ i τ (cid:17) (cid:16) ¯ n j − i ¯ µ j τ (cid:17) + (cid:16) ¯ m i − i ¯ µ i τ (cid:17) (cid:16) ¯ m j − i ¯ µ j τ (cid:17)(cid:111) + f (cid:88) i =2 (cid:26)(cid:16) ¯ n i − i ¯ µ i τ (cid:17) + (cid:16) ¯ m i − i ¯ µ i τ (cid:17) (cid:27)(cid:33)(cid:35) , (18)where the prime denotes that (cid:80) fi =2 (¯ n i − ¯ m i ) be a multiple of f . The integral over h along with the sum over n reduces to a complete Gaussian integral and the result is Z t ( µ , τ ) = 1 √ τ f ∞ (cid:88) (cid:48){ ¯ n i , ¯ m i } = −∞ exp (cid:34) − πτ (cid:32) f (cid:88) i =2 (cid:26)(cid:16) ¯ n i − i ¯ µ i τ (cid:17) + (cid:16) ¯ m i − i ¯ µ i τ (cid:17) (cid:27) − f f (cid:88) i,j =2 (cid:110)(cid:16) ¯ n i − i ¯ µ i τ (cid:17) (cid:16) ¯ n j − i ¯ µ j τ (cid:17) + (cid:16) ¯ m i − i ¯ µ i τ (cid:17) (cid:16) ¯ m j − i ¯ µ j τ (cid:17)(cid:111)(cid:33)(cid:35) . (19)The prime in the sum can be removed if we trade ¯ n f for ¯ k where¯ n f = f (cid:88) i =2 ¯ m i − f − (cid:88) i =2 ¯ n i + ¯ kf. (20)We change ¯ m i → − ¯ m i and define the (2 f − n t = (cid:0) ¯ m ¯ m · · · ¯ m f ¯ n ¯ n · · · ¯ n f − ¯ k (cid:1) . (21)Then statement (5) follows from (19). B. Particle number
We define particle numbers N i corresponding to the chemical potentials µ i as N i ( µ , τ ) = τ π ∂∂µ i ln Z t ( µ , τ ) . (22)Analogously to Eq. (9), we set ¯ N k ( µ , τ ) = N ( µ , τ ) − N k ( µ , τ ) for 2 ≤ k ≤ f . (23)In the infinite- τ limit, the infinite sums in Eq. (5) are dominated by n = which results in¯ N k ( µ , ∞ ) = ¯ µ k for 2 ≤ k ≤ f . (24)Since the partition function is independent of ¯ µ , ¯ N ( µ , τ ) = (cid:80) fi =1 N i ( µ , τ ) = 0 for all τ . C. Zero-temperature limit
In order to study the physics at zero temperature ( τ →
0) we setΩ = T t (cid:18) ¯Ω ¯Ω (cid:19) T ; Γ = 1 τ T − . (25)Then we can rewrite (5) using the Poisson summation formula as Z t ( µ , τ ) = 1 √ τ f τ f − ∞ (cid:88) k = −∞ exp (cid:104) − πτ (cid:0) k t Ω − k − k t T − s (cid:1)(cid:105) (26)with 1Ω = · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · − f , (27)where the block in the upper left corner has dimensions ( f − × ( f −
1) and the second block on the diagonal hasdimensions ( f − × ( f −
2) .For fixed ¯ µ k , the partition function in the zero-temperature limit is determined by minimizing the term k t Ω − k − k t T − s in the exponent in Eq. (26). Assuming in general that the minimum is M -fold degenerate,let S = { k ( i ) } i =1 ,...,M , k ( i ) ∈ Z f − , label these M minima. Then¯ N j ( µ ,
0) = 12 M M (cid:88) i =1 f − (cid:88) l =1 k ( i ) l − f − (cid:88) l = f k ( i ) l + k ( i ) j − − k ( i ) f + j − , ≤ j ≤ f − , (28)¯ N f ( µ ,
0) = 12 M M (cid:88) i =1 f − (cid:88) l =1 k ( i ) l − f − (cid:88) l = f k ( i ) l + k ( i ) f − . (29)If the minimum is non-degenerate (or if all k ( i ) individually result in the same ¯ N j ( µ , N j ( µ ,
0) assume integer or half-integer values at zero temperature. Since k ∈ Z f − and we only have ( f −
1) ¯ N j ( µ , N ( µ , τ ) = 0 for all τ ), there are in general many possibilities to obtain identical particle numbers from different k ’s. The zero-temperature phase boundaries in the ( f − µ ,...,f are determined by those ¯ µ ’s leading to degenerate minima with different ¯ N ’s. As we will see later, phases withdifferent particle numbers will be separated by first-order phase transitions.One can numerically determine the phase boundaries as follows: Having chosen one set for the traceless chemicalpotentials, one finds the traceless particle numbers at zero temperature (by numerically searching for the minimum) atseveral points in the traceless chemical potential space close to the initial one. We label the initial choice of chemicalpotentials by the number of different values one obtains for the traceless particle