2017/01/31 by Alexi Morin-Duchesne, Andreas Klümper, Paul A. Pearce +1
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Ansatz #Bethe ansatz #Boundary (topology) #Conformal map #Conformal symmetry #Lattice (music) #Logarithm #Nonlinear system #Partition function (quantum field theory) #Physics of Superconductivity and Magnetism #Quantum many-body systems #cond-mat.stat-mech #hep-th #math-ph #math.MP
paper · pdf · doi:10.1088/1742-5468/aa75e2
74 pages
openalex created_date 2017/02/10 · openalex publication_date 2017/08/01 · arxiv created 2017/08/11 · arxiv updated 2017/09/13 · openalex updated_date 2026/08/05
Abstract Using the planar Temperley–Lieb algebra, critical bond percolation on the square lattice can be reformulated as a loop model. In this form, it is incorporated as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script">L</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script">M</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:math> in the Yang–Baxter integrable family of logarithmic minimal models <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script">L</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script">M</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>′</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:math> . We consider this model of percolation in the presence of boundaries and with periodic boundary conditions. Inspired by Kuniba, Sakai and Suzuki, we rewrite the recently obtained infinite Y -system of functional equations. In this way, we obtain nonlinear integral equations in the form of a closed finite set of TBA equations described by a D 3 Dynkin diagram. Following the methods of Klümper and Pearce, we solve the TBA equations for the conformal finite-size corrections. For the ground states of the standard modules on the strip, these agree with the known central charge c = 0 and conformal weights <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> </mml:mstyle> </mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mo>⩾</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:mstyle> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mi>r</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mrow> <mml:mspace width="0pt"/> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>24</mml:mn> </mml:mstyle> </mml:math> . For the periodic case, the finite-size corrections agree with the conformal weights <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> </mml:mstyle> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> </mml:mstyle> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>s</mml:mi>