Abstract We investigate the bipartite fidelity for a lattice model described by a logarithmic CFT: the model of critical dense polymers. We define this observable in terms of a partition function on the pants geometry, where d defects enter at the top of the pants lattice and exit in one of the legs. Using the correspondence with the XX spin chain, we obtain an exact closed-form expression for and compute the leading terms in its 1/ N asymptotic expansion as a function of x = N A / N , where N is the lattice width at the top of the pants and N A is the width of the leg where the defects exit. We find an agreement with the results of Stéphan and Dubail for rational CFTs, with the central charge and conformal weights specialised to c = −2 and . We compute a second instance of the bipartite fidelity for d = 2 by imposing a different rule for the connection of the defects. In the conformal setting, this choice corresponds to inserting two boundary condition changing fields of weight that are logarithmic instead of primary. We compute the asymptotic expansion in this case as well and find a simple additive correction compared to , of the form . We confirm this lattice result with a CFT derivation and find that this correction term is identical for all logarithmic theories, independently of c and .
Abstract We investigate the bipartite fidelity for a lattice model described by a logarithmic CFT: the model of critical dense polymers. We define this observable in terms of a partition function on the pants geometry, where d defects enter at the top of the pants lattice and exit in one of the legs. Using the correspondence with the XX spin chain, we obtain an exact closed-form expression for and compute the leading terms in its 1/ N asymptotic expansion as a function of x = N A / N , where N is the lattice width at the top of the pants and N A is the width of the leg where the defects exit. We find an agreement with the results of Stéphan and Dubail for rational CFTs, with the central charge and conformal weights specialised to c = −2 and . We compute a second instance of the bipartite fidelity for d = 2 by imposing a different rule for the connection of the defects. In the conformal setting, this choice corresponds to inserting two boundary condition changing fields of weight that are logarithmic instead of primary. We compute the asymptotic expansion in this case as well and find a simple additive correction compared to , of the form . We confirm this lattice result with a CFT derivation and find that this correction term is identical for all logarithmic theories, independently of c and .
Abstract For a given statistical model, the bipartite fidelity F is computed from the overlap between the groundstate of a system of size N and the tensor product of the groundstates of the same model defined on two subsystems A and B , of respective sizes N A and N B with N = N A + N B . In this paper, we study F for critical lattice models in the case where the full system has periodic boundary conditions. We consider two possible choices of boundary conditions for the subsystems A and B , namely periodic and open. For these two cases, we derive the conformal field theory prediction for the leading terms in the 1/ N expansion of F , in a most general case that corresponds to the insertion of four and five fields, respectively. We provide lattice calculations of F , both exact and numerical, for two free-fermionic lattice models: the XX spin chain and the model of critical dense polymers. We study the asymptotic behaviour of the lattice results for these two models and find an agreement with the predictions of conformal field theory.
For a given statistical model, the bipartite fidelity \\({\\mathcal F}\\) is computed from the overlap between the groundstate of a system of size \\(N\\) and the tensor product of the groundstates of the same model defined on two subsystems \\(A\\) and \\(B\\), of respective sizes \\(N_A\\) and \\(N_B\\) with \\(N = N_A + N_B\\). In this paper, we study \\({\\mathcal F}\\) for critical lattice models in the case where the full system has periodic boundary conditions. We consider two possible choices of boundary conditions for the subsystems \\(A\\) and \\(B\\), namely periodic and open. For these two cases, we derive the conformal field theory prediction for the leading terms in the \\(1/N\\) expansion of \\({\\mathcal F}\\), in a most general case that corresponds to the insertion of four and five fields, respectively. We provide lattice calculations of \\({\\mathcal F}\\), both exact and numerical, for two free-fermionic lattice models: the XX spin chain and the model of critical dense polymers. We study the asymptotic behaviour of the lattice results for these two models and find an agreement with the predictions of conformal field theory.
A lattice model of critical dense polymers is solved exactly on a cylinder with finite circumference. The model is the first member of the Yang–Baxter integrable series of logarithmic minimal models. The cylinder topology allows for non-contractible loops with fugacity α that wind around the cylinder or for an arbitrary number of defects that propagate along the full length of the cylinder. Using an enlarged periodic Temperley–Lieb algebra, we set up commuting transfer matrices acting on states whose links are considered distinct with respect to connectivity around the front or back of the cylinder. These transfer matrices satisfy a functional equation in the form of an inversion identity. For even N , this involves a non-diagonalizable braid operator J and an involution R = − ( J 3 − 12 J )/16 = (−1) F with eigenvalues . This is reminiscent of supersymmetry with a pair of defects interpreted as a fermion. The number of defects thus separates the theory into Ramond ( even), Neveu–Schwarz ( odd) and ( odd) sectors. For the case of loop fugacity α = 2, the inversion identity is solved exactly sector by sector for the eigenvalues in finite geometry. The eigenvalues are classified according to the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity. The finite-size corrections are obtained from Euler–Maclaurin formula. In the scaling limit, we obtain the conformal partition functions as sesquilinear forms and confirm the central charge c = − 2 and conformal weights . Here and in the even sectors with Kac labels r = 1, 2, 3,...; s = 1, 2 while in the odd sectors. Strikingly, the odd sectors exhibit a -extended symmetry but the even sectors do not. Moreover, the naive trace summing over all even sectors does not yield a modular invariant.
