Chiral Potts Model Research Articles

The exotic ground state of the decorated honeycomb lattice

The study is focusing on the exploration of the magnetic properties of the frustrated decorated honeycomb lattices. The presence of geometrical frustration and C3 symmetry leads to an exotic ground state. Monte Carlo simulations and analytical calculations are used to analyze the system’s behavior. The dependence of the magnetization on the external field of the Ising model exhibits a step-like behavior, while the magnetization of the classical Heisenberg model has no plateau in the isotropic case. An efficient Hamiltonian is proposed to describe the properties of this system on the unfrustrated hexagonal lattice within the framework of the chiral Potts model. Within a specific range of fields, the state of the effective Hamiltonian aligns with that of the original Hamiltonian. The ground state configurations and degeneracy of the system are explored, revealing fractured stripe patterns separated by spins with opposite orientations. These findings contribute to the knowledge of the properties of decorated lattices, offering valuable insights for potential experimental and practical applications.

Real Space Renormalization Group of the Chiral Potts Model

Real Space Renormalization Group of the Chiral Potts Model

Explicit R-matrices for inhomogeneous 3D chiral Potts models: Integrability and the action formulation for IM

We construct the exact spectral parameter dependent vertex R-matrix for the classical 3D N-state chiral Potts models, convenient for considering the model in context of the Bethe ansatz. The R-matrix is defined on the N4 dimensional space VN⊗VN⊗VN⊗VN, appropriate for consideration by means of the cube-equations defined in Khachatryan et al. (2015). We present the 2D quantum spin Hamiltonians for general case and, at N=2, a fermionic lattice action representation corresponding to 3D Ising’s statistical model.

Identifying the Huse-Fisher universality class of the three-state chiral Potts model

Using the corner-transfer matrix renormalization group approach, we revisit the three-state chiral Potts model on the square lattice, a model proposed in the eighties to describe commensurate-incommensurate transitions at surfaces, and with direct relevance to recent experiments on chains of Rydberg atoms. This model was suggested by Huse and Fisher to have a chiral transition in the vicinity of the Potts point, a possibility that turned out to be very difficult to definitely establish or refute numerically. Our results confirm that the transition changes character at a Lifshitz point that separates a line of Pokrosky-Talapov transitions far enough from the Potts point from a line of direct continuous order-disorder transition close to it. Thanks to the accuracy of the numerical results, we have been able to base the analysis entirely on effective exponents to deal with the crossovers that have hampered previous numerical investigations. The emerging picture is that of a new universality class with exponents that do not change between the Potts point and the Lifshitz point, and that appear to be consistent with those of a self-dual version of the model, namely correlation lengths exponents νx=2/3 in the direction of the asymmetry and νy=1 perpendicular to it, an incommensurability exponent β¯=2/3, a specific heat exponent that keeps the value α=1/3 of the three-state Potts model, and a dynamical exponent z=3/2. These results are in excellent agreement with experimental results obtained on reconstructed surfaces in the nineties, and shed light on recent Kibble-Zurek experiments on the period-3 phase of chains of Rydberg atoms.

A coupled Temperley–Lieb algebra for the superintegrable chiral Potts chain

The Hamiltonian of the N-state superintegrable chiral Potts (SICP) model is written in terms of a coupled algebra defined by N − 1 types of Temperley–Lieb generators. This generalises a previous result for N = 3 obtained by Fjelstad and Månsson (2012 J. Phys. A: Math. Theor. 45 155208). A pictorial representation of a related coupled algebra is given for the N = 3 case which involves a generalisation of the pictorial presentation of the Temperley–Lieb algebra to include a pole around which loops can become entangled. For the two known representations of this algebra, the N = 3 SICP chain and the staggered spin-1/2 XX chain, closed (contractible) loops have weight and weight 2, respectively. For both representations closed (non-contractible) loops around the pole have weight zero. The pictorial representation provides a graphical interpretation of the algebraic relations. A key ingredient in the resolution of diagrams is a crossing relation for loops encircling a pole which involves the parameter ρ = e 2πi/3 for the SICP chain and ρ = 1 for the staggered XX chain. These ρ values are derived assuming the Kauffman bracket skein relation.

