Yang-Baxter Solution of Dimers as a Free-Fermion Six-Vertex Model. Paul A. Pearce & Alessandra Vittorini-Orgeas

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1 0-1 Yang-Baxter Solution of Dimers as a Free-Fermion Six-Vertex Model MATRIX, 7 July 2017 Paul A. Pearce & Alessandra Vittorini-Orgeas School of Mathematics and Statistics University of Melbourne PAP, A. Vittorini-Orgeas, arxiv:

2 0-2 Some History: Dimers & Dense Polymers 1961 Kasteleyn: Pfaffian solution of the dimer problem on the square lattice 1961 Temperley, Fisher: Independent solution on the square lattice 1967 Lieb: A non-yang-baxter transfer matrix method 2003 Izmailian, Oganesyan, Hu: Exact finite-size corrections of the free energy for the square lattice dimer model under different boundary conditions 2005 Izmailian, Priezzhev, Ruelle, Hu: Logarithmic conformal field theory and boundary effects in the dimer model 2007 Izmailian, Priezzhev, Ruelle: Non-local finite-size effects in the dimer model 2012 Rasmussen, Ruelle: Refined analysis of conformal spectra in the dimer model 2015 Nigro: Finite size corrections for dimers 2015 Morin-Duchesne, Rasmussen, Ruelle: Dimer representations of the Temperley-Lieb algebra 2016 Morin-Duchesne, Rasmussen, Ruelle: Integrability and conformal data of the dimer model 2007 Pearce, Rasmussen: Solution of critical dense polymers on the strip 2010 Pearce, Rasmussen, Villani: Solution of critical dense polymers on the cylinder 2013 Morin-Duchesne, Pearce, Rasmussen: Solution of critical dense polymers on the torus

3 0-3 Controversy/Approach The Big Question: Is the dimer model a c = 1 Gaussian free theory or is it a c = 2 logarithmic CFT? Strategy: Enumerate degrees of freedom (map to λ = π/2 six-vertex model). Introduce a spectral parameter (spatial anisotropy). Establish Yang-Baxter integrability (rotate faces by 45 degrees). Gain control to construct (r, s) type integrable/conformal boundary conditions on the strip.

4 0-4 Usual Periodic Tiling of Horizontal and Vertical Dimers The known Pfaffian solution (Kasteleyn/Temperley-Fisher 1961) for the number of periodic dimer configurations is Z α,β N/2 1 M N = n=0 M/2 1 m=0 Z M N = 2 1 1/2,1/2 ( Z M N + Z 0,1/2 M N + Z 1/2,0 M N ) ( 4 sin 2 2π(n+α) N +sin 2 2π(m+β) ), M,N = 2,4,6,... M Explicit counting on a M N square lattice yields ( Z M N ) = ,10 39, ,10 90,176 3,113,60 1,156 39,952 3,113,60 311,53, Z = 311,53,312 N,M = 2,4,6, dimer configurations for a 2 2 lattice

5 0-5 Six-Vertex, Particle and Dimer Representations Equivalent tiles: Vertex, particle and dimer (Korepin&Zinn-Justin 2000) representations: At free-fermion point: λ = π 2 a(u) = ρ sin(λ u) sinλ b(u) = ρ sinu sinλ = ρsinu = ρcosu c 1 (u) = ρg, c 2 (u) = ρ g, ρ R or Counting isotropic dimers: ρ = g = 2, u = λ 2 = π 4 } {{ } a(u) } {{ } b(u) }{{} c 1 (u) }{{} c 2 (u) c 1 (u) = 2, a(u) = b(u) = c 2 (u) = 1 The free fermion condition is satisfied at the free-fermion point λ = π 2 a(u) 2 +b(u) 2 = c 1 (u)c 2 (u) Particle lines are drawn if arrows point down or left. Tiles corresponding to a source of horizontal arrows (apricot) have a double degeneracy. Locally, the mapping is one-to-two for these faces. Sources and sinks of horizontal arrows appear in pairs so g is a gauge which we fix to g = e iu.

6 0-6 Lattice Configurations A typical periodic configuration on a 6 4 rectangular lattice: vertex, particle and (one of the 2 3 = ) possible dimer configurations: The boundary conditions are periodic so the left/right and top/bottom edges are identified. The excess of up arrows over down arrows along a row is conserved. Particles are conserved and move up and to the right around the torus but do not cross. The Z 2 up-down arrow symmetry translates into a particle/hole duality. The particle trajectories are non-local (logarithmic) degrees of freedom. An M N rectangular lattice is covered by MN dimers. Each dimer covers two bonds of the original square lattice.

