RAVEN was a project funded by the EU's Research Executive agency (REA). It supported an International Incoming Fellowship for Dr Nikolay Izmailyan to work with Professor Ralph Kenna (the PI) over a two-year period from 2013 to 2015. Here we summarise some details about the grant and what was achieved by the research it funded.
International Incoming Fellowships
International Incoming Fellowships (IIFs) are a Marie Curie Action specially designed to fund top-class researchers from Third Countries to work on projects in Europe. This helps to develop research cooperation between Europe and other parts of the world – to everybody’s benefit.
In the case of RAVEN, the funding supported a move by Nikolay Izmailyan from Armenia to the AMRC on 01.04.2013 to stay in the UK for 2 years working closely with Ralph Kenna.
The RAVEN proposal scored 87.90/100.00.
Abstract of the Proposal
A cornerstone in the study of phase transitions is the principal of universality. This maintains that entire families of systems behave identically in
the neighbourhood of criticality, such as the transition point between a liquid and a gas or at the Curie point in a magnet, at which two phases
become indistinguishable. Near the critical point, thermodynamic observables and critical exponents do not depend on the details of intermolecular
interactions. Instead they depend only on the range of interactions, symmetries and spatial dimensionality. This fact allows us to understand real
materials and systems through simplified mathematical models.
The universality concept is commonly stated together with the hypotheses of scaling and finite-size scaling. The associated theories have been
mostly successful in describing critical and non-critical properties, but significant discrepancies between them and experiments remain. To
understand the experiments, the theories have to be improved. This project seeks to increase our understanding by researching corrections to
scaling. Our proposal is to investigate statistical mechanical models in an attempt to place our theoretical understanding of critical phenomena
closer on firmer ground and to render them closer to experimental measurements.
We will especially target universality, scaling, and finite-size effects in two dimensional models of statistical mechanics as these can be tackled
using exact methods, as well as analytic and numerical ones. In addition, more challenging three dimensional models will be investigated.
Theories of critical phenomena in particular are crucial in our understanding of how everything depends on everything else in many disciplines
outside physics. It thus permeates all of natural sciences and even beyond. It is therefore a priority that this foundation stone be correct, exact and
Summary of Project
This project was designed to addresses discrepancies between theory and experiment in one of the foundation stones of modern physics: critical phenomena and phase transitions. Phase transitions occur when systems change from one state to another, such as from a liquid to a gas or from a paramagnet to a ferromagnet. Very few of models, which exhibit phase transitions, are solvable by exact mathematical methods. Most of those which are integrable are so only in one or two dimensions. Other models, including the most realistic and physically interesting ones, have to be approached using approximate methods, and this creates problems with matching to experiments.
The aim of this innovative project was to improve and develop scaling, finite-size scaling, and corrections-to-scaling theories of statistical physics, using an array of analytical, exact and numerical mathematical tools with a view to improving the match with experiment. This aim was be achieved through 3 specific objectives:
Specific Objective 1: The study of universality in analytic corrections to scaling in lattice models.
Specific Objective 2: The study of finite-size effects in two-dimensional lattice models.
Specific Objective 3: Investigation of spin models beyond two dimensions.
These objectives were to be met through four distinct but overlapping tasks which were as follows:
Task 1 (addressing SO1): Amplitude ratios in the bulk in 2D
Task 2 (addressing SO2): Finite-size effects in 2D spin models
Task 3 (addressing SO’s 1&2): Finite-size effects in 2D dimer models
Task 4 (addressing SO2): Beyond two dimensions
The Fellow was immediately identified as a perfect match for the Host and the Statistical Physics Group at Coventry University’s Applied Mathematics Research Centre. For these reasons, and due to excellent support by the University’s Business Development Support Office, integration of the Fellow into the Group and Centre was seamless and we were able to star up scientific collaboration immediately.
So far (end of May 2015) 17 publications have resulted from the project. Ten of these are jointly authored by the Fellow and the PI. Two are authored by the Fellow with other scientists. Five are authored by the PI with other scientists. Three further papers, jointly authored by the PI and Fellow are at various stages of preparedness: one is under review at Phys. Rev. E, one is shortly to be submitted to Phys. Rev. Lett. and research for one is still being carried out. All four tasks were addressed in these 20 publications. One publication was selected for IOP SELECT (articles chosen by Institute of Physics editors for their novelty, significance and potential impact on future research). The paper has already been cited 15 times since publication in 2014.
