Integrability of Limit Shape Phenomena in Six Vertex Model Ananth Sridhar UC Berkeley Physics asridhar@berkeley.edu June 19, 2015 1 / 59
Introduction Background • The six vertex model is can be reformulated as a random stepped surface called heights. 2 / 59
Introduction Background • The six vertex model is can be reformulated as a random stepped surface called heights. • In the thermodynamic limit, the limiting average height function becomes deterministic and can be found by solving a certain boundary value problem. 3 / 59
Introduction Background • The six vertex model is can be reformulated as a random stepped surface called heights. • In the thermodynamic limit, the limiting average height function becomes deterministic and can be found by solving a certain boundary value problem. • The six vertex model is quantum integrable in the sense that it admits commuting transfer matrices and can be solved by Bethe ansatz. 4 / 59
Introduction Background • The six vertex model is can be reformulated as a random stepped surface called heights. • In the thermodynamic limit, the limiting average height function becomes deterministic and can be found by solving a certain boundary value problem. • The six vertex model is quantum integrable in the sense that it admits commuting transfer matrices and can be solved by Bethe ansatz. • What does the quantum integrability imply for the PDE governing the limiting height function? 5 / 59
Introduction Outline of Talk • Quick Review of Six Vertex Model • Thermodynamic Limit • Integrability: • Transfer Matrices • Commuting Hamiltonians • Examples • Outlook 6 / 59
Review: Six Vertex Model Configurations and Weights T = ǫ Z 2 be the scaled square lattice • Let S T = [0 , T ] × [0 , 1], and let S ǫ centered inside S T . 7 / 59
Review: Six Vertex Model Configurations and Weights T = ǫ Z 2 be the scaled square lattice • Let S T = [0 , T ] × [0 , 1], and let S ǫ centered inside S T . • A configuration s of the six vertex model is a set of paths that only go right and up. w 1 w 1 w 2 w 2 w 3 w 3 8 / 59
Review: Six Vertex Model Configurations and Weights T = ǫ Z 2 be the scaled square lattice • Let S T = [0 , T ] × [0 , 1], and let S ǫ centered inside S T . • A configuration s of the six vertex model is a set of paths that only go right and up. w 1 w 1 w 2 w 2 w 3 w 3 • Each vertex has a weight v ( s ). • The Boltzmann weight of s : � w ( s ) = v ( s ) vertex v 9 / 59
Review: Six Vertex Model Boundary Conditions • The state of s at time t is the set of horizontal edges traversed by s at t . 10 / 59
Review: Six Vertex Model Boundary Conditions • The state of s at time t is the set of horizontal edges traversed by paths at t . • Fixed boundary conditions are choice initial and final states η 1 and η 2 . 11 / 59
Review: Six Vertex Model Boundary Conditions • The state of s at time t is the set of horizontal edges traversed by paths at t . • Fixed boundary conditions are choice initial and final states η 1 and η 2 . • The partition function and the normalized free energy are: � Z ǫ η 1 ,η 2 , T = w ( s ) s (0)= η 1 s (1)= η 2 � � η 1 ,η 2 , T = ǫ 2 log f ǫ Z η 1 ,η 2 12 / 59
Review: Six Vertex Model Height Function • A height function is a function on faces satisfying a gradient constraint: • 0 ≤ h ( x , y ) − h ( x + ǫ, y ) ≤ 1 • 0 ≤ h ( x , y + ǫ ) − h ( x , y ) ≤ 1 13 / 59
Review: Six Vertex Model Height Function • A height function is a function on faces satisfying a gradient constraint: • 0 ≤ h ( x , y ) − h ( x + ǫ, y ) ≤ 1 • 0 ≤ h ( x , y + ǫ ) − h ( x , y ) ≤ 1 • Height functions are in bijection with configurations; the level curves of h are the paths of the configuration. 3 3 3 2 2 2 2 2 1 1 1 0 14 / 59
Review: Six Vertex Model Height Function • A height function is a function on faces satisfying a gradient constraint: • 0 ≤ h ( x , y ) − h ( x + ǫ, y ) ≤ 1 • 0 ≤ h ( x , y + ǫ ) − h ( x , y ) ≤ 1 • Height functions are in bijection with configurations; the level curves of h are the paths of the configuration. 3 3 3 2 2 2 2 2 1 1 1 0 • The boundary conditions determine the height function at the boundary. 15 / 59
Review: Six Vertex Model Height Function • A height function is a function on faces satisfying a gradient constraint: • 0 ≤ h ( x , y ) − h ( x + ǫ, y ) ≤ 1 • 0 ≤ h ( x , y + ǫ ) − h ( x , y ) ≤ 1 • Height functions are in bijection with configurations; the level curves of h are the paths of the configuration. 3 3 3 2 2 2 2 2 1 1 1 0 • The boundary conditions determine the height function at the boundary. • The normalized height function ¯ h = ǫ h . The average height function � ¯ h � is the ensemble average of the normalized height function. 16 / 59