7 0-7 Fermionic Algebra In the particle representation, the face operators of the free-fermion six-vertex model decompose into a sum of contributions from six elementary tiles ( X j (u) = u = a(u) j j+1 + ) ( +b(u) + ) +c 1 (u) +c 2 (u) As operators, the elementary tiles E j act diagrammatically on an upper row particle configuration to produce a lower row particle configuration E j = n 00 j, n11 j, f j f j+1, f j+1 f j, n 10 j, n01 j The (diagonal) number operators n 1 j and n0 j are orthogonal projectors that count the single site occupancy and vacancies respectively at position j n ab j = n a j nb j+1, na j nb j = δ abn a j, n0 j +n1 j = I, n00 j +n 11 j +n 10 j +n 01 j = I, a,b = 0,1 In the hopping terms, f j and f j are (non-diagonal) single-site particle annihilation and creation operators respectively satisfying the CARs {f j,f k } = {f j,f k } = 0, {f j,f k } = δ jk, n 1 j = f j f j, n 0 j = f jf j = 1 f j f j

8 0- More Fermionic Algebra The tiles are expressed as combinations of bilinears in fermi operators = (1 f j f j)(1 f j+1 f j+1) = (1 f j f j)f j+1 f j+1 = f j f j+1 = f j f j f j+1 f j+1 = f j f j(1 f j+1 f j+1) = f j+1 f j Multiplication of tiles in the fermionic algebra is given diagrammatically: = = = = = = = =

9 0-9 From Fermionic Algebra to Temperley-Lieb Algebra The Temperley-Lieb (TL) algebra has generators I and e j and is defined by e 2 j = βe j e j e j±1 e j = e j e i e j = e j e i, i j 2 The TL algebra admits a planar diagrammatic representation consisting of monoids The monoids satisfy = I = e j = β = 2cosλ = loop fugacity =β = The free-fermion algebra gives a representation of the TL algebra (Gainutdinov EtAl 2014) I = + + +, e j = + +x +x 1 where x = e iλ = i and x+x 1 = 2cosλ = β = 0. With these definitions, the diagrammatic relations between fermionic tiles imply the defining relations of the TL algebra.

10 0-10 YBE and Inversion Relation In terms of the generators of the TL algebra, the face transfer operators of the free-fermion six vertex/dimer model take the form X j (u) = u = cosui +sinue j j j+1 This form of the face transfer operator is sufficient (Baxter 192) to guarantee that X j (u) satisfies the Yang-Baxter Equation and Inversion Relation X j (v u)x j+1 (v)x j (u) = X j+1 (u)x j (v)x j+1 (v u), X j (u)x j ( u) = ρ 2 cos 2 ui f e u v u a b d v = c f a v e d v u u b c d a u u c b = ρ 2 cos 2 u δ(a,d)δ(b,c) subject to the initial condition X j (0) = I.

11 0-11 Commuting Periodic Row Transfer Matrices YBE + Inversion [T(u),T(v)] = 0 Yang-Baxter Integrable T(u)T(v) = v v v v v v u u v u u u u u u u u u u = v u u v v v v v v = u u u u u u v v u v v v v v = T(v)T(u) Since T(u) T = T(λ u), the commuting row transfer matrices are normal and admit a common set of eigenvectors independent of u. So they are simultaneously diagonalizable by a similarity transformation. The eigenvalue spectra can be found by solving functional equations satisfied by T(u).

12 0-12 Periodic Row Transfer Matrices b 1 b 2 b N Z = TrT(u) M = T T(u) = u u u u u u a 1 a 2 a N In the six-vertex representation, the total magnetization is conserved under the action of the transfer matrix S z = N σ j = N, N +2,...,N 2,N So S z is a good quantum number separating the spectrum into sectors labelled by l= S z : Z 4 : N odd, l odd, Ramond : N even, 2 l even, Neveu-Schwarz : N even, 2 l odd The number of down arrows coincides with the number of particles d = N a j and is also conserved l = N 2d = S z = 0,2,4,...,N, N even 1,3,5,...,N, N odd The transfer matrix and the vector space of states thus decompose as T(u) = N d=0 T d (u) dimv (N) = N d=0 dimv (N) d = N d=0 ( ) N d = 2 N = dim(c 2 ) N