In addressing Task 1, for example, we solved exactly the Ising model in two dimensions with duality twisted boundary conditions and found the new set of universal amplitude ratios for that model. Regarding Task 2, for the first time we confirmed the conformal field prediction for the corner contribution to the free energy for the Ising model on the square lattice and triangular lattice with free boundaries. For Task 3, we confirmed the conformal field theory prediction for the corner free energy of the dimer and spanning tree models, for which the central charge is c = -2. In a body of work related to Tasks 2 and 3, we obtained a new expression for the two-point resistance between two arbitrary nodes of the resistor network, which is simpler and can be easier to use in practice. We also used an analytic approach to develop exact expressions for the two-point resistance between arbitrary nodes on certain non-regular resistor networks. This generalizes previous approaches, which only deliver results for networks of more regular geometry. For Task 4, we found that for ratios and combinations of amplitudes which are universal, Fisher renormalization is involuntary. We also investigated the generalized Potts model on a Bethe lattice with z neighbours and determined the number of invisible states required to manifest the equivalent Blume-Emery-Griffiths tri-criticality In the q=2 case. Finally, scaling and finite-size scaling above the upper critical dimension has been reformulated and hyperscaling and Fisher’s relation extended to that circumstance.
The Fellow accepted 6 invitations to present at international conferences including in China, Germany, Russia, the Czech Republic, and Greece. The PI gave related presentations in China and France. The Fellow accepted 4 invitations to give seminars, including in Leipzig and Freiburg. The PI gave related talks at Oxford, York and Lviv. The Fellow had extended visits (1 month long) to Yerevan, Leipzig, Freiburg and Dubna.
Management of the project was very straightforward and the Fellow had full access to all resources of Coventry University and was treated as a full employee. The Fellow had his own office, his own computer and was a fully-fledged member of the Applied Mathematics Research Centre, of the same status as other members. Co-location of the Fellow to the PI meant scientific dialogue and interaction was continuously facilitated.
Dissemination was primarily through (a) publications; (b) talks at conferences (many invited); (c) seminars at universities and research institutes; (d) extended scientific visits to other institutions. Publications are listed elsewhere in this Final Report. Here we list talks at conferences, seminars and visits.
The following talks at conferences, invited seminars at universities, and scientific visits have taken place within the framework of the project during the period from 1.04.2013 - 31.03.2015.
Talks at Conferences:
1. The Fellow gave an invited talk ”Universal Ratios among Correction Amplitudes in Ising Universality Class” at the International Workshop on Critical Behaviour in Lattice Models, April 1-5 2013, Beijing, China.
2. At the same conference, the PI gave an invited talk “Hyperscaling above the Upper Critical Dimension”. (The PI’s invitation came about as a result of that of the Fellow and through the contacts of the Fellow.)
3. The Fellow gave an invited talk titled ”2D Ising model with free boundary: numerical solution” at Comp-Phys13, 28-20 November 2013, Leipzig, Germany.
4. The Fellow delivered a talk ”2D Ising model with free boundary: numerical solution” at Problems of (Supersymmetric) Integrable Systems, Armenia-Dubna Workshop, 24-26 December 2013, Dubna, Russia.
5. A talk “Exact Finite-Size Corrections and Corner Free Energy for the Models in c=-2 Universality Class”, was given by the Fellow at the Integrable Systems and Quantum Symmetries (ISQS22), 2 3- 29 June 2014, Prague, Czech Republic.
6. A talk “Exact Finite-Size Corrections and Corner Free Energies For The c=-2 Universality Class” was given by the Fellow at the International conference on statistical physics (SigmaPhi2014), 07 – 11 July 2014, Rhodes, Greece.
7. The Fellow gave a talk “Corner contribution to free energy for 2D dimer model” at the 15th International NTZ-Workshop on New Developments in Computational Physics (CompPhys14), 27-29 November 2014, Leipzig, Germany.
8. The PI gave an invited talk “Scaling And Finite-size Scaling At Phase Transitions Above The Upper Critical Dimension” at the Statistical Physics and Low Dimensional Systems conference May 20th - May 22nd, 2015, Abbaye des Prémontrés, Pont-à-Mousson, France.
Seminar talks at universities or research institutes:
The Fellow delivered the following invited talks:
1. ”Universal Amplitude Ratios in c=1/2 and c=-2 Universality Classes” at Institut for Theoretische Physik, Leipzig, 05 December 2013.
