ONSAGERS SOLUTION OF THE ISING MODEL COULD HAVE BEEN GUESSED Manuel - PowerPoint PPT Presentation

If P n,m is the partition function for the n × m -torus, what happens for n, m → ∞ ? It diverges. Consider the free energy per site log ( P n,m ) f ( x ) := lim nm n,m →∞ This limit exists, and it knows everything about the system, for example: • “Internal energy”: U ( x ) = xf ′ ( x ) 8

If P n,m is the partition function for the n × m -torus, what happens for n, m → ∞ ? It diverges. Consider the free energy per site log ( P n,m ) f ( x ) := lim nm n,m →∞ This limit exists, and it knows everything about the system, for example: • “Internal energy”: U ( x ) = xf ′ ( x ) • “Specific heat”: C ( x ) = xf ′ ( x ) + x 2 f ′′ ( x ) 8

If P n,m is the partition function for the n × m -torus, what happens for n, m → ∞ ? It diverges. Consider the free energy per site log ( P n,m ) f ( x ) := lim nm n,m →∞ 2 This limit exists, and it knows everything f ( x ) about the system, for example: 1 • “Internal energy”: U ( x ) = xf ′ ( x ) • “Specific heat”: C ( x ) = xf ′ ( x ) + x 2 f ′′ ( x ) x 0 1 2 3 4 8

If P n,m is the partition function for the n × m -torus, what happens for n, m → ∞ ? It diverges. Consider the free energy per site log ( P n,m ) f ( x ) := lim nm n,m →∞ U ( x ) 2 This limit exists, and it knows everything f ( x ) about the system, for example: 1 • “Internal energy”: U ( x ) = xf ′ ( x ) • “Specific heat”: C ( x ) = xf ′ ( x ) + x 2 f ′′ ( x ) x 0 1 2 3 4 8

If P n,m is the partition function for the n × m -torus, what happens for n, m → ∞ ? It diverges. Consider the free energy per site log ( P n,m ) f ( x ) := lim nm n,m →∞ U ( x ) 2 This limit exists, and it knows everything f ( x ) about the system, for example: 1 • “Internal energy”: U ( x ) = xf ′ ( x ) C ( x ) • “Specific heat”: C ( x ) = xf ′ ( x ) + x 2 f ′′ ( x ) x 0 1 2 3 4 8

Theorem (Onsager 1944): 2 � x − x − 1 ∞ � � 2n � f ( x ) = log ( x + x − 1 ) − 1 1 2n � ( x + x − 1 ) 2 4 n n n = 1 9

Theorem (Onsager 1944): 2 � x − x − 1 ∞ � � 2n � f ( x ) = log ( x + x − 1 ) − 1 1 2n � ( x + x − 1 ) 2 4 n n n = 1 Proof: difficult. 9

Theorem (Onsager 1944): 2 � x − x − 1 ∞ � � 2n � f ( x ) = log ( x + x − 1 ) − 1 1 2n � ( x + x − 1 ) 2 4 n n n = 1 Proof: difficult. Claim: this formula can be found by guessing, taking into account two facts predating Onsager’s solution: 9

Theorem (Onsager 1944): 2 � x − x − 1 ∞ � � 2n � f ( x ) = log ( x + x − 1 ) − 1 1 2n � ( x + x − 1 ) 2 4 n n n = 1 Proof: difficult. Claim: this formula can be found by guessing, taking into account two facts predating Onsager’s solution: • A change of variables proposed by van der Waerden in 1941 9

Theorem (Onsager 1944): 2 � x − x − 1 ∞ � � 2n � f ( x ) = log ( x + x − 1 ) − 1 1 2n � ( x + x − 1 ) 2 4 n n n = 1 Proof: difficult. Claim: this formula can be found by guessing, taking into account two facts predating Onsager’s solution: • A change of variables proposed by van der Waerden in 1941 • The Kramers-Wannier duality from 1941 9

Van der Waerden’s change of variables (1941) Write � x + 2 + x − 1 nm � P n,m ( x ) = Z n,m ( w ) 2 with w = x − 1 x + 1 , and translate everything from P and x to Z and w . 10

