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Paul A. Pearce

Publications and source records attributed to Paul A. Pearce.

At least 37 records · Page 2Linked to original sources
arXiv2014

Kac boundary conditions of the logarithmic minimal models

We develop further the implementation and analysis of Kac boundary conditions in the general logarithmic minimal models ${\cal LM}(p,p')$ with $1\le p<p'$ and $p,p'$ coprime. Working in a strip geometry, we consider the $(r,s)$ boundary conditions, which are organized into infinitely extended Kac tables labeled by $r,s=1,2,3,...$. They are conjugate to Virasoro Kac representations with conformal dimensions $Δ_{r,s}$ given by the usual Kac formula. On a finite strip of width $N$, built from a square lattice, the associated integrable boundary conditions are constructed by acting on the vacuum $(1,1)$ boundary with an $s$-type seam of width $s-1$ columns and an $r$-type seam of width $ρ-1$ columns. The $r$-type seam contains an arbitrary boundary field $ξ$. The usual fusion construction of the $r$-type seam relies on the existence of Wenzl-Jones projectors restricting its application to $r\leρ<p'$. This limitation was recently removed by Pearce, Rasmussen and Villani who further conjectured that the conformal boundary conditions labeled by $r$ are realized, in particular, for $ρ=ρ(r)=\lfloor \frac{rp'}{p}\rfloor$. In this paper, we confirm this conjecture by performing extensive numerics on the commuting double row transfer matrices and their associated quantum Hamiltonian chains. Letting $[x]$ denote the fractional part, we fix the boundary field to the specialized values $ξ=\fracπ{2}$ if $[\fracρ{p'}]=0$ and $ξ=[\frac{ρp}{p'}]\fracπ{2}$ otherwise. For these boundary conditions, we obtain the Kac conformal weights $Δ_{r,s}$ by numerically extrapolating the finite-size corrections to the lowest eigenvalue of the quantum Hamiltonians out to sizes $N\le 32-ρ-s$. Additionally, by solving local inversion relations, we obtain general analytic expressions for the boundary free energies allowing for more accurate estimates of the conformal data.

hep-th↗
arXiv2014

Logarithmic Superconformal Minimal Models

The higher fusion level logarithmic minimal models LM(P,P';n) have recently been constructed as the diagonal GKO cosets (A_1^{(1)})_k oplus (A_1^{(1)})_n / (A_1^{(1)})_{k+n} where n>0 is an integer fusion level and k=nP/(P'-P)-2 is a fractional level. For n=1, these are the logarithmic minimal models LM(P,P'). For n>1, we argue that these critical theories are realized on the lattice by n x n fusion of the n=1 models. For n=2, we call them logarithmic superconformal minimal models LSM(p,p') where P=|2p-p'|, P'=p' and p,p' are coprime, and they share the central charges of the rational superconformal minimal models SM(P,P'). Their mathematical description entails the fused planar Temperley-Lieb algebra which is a spin-1 BMW tangle algebra with loop fugacity beta_2=x^2+1+x^{-2} and twist omega=x^4 where x=e^{i(p'-p)pi/p'}. Examples are superconformal dense polymers LSM(2,3) with c=-5/2, beta_2=0 and superconformal percolation LSM(3,4) with c=0, beta_2=1. We calculate the free energies analytically. By numerically studying finite-size spectra on the strip with appropriate boundary conditions in Neveu-Schwarz and Ramond sectors, we argue that, in the continuum scaling limit, these lattice models are associated with the logarithmic superconformal models LM(P,P';2). For system size N, we propose finitized Kac character formulas whose P,P' dependence only enters in the fractional power of q in a prefactor. These characters involve Motzkin and Riordan polynomials defined in terms of q-trinomial coefficients. Using the Hamiltonian limit, we argue that there exist reducible yet indecomposable representations for which the Virasoro dilatation operator L_0 exhibits rank-2 Jordan blocks confirming that these theories are indeed logarithmic. We relate these results to the N=1 superconformal representation theory.

