Phases of fuzzy eld theories

Transcription

1 Phases of fuzzy eld theories Department of Theoretical Physics Faculty of Mathematics, Physics and Informatics Comenius University, Bratislava Seminar aus Mathematischer Physik, University of Vienna

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6 Short outline

7 Short outline what are the noncommutative/fuzzy spaces properties of scalar eld theories on these spaces basics of matrix models matrix model corresponding to fuzzy scalar elds results for the triple point and phase diagram current research, challenges and outlook

8 Noncommutative spaces

9 NC spaces The usual sphere S 2 is given by coordinates x i x i = R 2, x i x j x j x i = 0 which generate the algebra of functions { ( ) f = a k1k 2k 3 x k i xi i x i = R 2} which is commutative by denition. Using this algebra, we can reconstruct all the information about the sphere. It turns out that any commutative algebra corresponds to some dierentiable manifold in a similar way.

10 NC spaces Noncommutative spaces are (by denition) spaces, which are in correspondence with noncommutative algebras [x i, x j ] = iθ ij The choice of Θ determines what the kind of space we have. x i are operators on a Hilbert space. If this space is nite dimensional, we call the space "fuzzy". It is a nite mode approximation to compact manifold.

11 NC spaces For the fuzzy sphere S 2 F x i x i = ρ 2, x i x j x j x i = iθε ijk x k This can be realized as a N = 2j + 1 dimensional representation of the SU(2) x i = 2r N 2 1 L 2r i, θ = N 2 1, ρ 2 = 4r2 N 2 j(j + 1) = r2 1 The coordinates x i still carry an action of SU(2) and thus the space still has the symmetry of the sphere. x i are N N matrices and functions on SF 2 are combinations of products a herminitian matrix M

12 NC spaces Hermitian N N matrix can be decomposed into M = N 1 m=l l=0 m= l c lm T lm matrices T l m are called polarization tensors and [L i, [L i, T lm ]] = l(l + 1)T lm Tr (T lm T l m ) = δ ll δ mm

13 NC spaces Hermitian N N matrix can be decomposed into M = N 1 m=l l=0 m= l c lm T lm there is an analogy with spherical harmonics on S 2 f(φ, θ) = m=l l=0 m= l a l my l m(φ, θ) algebra of functions on S 2 F is the algebra of functions on S2 with a basis cut o, such that functions with l > L a L = N 1 The multiplication rule has changed form a commutative pointwise multiplication to a noncommutative matrix multiplication. Limit of large N reproduces the original sphere S 2.

14 NC spaces By introducing a cut o on l we introduce a maximal momentum N, which in turn means a smallest possible distance 1/N θ. Number of extrema of Y l m is 2l 2, so one extremum corresponds to area 1/l 2 and function Y l m can give us information about distances 1/l. Points on S 2 are encoded in the algebra of functions as sequences of gaussians converging to the corresponding δ-function. But after the cuto we can not construct an arbitrarily narrow gaussian and we can no longer construct and arbitrarily precise approximation of a point. The points on S 2 F no longer exist.

15 Fuzzy eld theory

16 Scalar eld on S 2 F Scalar eld is a function on SF 2, i.e. an N N hermitian matrix. Derivatives become commutators with generators L i, integrals become traces and we can write and Euclidean eld theory action S = 1 2 Tr ([L i, M][L i, M]) rtr ( M 2) + V (M) = = 1 2 Tr (M[L i, [L i, M]]) rtr ( M 2) + V (M) The theory is given by functional correlation functions dm F [M]e S[M] F = dm e S[M] These can be computed using a noncommutative version of the Feynmann rules with a propagator c lm c l m = δ ll δ mm l(l + 1) + r Fuzzy scalar eld theory is a matrix model!

17 Scalar eld on CP n F Taking L i to be generators of SU(n + 1), we can obtain x i 's as coordinates on fuzzy version of CP n, with a corresponding eld theory action S = 1 2 Tr (M[L i, [L i, M]]) rtr ( M 2) + V (M) where i = 1,..., (n + 1) 2 1, polarization tensors [L i, [L i, T lm ]] = l(l + n)t lm Tr (T lm T l m ) = δ ll δ mm a more complicated dependence of N on l, but a theory which is still a particular matrix model.