We compute lattice correlation functions for the model of critical dense polymers on a semi-infinite cylinder of perimeter n n . In the lattice loop model, contractible loops have a vanishing fugacity whereas non-contractible loops have a fugacity \alpha \in (0,\infty) α ∈ ( 0 , ∞ ) . These correlators are defined as ratios Z(x)/Z_0 Z ( x ) / Z 0 of partition functions, where Z_0 Z 0 is a reference partition function wherein only simple half-arcs are attached to the boundary of the cylinder. For Z(x) Z ( x ) , the boundary of the cylinder is also decorated with simple half-arcs, but it also has two special positions 1 1 and x x where the boundary condition is different. We investigate two such kinds of boundary conditions: (i) there is a single node at each of these points where a long arc is attached, and (ii) there are pairs of adjacent nodes at these points where two long arcs are attached. We find explicit expressions for these correlators for finite n n using the representation of the enlarged periodic Temperley-Lieb algebra in the XX spin chain. The resulting asymptotics as n\to \infty n → ∞ are expressed as simple integrals that depend on the scaling parameter \tau = \frac {x-1} n \in (0,1) τ = x − 1 n ∈ ( 0 , 1 ) . For small \tau τ , the leading behaviours are proportional to \tau^{1/4} τ 1 / 4 , \tau^{1/4}\log \tau τ 1 / 4 log τ ,
A lattice model of critical dense polymers is solved exactly for finite strips. The model is the first member of the principal series of the recently introduced logarithmic minimal models. The key to the solution is a functional equation in the form of an inversion identity satisfied by the commuting double-row transfer matrices. This is established directly in the planar Temperley–Lieb algebra and holds independently of the space of link states on which the transfer matrices act. Different sectors are obtained by acting on link states with s −1 defects where s = 1,2,3,... is an extended Kac label. The bulk and boundary free energies and finite-size corrections are obtained from the Euler–Maclaurin formula. The eigenvalues of the transfer matrix are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields a selection rule for the physically relevant solutions to the inversion identity and explicit finitized characters for the associated quasi-rational representations. In particular, in the scaling limit, we confirm the central charge c = −2 and conformal weights Δ s = ((2− s ) 2 −1)/8 for s = 1,2,3,.... We also discuss a diagrammatic implementation of fusion and show with examples how indecomposable representations arise. We examine the structure of these representations and present a conjecture for the general fusion rules within our framework.
A lattice model of critical dense polymers is solved exactly on a cylinder with finite circumference. The model is the first member of the Yang–Baxter integrable series of logarithmic minimal models. The cylinder topology allows for non-contractible loops with fugacity α that wind around the cylinder or for an arbitrary number of defects that propagate along the full length of the cylinder. Using an enlarged periodic Temperley–Lieb algebra, we set up commuting transfer matrices acting on states whose links are considered distinct with respect to connectivity around the front or back of the cylinder. These transfer matrices satisfy a functional equation in the form of an inversion identity. For even N, this involves a non-diagonalizable braid operator J and an involution R = − (J3 − 12J)/16 = (−1)F with eigenvalues . This is reminiscent of supersymmetry with a pair of defects interpreted as a fermion. The number of defects thus separates the theory into Ramond ( even), Neveu–Schwarz ( odd) and ( odd) sectors. For the case of loop fugacity α = 2, the inversion identity is solved exactly sector by sector for the eigenvalues in finite geometry. The eigenvalues are classified according to the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity. The finite-size corrections are obtained from Euler–Maclaurin formula. In the scaling limit, we obtain the conformal partition functions as sesquilinear forms and confirm the central charge c = − 2 and conformal weights . Here and in the even sectors with Kac labels r = 1, 2, 3,...;s = 1, 2 while in the odd sectors. Strikingly, the odd sectors exhibit a -extended symmetry but the even sectors do not. Moreover, the naive trace summing over all even sectors does not yield a modular invariant.