13 0-13 Free Energy, Residual Entropy and Hamiltonian The bulk partition function per site ρκ(u) = ρexp( f bulk (u)) can be obtained by solving the inversion relation κ(u)κ( u) = cos 2 u (Baxter 192) or by using the Euler-Maclaurin formula. This gives the bulk free energy f bulk (u) = sinhutsinh( π 2 u)t tsinhπtcosh πt 2 dt = 1 2 log2 1 π π/2 0 log(cosect+sin2u)dt Setting ρ = 2 and u = π 4 gives the known (Fisher 1961) molecular freedom W and residual entropy S of dimers on the square lattice W = e S = 2exp( f bulk ( π 4 )) = exp(2g π ) = , S = 2G π = where W and Catalan s constant are W = 2κ( π 4 ) = lim (Z M N) MN, 1 G = 1 M,N 2 π/2 0 log(1+cosect)dt = The quantum Hamiltonian is given by the logarithmic derivative of the transfer matrix given by the u(1) symmetric XX model H = d du logt(u) = u=0 N e j = N (f j f j+1 +f j+1 f j)

14 0-14 Inversion Identities The free-fermion single row transfer matrix satisfies the inversion identities (Felderhof 73) T(u)T(u+λ) = ( cos 2N u sin 2N u ) I, N odd T d (u)t d (u+λ) = (cos N u+( 1) d sin N u) 2 I, N even To solve these functional equations we factorize the right side. For example, in the Z 4 sector, this factorization yields cos 2N u sin 2N u = e 2Niu 2 2N 1 N ( e 2iu +iǫ j tan (2j 1π) 4N )( e 2iu iǫ j tan (2j 1π) Sharing out the zeros between T(u) and T(u+λ) gives 2 N eigenvalues 4N ), ǫ j = ±1 T(u) = ǫ ( i)n/2 e Niu 2 N 1/2 N ( e 2iu +iǫ j tan (2j 1)π 4N ), Z 4 : N,l odd where ǫ = ( 1) (N l)/4. Similarly, the solution of the inversion identity in the N even sectors yields T(u) = ǫr ( i) N 2e Niu 2 N 1 N T(u) = ǫns ( i) N 2Ne Niu 2 N 1 ( e 2iu +iǫ j tan (2j 1)π 2N N j N/2 ( e 2iu +iǫ j tan jπ N ), R: N, l/2 even ), NS: N even, l/2 odd

15 0-15 Counting Rotated Periodic Dimer Configurations The exact counting of periodic dimer configurations on a finite M N square lattice, in the 45 degree rotated orientation, is given by where we set ρ = 2 and u = λ 2 = π 4. Z M N = Tr T (N)( π 4 ) M The explicit formulas are 2 MN+1 N s= N +2;4 N ǫ j =s ( 1) M(N s) 4 N cos M( ǫ j t j π 4), N odd Z M N = 2 MN N s= N s = 0 mod MN N N ǫ j = s s= N s = 2 mod 4 ( 1) M(2N+s) 4 N ǫ j = s N ( 1) M(2N+ s +2) 4 cos M( ǫ j t R j π 4 N ) cos M( ǫ j t NS j π 4), N even where s = S z, t j = (2j 1)π, t R j 4N (2j 1)π =, t NS j = 2N jπ N, j N 2 0, j = N 2

16 0-16 To obtain this formula, we use the trigonometric identities: 1+ǫ j tant j = cost j +ǫ j sint j cost j = 2 cos(ǫ jt j π 4 ) cost j, ǫ j = ±1 N cost j = 2 1/2 N, N cost R j = ( 1)N/2 2 1 N, N cost NS j = ( 1) N/2 N 2 1 N (Jolley 1961) The exact counting of rotated periodic dimer configurations on an M N rectangular lattice is easily obtained by coding the formulas in Mathematica: (Z M N ) = , ,624 15, ,624 26,752 20, ,0 15,616 20,32 5,00, , M,N = 1,2,3,... Z = 3,735,27,017,30,352 ( Z = 311,53,312) Z N,N Z 2N, 2N Z 2 2 = 24

17 0-17 Bulk CFT and Finite-Size Spectra The anisotropic partition function is Z N,M = TrT(u) M = T n (u) M = n 0 n 0 e ME n(u) Finite-size corrections from conformal invariance E 0 = Nf bulk (u) πc 6N sin2u, E n E 0 = 2πi N [ ( +k)e 2iu ( + k)e 2iu] The analytic results using Euler-Maclaurin are c = 2, c eff = 1, min = 1, j = j = j2 1 = 1, 0, 3, j = 0,1,2 In the scaling limit, the modular invariant conformal partition function is a sesquilinear form in u(1) characters Z(q) =, N, κ (q)κ ( q), q = exp(2πiτ), τ = M N e 2iu where N, = operator content = , q = modular nome κ (q) = q c/24 d (k)q +k k=0