18 UV/IR mixing The trademark property of the noncommutative eld theories is the UV/IR mixing, which arises as a consequence of the nonlocality of the theory. Quanta can not be compressed into arbitrarily small volume and if we try to do so in one direction, they will stretch in a dierent direction. Processes at high momenta contribute to processes at low momenta. In theories on noncompact spaces, e.g. Moyal plane or R 4, divergence of non-planar diagrams at small momentum Due to no clear separation of scales the theory is not renormalizable. The commutative limit of such noncommutative theory is (very) dierent than the commutative theory we started with.

19 UV/IR mixing To obtain a renormalizable theory with a well dened commutative limit, we have to modify the original action such that we remove the UV/IR divergence Highly nontrivial, several approaches B.P. Dolan, D. O'Connor and P. Pre²najder Matrix φ 4 models on the fuzzy sphere and their continuum limits [arxiv:/ ] H. Grosse and R. Wulkenhaar Renormalization of φ 4 theory on noncommutative R 4 in the matrix base [arxiv: ] R. Gurau, J. Magnen, V. Rivasseau and A. Tanasa A translation-invariant renormalizable non-commutative scalar model [arxiv: ]

20 UV/IR mixing In our setting the UV/IR mixing will arise as an extra phase in the phase diagram, not present in the commutative case. The commutative eld can oscillate around zero, in which case Tr (M) = 0 or around some nonzero value Tr (M), which is a minimum of the potential. Noncommutative theories will exhibit a third phase, where the eld does not oscilate around any xed value. This is a consequence of the non locality of the theory.

21 Random Matrix Models

22 Some basic results of random matrix theory The random variable is N N hermitian matrix M, expressed as M = N 1 l l=1 m= l c l mt l m We will expect SU(N) symmetric probability distribution [ 1 2 Z e N S[M], S[M] = 1 1 N 2 rtr ( ] M 2) N + g n Tr (M n ) The mean value of matrix function f(m) is f = 1 dm e N 2 S[M] f(m) Z where dm = i j drem ij dimm ij What is the distribution of the eigenvalues of the random matrix M? i<j n=3

23 Some basic results of random matrix theory The key is diagonalization, we write M = UΛU 1 for some U SU(N) and Λ = diag(λ 1,..., λ N ), the integration measure becomes ( ) N dm = du dλ i (λ i λ j ) 2 i=1 i<j The problem is now N in an eective potential (eigenvalue repulsion) [ S eff = 1 1 N 2 r λ 2 i + g ] λ 4 i 2 N 2 log(λ i λ j ) i i i<j

24 Some basic results of random matrix theory The key is diagonalization, we write M = UΛU 1 for some U SU(N) and Λ = diag(λ 1,..., λ N ), the integration measure becomes ( ) N dm = du dλ i (λ i λ j ) 2 i=1 i<j The problem is now N in an eective potential (eigenvalue repulsion) [ S eff = 1 1 N 2 r λ 2 i + g ] λ 4 i 2 N 2 log(λ i λ j ) i i i<j The eigenvalue distribution ρ(λ) = 1 N N δ(λ λ i ) i=1 will become continuous in the large N limit

25 Some basic results of random matrix theory In this limit g(x i ) = N dx g(λ)ρ(λ) So the eective action [ S eff = 1 1 N 2 r λ 2 i + g i i becomes S eff = i λ 4 i [ ] 1 dλ 2 rλ2 + gλ 4 ρ(λ) 2 ] 2 N 2 log(λ i λ j ) i<j dλ dλ log(λ λ )ρ(λ)ρ(λ )