Thermodynamic Limit Thermodynamic limit • Suppose we have a sequence of six vertex models S ǫ i t and boundary 2 with ǫ i → 0. height functions η ǫ i 1 , η ǫ i 17 / 59
Thermodynamic Limit Thermodynamic limit • Suppose we have a sequence of six vertex models S ǫ i t and boundary 2 with ǫ i → 0. height functions η ǫ i 1 , η ǫ i • The boundary conditions are said to be stabilizing if the normalized boundary height functions η ǫ 1 , η ǫ 2 converge to η 1 , η 2 : [0 , 1] → R in the uniform metric as ǫ → 0. 18 / 59
Thermodynamic Limit Thermodynamic limit • Suppose we have a sequence of six vertex models S ǫ i t and boundary 2 with ǫ i → 0. height functions η ǫ i 1 , η ǫ i • The boundary conditions are said to be stabilizing if the normalized boundary height functions η ǫ 1 , η ǫ 2 converge to η 1 , η 2 : [0 , 1] → R in the uniform metric as ǫ → 0. • In this case, there exist limiting free energy and limiting height function: ǫ → 0 f ǫ f η 1 ,η 2 , T = lim η 1 ,η 2 , T ǫ → 0 � h � ǫ � h � = lim 19 / 59
Thermodynamic Limit Variational Principle • The limiting free energy and average height function can be computed by variational principle. � 1 � T f η 1 ,η 2 , T = max σ w ( ∂ t h , ∂ y h ) dt dy h ∈H 0 0 where σ is called the surface tension function, and H is the set of limiting height functions, h : S t → R satisfying: h (0 , 0) = 0, monotonicity, and Lipschitz continuity with constant 1. 20 / 59
Thermodynamic Limit Variational Principle • The limiting free energy and average height function can be computed by variational principle. � 1 � T f η 1 ,η 2 , T = max σ w ( ∂ t h , ∂ y h ) dt dy h ∈H 0 0 where σ is called the surface tension function, and H is the set of limiting height functions, h : S t → R satisfying: h (0 , 0) = 0, monotonicity, and Lipschitz continuity with constant 1. • The limiting height function � h � is the maximizer. 21 / 59
Thermodynamic Limit Variational Principle • The limiting free energy and average height function can be computed by variational principle. � 1 � T f η 1 ,η 2 , T = max σ w ( ∂ t h , ∂ y h ) dt dy h ∈H 0 0 where σ is called the surface tension function, and H is the set of limiting height functions, h : S t → R satisfying: h (0 , 0) = 0, monotonicity, and Lipschitz continuity with constant 1. • The limiting height function � h � is the maximizer. • Euler Lagrange equations: ∂ 11 σ w ∂ 2 t h + 2 ∂ 12 σ w ∂ t ∂ y h + ∂ 22 σ w ∂ 2 y h = 0 22 / 59
Integrability of the Six Vertex Model Transfer Matrices • Let { e 0 , e 1 } be an orthonormal basis for C 2 , and let V = ( C 2 ) ⊗⌊ 1 /ǫ ⌋ . 23 / 59
Integrability of the Six Vertex Model Transfer Matrices • Let { e 0 , e 1 } be an orthonormal basis for C 2 , and let V = ( C 2 ) ⊗⌊ 1 /ǫ ⌋ . • A state s of the six vertex model corresponds to a basis vector | s � = e s 0 ⊗ e s 1 · · · e s N , where s i = 1 is the indicator of the i th edge. 24 / 59
Integrability of the Six Vertex Model Transfer Matrices • Let { e 0 , e 1 } be an orthonormal basis for C 2 , and let V = ( C 2 ) ⊗⌊ 1 /ǫ ⌋ . • A state s of the six vertex model corresponds to a basis vector | s � = e s 0 ⊗ e s 1 · · · e s N , where s i = 1 is the indicator of the i th edge. • Define the transfer matrix T w : V → V by its matrix elements: � s 1 | T w | s 2 � = Z s 1 , s 2 ,ǫ (ie. the partition function for just one column). 25 / 59
Integrability of the Six Vertex Model Transfer Matrices • Let { e 0 , e 1 } be an orthonormal basis for C 2 , and let V = ( C 2 ) ⊗⌊ 1 /ǫ ⌋ . • A state s of the six vertex model corresponds to a basis vector | s � = e s 0 ⊗ e s 1 · · · e s N , where s i = 1 is the indicator of the i th edge. • Define the transfer matrix T w : V → V by its matrix elements: � s 1 | T w | s 2 � = Z s 1 , s 2 ,ǫ (ie. the partition function for just one column). • Then: Z η 1 ,η 2 , t = � η 1 | T ⌊ t /ǫ ⌋ | η 2 � w 26 / 59
Integrability of the Six Vertex Model Hamiltonian Formulation of Variational Principle • Recast the variational problem in the Hamiltonian formulation by Legendre transform: H w ( π, t ) = max π s − σ w ( s , t ) s The new variables are h and π , where π is conjugate to ∂ t h . 27 / 59
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