hep-th↗
arXiv2014

Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models

A Temperley-Lieb (TL) loop model is a Yang-Baxter integrable lattice model with nonlocal degrees of freedom. On a strip of width N, the evolution operator is the double-row transfer tangle D(u), an element of the TL algebra TL_N(beta) with loop fugacity beta=2cos(lambda). Similarly on a cylinder, the single-row transfer tangle T(u) is an element of the enlarged periodic TL algebra. The logarithmic minimal models LM(p,p') comprise a subfamily of the TL loop models for which the crossing parameter lambda=(p'-p)pi/p' is parameterised by coprime integers 0 2 takes the form of functional relations for D(u) and T(u) of polynomial degree p'. These derive from fusion hierarchies of commuting transfer tangles D^{m,n}(u) and T^{m,n}(u) where D(u)=D^{1,1}(u) and T(u)=T^{1,1}(u). The fused transfer tangles are constructed from (m,n)-fused face operators involving Wenzl-Jones projectors P_k on k=m or k=n nodes. Some projectors P_k are singular for k>p'-1, but we argue that D^{m,n}(u) and T^{m,n}(u) are well defined for all m,n. For generic lambda, we derive the fusion hierarchies and the associated T- and Y-systems. For the logarithmic theories, the closure of the fusion hierarchies at n=p' translates into functional relations of polynomial degree p' for D^{m,1}(u) and T^{m,1}(u). We also derive the closure of the Y-systems for the logarithmic theories. The T- and Y-systems are the key to exact integrability and we observe that the underlying structure of these functional equations relate to Dynkin diagrams of affine Lie algebras.

math-ph↗
arXiv2013

Coset construction of logarithmic minimal models: branching rules and branching functions

Working in the Virasoro picture, it is argued that the logarithmic minimal models LM(p,p')=LM(p,p';1) can be extended to an infinite hierarchy of logarithmic conformal field theories LM(p,p';n) at higher fusion levels n=1,2,3,.... From the lattice, these theories are constructed by fusing together n x n elementary faces of the appropriate LM(p,p') models. It is further argued that all of these logarithmic theories are realized as diagonal cosets (A_1^{(1)})_k \oplus (A_1^{(1)})_n / (A_1^{(1)})_{k+n} where n is the integer fusion level and k=np/(p'-p)-2 is a fractional level. These cosets mirror the cosets of the higher fusion level minimal models of the form M(m,m';n), but are associated with certain reducible representations. We present explicit branching rules for characters in the form of multiplication formulas arising in the logarithmic limit of the usual Goddard-Kent-Olive coset construction of the non-unitary minimal models M(m,m';n). The limiting branching functions play the role of Kac characters for the LM(p,p';n) theories.

hep-th↗
arXiv2013

Modular invariant partition function of critical dense polymers

A lattice model of critical dense polymers is solved exactly for arbitrary system size on the torus. More generally, an infinite family of lattice loop models is studied on the torus and related to the corresponding Fortuin-Kasteleyn random cluster models. Starting with a cylinder, the commuting periodic single-row transfer matrices are built from the periodic Temperley-Lieb algebra extended by the shift operators Omega and Omega^{-1}. In this enlarged algebra, the non-contractible loop fugacity is alpha and the contractible loop fugacity is beta. The torus is formed by gluing the top and bottom of the cylinder. This gives rise to a variety of non-contractible loops winding around the torus. Because of their nonlocal nature, the standard matrix trace does not produce the proper geometric torus. Instead, we introduce a modified matrix trace for this purpose. This is achieved by using a representation of the enlarged periodic Temperley-Lieb algebra with a parameter v that keeps track of the winding of defects on the cylinder. The transfer matrix representatives and their eigenvalues thus depend on v. The modified trace is constructed as a linear functional on planar connectivity diagrams in terms of matrix traces Tr_d (with a fixed number of defects d) and Chebyshev polynomials of the first kind. For critical dense polymers, where beta=0, the transfer matrix eigenvalues are obtained by solving a functional equation in the form of an inversion identity. The solution depends on d and is subject to selection rules which we prove. Simplifications occur if all non-contractible loop fugacities are set to alpha=2 in which case the traces are evaluated at v=1. In the continuum scaling limit, the corresponding conformal torus partition function obtained from finite-size corrections agrees with the known modular invariant partition function of symplectic fermions.

hep-th↗
arXiv2013

Infinitely extended Kac table of solvable critical dense polymers

Solvable critical dense polymers is a Yang-Baxter integrable model of polymers on the square lattice. It is the first member LM(1,2) of the family of logarithmic minimal models LM(p,p'). The associated logarithmic conformal field theory admits an infinite family of Kac representations labelled by the Kac labels r,s=1,2,.... In this paper, we explicitly construct the conjugate boundary conditions on the strip. The boundary operators are labelled by the Kac fusion labels (r,s)=(r,1) x (1,s) and involve a boundary field xi. Tuning the field xi appropriately, we solve exactly for the transfer matrix eigenvalues on arbitrary finite-width strips and obtain the conformal spectra using the Euler-Maclaurin formula. The key to the solution is an inversion identity satisfied by the commuting double-row transfer matrices. The transfer matrix eigenvalues are classified by 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 which takes the form of a decomposition into irreducible blocks corresponding combinatorially to finitized characters given by generalized q-Catalan polynomials. This decomposition is in accord with the decomposition of the Kac characters into irreducible characters. In the scaling limit, we confirm the central charge c=-2 and the Kac formula for the conformal weights Delta_{r,s}=((2r-s)^2-1)/8 for r,s=1,2,3,... in the infinitely extended Kac table.