26 Some basic results of random matrix theory In this limit g(x i ) = N dx g(λ)ρ(λ) So the eective action becomes [ ] 1 S eff = dλ 2 rλ2 + gλ 4 ρ(λ) 2 i dλ dλ log(λ λ )ρ(λ)ρ(λ ) Thanks to the factor N 2 the large N average f will be given by the conguration of eigenvalues which minimizes this action f = f(x i ) = dλ f(λ)ρ(λ) i Euler-Lagrange equation for ρ is rλ + 4gλ 3 = 2P dλ ρ(λ ) λ λ which can be solved by complex analysis, with an additional assumption on the support of the distribution

27 Some basic results of random matrix theory Assuming the support to be only a single interval ( a, a), we get the distribution (normalized to 1, not N) ρ(λ) = 1 ( ) 1 a2 π 2 r + ga2 + 2gλ 2 λ 2 a 2 = 1 ( r2 + 48g r) 6g

28 Some basic results of random matrix theory Assuming the support to be only a single interval ( a, a), we get the distribution (normalized to 1, not N) ρ(λ) = 1 ( ) 1 a2 π 2 r + ga2 + 2gλ 2 λ 2 a 2 = 1 ( r2 + 48g r) 6g for g = 0 the Wigner semicircle

29 Some basic results of random matrix theory Assuming the support to be only a single interval ( a, a), we get the distribution (normalized to 1, not N) for nonzero g ρ(λ) = 1 ( ) 1 a2 π 2 r + ga2 + 2gλ 2 λ 2 a 2 = 1 ( r2 + 48g r) 6g

30 Some basic results of random matrix theory Assuming the support to be only a single interval ( a, a), we get the distribution (normalized to 1, not N) ρ(λ) = 1 ( ) 1 a2 π 2 r + ga2 + 2gλ 2 λ 2 a 2 = 1 ( r2 + 48g r) 6g for negative r the potential has two wells

31 Some basic results of random matrix theory Assuming the support to be only a single interval ( a, a), we get the distribution (normalized to 1, not N) ρ(λ) = 1 ( ) 1 a2 π 2 r + ga2 + 2gλ 2 λ 2 a 2 = 1 ( r2 + 48g r) 6g for negative r less dense around origin

32 Some basic results of random matrix theory If we allow negative r, the dip in the potential can be so deep that it splits the eigenvalues completely.

33 Some basic results of random matrix theory If we allow negative r, the dip in the potential can be so deep that it splits the eigenvalues completely.

34 Some basic results of random matrix theory If we allow negative r, the dip in the potential can be so deep that it splits the eigenvalues completely. At that point, the formula ρ(λ) = 1 ( ) 1 a2 π 2 r + ga2 + 2gλ 2 λ 2 becomes negative and looses the interpretation of a probability distribution This happens, if r 4 g We need to change the one-cut assumption and look for a distribution supported on two disjoint intervals ( a, b) (b, a) with the solution ρ(λ) = 2g λ (a + λ)(a λ)(λ + b)(λ b) π a 2 = r 4g + 1, b 2 = r g 4g 1 g We thus obtain the phase diagram.

35 Some basic results of random matrix theory

36 Asymmetric eigenvalue distributions What if we start with probability distribution, which is not symmetric, eg. S[M] = 1 [ 1 N 2 rtr ( M 2) + htr ( M 3) + gtr ( M 4) ] we still get the one-cut and the two-cut distributions, but qualitatively dierent third possibility - an asymmetric one-cut distribution. This happens if one of the wells of the potential is deep enough.

37 Fuzzy eld theory matrix model

38 As we have seen, action of the fuzzy CP n is given by S = 1 [ 1 N 2 Tr (M[L i, [L i, M]]) + 1 ( 2 rtr M 2) + gtr ( M 4)] = S kin [M] + 1 [ 1 ( N 2 rtr M 2) + gtr ( M 4)] and the correlation functions are computed as F = dm F [M]e N 2 S[M] dm e N 2 S[M] = dλf (λ)ρ(λ) the same kind of integrals as for a matrix model before