18 0-1 Spectra: Sector-by-Sector Sector-by-sector Inversion Identity and patterns of zeros in complex u-plane: (l = S z ) T(u)T(u+ π 2 ) = cos 2N u sin 2N u, N,l odd, { Z 4 Sectors ( cos N u+( 1) (N l)/2 sin N u ) 2, N,l even, Ramond (l/2 even) Neveu-Schwarz (l/2 odd) y 1 y 1 y 2 y 2 y 3 y 4 y 5 y 3 y 4 y 5 π 4 y 5 y 4 y 3 π 4 π 2 3π 4 π 4 y 5 y 4 y 3 π 4 π 2 3π 4 y 2 y 2 y 1 y 1 N,l odd N,l even The y-ordinates of 1-strings u j and 1-string energies E j are y j = 1 2 logtane jπ N, E j = 1 2 (j 1 2 ), j = 1,2,...,N; Z 4 j 2 1, j = 1,2,...,N/2; Ramond j, j = 1,2,...,N/2 1; Neveu-Schwarz

19 0-19 Physical Combinatorics: Ramond Sectors The building blocks of the spectra in the Ramond sectors consist of the q-binomials [ n m] q = [ n n/2 σ ]q = q 1 2 σ2 double columns for fixed σ q j m je j, σ = n/2 m = #right #left E j = j 1 2 j = 3 j = 2 j = 1 [ 6 2] q = 1 + q + 2q2 + 2q 3 + 3q 4 + 2q 5 + 2q 6 + q 7 + q (σ = 1) ] ] As q-binomials, [ n m q = [ n n m q In a given l sector, the quantum numbers of the groundstate satisfy, but they have different combinatorial interpretations. σ = σ = l/4, l = 0,4,,... E(σ)+E( σ) = l2 16 Excitations are generated either by inserting a left-right pair of 1-strings at position j = 1 or incrementing the position j of a 1-string by 1 unit. The selection rules are σ + σ = l/2, 1 2 (σ σ) Z

20 0-20 Physical Combinatorics: Neveu-Schwarz Sectors The building blocks of the spectra in the Neveu-Schwarz sectors consist of the q-binomials [ n m] q = [ n n/2 σ ]q = q 1 2 σ(σ+1) double columns for fixed σ q j m je j, σ = n/2 m, #right #left = σ,σ +1 E j = j j = 3 j = 2 j = 1 [ 7 2] q = 1 + q + 2q2 + 2q 3 + 3q 4 + 3q 5 + 3q 6 + 2q 7 + 2q + q 9 + q 10 (σ = 1) As q-binomials, [ ] n m q = [ ] n n m, but they have different combinatorial interpretations. q In a given l sector, the quantum numbers of the groundstate satisfy σ = σ = (l 2)/4, l = 2,6,10,... E(σ)+E( σ) = l Excitations are generated either by inserting a right or left 1-string at position j = 1 or incrementing the position j of a 1-string by 1 unit. The selection rules are ( ) σ + σ = (l 2)/2, 1 2 (σ σ) Z

21 0-21 Finitized Modular Invariant Partition Function (N, l Even) Ramond sectors (l/2 even) Z (N) l (q) = (q q) c/24 [ q 2k+l/2 k Z Neveu-Schwarz sectors (l/2 odd) Z (N) l (q) = (q q) c/24 [ q 2k+l/2 k Z 2 N+2 4 N+2 l 4 k 2 N 4 +1 N+2 l 4 k Finitized Modular Invariant Partition Function ] ] q 2k l/2 q q 2k l/2 q [ 2 N ] 4 N l 4 +k q [ 2 N N l 4 +k ] q We find that Z N (q) = Z (N) 0 +2 l N l 4N Z (N) l (q)+2 l N l 4N 2 Z (N) l (q) Z N (q) = 1 2 (q q) c 24 1 [ N+2 4 n=1 +2(q q) c 24 (1+q n 1 2) 2 N 4 n=1 N 4 n=1 (1+q n ) 2 (1+ q n 1 2) 2 + N 2 4 n=1 (1+ q n ) 2 N+2 4 (1 q n 1 2) 2 N 4 n=1 n=1 (1 q n 1 2) 2 ]