numbers in its small neighborhoodand this enables us to trace the phase boundaries. Whereas this method works in general, it is possible to performcertain orthogonal changes of variables in the space of traceless chemical potentials and obtain expressions equivalentto (26) that are easier to deal with when tracing the phase boundaries. Such equivalent expressions for the case of f = 3 and f = 4 are provided in Appendix A.Consider the system at high temperature with a certain choice of traceless chemical potentials which results inaverage values for the traceless particle numbers equal to the choice as per (24). The system will show typical thermalfluctuations as one cools the system but the thermal fluctuations will only die down and produce a uniform distributionof traceless particle numbers if the initial choice of traceless chemical potentials did not lie at a point in the phaseboundary. Tuning the traceless chemical potentials to lie at a point in the phase boundary will result in a system atzero temperature with several co-existing phases. In other words, the system will exhibit spatial inhomogeneities. Wewill demonstrate this for f = 2 , , D. Quasi-periodicity
Consider the change of variables s (cid:48) = s + T Ω − m (30)with m ∈ Z f − . Since s is of the special form (8), there is a restriction on m . From Eq. (27), we find that m hasto satisfy m f − k = m f − − m k ≤ k ≤ f − ,m f − = − f m f − ∈ Z . (31)This corresponds to ¯ µ (cid:48) k +1 = ¯ µ k +1 + m k − f m f − + f − (cid:88) i =1 m i ≤ k ≤ f − , (32)and Z t ( µ (cid:48) , τ ) = Z t ( µ , τ ) e πτ ( m t Ω − m +2 m t T − s ) . (33)The particle numbers under this shift are related by¯ N k +1 ( µ (cid:48) , τ ) = ¯ N k +1 ( µ , τ ) + m k − f m f − + f − (cid:88) i =1 m i , (34)which is the same as the shift in ¯ µ as defined in (32). III. RESULTSA. Phase structure for f = 2 We reproduce the results in [6] in this subsection. The condition on integer shifts in (31) reduces to m = − m and the shift in chemical potential is given by ¯ µ (cid:48) = ¯ µ + m . From Eq. (26) for f = 2, we obtain¯ N = (cid:80) ∞ k = −∞ ke − πτ ( k − ¯ µ ) (cid:80) ∞ k = −∞ e − πτ ( k − ¯ µ ) , (35)and this is plotted in Fig. 1. The quasi-periodicity under ¯ µ (cid:48) = ¯ µ + m is evident. For small τ , the dominating termin the infinite sum is obtained when k assumes the integer value closest to ¯ µ . Therefore, ¯ N (¯ µ ) approaches a stepfunction in the zero-temperature limit (see Fig. 1). Taking into account the first sub-leading term, we obtain (fornon-integer ¯ µ ) ¯ N = (cid:98) ¯ µ (cid:99) + 12 (cid:20) (cid:18) πτ (cid:20) ¯ µ − (cid:98) ¯ µ (cid:99) − (cid:21)(cid:19)(cid:21) + . . . (36)At zero temperature, first-order phase transitions occur at all half-integer values of ¯ µ , separating phases which arecharacterized by different (integer) values of ¯ N .If a system at high temperature is described in the path-integral formalism by fluctuations (as a function of the twoEuclidean spacetime coordinates) of ¯ N around a half-integer value, the corresponding system at zero temperaturewill have two coexisting phases (fluctuations are amplified when τ is decreased). On the other hand, away from thephase boundaries, the system will become uniform at zero temperature (fluctuations are damped when τ is decreased).Fig. 2 shows spatial inhomogeneities develop in a system with ¯ µ chosen at the phase boundary as it is cooled andFig. 3 shows thermal fluctuations dying down in a system with ¯ µ chosen away from the phase boundary. The squaregrid with many cells can either be thought of as an Euclidean spacetime grid or a sampling of several identical systems(in terms of the choice of ¯ µ and τ ). Μ N FIG. 1: For f = 2, plot of ¯ N as a function of ¯ µ for τ = 1 . τ = 0 . τ = 0 . τ = 0 .