39 A clear problem is that the diagonalization trick no longer works dm e N 2 S[M] = dλ 2 e NTr( 1 2 rm 2 +gm 4 ) du e N 2 S kin [U ΛU] As long as the function F is invariant, we can write du e N 2 S kin [U ΛU] = e N 2 S eff [Λ] and the model becomes a quartic model with new terms [ S[Λ] = S eff [Λ] N 2 r λ 2 i + g ] λ 4 i 2 N 2 log(λ i λ j ) i i i<j the new terms change the EOM for the eigenvalue distribution

40 Numerical results dm integral can be treated numerically and computed by Monte-Carlo. Most recently by B. Ydri [arxiv: ] The following picture from F. Garcia Flores, X. Martin and D. O'Connor [arxiv: ]

41 Numerical results

42 Numerical results

43 Numerical results

44 Perturbative calculation One possible analytic treatment due to D. O'Connor and Ch. Sämann [arxiv: ,arxiv: ] Expand the exponential in powers of kinetic term e atr(m[li,[li,m]]) = 1 atr (M[L i, [L i, M]])+ 1 2 a2 [Tr (M[L i, [L i, M]])] and perform the du integral explicitly. For the fuzzy sphere, the result is (up to third order in a) where S kin = a 1 2 ( c2 c 2 1) a 2 1 ( c2 c ) + a 3 1 ( c3 3c 1 c 2 + 2c ) c n = Tr (M n ) Note, that we obtain a multi-trace model, since the eective action contains powers of traces of M. We will see how odd multi-traces generate asymmetric solutions. The terms become very nasty and dicult to handle beyond the third order.

45 Bootstrap Most recently, Ch. Sämann [arxiv: ] introduced a method using a version of Schwinger-Dyson equations to compute the eective action up to fourth order in a, with and extra term (in the symmetric c 1 = 0 case) a m4 2 a 4 1 ( m4 2m ) From here on, we rescale the matrix M and couplings r, g with N in such a way, that all the terms in the eective action contribute in the large N limit, i.e. are of order N 2.

46 Dierent Bootstrap Finaly, A. Polychronakos [arxiv: ] used the following method to compute a part of the eective action non-perturbatively. It starts from previous result, that the free theory g = 0 eigenvalue distribution is still a semicircle, with a rescaled radius. The eective action can thus be split into two parts S eff = 1 2 F (m 2) + R[m n t n ] where m n are the symmetrized moments of the distribution m n = Tr (M 1N ) n Tr (M) and t n are the (symmetrized) moments of the semicircle distribution and odd vanish. t 2n = C n m n 2

47 Computation of F (t) We will concentrate on F part, i.e. we work with the action ( ) [ S[Λ] = 1 2 F λ 2 i N 2 r λ 2 i + g ] λ 4 i 2 N 2 log(λ i λ j ) i i i i<j = 1 2 F (c 2) + 1 [ ] 1 N 2 rc 2 + gc 4 2 N 2 log(λ i λ j ) (1) we have assumed a symmetric distribution c 1 = 0, keeping this in mind later. The corresponding EOM for the minimizing eigenvalue distribution is ( ) 1 2 F dc2 (c 2 ) + 1 ( ) ( ) dλ i 2 r dc2 dc4 + g = [F (c 2 ) + r] λ i + 4gλ 3 i dλ i dλ i = 2 1 λ j i λ j i<j

48 Computation of F (t) [F (c 2 ) + r] λ i + 4gλ 3 i = 2 j 1 λ i λ j We see, that the multi-trace term containing c 2 acts as an eective mass, which depends on the second moment of the distribution. In the same way, terms containing c 1 and c 3 will introduce terms into the eective potential that break the symmetry. One can use this equation to derive the equation for the radius of the g = 0 semicircle. Then, we need to make sure that the resulting distribution yields a correct second moment. We get two equations that determine the F (t) function completely.