math-ph↗
arXiv2012

Off-Critical Logarithmic Minimal Models

We consider the integrable minimal models ${\cal M}(m,m';t)$, corresponding to the $φ_{1,3}$ perturbation off-criticality, in the {\it logarithmic limit\,} $m, m'\to\infty$, $m/m'\to p/p'$ where $p, p'$ are coprime and the limit is taken through coprime values of $m,m'$. We view these off-critical minimal models ${\cal M}(m,m';t)$ as the continuum scaling limit of the Forrester-Baxter Restricted Solid-On-Solid (RSOS) models on the square lattice. Applying Corner Transfer Matrices to the Forrester-Baxter RSOS models in Regime III, we argue that taking first the thermodynamic limit and second the {\it logarithmic limit\,} yields off-critical logarithmic minimal models ${\cal LM}(p,p';t)$ corresponding to the $φ_{1,3}$ perturbation of the critical logarithmic minimal models ${\cal LM}(p,p')$. Specifically, in accord with the Kyoto correspondence principle, we show that the logarithmic limit of the one-dimensional configurational sums yields finitized quasi-rational characters of the Kac representations of the critical logarithmic minimal models ${\cal LM}(p,p')$. We also calculate the logarithmic limit of certain off-critical observables ${\cal O}_{r,s}$ related to One Point Functions and show that the associated critical exponents $β_{r,s}=(2-α)\,Δ_{r,s}^{p,p'}$ produce all conformal dimensions $Δ_{r,s}^{p,p'}<{(p'-p)(9p-p')\over 4pp'}$ in the infinitely extended Kac table. The corresponding Kac labels $(r,s)$ satisfy $(p s-p' r)^2< 8p(p'-p)$. The exponent $2-α={p'\over 2(p'-p)}$ is obtained from the logarithmic limit of the free energy giving the conformal dimension $Δ_t={1-α\over 2-α}={2p-p'\over p'}=Δ_{1,3}^{p,p'}$ for the perturbing field $t$. As befits a non-unitary theory, some observables ${\cal O}_{r,s}$ diverge at criticality.

hep-th↗
arXiv2011

Geometric Exponents of Dilute Logarithmic Minimal Models

The fractal dimensions of the hull, the external perimeter and of the red bonds are measured through Monte Carlo simulations for dilute minimal models, and compared with predictions from conformal field theory and SLE methods. The dilute models used are those first introduced by Nienhuis. Their loop fugacity is beta = -2cos(pi/barkappa}) where the parameter barkappa is linked to their description through conformal loop ensembles. It is also linked to conformal field theories through their central charges c = 13 - 6(barkappa + barkappa^{-1}) and, for the minimal models of interest here, barkappa = p/p' where p and p' are two coprime integers. The geometric exponents of the hull and external perimeter are studied for the pairs (p,p') = (1,1), (2,3), (3,4), (4,5), (5,6), (5,7), and that of the red bonds for (p,p') = (3,4). Monte Carlo upgrades are proposed for these models as well as several techniques to improve their speeds. The measured fractal dimensions are obtained by extrapolation on the lattice size H,V -> infinity. The extrapolating curves have large slopes; despite these, the measured dimensions coincide with theoretical predictions up to three or four digits. In some cases, the theoretical values lie slightly outside the confidence intervals; explanations of these small discrepancies are proposed.

cond-mat.stat-mech↗
arXiv2011

Coset Graphs in Bulk and Boundary Logarithmic Minimal Models

The logarithmic minimal models are not rational but, in the W-extended picture, they resemble rational conformal field theories. We argue that the W-projective representations are fundamental building blocks in both the boundary and bulk description of these theories. In the boundary theory, each W-projective representation arising from fundamental fusion is associated with a boundary condition. Multiplication in the associated Grothendieck ring leads to a Verlinde-like formula involving A-type twisted affine graphs A^{(2)}_{p} and their coset graphs A^{(2)}_{p,p'}=A^{(2)}_{p} x A^{(2)}_{p'}/Z_2. This provides compact formulas for the conformal partition functions with W-projective boundary conditions. On the torus, we propose modular invariant partition functions as sesquilinear forms in W-projective and rational minimal characters and observe that they are encoded by the same coset fusion graphs.