Van der Waerden’s change of variables (1941) Write � x + 2 + x − 1 nm � P n,m ( x ) = Z n,m ( w ) 2 with w = x − 1 x + 1 , and translate everything from P and x to Z and w . Note: 2 log ( Z n,n ( w )) � � f ( x ) = log + lim 1 − w 2 n 2 n →∞ 10

Van der Waerden’s change of variables (1941) Write � x + 2 + x − 1 nm � P n,m ( x ) = Z n,m ( w ) 2 with w = x − 1 x + 1 , and translate everything from P and x to Z and w . Note: log ( Z n,n ( w )) g ( w ) := lim n 2 n →∞ 10

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense 3 w 3 + w 4 + · · · 2 n = 3 : 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense 3 w 3 + w 4 + · · · 2 n = 3 : 2 w 4 + 0w 5 + · · · 3 n = 4 : 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense 3 w 3 + w 4 + · · · 2 n = 3 : 2 w 4 + 0w 5 + · · · 3 n = 4 : w 4 + 2 5 w 5 + 2w 6 + · · · n = 5 : 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense 3 w 3 + w 4 + · · · 2 n = 3 : 2 w 4 + 0w 5 + · · · 3 n = 4 : w 4 + 2 5 w 5 + 2w 6 + · · · n = 5 : w 4 + 0w 5 + 7 3 w 6 + 0w 7 + · · · n = 6 : 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense 3 w 3 + w 4 + · · · 2 n = 3 : 2 w 4 + 0w 5 + · · · 3 n = 4 : w 4 + 2 5 w 5 + 2w 6 + · · · n = 5 : w 4 + 0w 5 + 7 3 w 6 + 0w 7 + · · · n = 6 : w 4 + 0w 5 + 2w 6 + 2 7 w 7 + 9 2 w 8 + · · · n = 7 : 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense 3 w 3 + w 4 + · · · 2 n = 3 : 2 w 4 + 0w 5 + · · · 3 n = 4 : w 4 + 2 5 w 5 + 2w 6 + · · · n = 5 : w 4 + 0w 5 + 7 3 w 6 + 0w 7 + · · · n = 6 : w 4 + 0w 5 + 2w 6 + 2 7 w 7 + 9 2 w 8 + · · · n = 7 : w 4 + 0w 5 + 2w 6 + 0w 7 + 19 4 w 8 + 0w 9 + · · · n = 8 : 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense 3 w 3 + w 4 + · · · 2 n = 3 : 2 w 4 + 0w 5 + · · · 3 n = 4 : w 4 + 2 5 w 5 + 2w 6 + · · · n = 5 : w 4 + 0w 5 + 7 3 w 6 + 0w 7 + · · · n = 6 : w 4 + 0w 5 + 2w 6 + 2 7 w 7 + 9 2 w 8 + · · · n = 7 : w 4 + 0w 5 + 2w 6 + 0w 7 + 19 4 w 8 + 0w 9 + · · · n = 8 : w 4 + 0w 5 + 2w 6 + 0w 7 + 9 2 w 8 + 2 9 w 9 + 12w 10 + · · · n = 9 : 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense 3 w 3 + w 4 + · · · 2 n = 3 : 2 w 4 + 0w 5 + · · · 3 n = 4 : w 4 + 2 5 w 5 + 2w 6 + · · · n = 5 : w 4 + 0w 5 + 7 3 w 6 + 0w 7 + · · · n = 6 : w 4 + 0w 5 + 2w 6 + 2 7 w 7 + 9 2 w 8 + · · · n = 7 : w 4 + 0w 5 + 2w 6 + 0w 7 + 19 4 w 8 + 0w 9 + · · · n = 8 : w 4 + 0w 5 + 2w 6 + 0w 7 + 9 2 w 8 + 2 9 w 9 + 12w 10 + · · · n = 9 : w 4 + 0w 5 + 2w 6 + 0w 7 + 9 2 w 8 + 0w 9 + 61 5 w 10 + 0w 11 + · · · n = 10 : 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense 3 w 3 + w 4 + · · · 2 n = 3 : 2 w 4 + 0w 5 + · · · 3 n = 4 : w 4 + 2 5 w 5 + 2w 6 + · · · n = 5 : w 4 + 0w 5 + 7 3 w 6 + 0w 7 + · · · n = 6 : w 4 + 0w 5 + 2w 6 + 2 7 w 7 + 9 2 w 8 + · · · n = 7 : w 4 + 0w 5 + 2w 6 + 0w 7 + 19 4 w 8 + 0w 9 + · · · n = 8 : w 4 + 0w 5 + 2w 6 + 0w 7 + 9 2 w 8 + 2 9 w 9 + 12w 10 + · · · n = 9 : w 4 + 0w 5 + 2w 6 + 0w 7 + 9 2 w 8 + 0w 9 + 61 5 w 10 + 0w 11 + · · · n = 10 : w 4 + 0w 5 + 2w 6 + 0w 7 + 9 2 w 8 + 0w 9 + 12w 10 + 2 11 w 11 + · · · n = 11 : 