49 Computation of F (t) This approach is applicable for a more general theory, where S kin [M] = 1 Tr (MKM) 2 for which the kinetic action is diagonal, i.e. KT lm = K(l)T lm

50 Computation of F (t) The nal result is where F (t) + t = N 2 f(z) = l f(t) dim(l) K(l) + z For the fuzzy sphere, we can compute F (t) in closed form ( ) t F (t) = log 1 e t For a general CP n, f(z) is given in terms of hypergeometric function and the equation is transcendental. We are left only with a perturbative calculation F (t) = A 1 t + A 2 t 2 + A 3 t

51 Computation of F (t) We can compare the coecients A n with the perturbative result S kin = a 1 2 ( c2 c 2 1) a 2 1 ( c2 c ) + a 4 1 ( c4 2c ) without too much surprise they do agree. Also for CP 2 and CP 3, which involve also a a 3 term.

52 Phase transition We can now compute the phase transition line. The two conditions we have are This gives vanishing of the distribution F (t) + r = 4 g consistency on the second moment t = 1 g For the fuzzy sphere this becomes r(g) = 4 g F (1/ g) r(g) = 5 g 1 1 e 1/ g For higher CP n only a perturbative expansion in powers of 1/ g.

53 Phase transition The series from r(g) = 4 g F (1/ g) is asymptotic for small g. We could use Borel summation to resum the series, but not enough terms are known for practical purposes. To obtain numerical results, we use Pade approximation by a rational function. But this teaches us a lesson. The phase transition lines and other expressions we are going to encounter are not analytic around g = 0 and we need to be very careful when interpreting the results of perturbative computations. For example for the above formula r(g) = 4 g g 1 720g 3/ g 5/2 +...

54 Asymmetric distributions To introduce asymmetry, we recall that we should have F (t) F (c 2 c 2 1) yielding an equation for the distribution F (c 2 c 2 1)(λ i c 1 ) + rλ i + 4gλ 3 i = 2 j 1 λ i λ j We can think of this as an eective potential ] V (λ) = [F (c 2 c 2 1)c 1 λ + 1 [ ] F (c 2 c 2 2 1) + r λ 2 + gλ 4 i.e. the advertised asymmetry. Note that for symmetric distributions the asymmetric term vanishes, so the one-cut/two-cut phase transition is not modied. We now look for the solution in the form of an asymmetric one cut, supported on one interval (b, a). We again need to impose self consistency equations on the rst and second moments of the distribution.

55 Asymmetric distributions We write the interval as (D δ, D + δ). The solution then is ρ(λ) = [ 4D 2 g + 2δg + r + F ( δ 4 + δ3 g 1 δ (D λ) 2 2π 16 9D2 δ 4 g 2 16 The conditions on D, δ can be solved only perturbatively. ) + 4Dgλ + 4gλ 2 ] To nd the domain of existence of the asymmetric one cut solution, we seek the values of parameters when this distribution becomes negative.

56 Asymmetric phase transition We can do this perturbatively r = 2 15 g + 4A ( 1 135A A ( 2025A A 1 A A where A i 's from F (t) = A 1 t + A 2 t 2 + A 3 t ) 1 g + ) 1 g +... Note, that we again obtain a series in 1/ g, but this time the series is convergent.

57 Asymmetric phase transition

58 Asymmetric phase transition We compute the intersection of the two curves The numerical result is r C = 0.73, g C = 0.022

59 Asymmetric phase transition We compute the intersection of the two curves The numerical result is r C = 0.73, g C = [arxiv: ], r C = 0.8 ± 0.08, g C = ± 0.013

60 Asymmetric phase transition We compute the intersection of the two curves The numerical result is r C = 0.73, g C = [arxiv: ], r C = 0.8 ± 0.08, g C = ± [arxiv: ], r C = 2.3 ± 0.2, g C = 0.13 ± 0.005

61 Asymmetric phase transition We compute the intersection of the two curves The numerical result is r C = 0.73, g C = [arxiv: ], r C = 0.8 ± 0.08, g C = ± [arxiv: ], r C = 2.3 ± 0.2, g C = 0.13 ± [arxiv: ], r C = 2.49 ± 0.07, g C = ± What is the source of discrepancy?