Freely and openly

In the autumn of 1990, I took a month off between the end of my postdoc at the Institute of Space and Astronautical Science in Sagamihara, Japan, and the start of my next job at NASA’s Goddard Space Flight Center in Greenbelt, Maryland. During that sojourn I visited the Institute of Astronomy at Cambridge University, where I had done my PhD.Among the new people I met there was Ian Stevens, who coincidentally had just finished a postdoc at Goddard. I no longer remember the circumstances of our encounter. Most likely it was during the 11:00am coffee break, when almost all of the institute’s members gather in the library to converse over freshly brewed coffee. However, I can’t forget the outcome. Stevens and I decided to work together. Our paper, “An x-ray excited wind in Centaurus X-3,” appeared just over two years later.11. C. S. R. Day, I. R. Stevens, Astrophys. J. 403, 322 (1993). https://doi.org/10.1086/172205Collaboration pervades scientific research. Of the 150 or so news stories I’ve written for Physics Today’s Search and Discovery department, just one was about a paper by a single author, Rodney Baxter (“Order parameter of the chiral Potts model succumbs at last to exact solution,” November 2005, page 19). Sometimes the work I reported on resulted from a collaboration among the members of a single group. Other times it resulted from a collaboration between groups.One of my favorite examples of intergroup collaboration is the first observation of the quantum spin Hall effect (QSHE) in a heterostructure made of a layer of mercury telluride of just the right thickness sandwiched between layers of mercury cadmium telluride (see Physics Today, January 2008, page 19).The intricate setting has an interesting back story. Condensed-matter theorists began investigating the QSHE in the early 2000s. It arises when an insulator’s band structure includes two spin-polarized edge states. Because of the states’ topology, the corresponding edge currents flow without dissipation. The QSHE is not only interesting; it also has potential applications. The effect proved difficult to measure. It was too puny in the first prospective material, graphene. A proposal to realize it in a mechanically strained semiconductor was experimentally infeasible.A hint of a way forward arrived when one of the theorists who elucidated the QSHE, Shoucheng Zhang of Stanford University, met one of the experimenters who eventually observed it, Laurens Molenkamp of the University of Würzburg, at a conference in South Korea. Molenkamp told Zhang about the narrow-bandgap semiconductor, HgTe. Spin-orbit coupling is so strong in HgTe that it splits the valence band and raises the higher of the split bands above what would otherwise be the conduction band. Could band inversion be a way to create the QSHE? the experimenter asked the theorist. When they met again a year later in Singapore, they agreed to collaborate.A crucial step was taken from a general search for the QSHE in HgTe to a specific one in a heterostructure of HgCdTe/HgTe/HgCdTe when Molenkamp gave Zhang an electronic copy of the doctoral thesis of Alexander Pfeuffer-Jeschke. In 2000 Pfeuffer-Jeschke had calculated the threshold thickness below which the bandgap of the HgTe layer becomes zero and then becomes positive under the increasingly strong influence of the HgCdTe layers. He wrote his thesis in German, a language that Zhang had acquired as an undergraduate at the Free University of Berlin. Zhang read the thesis on the flight back from Singapore.The story of the discovery of the QSHE is not only one of collaboration. It’s also one of the free movement of people and ideas between countries. Molenkamp works in Germany and is from the Netherlands. The experiment that he and his collaborators carried out was sketched out in a Science paper22. B. A. Bernevig, T. L. Hughes, S.-C. Zhang, Science 314, 1757 (2006). https://doi.org/10.1126/science.1133734 written by Zhang and two of his then students, B. Andrei Bernevig, who is from Romania, and Taylor Hughes, who is from the US. Zhang grew up in Shanghai.I asked Molenkamp to fact-check the first draft of this editorial. His emailed reply closed with the observation that physics has been an international undertaking for a century. “There is just a rather limited number of professionals in the field,” he wrote. “And you really only get things done when you exchange ideas. Freely and openly. And respecting scientific discourse.”ReferencesSection:ChooseTop of pageReferences <<CITING ARTICLES1. C. S. R. Day, I. R. Stevens, Astrophys. J. 403, 322 (1993). https://doi.org/10.1086/172205, Google ScholarCrossref2. B. A. Bernevig, T. L. Hughes, S.-C. Zhang, Science 314, 1757 (2006). https://doi.org/10.1126/science.1133734, Google ScholarCrossref© 2020 American Institute of Physics.

Energy spectrum and critical exponents of the free parafermion ZN spin chain

Results are given for the ground state energy and excitation spectrum of a simple N-state ZN spin chain described by free parafermions. The model is non-Hermitian for with a real ground state energy and a complex excitation spectrum. Although having a simpler Hamiltonian than the superintegrable chiral Potts model, the model is seen to share some properties with it, e.g. the specific heat exponent and the anisotropic correlation length exponents and .

CSOS models descending from chiral Potts models: degeneracy of the eigenspace and loop algebra

Monodromy matrices of the model are known to satisfy a Yang–Baxter equation with a six-vertex R-matrix as the intertwiner. The commutation relations of the elements of the monodromy matrices are completely determined by this R-matrix. We show the reason why in the superintegrable case the eigenspace is degenerate, but not in the general case. We then show that the eigenspaces of special CSOS models descending from the chiral Potts model are also degenerate. The existence of an quantum loop algebra (or subalgebra) in these models is established by showing that the Serre relations hold for the generators. The highest weight polynomial (or the Drinfeld polynomial) of the representation is obtained by using the method of Baxter for the superintegrable case. As a byproduct, the eigenvalues of all such CSOS models are given explicitly.