025 (black, solid). N (cid:72) Τ(cid:61)(cid:165) (cid:76) N (cid:72) Τ(cid:61) (cid:76) FIG. 2: For f = 2, a spacetime grid with small fluctuations around ¯ N = 1 / τ (left panel) results in two coexistingphases (characterized by ¯ N = 0 and ¯ N = 1) at zero temperature (right panel). B. Phase structure for f = 3 We determine the phase boundaries, separating cells with different ( ¯ N , ¯ N ) as described in Sec. II C. As explainedin Sec. II C it is also instructive to use a different coordinate system for the chemical potentials, obtained from( µ , µ , µ ) by an orthonormal transformation: ˜ µ ˜ µ ˜ µ = √ √ √ √ − √ √ √ − √ µ µ µ , (37)i.e., ˜ µ = ¯ µ / √ µ = ( − ¯ µ + 2¯ µ ) / √
6. We denote the corresponding particle numbers by ˜ N and ˜ N . Analternative representation of the partition function, which simplifies the determination of vertices in terms of thecoordinates ˜ µ i , is given in appendix A. In these coordinates, the phase structure is symmetric under rotations by π/ N (cid:72) Τ(cid:61)(cid:165) (cid:76) N (cid:72) Τ(cid:61) (cid:76) FIG. 3: For f = 2, a spacetime grid with random fluctuations around ¯ N = 2 / τ (left panel) results in a uniformparticle number ( ¯ N = 0) at τ = 0 (right panel). hexagons, which are identical up to rotations. Figure 4 shows the phase boundaries at zero temperature in bothcoordinate systems.The condition on the integers m as given in (31) reduce to m = m − m and m = − m . Therefore, we require m to be even and write it as 2 l . From Eq. (32) we see that the boundaries in the (¯ µ , ¯ µ ) plane are periodic undershifts (cid:18) ¯ µ (cid:48) ¯ µ (cid:48) (cid:19) = (cid:18) ¯ µ ¯ µ (cid:19) + m (cid:18) (cid:19) − l (cid:18) − (cid:19) m , l ∈ Z . (38)The shift symmetry (38) is obvious in Fig. 4. (cid:45) (cid:45) Μ (cid:45) (cid:45) Μ (cid:45) (cid:45) Μ(cid:142) (cid:45) (cid:45) Μ(cid:142) FIG. 4: Phase boundaries at zero temperature for f = 3 in the ¯ µ plane (left) and the ˜ µ plane (right). All ¯ µ ’s inside a given hexagonal cell result in identical ¯ N as τ →
0, given by the coordinates of the center of the cell.For example, ¯ µ ’s in the central hexagonal cell lead to ¯ N , = (0 ,
0) at τ = 0, the six surrounding cells are characterizedby ¯ N , = ± (1 , ), ¯ N , = ± ( , N , = ± ( − , ). Every vertex is common to three cells. The coordinatesof the vertices between the central cell and the six surrounding cells are ± ( , ), ± (0 , ), ± ( , ± (1 , ± (0 , ± (1 , µ plane can be generated by shifts of the form (38).First-order phase transitions occur between neighboring cells with different particle numbers ¯ N , at τ = 0. At theedges of the hexagonal cells, two phases can coexist, and at the vertices, three phases can coexist at zero temperature. N (cid:72) Τ(cid:61) (cid:76) N (cid:72) Τ(cid:61) (cid:76) N (cid:72) Τ(cid:61) (cid:76) FIG. 5: Left panel shows, for f = 3, the result of cooling a spacetime grid with random fluctuations around ¯ N ≡ ( ¯ N , ¯ N ) =( , ) at large τ to τ = 0, where three phases coexist: ¯ N = (0 ,
0) (red squares), ¯ N = ( ,
1) (dark-blue squares), and ¯ N = (1 , )(light-green squares). Center panel shows result starting from ¯ N = ( , ) at high τ , which results in two coexisting phases( ¯ N = ( ,
1) and ¯ N = (1 , )) at τ = 0. Right panel shows results starting from ¯ N = ( , ) at high τ , resulting in a single phase(characterized by ¯ N = (0 , τ = 0. In analogy to the two-flavor case (cf. Fig. 2), a high-temperature system with small fluctuations (as a function ofEuclidean spacetime) of ¯ µ , can result in two or three phases coexisting or result in a pure state as τ → µ , (see Fig. 5 for examples of all three cases). Figure 6 shows the flow of ( ˜ N ( τ ) , ˜ N ( τ )) from τ = ∞ to τ = 0 at fixed (˜ µ , ˜ µ ) = ( ˜ N ( τ = ∞ ) , ˜ N ( τ = ∞ )). The zero-temperature limit ( ˜ N (0) , ˜ N (0)) is given by thecoordinates of the center of the respective hexagonal cell. C. f=4
