Kento SUGIYAMA (Shizuoka Univ.) “Strings and Fields 2019” in YITP (Aug.19-23) Hermitian Matrix Model with Cusp Potential Kento Sugiyama (Shizuoka Univ.) in collaboration with Takeshi Morita (Ongoing work)
Kento SUGIYAMA (Shizuoka Univ.) “Strings and Fields 2019” in YITP (Aug.19-23) (Non-polynomial) various aspects of gauge theories and string theories. (1/14) Hermitian matrix models (HMMs) with cusp potentials at large-N. Ex.) One-HMMs in 0dim. In the case of ordinary smooth potentials, the general analysis are well known. [Brezin,Itzykson,Parisi,Zuber’78]et.a l But in the case of singular ones, the results have not been analyzed yet. So as a trial, we consider the following singular potential. ＊ Phase transitions of matrix models at large-N are related to (M:N×N Hermitian matrix, V(M):Potential) Gaussian <latexit sha1_base64="GVzJ+JKjr/H9a2Vc1jUT065gi4=">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</latexit> Singular term ＊ Summary ＊ In this study , we investigate 0 and 1 dimensional = x 2 2 − g | x | + g 2 2
Kento SUGIYAMA (Shizuoka Univ.) “Strings and Fields 2019” in YITP (Aug.19-23) (2/14) [Gross,Witten’80],[Wadia’80] et.al [Brezin,Itzykson,Parisi,Zuber’78] et.al this looks similar to a quartic anharmonic potential. Although this cusp potential is singular at x=0, Let us compare a ordinary quartic anharmonic potential and our cusp potential. ＊ Summary Ordinary potential case ： V(x)=mx 2 +g 4 x 4 (g 4 >0:fixed) -m<<0 |m|>>0 Our cusp potential case ： V(x)=x 2 /2-g|x|+g 2 /2 g>>0 g>0 g<0 g=0 -g g -g g
Kento SUGIYAMA (Shizuoka Univ.) “Strings and Fields 2019” in YITP (Aug.19-23) (2/14) [Gross,Witten’80],[Wadia’80] et.al [Brezin,Itzykson,Parisi,Zuber’78] et.al A Similar phase structure is anticipated in our cusp potential case. In general, the phase structures are characterized by the eigenvalue density ρ(x). ＊ Summary Ordinary potential case ： V(x)=mx 2 +g 4 x 4 (g 4 >0:fixed) ρ (x) ： eigenvalue density |m|<|m c | -m<<0 |m|=|m c | |m|>|m c | 3rd order transition two-cut one-cut at m=m c : finite Our cusp potential case ： V(x)=x 2 /2-g|x|+g 2 /2 g>>0 g>0 g<0 g=0 -g g -g g
(2/14) However this expectation is NOT true. In general, the phase structures are characterized by the eigenvalue density ρ(x). g>0 is always in two-cut phase. Difference： [Gross,Witten’80],[Wadia’80] et.al [Brezin,Itzykson,Parisi,Zuber’78] et.al We will show that large N phase transitions in these models are quite different from the 3rd order phase transitions. ＊ Summary Ordinary potential case ： V(x)=mx 2 +g 4 x 4 (g 4 >0:fixed) ρ (x) ： eigenvalue density |m|<|m c | -m<<0 |m|=|m c | |m|>|m c | 3rd order transition two-cut one-cut at m=m c : finite Our cusp potential case ： V(x)=x 2 /2-g|x|+g 2 /2 ρ (x) ： eigenvalue density g>>0 g>0 g<0 g=0 NOT GWW-type one-cut two-cut transition at g=0
Ordinary Potentials transition at g=g c Kento SUGIYAMA (Shizuoka Univ.) “Strings and Fields 2019” in YITP (Aug.19-23) This talk: We show these results! these phase structures are also different. ⇒ Amazingly, we find that in the case of cusp potentials, We investigate the 0dim. cases and 1dim. cases with cusp potentials. ＊ Table of our results (NOT GWW-type) 2nd order phase Cusp Potentials 1dim.HMMs No transition in g>0 [Gross,Witten’80],[Wadia’80] et.al [Brezin,Itzykson,Parisi,Zuber’78] et.al at g=g c 3rd order phase transition GWW-type 0dim.HMMs (3/14) ＊ Summary
＊ Summary ＊ 0dim. HMM with cusp potential ＊ 1dim. HMM with cusp potential ＊ Conclusions Kento SUGIYAMA (Shizuoka Univ.) “Strings and Fields 2019” in YITP (Aug.19-23) Plan of my talk