2. ”Universal Amplitude Ratios in c=1/2 and c=-2 Universality Classes” at Theoretical Polymer Physics, University of Freiburg, 16 December 2013.
3. “Corner contribution to free energy in 2D models”, at Albert-Ludwigs-University-Freiburg, Germany, 08 December 2014.
4. “Spin-spin correlation function on the Bethe lattice”, was given at Albert-Ludwigs-University-Freiburg, Germany, 18 December 2014.
Additionally, the PI gave talks on Task 4 (scaling in high dimensions) at the Ising Lectures in Lviv (2013), the Condensed Matter Physics Seminar Series in Oxford (2014) and the Mathematical Physics Seminar Series in the University of York (2015).
Scientific visits to universities or research institutes:
The Fellow undertook the following scientific visits during the course of the project:
1. A scientific visit to the Yerevan Physics Institute, Yerevan, Armenia for scientific cooperation with the group of Prof. N. Ananikyan in July 2013 and October 2013.
2. A scientific visit to the Institut for Theoretische Physik, Leipzig, Germany for scientific cooperation with the group of Prof. W. Janke in December 2013.
3. A scientific visit to the Theoretical Polymer Physics, University of Freiburg, Germany for scientific cooperation with the group of Prof. A. Blumen in December 2013.
4. A scientific visit to the Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia for scientific cooperation with the group of Prof. V.B. Priezzhev in December 2013.
5. A scientific visit to the Theoretical Polymer Physics, University of Freiburg, Germany for scientific cooperation with the group of Prof. A. Blumen in December 2014.
A summary of progress towards objectives and details for each task
The following results have been obtained in the research conducted within the framework of the project for each task during the period from 01.04.2013 - 31.03.2015.
Task 1: In task 1 of our project we proposed to study amplitude ratios in the bulk in 2D. The following results have been obtained in the research conducted in the frame of the project for task 1 during the period from 1.04.2013 - 31.03.2015.
1. Universal amplitude ratios for constrained critical systems: The critical properties of systems under constraint differ from their ideal counterparts through Fisher renormalization. The mathematical properties of Fisher renormalization applied to critical exponents are well known: the renormalized indices obey the same scaling relations as the ideal ones and the transformations are involutions in the sense that re-renormalizing the critical exponents of the constrained system delivers their original, ideal counterparts. We examine Fisher renormalization of critical amplitudes and show that, unlike for critical exponents, the associated transformations are not involutions. However, for ratios and combinations of amplitudes which are universal, Fisher renormalization is involuntary. The results are included in papers [1,6]. (Citations refer to Section 2 – Project Objectives)
2. The Ising model in two dimensions with special toroidal, defect boundary conditions: We studied the Ising model with special boundary conditions, which we call the duality twisted boundary conditions. This system may be interpreted as inserting a specific defect along non-contractible cycles of a cylinder, before closing it into a torus. We derive exact expressions for the eigenvalues of the transfer matrix for that model. As a result we have obtain analytically the partition function for the Ising model with such boundary conditions. For the case of the infinitely long cylinder of finite circumference with duality twisted boundary conditions, we obtain the asymptotic expansion of the free energy and the inverse correlation lengths. We find new infinite sets of universal amplitude ratios for the finite-size correction terms and show that such universal behaviour are correctly reproduced by the conformal perturbative approach. Besides the Fellow and PI, the work is done together with our joint PhD candidate A. Poghosian. The results are included in the paper  which we will submit to Phys. Rev. Lett. In June 2015.
Task 2: In task 2, we proposed to study finite-size effects in 2D Ising model. The following results have been obtained.
1. The bond-propagation algorithm for the specific heat of the two dimensional Ising model was developed. Using these algorithms, we study the critical specific heat of the model on the square lattice and triangular lattice with free boundaries. The exact values of all edge and corner terms are conjectured. The accurate forms of finite-size scaling for the specific heat are determined for the rectangle-shaped square lattice with various aspect ratios and various shaped triangular lattices. For the rectangle-shaped square and triangular lattices and the triangle-shaped triangular lattice, there is no logarithmic correction terms of order higher than the reciprocal of the area of the system. For the triangular lattice in rhombus, trapezoid and hexagonal shapes, there exist logarithmic correction terms of all orders for the specific heat. The results are included in paper .