Discrete holomorphicity in the chiral Potts model

We construct lattice parafermions for the Z(N) chiral Potts model in terms of quasi-local currents of the underlying quantum group. We show that the conservation of the quantum group currents leads to twisted discrete-holomorphicity (DH) conditions for the parafermions. At the critical Fateev–Zamolodchikov point the parafermions are the usual ones, and the DH conditions coincide with those found previously by Rajabpour and Cardy. Away from the critical point, we show that our twisted DH conditions can be understood as deformed lattice current conservation conditions for an underlying perturbed conformal field theory in both the general and N = 2 Ising cases.

On the Form Factors of Local Operators in the Bazhanov–Stroganov and Chiral Potts Models

We consider general cyclic representations of the six-vertex Yang–Baxter algebra and analyze the associated quantum integrable systems, the Bazhanov–Stroganov model and the corresponding chiral Potts model on finite size lattices. We first determine the propagator operator in terms of the chiral Potts transfer matrices and we compute the scalar product of separate states (including the transfer matrix eigenstates) as a single determinant formulae in the framework of Sklyanin’s quantum separation of variables. Then, we solve the quantum inverse problem and reconstruct the local operators in terms of the separate variables. We also determine a basis of operators whose form factors are characterized by a single determinant formulae. This implies that the form factors of any local operator are expressed as finite sums of determinants. Among these form factors written in determinant form are in particular those which will reproduce the chiral Potts order parameters in the thermodynamic limit. The results presented here are the generalization to the present models associated to the most general cyclic representations of the six-vertex Yang–Baxter algebra of those we derived for the lattice sine–Gordon model.

The τ2 model and parafermions

Paul Fendley has recently found a ‘parafermionic’ way to diagonalize a simple solvable Hamiltonian associated with the chiral Potts model. Here we indicate how this method generalizes to the τ2 model with open boundaries and make some comments.

Trigonometrical sums connected with the chiral Potts model, Verlinde dimension formula, two-dimensional resistor network, and number theory

We have recently developed methods for obtaining exact two-point resistance of the complete graph minus N edges. We use these methods to obtain closed formulas of certain trigonometrical sums that arise in connection with one-dimensional lattice, in proving Scott’s conjecture on permanent of Cauchy matrix, and in the perturbative chiral Potts model. The generalized trigonometrical sums of the chiral Potts model are shown to satisfy recursion formulas that are transparent and direct, and differ from those of Gervois and Mehta. By making a change of variables in these recursion formulas, the dimension of the space of conformal blocks of SU(2) and SO(3) WZW models may be computed recursively. Our methods are then extended to compute the corner-to-corner resistance, and the Kirchhoff index of the first non-trivial two-dimensional resistor network, 2×N. Finally, we obtain new closed formulas for variant of trigonometrical sums, some of which appear in connection with number theory.

PT symmetry in classical and quantum statistical mechanics

-symmetric Hamiltonians and transfer matrices arise naturally in statistical mechanics. These classical and quantum models often require the use of complex or negative weights and thus fall outside the conventional equilibrium statistical mechanics of Hermitian systems. -symmetric models form a natural class where the partition function is necessarily real, but not necessarily positive. The correlation functions of these models display a much richer set of behaviours than Hermitian systems, displaying sinusoidally modulated exponential decay, as in a dense fluid, or even sinusoidal modulation without decay. Classical spin models with -symmetry include Z( N ) models with a complex magnetic field, the chiral Potts model and the anisotropic next-nearest-neighbour Ising model. Quantum many-body problems with a non-zero chemical potential have a natural -symmetric representation related to the sign problem. Two-dimensional quantum chromodynamics with heavy quarks at non-zero chemical potential can be solved by diagonalizing an appropriate -symmetric Hamiltonian.

Superconducting proximity effect on the edge of fractional topological insulators

We study the superconducting proximity effect on the helical edge states of time-reversal-symmetric fractional topological insulators (FTI). The Cooper pairing of physical electrons results in many-particle condensation of the fractionalized excitations on the edge. We find localized zero-energy modes emerge at interfaces between superconducting regions and magnetically insulating regions, which are responsible for the topological degeneracy of the ground states. By mapping the low-energy effective Hamiltonian to the quantum chiral Potts model, we determine the operator algebra of the zero modes and show that they exhibit nontrivial braiding properties. We then demonstrate that the Josephson current in the junction between superconductors mediated by the edge states of the FTI exhibit fractional Josephson effect with period as multiples of $4\ensuremath{\pi}$.

