Partition functions for complex fugacity Part I Barry M. McCoy CN - PowerPoint PPT Presentation

Partition functions for complex fugacity Part I Barry M. McCoy CN Yang Institute of Theoretical Physics State University of New York, Stony Brook, NY, USA Partition functions for complex fugacity – p.1/51

In collaboration with Michael Assis, Stony Brook University Jesper Jacobsen, ENS Paris Iwan Jensen, University of Melbourne Jean-Marie Maillard, University of Paris VI Based in part on “Hard hexagon partition function for complex fugacity” arXiv: 1306.6389 Partition functions for complex fugacity – p.2/51

Motivation In 1999-2000 Nickel discovered and in 2001 Orrick, Nickel, Guttmann and Perk extensively analyzed the evidence for a natural boundary in the susceptibility of the Ising model in the complex temperature plane. This present study is an attempt to understand the implications of this discovery. Partition functions for complex fugacity – p.3/51

Outline 1. Problems for complex fugacity 2. Preliminaries for hard hexagons and squares 3. Hard hexagon analytic results 4. Hard hexagon equimodular curves 5. Hard hexagon partition function zeros 6. Hard square zeros 7. Ising in a field 8. Further open questions 9. Conclusion Partition functions for complex fugacity – p.4/51

1. Problems for complex fugacity 1. Existence of a shape independent partition function per site. 2. Equimodular curves versus partition function zeros 3. Areas versus curves of zeros 4. Analytic continuation versus natural boundaries 5. Integrable versus generic non-integrable systems Partition functions for complex fugacity – p.5/51

2. Preliminaries for hard hexagons and squares 1. Grand partition function on an L v × L h lattice Z L v ,L h ( z ) = � ∞ N =0 g ( N ) · z N where g ( N ) is the number of allowed configurations. 2. Transfer matrices T { b 1 , ··· b Lh } , { a 1 , ··· .a Lh } = � L h j =1 W ( a j , a j +1 ; b j , b j +1 ) where the occupation numbers a j , b j take the values 0 , 1 with 3. Boltzmann weights b b j j+1 a a j j+1 Partition functions for complex fugacity – p.6/51

For hard squares W ( a j , a j +1 ; b j , b j +1 ) = 0 for a j a j +1 = b j b j +1 = a j b j = a j +1 b j +1 = 1 , and otherwise: W ( a j , a j +1 ; b j , b j +1 ) = z ( a j + a j +1 + b j + b j +1 ) / 4 For hard hexagons W ( a j , a j +1 ; b j , b j +1 ) = 0 for a j a j +1 = b j b j +1 = a j b j = a j +1 b j +1 = a j +1 b j = 1 , and otherwise: W ( a j , a j +1 ; b j , b j +1 ) = z ( a j + a j +1 + b j + b j +1 ) / 4 Partition functions for complex fugacity – p.7/51

4. Partition functions from transfer matrices eigenval- ues For toroidal boundary conditions k λ L v Z T L v ,L h ( z ) = Tr T L v ( z ; L h ) = � k ( z ; L h ) For cylindrical boundary conditions k λ L v Z C L v ,L j h ( z ) = � v | T L v ( z ; L h ) | v � = � k ( z ; L h ) c k with j =1 z a j / 2 and v ( a 1 , a 2 . · · · , a L h ) = � L h c k = ( v · v k )( v k · v ) where λ k are eigenvalues and v k are eigenvectors For hard squares T = T t ; λ k real for real z For hard hexagons T � = T t ; some λ k complex for real z Partition functions for complex fugacity – p.8/51

5. Thermodynamic limit For thermodynamics to be valid we must have F/k B T = lim L v ,L h →∞ ( L v L h ) − 1 ln Z L v ,L h ( z ) independent of the aspect ratio L v /L h . In terms of the transfer matrix eigenvalues lim L v →∞ L − 1 v ln Z L v ,L h ( z ) = ln λ max ( z ; L h ) Therefore if lim L h →∞ L − 1 h lim L v →∞ L − 1 v ln Z L v ,L h ( z ) = lim L v ,L h →∞ ( L v L h ) − 1 ln Z L v ,L h ( z ) then − F/k B T = lim L h →∞ L − 1 h ln λ max ( z ; L h ) For z ≥ 0 this independence is rigorously true in general. For complex z there is no general proof and for hard squares for z = − 1 it is known to be false. Partition functions for complex fugacity – p.9/51

