There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

Quantum inverse scattering method and correlation functions

A review is given on the foundations and applications of non-Hermitian classical and quantum physics. First, key theorems and central concepts in non-Hermitian linear algebra, including Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity and parity-time symmetry, are delineated in a pedagogical and mathematically coherent manner. Building on these, we provide an overview of how diverse classical systems, ranging from photonics, mechanics, electrical circuits, acoustics to active matter, can be used to simulate non-Hermitian wave physics. In particular, we discuss rich and unique phenomena found therein, such as unidirectional invisibility, enhanced sensitivity, topological energy transfer, coherent perfect absorption, single-mode lasing, and robust biological transport. We then explain in detail how non-Hermitian operators emerge as an effective description of open quantum systems on the basis of the Feshbach projection approach and the quantum trajectory approach. We discuss their applications to physical systems relevant to a variety of fields, including atomic, molecular and optical physics, mesoscopic physics, and nuclear physics with emphasis on prominent phenomena/subjects in quantum regimes, such as quantum resonances, superradiance, continuous quantum Zeno effect, quantum critical phenomena, Dirac spectra in quantum chromodynamics, and nonunitary conformal field theories. Finally, we introduce the notion of band topology in complex spectra of non-Hermitian systems and present their classifications by providing the proof, firstly given by this review in a complete manner, as well as a number of instructive examples. Other topics related to non-Hermitian physics, including nonreciprocal transport, speed limits, nonunitary quantum walk, are also reviewed.

Non-Hermitian Physics

http://people.physics.anu.edu.au/~drt105/papers/1503.07584.pdf

Boundary algebras and Kac modules for logarithmic minimal models

We show how to use the link representation of the transfer matrix DN of loop models on the lattice to calculate partition functions, at criticality, of the Fortuin–Kasteleyn model with various boundary conditions and parameter and, more specifically, partition functions of the corresponding Q-Potts spin models, with Q = β2. The braid limit of DN is shown to be a central element FN(β) of the Temperley–Lieb algebra TLN(β), its eigenvalues are determined and, for generic β, a basis of its eigenvectors is constructed using the Wenzl–Jones projector. With any element of this basis is associated a number of defects d, 0 ≤ d ≤ N, and the basis vectors with the same d span a sector. Because components of these eigenvectors are singular when and , the link representations of FN and DN are shown to have Jordan blocks between sectors d and d' when d − d' d'). When a and b do not satisfy the previous constraint, DN is diagonalizable.

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The Jordan structure of two-dimensional loop models

Among the lattice loop models defined by Pearce, Rasmussen and Zuber (2006), the model corresponding to critical dense polymers ($\beta = 0$) is the only one for which an inversion relation for the transfer matrix $D_N(u)$ was found by Pearce and Rasmussen (2007). From this result, they identified the set of possible eigenvalues for $D_N(u)$ and gave a conjecture for the degeneracies of its relevant eigenvalues in the link representation, in the sector with $d$ defects. In this paper, we set out to prove this conjecture, using the homomorphism of the $TL_N (\beta)$ algebra between the loop model link representation and that of the XXZ model for $\beta = -(q+q^{-1})$.

A Proof of Selection Rules for Critical Dense Polymers

In the presence of a conserved quantity, symmetry-resolved entanglement entropies are a refinement of the usual notion of entanglement entropy of a subsystem. For critical 1d quantum systems, it was recently shown in various contexts that these quantities generally obey entropy equipartition in the scaling limit, i.e. they become independent of the symmetry sector. 
In this paper, we examine the finite-size corrections to the entropy equipartition phenomenon, and show that the nature of the symmetry group plays a crucial role. In the case of a discrete symmetry group, the corrections decay algebraically with system size, with exponents related to the operators' scaling dimensions. In contrast, in the case of a U(1) symmetry group, the corrections only decay logarithmically with system size, with model-dependent prefactors. We show that the determination of these prefactors boils down to the computation of twisted overlaps.

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Finite-size corrections in critical symmetry-resolved entanglement

We show how to use the link representation of the transfer matrix $D_N$ of loop models on the lattice to calculate partition functions, at criticality, of the Fortuin-Kasteleyn model with various boundary conditions and parameter $\beta = 2 \cos(\pi(1-a/b)), a,b\in \mathbb N$ and, more specifically, partition functions of the corresponding $Q$-Potts spin models, with $Q=\beta^2$. The braid limit of $D_N$ is shown to be a central element $F_N(\beta)$ of the Temperley-Lieb algebra $TL_N(\beta)$, its eigenvalues are determined and, for generic $\beta$, a basis of its eigenvectors is constructed using the Wenzl-Jones projector. To any element of this basis is associated a number of defects $d$, $0\le d\le N$, and the basis vectors with the same $d$ span a sector. Because components of these eigenvectors are singular when $b \in \mathbb{Z}^*$ and $a \in 2 \mathbb{Z} + 1$, the link representations of $F_N$ and $D_N$ are shown to have Jordan blocks between sectors $d$ and $d'$ when $d-d' d'$). When $a$ and $b$ do not satisfy the previous constraint, $D_N$ is diagonalizable.

Alexi Morin-Duchesne

Papers

Boundary algebras and Kac modules for logarithmic minimal models

The Jordan structure of two-dimensional loop models

A Proof of Selection Rules for Critical Dense Polymers

Finite-size corrections in critical symmetry-resolved entanglement

The Jordan Structure of Two Dimensional Loop Models