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Idea: Compute many terms and guess the result 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Idea: Compute many terms and guess the result • Enumerating all 2 n 2 configurations is feasible for n ≤ 5 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Idea: Compute many terms and guess the result • Enumerating all 2 n 2 configurations is feasible for n ≤ 5 • We can use transfer matrices to compute P n,m ( x ) 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Idea: Compute many terms and guess the result • Enumerating all 2 n 2 configurations is feasible for n ≤ 5 • We can use transfer matrices to compute P n,m ( x ) x 3 x − 1 x − 1 x − 1  1 1 1 1  x − 1 x − 1 x − 2 x 2 x 1 1 x   x 2 x − 1 x − 1 x − 2   x 1 1 x    x − 2 x − 1 x − 1 x 2  x 1 1 x    x 2 x − 1 x − 1 x − 2  x 1 1 x     x − 2 x − 1 x − 1 x 2 x 1 1 x     x − 2 x − 1 x − 1 x 2 x 1 1 x   x − 1 x − 1 x − 1 x 3 1 1 1 1 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Idea: Compute many terms and guess the result • Enumerating all 2 n 2 configurations is feasible for n ≤ 5 • We can use transfer matrices to compute P n,m ( x ) • For a suitable 2 n × 2 n matrix T we have P n,m ( x ) = Tr ( T m ) 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Idea: Compute many terms and guess the result • Enumerating all 2 n 2 configurations is feasible for n ≤ 5 • We can use transfer matrices to compute P n,m ( x ) • For a suitable 2 n × 2 n matrix T we have P n,m ( x ) = Tr ( T m ) • Using the structure of T , we can compute Tr ( T m ) efficiently 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Idea: Compute many terms and guess the result • Enumerating all 2 n 2 configurations is feasible for n ≤ 5 • We can use transfer matrices to compute P n,m ( x ) • For a suitable 2 n × 2 n matrix T we have P n,m ( x ) = Tr ( T m ) • Using the structure of T , we can compute Tr ( T m ) efficiently • This approach is feasible for n ≤ 12 , which is not enough 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Feature: Z n,m ( w ) is a polynomial with integer coefficients. 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Feature: Z n,m ( w ) is a polynomial with integer coefficients. The coefficient of w k counts how many polygonal shapes with k edges of a certain type fit on the n × m -torus. 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Feature: Z n,m ( w ) is a polynomial with integer coefficients. The coefficient of w k counts how many polygonal shapes with k edges of a certain type fit on the n × m -torus. 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Feature: Z n,m ( w ) is a polynomial with integer coefficients. The coefficient of w k counts how many polygonal shapes with k edges of a certain type fit on the n × m -torus. 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Feature: Z n,m ( w ) is a polynomial with integer coefficients. The coefficient of w k counts how many polygonal shapes with k edges of a certain type fit on the n × m -torus. 