62 Asymmetric phase transition

63 Some topics of current research

64 Fourth order contribution We can use Sämann's result for the eective action for the computation. S kin =a 1 ( c2 c 2 2 1) a 2 1 ( c2 c ) + a 3 1 ( c3 3c 1 c 2 + 2c 3 ) a (c 2 c 2 1) 4 a ( (c4 4c 1 c 3 + c 2 c 3 1 3c 4 1) 2(c 2 c 1 ) 2) 2 The main distinction is the presence of c 2 4 term, which acts as an eective coupling. The properties of the asymmetric transition line are not altered dramatically. But the symmetric transition line gets shifted quite drastically, moreover loosing Borel summability! Also Pade approximation is quite involved. The overall eect is to shift the triple point to even lower values of r and g, further away from the numerical result.

65 Fourth order contribution The fourth order eective coupling has another interesting feature. It can be expressed as ( g eff = g + 2F 1 c40 2c 2 ) 20 with F 1 negative. So for distributions that give negative second term, one can have g eff > 0 even for g < 0. Such distributions are "spread out" more than semicircle, so for a negative r eff, we might have a region of parameter space where we obtain a stable solution even for negative coupling. Two caveats seems that this is not the case for fuzzy sphere, might be only a perturbative eect and the full eective action might remove this eect.

66 Free energies The sole existence of the asymmetric one-cut solution does not guarantee, that the solution is going to be realized. Even with no kinetic term, asymmetric solution exists as long as r 2 15 g but is never realized since it has higher free energy as the double-cut.

67 Free energies The sole existence of the asymmetric one-cut solution does not guarantee, that the solution is going to be realized. Even with no kinetic term, asymmetric solution exists as long as r 2 15 g but is never realized since it has higher free energy as the double-cut. The result is still interesting, because it gives a top bound on the triple point, but for a complete result we should compute the free energy of the distributions. Or do something else.

68 Free energies The sole existence of the asymmetric one-cut solution does not guarantee, that the solution is going to be realized. Even with no kinetic term, asymmetric solution exists as long as r 2 15 g but is never realized since it has higher free energy as the double-cut. The result is still interesting, because it gives a top bound on the triple point, but for a complete result we should compute the free energy of the distributions. Or do something else.

69 Outlook

70 What is next? Complete treatment of the eective action. What is the non-perturbatve formula for the kinetic term eective action? Can it be computed?

71 What is next? Complete treatment of the eective action. Dierent sources of asymmetry Contributions of nontrivial congurations to the saddle point, instantons.

72 What is next? Complete treatment of the eective action. Dierent sources of asymmetry Limit CP n R 2n. Phase diagram in the non-compact case, phase structure of the theory on non-commutative plane, etc.

73 What is next? Complete treatment of the eective action. Dierent sources of asymmetry Limit CP n R 2n. Phase diagram of the theories with no UV/IR mixing. Does the non-uniform order phase disapear from the diagram? What is the mechanism? What are the lessons to be learned

74 What is next? Complete treatment of the eective action. Dierent sources of asymmetry Limit CP n R 2n. Phase diagram of the theories with no UV/IR mixing. Phase diagram nonenormalizable theories. Is there anything we can learn from the phase structure of theories, which are not renormalizable even at the commutative level, eg. φ 6 on R 4 or φ 4 on R 6.

75 What is next? Complete treatment of the eective action. Dierent sources of asymmetry Limit CP n R 2n. Phase diagram of the theories with no UV/IR mixing. Phase diagram nonenormalizable theories. Theory on R S 2 F. Theory with a commutative time. There are similar perturbative and numerical results available to compare with.

76 What is next? Complete treatment of the eective action. Dierent sources of asymmetry Limit CP n R 2n. Phase diagram of the theories with no UV/IR mixing. Phase diagram nonenormalizable theories. Theory on R S 2 F. Theory on S 4. There is a realization of the theory on fuzzy S 4 that is suitable for our approach.

77 What is next? Complete treatment of the eective action. Dierent sources of asymmetry Limit CP n R 2n. Phase diagram of the theories with no UV/IR mixing. Phase diagram nonenormalizable theories. Theory on R SF 2. Theory on S 4.

78 Thanks for your attention!