2. We considered the critical two-dimensional Ising model with four types boundary conditions: free, fixed ferromagnetic, fixed antiferromagnetic, and fixed double antiferromagnetic. Using bond propagation algorithms with surface fields, we obtain the free energy, internal energy, and specific heat numerically on square lattices with a square shape and various combinations of the four types of boundary conditions. The calculations are carried out on the square lattices with size N × N with 30 < N < 1000. The numerical data are analyzed with finite-size scaling. The bulk, edge, and corner terms are extracted very accurately. The exact results are conjectured for the corner logarithmic term in the free energy, the edge logarithmic term in the internal energy, and the corner logarithmic term in the specific heat. The corner logarithmic terms in the free energy agree with the conformal
field theory very well. The results are included in paper .
Task 3: In task 3 of our project we proposed to study finite-size effects in 2D dimer models. The following results have been obtained.
1. We consider the partition functions of the anisotropic dimer model on the rectangular lattice with (a) free and (b) cylindrical boundary conditions with a single monomer residing on the boundary. We express (a) and (b) in terms of a principal partition function with twisted boundary conditions. Based on these expressions, we derive the exact asymptotic expansions of the free energy for both cases (a) and (b). We confirm the conformal field theory prediction for the corner free energy of these models, and find the central charge is c = -2. We also show that the dimer model on the cylinder with an odd number of sites on the perimeter exhibits the same finite-size corrections as on the plane. The results are included in the paper .
2. We consider finite size corrections in the spanning tree model , which is closely related to the dimer model. We express the partition functions of the spanning tree on finite square lattices under five different sets of boundary conditions in terms of a principal partition function with twisted-boundary conditions. Based on these expressions, we derive the exact asymptotic expansions of the logarithm of the partition function for each case. We have also established several groups of identities relating spanning-tree partition functions for the different boundary conditions. We also explain an apparent discrepancy between logarithmic correction terms in the free energy for a two-dimensional spanning-tree model with periodic and free-boundary conditions and conformal field theory predictions. We have obtained corner free energy for the spanning tree under free-boundary conditions in full agreement with conformal field theory predictions.The results are included in paper .
3. We present the exact closed-form expression for the partition function of a dimer model on a generalized finite checkerboard rectangular lattice under periodic boundary conditions. We investigate three different set of dimer weights, denoted as A, B and C, each with different critical behavior. We then consider different limits for model on the three lattices. In one limit the model for each of the three lattices is reduced to the dimer model on a rectangular lattice, which belongs to the c=-2 universality class. In another limit, two of the lattices reduce to the anisotropic Kasteleyn model on a honeycomb lattice, the universality class of which is given by c=1. Thus the dimer model on a generalized checkerboard rectangular lattice can manifest different critical behaviour depending on the parameters (weights) characterising the model. The results are included in the paper  which has been submitted to Phys. Rev. E.
Tasks 2 and 3 (overlapping or related): Tasks 2 & 3 concern finite-size effects in 2D lattice spin models. These are closely related to the problem of determining two-point resistances in resistor networks. E.g., the random cluster model represents a unification of the problems of Ising and Potts models, along with percolation, with the electrical networks problems. The latter has received a great deal of attention in the past couple of years and we made significant contributions in the context of this project. In this context, we considered the two-point resistance in a resistor network . By formulating the problem differently, we obtained a new expression for the two-point resistance between two arbitrary nodes which is simpler and can be easier to use in practice. We applied the new formulation to the cobweb resistor network in an analysis which also solves the spanning tree problem on that substrate. The resulting paper  was selected for IOP Select (articles from the last 12 months that have been chosen by Institute of Physics editors for their novelty, significance and potential impact on future research). The paper has already been cited 15 times since publication in 2014. We also considered the problem on a fan-resistor network, which is a segment of the cobweb network, and obtained the exact resistance between two arbitrary nodes on such a network (and the solution of the spanning tree problem) .
Inspired by these successes, we considered finite size corrections for the resistance between the centre node and a node on the boundary of a cobweb network of resistors which are different in the two spatial directions. We derived the exact asymptotic expansions of the resistance between the centre node and a node on the boundary . This was extended to certain non-regular resistor networks in . This generalizes previous approaches, which only deliver results for networks of more regular geometry. The new approach exploits the second minor of the Laplacian matrix associated with the given network to obtain the resistance in terms of its eigenvalues and eigenvectors. The method is illustrated by application to the resistor network on the globe lattice, for which the resistance between two arbitrary nodes is obtained in the form of single summation . Finally we teamed up with John Essam (Royal Holloway College, University of London) and Zhi-Zhong Tan (Nantong University, chhina) to write a review-type paper outlining the considerable progress which has recently been made in the development of techniques to exactly determine two-point resistances in networks of various topologies..