We use Eq. (26) to identify the phase structure in the (¯ µ , ¯ µ , ¯ µ ) space, which is divided into three-dimensionalcells characterized by identical particle numbers ¯ N , , at zero temperature. At the boundaries of these cells, multiplephases can coexist at zero temperature (see Fig. 7 for examples). We find different types of vertices (corners of thecells), where four and six phases can coexist. At all edges, three phases can coexist.We set l = m + m − m , l = m and l = m . From Eq. (32) for f = 4, we see that the phase structure isperiodic under ¯ µ ¯ µ ¯ µ → ¯ µ ¯ µ ¯ µ + l + l + l l , , ∈ Z . (39)As in the three flavor case, we observe that the phase structure exhibits higher symmetry in coordinates ˜ µ whichare related to µ through an orthonormal transformation. A particularly convenient choice for f = 4 turns out to begiven by ˜ µ ˜ µ ˜ µ ˜ µ = 12 (cid:18) − (cid:19) ⊗ (cid:18) − (cid:19) µ µ µ µ , (40)since the phase structure becomes periodic under shifts parallel to the coordinate axes: ˜ µ ˜ µ ˜ µ → ˜ µ ˜ µ ˜ µ + l + l + l l , , ∈ Z (41)as obtained from Eq. (39). An alternative representation of the partition function in these coordinates is given inEq. (A8). At zero temperature the ˜ µ , , space is divided into two types of cells which are characterized by identical0 (cid:45) (cid:45) N (cid:142) (cid:45) (cid:45) N (cid:142) FIG. 6: Visualization of the ˜ N evolution with decreasing τ starting from randomly scattered initial points at τ = ∞ (indicatedby dots in the plot). particle numbers (see Fig. 8 for visualizations). We can think of the first type as a cube (centered at the origin, withside lengths 1 and parallel to the coordinate axes) where all the edges have been cut off symmetrically. The originalfaces are reduced to smaller squares (perpendicular to the coordinate axes) with corners at ˜ µ , , = ( ± , ± , ± )(permutations and sign choices generate the six faces). This determines the coordinates of the remaining 8 cornersto be located at ( ± , ± , ± ). The shift symmetry (41) tells us that these “cubic” cells are stacked together face toface. The remaining space (around the edges of the original cube) is filled by cells of the second type (in the followingreferred to as “edge” cells), which are identical in shape and are oriented parallel to the three coordinate axes.This leads to different kinds of vertices (at the corners of the cells described above) where multiple phases cancoexist at zero temperature. There are corners which are common points of two cubic and two edge cells (coexistenceof 4 phases, for example at ( ± , ± , ± )), there are corners which are common points of one cubic and three edgecells (coexistence of 4 phases, for example at ( ± , ± , ± )), and there are corners which are common points of sixedge cells (coexistence of six phases, for example at ˜ µ , , = ( ± , ± , ± ). Any edge between two of these vertices iscommon to three cells. D. Phase structure for f > For f = 3 and f = 4, we find that the coordinates (¯ µ , . . . , ¯ µ f ) of all vertices (corners of the cells in the ¯ µ spaceresulting in identical particle numbers at zero temperature) are multiples of f . In general, two special vertices arelocated at ¯ µ i = 1 for all 2 ≤ i ≤ f and ¯ µ i = 1 − f for all 2 ≤ i ≤ f .If ¯ µ i = 1 − f for all 2 ≤ i ≤ f , we find that f phases can coexist at zero temperature. These have particle numbers1 N (cid:72) Τ(cid:61) (cid:76) N (cid:72) Τ(cid:61) (cid:76) N (cid:72) Τ(cid:61) (cid:76) FIG. 7: Left panel shows, for f = 4, the result of cooling a spacetime grid with random fluctuations around ¯ N ≡ ( ¯ N , ¯ N , ¯ N ) =( , , ) at large τ to τ = 0, where four phases coexist: ¯ N = (0 , , N = ( , , N = ( , , ), and ¯ N = (1 , , ). Differentcolors are assigned to different phases. Center panel shows result starting from ¯ N = ( , , ) at high τ , which results inthree coexisting phases ( ¯ N = ( , , N = ( , , ), and ¯ N = (1 , , )) at τ = 0. Right panel shows results startingfrom ¯ N = (1 , ,