hep-th↗
arXiv2010

Solvable Critical Dense Polymers on the Cylinder

A lattice model of critical dense polymers is solved exactly on a cylinder with finite circumference. The model is the first member LM(1,2) of the Yang-Baxter integrable series of logarithmic minimal models. The cylinder topology allows for non-contractible loops with fugacity alpha that wind around the cylinder or for an arbitrary number ell 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^3-12J)/16=(-1)^{F} with eigenvalues R=(-1)^{ell/2}. The number of defects ell separates the theory into sectors. For the case of loop fugacity alpha=2, the inversion identity is solved exactly for the eigenvalues in finite geometry. The eigenvalues are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane yielding selection rules. The finite-size corrections are obtained from Euler-Maclaurin formulas. In the scaling limit, we obtain the conformal partition functions and confirm the central charge c=-2 and conformal weights Delta_t=(t^2-1)/8. Here t=ell/2 and t=2r-s in the ell even sectors with Kac labels r=1,2,3,...; s=1,2 while t is half-integer in the ell odd sectors. Strikingly, the ell/2 odd sectors exhibit a W-extended symmetry but the ell/2 even sectors do not. Moreover, the naive trace summing over all ell even sectors does not yield a modular invariant.

hep-th↗
arXiv2009

Grothendieck ring and Verlinde-like formula for the W-extended logarithmic minimal model WLM(1,p)

We consider the Grothendieck ring of the fusion algebra of the W-extended logarithmic minimal model WLM(1,p). Informally, this is the fusion ring of W-irreducible characters so it is blind to the Jordan block structures associated with reducible yet indecomposable representations. As in the rational models, the Grothendieck ring is described by a simple graph fusion algebra. The 2p-dimensional matrices of the regular representation are mutually commuting but not diagonalizable. They are brought simultaneously to Jordan form by the modular data coming from the full (3p-1)-dimensional S-matrix which includes transformations of the p-1 pseudo-characters. The spectral decomposition yields a Verlinde-like formula that is manifestly independent of the modular parameter $τ$ but is, in fact, equivalent to the Verlinde-like formula recently proposed by Gaberdiel and Runkel involving a $τ$-dependent S-matrix.

hep-th↗
arXiv2009

Physical Combinatorics and Quasiparticles

We consider the physical combinatorics of critical lattice models and their associated conformal field theories arising in the continuum scaling limit. As examples, we consider A-type unitary minimal models and the level-1 sl(2) Wess-Zumino-Witten (WZW) model. The Hamiltonian of the WZW model is the $U_q(sl(2))$ invariant XXX spin chain. For simplicity, we consider these theories only in their vacuum sectors on the strip. Combinatorially, fermionic particles are introduced as certain features of RSOS paths. They are composites of dual-particles and exhibit the properties of quasiparticles. The particles and dual-particles are identified, through an energy preserving bijection, with patterns of zeros of the eigenvalues of the fused transfer matrices in their analyticity strips. The associated (m,n) systems arise as geometric packing constraints on the particles. The analyticity encoded in the patterns of zeros is the key to the analytic calculation of the excitation energies through the Thermodynamic Bethe Ansatz (TBA). As a by-product of our study, in the case of the WZW or XXX model, we find a relation between the location of the Bethe root strings and the location of the transfer matrix 2-strings.

hep-th↗
arXiv2008

Geometric Exponents, SLE and Logarithmic Minimal Models

In statistical mechanics, observables are usually related to local degrees of freedom such as the Q < 4 distinct states of the Q-state Potts models or the heights of the restricted solid-on-solid models. In the continuum scaling limit, these models are described by rational conformal field theories, namely the minimal models M(p,p') for suitable p, p'. More generally, as in stochastic Loewner evolution (SLE_kappa), one can consider observables related to nonlocal degrees of freedom such as paths or boundaries of clusters. This leads to fractal dimensions or geometric exponents related to values of conformal dimensions not found among the finite sets of values allowed by the rational minimal models. Working in the context of a loop gas with loop fugacity beta = -2 cos(4 pi/kappa), we use Monte Carlo simulations to measure the fractal dimensions of various geometric objects such as paths and the generalizations of cluster mass, cluster hull, external perimeter and red bonds. Specializing to the case where the SLE parameter kappa = 4p'/p is rational with p < p', we argue that the geometric exponents are related to conformal dimensions found in the infinitely extended Kac tables of the logarithmic minimal models LM(p,p'). These theories describe lattice systems with nonlocal degrees of freedom. We present results for critical dense polymers LM(1,2), critical percolation LM(2,3), the logarithmic Ising model LM(3,4), the logarithmic tricritical Ising model LM(4,5) as well as LM(3,5). Our results are compared with rigourous results from SLE_kappa, with predictions from theoretical physics and with other numerical experiments. Throughout, we emphasize the relationships between SLE_kappa, geometric exponents and the conformal dimensions of the underlying CFTs.