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Feature: Z n,m ( w ) is a polynomial with integer coefficients. The coefficient of w k counts how many polygonal shapes with k edges of a certain type fit on the n × m -torus. Feature: For n, m > k , we have [ w k ] Z n,m = poly k ( nm ) 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Feature: Z n,m ( w ) is a polynomial with integer coefficients. The coefficient of w k counts how many polygonal shapes with k edges of a certain type fit on the n × m -torus. Feature: For n, m > k , we have [ w k ] Z n,m = poly k ( nm ) [ w 2 ] Z n,m = 0 [ w 3 ] Z n,m = 0 [ w 4 ] Z n,m = 1nm [ w 5 ] Z n,m = 0 [ w 6 ] Z n,m = 2nm [ w 7 ] Z n,m = 0 [ w 8 ] Z n,m = 9 2 ( nm ) 2 [ w 9 ] Z n,m = 0 2 nm + 1 [ w 10 ] Z n,m = 6nm + ( nm ) 2 [ w 11 ] Z n,m = 0 2 ( nm ) 2 + 1 [ w 12 ] Z n,m = 112 3 nm + 13 6 ( nm ) 3 [ w 13 ] Z n,m = 0 [ w 14 ] Z n,m = 130nm + 21 ( nm ) 2 + ( nm ) 3 [ w 14 ] Z n,m = 0 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Feature: Z n,m ( w ) is a polynomial with integer coefficients. The coefficient of w k counts how many polygonal shapes with k edges of a certain type fit on the n × m -torus. Feature: For n, m > k , we have [ w k ] Z n,m = poly k ( nm ) ∞ log Z n,n ( w ) � � [ X 1 ] poly k ( X ) � w k Feature: g ( w ) = lim = n 2 n →∞ k = 0 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Feature: Z n,m ( w ) is a polynomial with integer coefficients. The coefficient of w k counts how many polygonal shapes with k edges of a certain type fit on the n × m -torus. Feature: For n, m > k , we have [ w k ] Z n,m = poly k ( nm ) [ w 2 ] Z n,m = 0 [ w 3 ] Z n,m = 0 [ w 4 ] Z n,m = 1nm [ w 5 ] Z n,m = 0 [ w 6 ] Z n,m = 2nm [ w 7 ] Z n,m = 0 [ w 8 ] Z n,m = 9 2 ( nm ) 2 [ w 9 ] Z n,m = 0 2 nm + 1 [ w 10 ] Z n,m = 6nm + ( nm ) 2 [ w 11 ] Z n,m = 0 2 ( nm ) 2 + 1 [ w 12 ] Z n,m = 112 3 nm + 13 6 ( nm ) 3 [ w 13 ] Z n,m = 0 [ w 14 ] Z n,m = 130nm + 21 ( nm ) 2 + ( nm ) 3 [ w 14 ] Z n,m = 0 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Feature: Z n,m ( w ) is a polynomial with integer coefficients. The coefficient of w k counts how many polygonal shapes with k edges of a certain type fit on the n × m -torus. Feature: For n, m > k , we have [ w k ] Z n,m = poly k ( nm ) ∞ log Z n,n ( w ) � � [ X 1 ] poly k ( X ) � w k Feature: g ( w ) = lim = n 2 n →∞ k = 0 = 1w 4 + 2w 6 + 9 2 w 8 + 6w 10 + 112 3 w 12 + 130w 14 + · · · 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Feature: Z n,m ( w ) is a polynomial with integer coefficients. The coefficient of w k counts how many polygonal shapes with k edges of a certain type fit on the n × m -torus. Feature: For n, m > k , we have [ w k ] Z n,m = poly k ( nm ) ∞ log Z n,n ( w ) � � [ X 1 ] poly k ( X ) � w k Feature: g ( w ) = lim = n 2 n →∞ k = 0 = 1w 4 + 2w 6 + 9 2 w 8 + 6w 10 + 112 3 w 12 + 130w 14 + · · · Using interpolation and some combinatorial considerations, we man- age to compute poly k for all k ≤ 32 . 11