Kento SUGIYAMA (Shizuoka Univ.) “Strings and Fields 2019” in YITP (Aug.19-23) Ordinary Potentials Cusp Potentials 0dim.HMMs GWW-type 3rd order phase transition at g=g c [Brezin,Itzykson,Parisi,Zuber’78] et.al [Gross,Witten’80],[Wadia’80] et.al No transition in g>0 1dim.HMMs 2nd order phase transition at g=g c (NOT GWW-type) ＊ 0dim. HMM with cusp potential
＊ Review on 0dim. HMMs at large-N ・Partition function (M: N×N Hermitian matrix) Kento SUGIYAMA (Shizuoka Univ.) “Strings and Fields 2019” in YITP (Aug.19-23) So we solve the equation (☆) in order to evaluate ρ(x) at large-N. be calculated by using the saddle point approximation. When ρ(x) is obtained at large-N, the free energy can [Brezin,Itzykson,Parisi,Zuber’78] gauge fixing ･･･ (4/14) ・Saddle point equation at large-N ＊ 0dim. HMM with cusp potential U: N × N unitary matrix Λ =diag(x 1 ,x 2 ,…,x N ) ρ (x) V(x) repulsive force between eigenvalues potential interactions x x 1 x 2 x N large-N potential ･･･ ( ☆ ) large-N large-N def.) eigenvalue density N ρ ( x ) := 1 plateau X C: support of ρ (w) δ ( x − x i ) ≥ 0 N i =1 filling x
＊ Consider the cusp potential case. g<0 Kento SUGIYAMA (Shizuoka Univ.) “Strings and Fields 2019” in YITP (Aug.19-23) makes negativity of ρ(x) near x=0. When g>0, the logarithmic divergence ∵ ρ(x) must be positive by definition. Actually, it is always “wrong” in g>0. ii) g>0 It is consistent on the cut [-b,b]. ・One-cut solution <latexit sha1_base64="O6b28MzK+hb/ZaZBAXp+2BGUCxY=">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</latexit> given by pareameter g) (b: the end points (5/14) i) ＊ 0dim. HMM with cusp potential | {z } | {z } New term The ordinary It has a logarithmic ρ ( x ) | x =0 → − g semi-circle term 2 π log ( ∞ ) singularity at x=0. N ρ ( x ) := 1 X δ ( x − x i ) ≥ 0 N ρ (x) at g<0 ρ (x) at g>0 ~ -g log(1/x) ⇒ + ∞ i =1 breakdown x -b b ~ -g log(1/x) ⇒ - ∞ x -b b
(6/14) ρ(x) ＆ V(x) ｂ different from the GWW-type transition. a large-N phase transition, it is obviously If the phenomenon at g=0 is regarded as Kento SUGIYAMA (Shizuoka Univ.) “Strings and Fields 2019” in YITP (Aug.19-23) (a,b: the end points given by pareameter g) ・Two-cut solution at g>0 ＊ Consider the cusp potential case. ＊ Phase structure ＊ 0dim. HMM with cusp potential ✓ a 2 x 2 , a 2 ✓ a 2 a, a 2 ◆ ◆� g x 2 , sin − 1 b p ρ ( x ) = ( x 2 − a 2 )( b 2 − x 2 ) Im Π − Π π 2 bx b 2 b 2 -a -b a ρ (x) ＆ V(x) one-cut transition two-cut
ρ(x) ＆ V(x) ・In this case, the gap of each cuts is (a,b: the end points given by pareameter g) Kento SUGIYAMA (Shizuoka Univ.) “Strings and Fields 2019” in YITP (Aug.19-23) ρ(x) ｂ different from the ordinary ones. suggest that this transition may be ⇒The strange behaviors of the end point exponentially small e -π/g near g=0. (7/14) near critical points are order (g-g c ) # . We investigate the end points near g=0. The two-cut solution is consistent in g>0. ・Two-cut solution at g>0 ＊ Consider the cusp potential case. ・Normally, behaviors of closing to cuts ＊ 0dim. HMM with cusp potential ✓ a 2 x 2 , a 2 ✓ a 2 a, a 2 ◆ ◆� g x 2 , sin − 1 b p ρ ( x ) = ( x 2 − a 2 )( b 2 − x 2 ) Im Π − Π π 2 bx b 2 b 2 | {z -a -b a } g=0 g=0.5 g=1 g=2
＊ Phase structure Our Claim：0dim.HMMs with cusp potentials might have NO large-N phase transition at finite couplings. ⇒ If potentials have cusp singularities, Eigenvalues cannot be located at singular points. Kento SUGIYAMA (Shizuoka Univ.) “Strings and Fields 2019” in YITP (Aug.19-23) If the phenomenon at g=0 is regarded as a large-N phase transition, it is obviously different from the GWW-type transition. (8/14) ＊ 0dim. HMM with cusp potential ρ (x) ＆ V(x) one-cut transition two-cut
Recommend
More recommend