Task 4: In task 4 of our project we proposed to study finite-size effects in the models beyond two dimensions. The following results have been obtained.
1. We considered the Potts models with invisible states on general Bethe lattices. We investigated the number of so-called invisible states which need to be added to the q-state Potts model to transmute its phase transition from continuous to first order. In the q = 2 case, a Bragg-Williams (mean-field) approach necessitates four such invisible states while a 3-regular random graph formalism requires seventeen. In both of these cases, the change over from second- to first-order behaviour induced by the invisible states is identified through the tri-critical point of an equivalent Blume-Emery-Griffiths model. We investigated the generalized Potts model on a Bethe lattice with z neighbours and determined the number of invisible states required to manifest the equivalent Blume-Emery-Griffiths tri-criticality .
2. It is well known that the imposition of a constraint can transform the properties of critical systems. We investigated the involutory nature of transformations between the critical parameters describing ideal and constrained systems, paying particular attention to matters relating to universality. The results are included into the papers [1,6] (which are also of relevance to Task 1 – see above).
3. The nature of critical phenomena above the upper critical dimension: is an old problem and has long been considered settled. However, recent investigations have uncovered new, very important, fundamental aspects that were hitherto misunderstood. These are of foundational importance for theories of critical phenomena. In a series or papers by the PI, but with solid backing through discussions with the Fellow, a new finite-size scaling theory for high dimensional systems has been developed [13-15]. In particular, new scaling relations have been developed, both at the upper critical dimension itself and beyond.
4. In Task 4 we hypothesised that the locus of zeros of Fisher renormalized partition functions should be identical to the Fisher renormalized locus of zeros. This work connects with the start made in Papers [1,6]. Significant progress has been made to address this quite tricky problem. However, we are not yet quite at publication stage. We estimate that another few months are required to complete our understanding of this adventurous undertaking (Paper 20).
Through this project, the PI and the host institution has come to develop new contacts with Armenia, the home country of the Fellow. This has resulted in new collaborations and a paper on the zeros of the spin-1/2 Ising-Heisenberg model on a diamond chain has already resulted . Also, in , the Heisenberg model in three dimensions was analysed. Again, this paper was born with the help of continuing discussions with the Fellow.
Summary of the progress of the researcher training activities/transfer of knowledge activities/integration activities (as it applies for the MC action)
To facilitate knowledge transfer, Dr Izmailyan was embedded in the Department of Mathematical Sciences. He had office space in proximity to the other statistical physicists and postgraduate students. His expertise was showcased within a month of arrival through a seminar, open to all Departmental members. He helped run the AMRC’s seminar series in order to facilitate his interaction with other visiting scientists and the wider statistical physics community, both in the UK and in the wider EU.
In what is hoped to be a signal of a reversal of the decline of physics and applied mathematics courses in the UK over the past decades, Coventry University initiated a new applied mathematics and theoretical physics undergraduate degree, as planned, commencing in 2012-2013. Also as planned, Dr Izmailyan contributed to the design of this degree and one of its modules. He was also involved (i.e. assisted) in the training of 3 postgraduate students.
The Fellow visited the University of Leipzig for a month as planned. However, his planned visit to Nancy did not materialise due to visa issues. Instead he had two visits to the Blumen Group in Freiburg.
As planned, Dr Izmailian gave three seminars locally in Coventry – covering his recent research and of direct interest to the broader department. He also interacted fully with our various visitors and seminar speakers (dozens of scientists from the UK and abroad) an activity which gave him significant prominence and boosted the reputation and visibility of our Group.
Highlights of clearly significant results
- 1. We solved exactly the Ising model in two dimensions with the duality twisted boundary conditions and find the new set of universal amplitude ratios for that model (Task 1).
- 2. We found that for ratios and combinations of amplitudes which are universal, Fisher renormalization is involuntary (Task 4).
- 3. For the first time we confirmed the conformal field prediction for the corner contribution to the free energy for the Ising model on the square lattice and triangular lattice with free boundaries. (Task 2)
- 4. We also confirmed the conformal field theory prediction for the corner free energy of the dimer and spanning tree models, for which the central charge is c = -2. (Task 3.)