1 Feature: n 2 log ( Z n,n ( w )) converges in the power series sense Feature: Z n,m ( w ) is a polynomial with integer coefficients. The coefficient of w k counts how many polygonal shapes with k edges of a certain type fit on the n × m -torus. Feature: For n, m > k , we have [ w k ] Z n,m = poly k ( nm ) ∞ log Z n,n ( w ) � � [ X 1 ] poly k ( X ) � w k Feature: g ( w ) = lim = n 2 n →∞ k = 0 = 1w 4 + 2w 6 + 9 2 w 8 + 6w 10 + 112 3 w 12 + 130w 14 + · · · Using interpolation and some combinatorial considerations, we man- age to compute poly k for all k ≤ 32 . Unfortunately, 32 terms of g ( w ) are still not enough. 11

Kramers-Wannier Duality 12

log P n,n ( x ) log Z n,n ( w ) Recall: f ( x ) = lim , g ( w ) = lim n 2 n 2 n →∞ n →∞ 13

log P n,n ( x ) log Z n,n ( w ) Recall: f ( x ) = lim , g ( w ) = lim n 2 n 2 n →∞ n →∞ Theorem (Kramers-Wannier 1941): f ( x ) − log ( x + x − 1 ) = f ( x ∗ ) − log ( x ∗ + ( x ∗ ) − 1 ) with x ∗ = x + 1 x − 1 . 13

log P n,n ( x ) log Z n,n ( w ) Recall: f ( x ) = lim , g ( w ) = lim n 2 n 2 n →∞ n →∞ Theorem (Kramers-Wannier 1941): f ( x ) − log ( x + x − 1 ) = f ( x ∗ ) − log ( x ∗ + ( x ∗ ) − 1 ) with x ∗ = x + 1 x − 1 . This equation connects the behaviour at low temperature ( x → 1 + ) with the behaviour at high temperature ( x → ∞ ) 13

log P n,n ( x ) log Z n,n ( w ) Recall: f ( x ) = lim , g ( w ) = lim n 2 n 2 n →∞ n →∞ Theorem (Kramers-Wannier 1941): f ( x ) − log ( x + x − 1 ) = f ( x ∗ ) − log ( x ∗ + ( x ∗ ) − 1 ) with x ∗ = x + 1 x − 1 . This equation connects the behaviour at low temperature ( x → 1 + ) with the behaviour at high temperature ( x → ∞ ) Idea 1: consider f ( x ) − log ( x + x − 1 ) instead of f ( x ) 13

log P n,n ( x ) log Z n,n ( w ) Recall: f ( x ) = lim , g ( w ) = lim n 2 n 2 n →∞ n →∞ Theorem (Kramers-Wannier 1941): f ( x ) − log ( x + x − 1 ) = f ( x ∗ ) − log ( x ∗ + ( x ∗ ) − 1 ) with x ∗ = x + 1 x − 1 . This equation connects the behaviour at low temperature ( x → 1 + ) with the behaviour at high temperature ( x → ∞ ) Idea 1: consider f ( x ) − log ( x + x − 1 ) instead of f ( x ) Idea 2: change to a new variable which is invariant under x ↔ x ∗ 13

We search for a symmetric function z = rat ( x, x ∗ ) such that expressing w = x − 1 x + 1 in terms of z gives a series of positive order with only even exponents. 14

We search for a symmetric function z = rat ( x, x ∗ ) such that expressing w = x − 1 x + 1 in terms of z gives a series of positive order with only even exponents. Such rational functions can be easily found using Gr¨ obner bases. 14

We search for a symmetric function z = rat ( x, x ∗ ) such that expressing w = x − 1 x + 1 in terms of z gives a series of positive order with only even exponents. Such rational functions can be easily found using Gr¨ obner bases. The smallest solution turns out to be z = cx ( x 2 − 1 ) ( 1 + x 2 ) 2 = cw ( 1 − w 2 ) ( 1 + w 2 ) 2 , where c is an arbitrary nonzero constant. 14

We search for a symmetric function z = rat ( x, x ∗ ) such that expressing w = x − 1 x + 1 in terms of z gives a series of positive order with only even exponents. Such rational functions can be easily found using Gr¨ obner bases. The smallest solution turns out to be z = cx ( x 2 − 1 ) ( 1 + x 2 ) 2 = cw ( 1 − w 2 ) ( 1 + w 2 ) 2 , where c is an arbitrary nonzero constant. Let’s take c = 1 . 14

f ( x ) − log ( x + x − 1 ) 15