The τ2-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin’s SOV method

The most general cyclic representations of the quantum integrable τ2-model are analyzed. The complete characterization of the τ2-spectrum (eigenvalues and eigenstates) is achieved in the framework of Sklyanin’s separation of variables (SOV) method by extending and adapting previous results of one of the authors: (i) the determination of the τ2-spectrum is reduced to the classification of the solutions of a given functional equation in a class of polynomials; (ii) the determination of the τ2-eigenstates is reduced to the classification of the solutions of an associated Baxter equation. These last solutions are proved to be polynomials for a quite general class of τ2-self-adjoint representations and the completeness of the associated Bethe ansatz type equations is derived. Finally, the following results are derived for the inhomogeneous chiral Potts model: (i) simplicity of the spectrum, for general representations; (ii) complete characterization of the chiral Potts spectrum (eigenvalues and eigenstates) and completeness of Bethe ansatz type equations, for the self-adjoint representations of the τ2-model on the chiral Potts algebraic curves.

Exotic quantum criticality in one-dimensional coupled dipolar bosons tubes

The competition between intertube hopping processes and density-density interactions is investigated in one-dimensional quantum dipolar bosons systems of N coupled tubes at zero temperature. Using a phenomenological bosonization approach, we show that the resulting competition leads to an exotic quantum phase transition described by a U(1) x Z_N conformal field theory with a fractional central charge. The emerging Z_N parafermionic critical degrees of freedom are highly nontrivial in terms of the original atoms or polar molecules of the model. We further determine the main physical properties of the quantum critical point in a double-tube system which has central charge c=3/2. In triple-tube systems, we show that the competition between the two antagonistic processes is related to the physics of the two-dimensional Z_3 chiral Potts model. This work opens the possibility to study the exotic properties of the Z_N parafermions in the context of ultracold quantum Bose gases.

New symmetries of the chiral Potts model

In this paper a hitherto unknown spectrum generating algebra, consisting of two coupled Temperley–Lieb algebras, is found in the three-state chiral Potts model. From this, we can construct new Onsager integrable models. One realization is in terms of a staggered isotropic XY spin chain. Further we investigate the importance of the algebra for the existence of mutually commuting conserved charges. This leads us to a natural generalization of the boost operator, which generates the charges.

Quark–pion coupling strength and ground state baryon spectra in a chiral potential model

Chiral symmetry is incorporated into a relativistic independent-quark model with a square root confining potential and the quark–pion coupling strength, Gqqπ for quarks in a nucleon is estimated. The values of (Gqqπ2/4π) obtained with the approximation of a point pion as well as finite size pion are quite comparable with those estimated earlier by other researchers. Under the assumption that the baryon is an assembly of independent quarks confined in a first approximation by the square root potential that represents the nonperturbative gluon interactions, the model is also adopted to study the ground-state baryon spectrum taking into account perturbatively the contribution of the residual quark–pion coupling and that of the quark–gluon coupling in the baryon over and above the necessary center-of-mass correction. The physical masses of the ground state baryons calculated with the quark–gluon coupling constant αc = 0.227 agree reasonably well with experimental data.

Spontaneous magnetization of the integrable chiral Potts model

We show how Z-invariance in the chiral Potts model provides a strategy to calculate the pair correlation in the general integrable chiral Potts model using only the superintegrable eigenvectors. When the distance between the two spins in the correlation function becomes infinite it becomes the square of the order parameter. In this way, we show that the spontaneous magnetization can be expressed in terms of the inner products of the eigenvectors of the N asymptotically degenerate maximum eigenvalues. Using our previous results on these eigenvectors, we are able to obtain the order parameter as a sum almost identical to the one given by Baxter. This gives the known spontaneous magnetization of the chiral Potts model by an entirely different approach.

Proof of the determinantal form of the spontaneous magnetization of the superintegrable chiral Potts model

The superintegrable chiral Potts model has many resemblances to the Ising model, so it is natural to look for algebraic properties similar to those found for the Ising model by Onsager, Kaufman and Yang. The spontaneous magnetization $\mathcal{M}_r$ can be written in terms of a sum over the elements of a matrix $S_r$. The author conjectured the form of the elements, and this conjecture has been verified by Iorgov et al. The author also conjectured in 2008 that this sum could be expressed as a determinant, and has recently evaluated the determinant to obtain the known result for $\mathcal{M}_r$. Here we prove that the sum and the determinant are indeed identical expressions. Since the order parameters of the superintegrable chiral Potts model are also those of the more general solvable chiral Potts model, this completes the algebraic calculation of $\mathcal{M}_r$ for the general model. doi:10.1017/S1446181110000787