- 5. We obtain a new expression for the two-point resistance between two arbitrary nodes of the resistor network, which is simpler and can be easier to use in practice. (Tasks 2 & 3 and a paper which was selected for IOP Select and which has already been cited 15 times in under a year.)
- 6. A new analytic approach is presented to developing exact expressions for the two-point resistance between arbitrary nodes on certain non-regular resistor networks. This generalizes previous approaches, which only deliver results for networks of more regular geometry.
- 7. We investigate the generalized Potts model on a Bethe lattice with z neighbours and determined the number of invisible states required to manifest the equivalent Blume-Emery-Griffiths tri-criticality In the q=2 case.
- 8. Scaling and finite-size scaling above the upper critical dimension has been reformulated and hyperscaling and Fisher’s relation extended to that circumstance.
N. Ananikian, N.Sh. Izmailyan, D.A. Johnston, R. Kenna and R.P.K.C.M. Ranasinghe,
Potts Models with Invisible States on General Bethe Lattices,
J. Phys. A: Math. Theor. 46, 385002 (2013).
N.Sh. Izmailian, R. Kenna and F.Y. Wu,
The two-point resistance of a resistor network: A new formulation and application to the cobweb network,
J. Phys. A: Math. Theor. 47, 035003 (2014).
(Selected for IOP SELECT, this paper has already gathered 15 citations.)
N.Sh. Izmailian, R. Kenna, X. Wu and W. Guo,
Exact finite-size corrections and corner free energies for the c=-2 universality class,
Nucl. Phys. B 884, 157 (2014).
N.Sh. Izmailian and R. Kenna,
Universal Amplitude Ratios for Constrained Critical Systems,
J. Stat. Mech. P07011 (2014).
N. Izmailian and R. Kenna,
Exact asymptotic expansion for the resistance
between center node and a node on the cobweb
Cond. Mat. Phys. 17, 33008 (2014).
N. Izmailian and R. Kenna,
Critical Phenomena for Systems under Constraint,
Cond. Mat. Phys. 17, 33602 (2014).
N.Sh. Izmailian and R. Kenna,
A generalised formulation of the Laplacian approach to resistor
J. Stat. Mech. P09016 (2014).
N.Sh. Izmailian and R. Kenna,
The two-point resistance of fan networks,
Chin. J. Phys. 53, 040703 (2015).
N.Sh. Izmailian and R. Kenna,
Exact finite-size corrections for the spanning-tree model under
different boundary conditions,
Phys. Rev. E 91, 022129 (2015).
John W. Essam, Nikolay Sh. Izmailyan, Ralph Kenna and Zhi-Zhong Tan,
Comparison of methods to determine point-to-point resistance in nearly rectangular network with application to a hammock network,
Royal Society Open Science 2, 140420 (2015).
Xintian Wu, Ru Zheng, Nickolay Izmailian and Wenan Guo,
Accurate expansions of internal energy and specific heat of critical two-dimensional Ising model with free boundaries,
J. Stat. Phys. 155, 106 (2014).
Xintian Wu and Nickolay Izmailyan,
Critical two-dimensional Ising model with free, fixed ferromagnetic, fixed antiferromagnetic, and double antiferromagnetic boundaries,
Phys. Rev. E 91, 012102 (2015).
E.J. Flores-Sola, B. Berche, R. Kenna, M. Weigel,
Finite-size scaling above the upper critical dimension in Ising models with long-range interactions,
Eur. Phys. J. B 88 (2015) 28 [8 pages]
R. Kenna and B. Berche, Fisher's scaling relation above the upper critical dimension,
EPL 105 (2014) 26005.
N.S. Ananikian, V.V. Hovhannisyan and R. Kenna,
The partition function zeros of the spin-1/2 Ising-Heisenberg model on the diamond chain,
Physica A 396 (2014) 51-60.
A. Gordillo-Guerrero, R. Kenna and J.J. Ruiz-Lorenzo,
Scaling Behavior of the Heisenberg Model in Three Dimensions,
Phys. Rev. E 88 (2013) 062117.
Armen Poghosyan, Ralph Kenna and Nikolay Izmailian,
Critical Ising model on torus with specific defect line,
To be submitted to Phys. Rev. Lett. (expected to be submitted in June 2015).
N.Sh. Izmailian, C. K. Hu and R. Kenna,
Exact solution of the dimer model on the generalized finite checkerboard lattice,
submitted to Phys. Rev. E
N.Sh. Izmailian and R. Kenna, Fisher renormalization of Fisher zeros,