1) at high τ , resulting in six coexisting phases characterized by ¯ N = ( , , N = ( , , ), ¯ N = (1 , , ),¯ N = ( , , N = ( , , ), and ¯ N = (1 , , ). ¯ N ,...,f = (0 , . . . ,
0) and all ( f −
1) distinct permutations of (1 , , . . . , ).If ¯ µ i = 1 for all 2 ≤ i ≤ f , we find that (cid:0) f (cid:1) phases can coexist at zero temperature. The corresponding parti-cle numbers are given by the ( f −
1) distinct permutations of (1 , , . . . , ) and the (cid:0) f − (cid:1) distinct permutations of( , , , . . . , f = 5, we find only up to (cid:0) (cid:1) coexisting phases, we find up to (cid:0) (cid:1) coexisting phases for f = 6 (for exampleat ¯ µ ,..., = (1 , , , , (cid:0) (cid:1) coexisting phases for f = 8 (for example at ¯ µ ,..., = (1 , , , , , , (cid:0) f (cid:98) f/ (cid:99) (cid:1) , increasing exponentiallyfor large f . IV. CONCLUSIONS
Multiflavor QED in two dimensions with flavor-dependent chemical potentials exhibits a rich phase structure atzero temperature. We studied massless multiflavor QED on a two-dimensional tours. The system is always in a statewith a net charge of zero in the Euclidean formalism due to the integration over the toron variables. The toronvariables completely dominate the dependence on the chemical potentials and the resulting partition function has arepresentation in the form of a multidimensional theta function. We explicitly worked out the two-dimensional phasestructure for the three flavor case and the three-dimensional phase structure for the four flavor case. The differentphases at zero temperature are characterized by certain values of the particle numbers and separated by first-orderphase transitions. We showed that two or three phases can coexist in the case of three flavors. We also showed thattwo, three, four and six phases can coexist in the case of four flavors. Based on our exhaustive studies of the threeand four flavor case and an exploratory investigation of the five, six, and eight flavor case we conjecture that up to (cid:0) f (cid:98) f/ (cid:99) (cid:1) phases can coexist in a theory with f flavors. Appendix A: Alternative representations of the partition function
There are many equivalent representations of the partition function Z t ( µ , τ ), related by variable changes of theinteger summation variables in (4), (5) or (26). Here we present the result obtained by an orthonormal variable changeat the level of Eq. (4), splitting the chemical potentials µ , . . . , µ f in one flavor-independent and ( f −
1) tracelesscomponents according to ˜ µ = 1 √ f f (cid:88) i =1 µ i , (A1)2 FIG. 8: Cells defining the zero-temperature phase structure for f = 4 in the ˜ µ coordinates as described in the text. The topleft figure shows the central “cubic” cell, the top right figure a single “edge” cell. The bottom right figure shows the cubic celltogether with all 12 attaching edge cells. ˜ µ j = 1 (cid:112) j ( j − (cid:32) j − (cid:88) i =1 µ i − ( j − µ j (cid:33) , ≤ j ≤ f . (A2)The induced variable change in the 2 f integer summation variables in Eq. (4) is non-trivial and requires successivetransformations of the form ∞ (cid:88) k,l = −∞ f ( k − l, M k + l ) = M (cid:88) q =0 ∞ (cid:88) m,n = −∞ f (( M + 