cond-mat.stat-mech↗
arXiv2008

W-Extended Fusion Algebra of Critical Percolation

Two-dimensional critical percolation is the member LM(2,3) of the infinite series of Yang-Baxter integrable logarithmic minimal models LM(p,p'). We consider the continuum scaling limit of this lattice model as a `rational' logarithmic conformal field theory with extended W=W_{2,3} symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture. We find that the representation content of the ensuing closed fusion algebra contains 26 W-indecomposable representations with 8 rank-1 representations, 14 rank-2 representations and 4 rank-3 representations. We identify these representations with suitable limits of Yang-Baxter integrable boundary conditions on the lattice and obtain their associated W-extended characters. The latter decompose as finite non-negative sums of W-irreducible characters of which 13 are required. Implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit Cayley table. The closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry.

hep-th↗
arXiv2008

Integrable Boundary Conditions and W-Extended Fusion in the Logarithmic Minimal Models LM(1,p)

We consider the logarithmic minimal models LM(1,p) as `rational' logarithmic conformal field theories with extended W symmetry. To make contact with the extended picture starting from the lattice, we identify 4p-2 boundary conditions as specific limits of integrable boundary conditions of the underlying Yang-Baxter integrable lattice models. Specifically, we identify 2p integrable boundary conditions to match the 2p known irreducible W-representations. These 2p extended representations naturally decompose into infinite sums of the irreducible Virasoro representations (r,s). A further 2p-2 reducible yet indecomposable W-representations of rank 2 are generated by fusion and these decompose as infinite sums of indecomposable rank-2 Virasoro representations. The fusion rules in the extended picture are deduced from the known fusion rules for the Virasoro representations of LM(1,p) and are found to be in agreement with previous works. The closure of the fusion algebra on a finite number of representations in the extended picture is remarkable confirmation of the consistency of the lattice approach.

hep-th↗
arXiv2007

Fusion Algebras of Logarithmic Minimal Models

We present explicit conjectures for the chiral fusion algebras of the logarithmic minimal models LM(p,p') considering Virasoro representations with no enlarged or extended symmetry algebra. The generators of fusion are countably infinite in number but the ensuing fusion rules are quasi-rational in the sense that the fusion of a finite number of representations decomposes into a finite direct sum of representations. The fusion rules are commutative, associative and exhibit an sl(2) structure but require so-called Kac representations which are reducible yet indecomposable representations of rank 1. In particular, the identity of the fundamental fusion algebra is in general a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the results of Gaberdiel and Kausch for p=1 and with Eberle and Flohr for (p,p')=(2,5) corresponding to the logarithmic Yang-Lee model. In the latter case, we confirm the appearance of indecomposable representations of rank 3. We also find that closure of a fundamental fusion algebra is achieved without the introduction of indecomposable representations of rank higher than 3. The conjectured fusion rules are supported, within our lattice approach, by extensive numerical studies of the associated integrable lattice models. Details of our lattice findings and numerical results will be presented elsewhere. The agreement of our fusion rules with the previous fusion rules lends considerable support for the identification of the logarithmic minimal models LM(p,p') with the augmented c_{p,p'} (minimal) models defined algebraically.

hep-th↗
arXiv2007

Fusion algebra of critical percolation

We present an explicit conjecture for the chiral fusion algebra of critical percolation considering Virasoro representations with no enlarged or extended symmetry algebra. The representations we take to generate fusion are countably infinite in number. The ensuing fusion rules are quasi-rational in the sense that the fusion of a finite number of these representations decomposes into a finite direct sum of these representations. The fusion rules are commutative, associative and exhibit an sl(2) structure. They involve representations which we call Kac representations of which some are reducible yet indecomposable representations of rank 1. In particular, the identity of the fusion algebra is a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the recent results of Eberle-Flohr and Read-Saleur. Notably, in agreement with Eberle-Flohr, we find the appearance of indecomposable representations of rank 3. Our fusion rules are supported by extensive numerical studies of an integrable lattice model of critical percolation. Details of our lattice findings and numerical results will be presented elsewhere.

hep-th↗