22 0-22 Modular Invariant Partition Function Taking the thermodynamic limit N gives the modular invariant partition function Z 0 (q)+2 l 4N 2 l 4N 2 Z(q) = Z 0 (q)+2 l 2N Z l (q) = ϑ 0,2(q) 2 + ϑ 2,2 (q) 2 η(q) 2 = κ 2 0 (q) 2 + κ 2 2 (q) 2 Z l (q) = ϑ 1,2(q) 2 + ϑ 3,2 (q) 2 η(q) 2 = 2 ϑ 1,2(q) 2 η(q) 2 Z l (q) = 1 η(q) 2 3 j=0 = 2 κ 2 1 (q) 2 ϑ j,2 (q) 2 = κ 2 0 (q) 2 +2 κ 2 1 (q) 2 + κ 2 2 (q) 2 The u(1) characters are κ n j (q) = 1 η(q) ϑ j,n(q), j = 0,1,2 where the Dedekind eta and theta functions are η(q) = q 1/24 n=1(1 q n ), ϑ j,n (q) = k Z q (j+2kn)2 4n The dimer modular invariant partition function Z(q) is the same as in the usual orientation. It also precisely coincides with the MIPF of critical dense polymers (MDPR 2013). The latter coincidence is nontrivial because critical dense polymers requires implementation of a modified (Markov) trace.

23 0-23 Vacuum Boundary Condition on the Strip x = i x 1 1 x 1

24 0-24 Jordan Cells The Hamiltonian for dimers with the (r,s) = (1,1) vacuum boundary condition (no seam) on the strip coincides with the U q (sl(2))-invariant u(1) symmetric XX Hamiltonian H = = N 1 N 1 e j = 1 2 N 1 (σ x j σx j+1 +σy j σy j+1 ) 1 2 i(σz 1 σz N ) (f j f j+1 +f j+1 f j) i(f 1 f 1 f N f N) where σ x,y,z j are Pauli matrices and f j = σj x iσy j, f j = σx j +iσy j. This Hamiltonian is manifestly not Hermitian but the eigenvalues are real (MDRRSA2016). The Jordan canonical forms for N = 2 and N = 4 are 0 0 ( ) ( ) ( ) ( 2) ( ) ( 2) 2 ( ) 2 In the continuum scaling limit, the Hamiltonian gives the Virasoro dilatation operator L 0. Assuming that the Jordan cells persist in this scaling limit, the representation is reducible yet indecomposable and so, as a CFT, dimers is logarithmic! For dimers with (1, s) boundary conditions the conformal weights are 1,s = (2 s)2 1 = 0, 1,0,3,1,15,... s = 1,2,3,4,5,6,...

25 0-25 Summary and Outlook The anisotropic dimer model on the square lattice, with 45 degree rotated dimers, has been solved exactly on a torus using Yang-Baxter integrability. Explicit formulas are found for the counting of dimer configurations on a periodic M N rectangular lattice. The modular invariant partition function precisely coincides with critical dense polymers. Since 1,2 = 1 and the six-vertex model with λ = π 2 on the strip with vacuum boundary conditions exhibits Jordan cells (e.g. Gainutdinov, Nepomechie et al 2015), we argue that dimers is nonunitary and logarithmic with central charge c = 2 and c eff = 1. Yang-Baxter methods can now be applied to study dimers on a strip with many different boundary conditions. General (r, s) boundary conditions are under construction (with Rasmussen). Some insight may also be gained for Aztec diamonds and the six vertex model with domain wall boundary conditions. The inversion identity is the Y-system for dimers. The analogous Y-system for critical bond percolation can be solved analytically (Morin-Duchesne, Klümper, Pearce 2017). Remarkably, the same two-column symplectic binomial building blocks reappear in the patterns of zeros.

26 0-26 Critical Dense Polymer LM(1, 2) Kac Table Central charge: (p,p ) = (1,2) c = 1 6(p p ) 2 pp = 2 s Infinitely extended Kac table of conformal weights: r,s = (p r ps) 2 (p p ) 2 4pp = (2r s)2 1, r,s = 1,2,3,... Kac representation characters: χ r,s (q) = q c/24 q r,s(1 q rs ) n=1 (1 q n ) Irreducible representations are marked by r

27 0-27 Critical Dense Polymers Logarithmic Minimal Models: Yang-Baxter integrable loop models on the square lattice. Face operators defined in diagrammatic planar Temperley-Lieb algebra (Jones1999) X(u) = u = sin(λ u) + sinu 1 p < p coprime integers, λ = (p p)π p u = spectral parameter, = crossing parameter β = 2cosλ = nonlocal loop fugacity Critical Dense Polymers: (p,p ) = (1,2), λ = π 2 Z = loop configs cos N 1u sin N 2u, β = 0 no closed loops space filling dense polymer d SLE path = 2 2 1,1 = 2 There are no local degrees of freedom only nonlocal degrees of freedom in the form of extended polymer segments!