6. Partition function zeros versus equimodular curves We begin with the simplest case where L v → ∞ with L h fixed where the aspect ratio L v /L h → ∞ . The zeros will lie on curves where two or more transfer matrix eigenvalues have equal modulus | λ 1 ( z ; L h ) | = | λ 2 ( z ; L h ) | On this curve λ 1 ( z ; L h ) λ 2 ( z ; L h ) = e iφ ( z ) with φ ( z ) real . The density of zeros on this curve is proportional to dφ ( z ) /dz The cases of cylindrical and toroidal boundary conditions have distinct features which must be treated separately. Partition functions for complex fugacity – p.10/51

Cylindrical boundary conditions Because the boundary vector v is translationally invariant only eigenvectors in the sector P = 0 will have non vanishing scalar products ( v · v k ) . All equimodular curves have only two equimodular eigenvalues. Toroidal boundary conditions In this case all eigenvalues contribute. The eigenvalues for P and − P have equal modulus because of translational invariance and thus on equimodular curves there can be either 2, 3, or 4 equimodular eigenvalues. Partition functions for complex fugacity – p.11/51

3. Hard hexagon analytic results Baxter in 1980 has computed the fugacity z and the partition function per site κ ± ( z ) = lim L h →∞ λ max ( z ; L h ) 1 /L h for positive z terms of an auxiliary variable x using the functions G ( x ) = � ∞ 1 n =1 (1 − x 5 n − 4 )(1 − x 5 n − 1 ) H ( x ) = � ∞ 1 n =1 (1 − x 5 n − 3 )(1 − x 5 n − 2 ) Q ( x ) = � ∞ n =1 (1 − x n ) . 0 ≤ z ≤ z c ≤ z ≤ ∞ where There are two regimes √ z c = 11+5 5 = 11 . 090169 · · · 2 Both κ ± ( z ) have singularities only at z c , z d = − 1 /z c = − 0 . 090169 · · · , ∞ . Partition functions for complex fugacity – p.12/51

Partition functions per site High density z c < z < ∞ H ( x ) ) 5 x · ( G ( x ) z = 1 (1 − x 3 n − 2 )(1 − x 3 n − 1 ) x 1 / 3 · G 3 ( x ) Q 2 ( x 5 ) · � ∞ 1 κ + = n =1 (1 − x 3 n ) 2 H 2 ( x ) where, as x increases from 0 to 1 , the value of z − 1 increases from 0 to z − 1 c . Low density 0 < z < z c G ( x ) ) 5 z = − x · ( H ( x ) (1 − x 6 n − 4 )(1 − x 6 n − 3 ) 2 (1 − x 6 n − 2 ) κ − = H 3 ( x ) Q 2 ( x 5 ) · � ∞ (1 − x 6 n − 5 )(1 − x 6 n − 1 )(1 − x 6 n ) 2 , n =1 G 2 ( x ) where, as x decreases from 0 to − 1 , the value of z increases from 0 to z c . Partition functions for complex fugacity – p.13/51

Algebraic equation for κ + ( z ) Both κ ± ( z ) are algebraic functions of z . Joyce in 1987 obtained the equation for κ + ( z ) using the polynomials Ω 1 ( z ) = 1 + 11 z − z 2 Ω 2 ( z ) = z 4 + 228 z 3 + 494 z 2 − 228 z + 1 Ω 3 ( z ) = ( z 2 +1) · ( z 4 − 522 z 3 − 10006 z 2 +522 z +1) . f + ( z, κ + ) = � 4 k =0 C + k ( z ) κ 6 k + = 0 , where 0 ( z ) = − 3 27 z 22 C + 1 ( z ) = − 3 19 z 16 · Ω 3 ( z ) C + 2 ( z ) = − 3 10 z 10 · [Ω 2 C + 3 ( z ) − 2430 z · Ω 5 1 ( z )] 3 ( z ) = − z 4 · Ω 3 ( z ) · [Ω 2 C + 3 ( z ) − 1458 z · Ω 5 1 ( z )] C + 4 ( z ) = Ω 10 1 ( z ) . Partition functions for complex fugacity – p.14/51