1) m + q, ( M + 1) n − q ) , M ∈ N + . (A3)In this way, it is possible to write the partition function as a product of 2 f − f − f − µ i (with 2 ≤ i ≤ f ). However, the arguments of the theta functions are notindependent since they involve a number of finite summation variables resulting from variable changes of the form(A3) and the partition function does not factorize. The final result reads Z t ( µ , τ ) ∝ f (cid:89) j =1 1 (cid:88) k j =0 f (cid:89) j =2 j − (cid:88) q j =0 j − (cid:88) p j =0 δ , ( (cid:80) fj =2 p j + (cid:80) fj =1 k j ) mod 2 f × f (cid:89) j =2 h τj ( j − (cid:32) j ( j − j − (cid:88) i =2 q i − j q j + 1 (cid:112) j ( j − (cid:32) ˜ k j − iτ ˜ µ j (cid:33)(cid:33) × f (cid:89) j =2 h τj ( j − (cid:32) j ( j − j − (cid:88) i =2 p i − j p j + 1 (cid:112) j ( j −
1) ˜ k j (cid:33) , (A4)where ˜ k j = √ j ( j − (cid:16)(cid:80) j − i =1 k i − ( j − k j (cid:17) and h α ( z ) ≡ ∞ (cid:88) n = −∞ e − πα ( n + z ) . (A5)Permuting indices in variable changes of the form (A2) shows that the ( f − (cid:81) j (cid:80) p j willresult in an expression that depends only on (cid:80) fj =1 k j .To study the zero-temperature properties, we can apply the Poisson summation formula for each factor of h α ( z ) inEq. (A4).
1. Explicit form for f = 3 For f = 3, the Poisson-resummed version of (A4) can be simplified to Z t ( µ , τ ) ∝ ∞ (cid:88) m ,m ,l ,l = −∞ δ , ( m + l ) mod 2 δ , ( m + l ) mod 2 e − π τ ( ( m + m ) +3( m − m ) +( l + l ) + ( l − l ) ) × e πτ ( ( m + m ) √ µ +( m − m ) √ µ ) . (A6)For τ →
0, the sums over l , become trivial and we obtain Z t ( µ , τ ) → ∞ (cid:88) m ,m = −∞ e − πτ ( ( m + m ) + ( m − m ) − ( m + m ) √ µ +( m − m ) √ µ + ( − δ ,m δ ,m )) . (A7)The particle numbers ˜ N and ˜ N at zero temperature are determined by those integer pairs ( m , m ) dominatingthe sum in Eq. (A7). Compared to the general expression in Eq. (26), we have reduced the number of summationvariables from four to two, which simplifies the search for vertices where multiple phases coexist. Furthermore, thereis a one-to-one map from ( ˜ N , ˜ N ) to ( m , m ) inside any given cell in the zero-temperature phase-structure. Oncewe have located neighboring cells in terms of ( m , m ), we can immediately read off the ˜ µ , coordinates of thecorresponding vertices/edges between them (by requiring that the contributions to the sum (A7) are identical).
2. Explicit form for f = 4 Following the general procedure described above, we can write the partition function for f = 4 in the coordinatesdefined in Eq. (40) as Z t ( µ , τ ) ∝ ∞ (cid:88) m ,m ,m ,n ,n ,n = −∞ δ , ( m + m + m ) mod 2 δ , ( m + n + n ) mod 2 δ , ( m + n + n ) mod 2 e − π τ (cid:80) j =2 ( m j + n j − m j ˜ µ j ) . (A8)Similarly to the three flavor case, the sum over n , , becomes trivial in the τ → m mod 2and m mod 2. The remaining summation variables m , , directly determine the particle numbers in the differentphases at zero temperature and the vertices can be found analogously to the three flavor case.4 Acknowledgments
The authors acknowledge partial support by the NSF under grant numbers PHY-0854744 and PHY-1205396. RLwould like to acknowledge the theory group at BNL for pointing out that we are computing traceless particle numbersand not traceless number densities. [1] I. Sachs, A. Wipf and A. Dettki, Phys. Lett. B , 545 (1993) [hep-th/9308130].[2] I. Sachs and A. Wipf, Annals Phys. , 380 (1996) [hep-th/9508142].[3] I. Sachs and A. Wipf, Helv. Phys. Acta , 652 (1992) [arXiv:1005.1822 [hep-th]].[4] D. J. Gross, R. D. Pisarski and L. G. Yaffe, Rev. Mod. Phys. , 43 (1981).[5] R. Narayanan, Phys. Rev. D , 087701 (2012) [arXiv:1206.1489 [hep-lat]].[6] R. Narayanan, Phys. Rev. D86