Algebraic equation for κ − ( z ) For low density we have obtained by means of a Maple computation the algebraic equation for κ − ( z ) f − ( z, κ − ) = � 12 k =0 C − k ( z ) · κ 2 k − = 0 , where 0 ( z ) = − 2 32 · 3 27 · z 22 C − C − 1 ( z ) = 0 2 ( z ) = 2 26 · 3 23 · 31 · z 18 · Ω 2 ( z ) , C − 3 ( z ) = 2 26 · 3 19 · 47 · z 16 · Ω 3 ( z ) , C − 4 ( z ) = − 2 17 · 3 18 · 5701 · z 14 · Ω 2 C − 2 ( z ) , 5 ( z ) = − 2 16 · 3 14 · 7 2 · 19 · 37 · z 12 · Ω 2 ( z ) Ω 3 ( z ) , C − 6 ( z ) = − 2 10 · 3 10 · 7 · z 10 · [273001 · Ω 2 3 ( z ) + 2 6 · 3 5 · 5 · 4933 · z · Ω 5 C − 1 ( z )] , 7 ( z ) = − 2 9 · 3 10 · 11 · 13 · 139 · z 8 · Ω 3 ( z ) Ω 2 C − 2 ( z ) , 8 ( z ) = − 3 5 · z 6 · Ω 2 ( z ) · [7 · 1028327 · Ω 2 3 ( z ) − 2 6 · 3 4 · 11 · 419 · 16811 · z · Ω 5 C − 1 ( z )] , 9 ( z ) = − z 4 · Ω 3 ( z ) · [37 · 79087 Ω 2 3 ( z ) + 2 6 · 3 6 · 5150251 · z · Ω 5 C − 1 ( z )] , 10 ( z ) = − z 2 · Ω 2 3 ( z ) − 2 · 3 6 · 151 · 317 · z · Ω 5 2 ( z ) · [19 · 139Ω 2 C − 1 ( z )] 11 ( z ) = − Ω 2 ( z ) Ω 3 ( z ) · [Ω 2 3 ( z ) − 2 · 613 · z · Ω 5 C − 1 ( z )] , 12 ( z ) = Ω 10 C − 1 ( z ) . Partition functions for complex fugacity – p.15/51

Analyticity of κ ± ( z ) High density κ + ( z ) is real and positive for z c < z < ∞ For z → ∞ κ + ( z ) = z 1 / 3 + 1 3 z − 2 / 3 + 5 9 z − 5 / 3 + · · · κ + ( z ) is analytic in the plane cut from −∞ < z < z c On the segment −∞ < z < z d κ + ( z ) has the phase e ± πi/ 3 for Im z = ± ǫ → 0 . Low density κ − is real and positive for z d < z < z c κ − is analytic in the plane cut from z c < z < ∞ and −∞ < z < z d Partition functions for complex fugacity – p.16/51

Values of κ ± ( z ) at z c and z d At z c ( w c + + 3 9 ) 3 = 0 w c + = − (5 5 / 2 /z c ) 3 κ 6 with + ( z c ) ( w c − + 2 4 ) 2 · ( w c − − 3 3 ) 3 · ( w c − − 2 4 · 3 3 ) 6 = 0 w c − = 5 5 / 2 κ 2 with − ( z c ) /z c At z = z d ( w d + + 3 9 ) 3 = 0 w d + = − (5 5 / 2 /z d ) 3 κ 6 with + ( z d ) ( w d − − 2 4 ) 2 · ( w d − + 3 3 ) 3 · ( w d − + 2 4 · 3 3 ) 6 = 0 w d − = 5 5 / 2 κ 2 − ( z d ) 2 /z d with Thus using appropriate boundary conditions κ + ( z c ) = κ − ( z c ) = (3 3 · 5 − 5 / 2 z c ) 1 / 2 = 2 . 3144003 · · · κ + ( z d ) = e ± πi/ 3 0 . 208689 , κ − ( z d ) = 4 | κ + ( z d ) | Partition functions for complex fugacity – p.17/51