Union Jack Lattice Research Articles

Universal quantum computing using(Zd)3symmetry-protected topologically ordered states

Measurement-based quantum computation describes a scheme where entanglement of resource states is utilized to simulate arbitrary quantum gates via local measurements. Recent works suggest that symmetry-protected topologically non-trivial, short-ranged entanged states are promising candidates for such a resource. Miller and Miyake [NPJ Quantum Information 2, 16036 (2016)] recently constructed a particular $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry-protected topological state on the Union-Jack lattice and established its quantum computational universality. However, they suggested that the same construction on the triangular lattice might not lead to a universal resource. Instead of qubits, we generalize the construction to qudits and show that the resulting $(d-1)$ qudit nontrivial $\mathbb{Z}_d \times \mathbb{Z}_d \times \mathbb{Z}_d$ symmetry-protected topological states are universal on the triangular lattice, for $d$ being a prime number greater than 2. The same construction also holds for other 3-colorable lattices, including the Union-Jack lattice.

Percolation on hypergraphs with four-edges

We study percolation on self-dual hypergraphs that contain hyperedges with four bounding vertices, or ‘four-edges’, using three different generators, each containing bonds or sites with three distinct probabilities p, r, and t connecting the four vertices. We find explicit values of these probabilities that satisfy the self-duality conditions discussed by Bollobás and Riordan. This demonstrates that explicit solutions of the self-duality conditions can be found using generators containing bonds and sites with independent probabilities. These solutions also provide new examples of lattices where exact percolation critical points are known. One of the generators exhibits three distinct criticality solutions (p, r, t). We carry out Monte-Carlo simulations of two of the generators on two different hypergraphs to confirm the critical values. For the case of the hypergraph and uniform generator studied by Wierman et al, we also determine the threshold p = 0.441 374 ± 0.000 001, which falls within the tight bounds that they derived. Furthermore, we consider a generator in which all or none of the vertices can connect, and find a soluble inhomogeneous percolation system that interpolates between site percolation on the union-jack lattice and bond percolation on the square lattice.

Spontaneously magnetized Tomonaga-Luttinger liquid in frustrated quantum antiferromagnets

We develop a theory of spontaneously magnetized Tomonaga-Luttinger (TLL) liquid in geometrically frustrated quasi-one-dimensional quantum magnets by taking an $S=1/2$ ferrimagnet on a union-jack lattice as an example. We show that a strong frustration leads to a spontaneous magnetization because of the ferrimagnetic nature of lattice. Due to the ferrimagnetic order, the local magnetization has an incommensurate oscillation with the position. We show that the spontaneously magnetized TLL is smoothly connected to the existence of a Nambu-Goldstone boson in the canted ferrimagnetic phase of a two-dimensional frustrated antiferromagnet.

Critical manifold of the Potts model: Exact results and homogeneity approximation

The q-state Potts model has stood at the frontier of research in statistical mechanics for many years. In the absence of a closed-form solution, much of the past effort has focused on locating its critical manifold, trajectory in the parameter (q,e(J)) space where J is the reduced interaction, along which the free energy is singular. However, except in isolated cases, antiferromagnetic (AF) models with J<0 have been largely neglected. In this paper we consider the Potts model with AF interactions focusing on obtaining its critical manifold in exact and/or closed-form expressions. We first reexamine the known critical frontiers in light of AF interactions. For the square lattice we confirm the Potts self-dual point to be the sole critical frontier for J>0. We also locate its critical frontier for J<0 and find it to coincide with a solvability condition observed by Baxter in 1982 [R. J. Baxter, Proc. R. Soc. London Ser. A 388, 43 (1982)]. For the honeycomb lattice we show that the known critical frontier holds for all J, and determine its critical q(c) = 1/2(3 + sqrt[5]) = 2.61803 beyond which there is no transition. For the triangular lattice we confirm the known critical frontier to hold only for J>0. More generally we consider the centered-triangle (CT) and Union-Jack (UJ) lattices consisting of mixed J and K interactions, and deduce critical manifolds under homogeneity hypotheses. For K = 0 the CT lattice is the diced lattice, and we determine its critical manifold for all J and find q(c) = 3.32472. For K = 0 the UJ lattice is the square lattice and from this we deduce both the J > 0 and J < 0 critical manifolds and q(c) = 3. Our theoretical predictions are compared with recent numerical results.

Phase diagram of the frustrated Heisenberg antiferromagnet on the union jack lattice

Phase diagram of the frustrated Heisenberg antiferromagnet on the union jack lattice

Quantum spin model with frustration on the union jack lattice

A quantum spin model with frustration, the Heisenberg antiferromagnet on the union jack lattice, is analyzed using spin-wave theory. For small values of the frustrating coupling $\ensuremath{\alpha}$, the system is N\'eel ordered as usual, while for large $\ensuremath{\alpha}$ the frustration is found to induce a canted phase. The possibility of an intermediate spin-liquid phase is discussed.

Exact solution of the Ising model on a pentagonal lattice

It is shown that the Ising model on a two-dimensional lattice with pentagonal tiling and nearest-neighbour interactions is equivalent to the Ising model on the Union-Jack lattice with nearest- and second nearest-neighbour non-crossing interactions. Besides the critical temperature, exact expressions are obtained for the spontaneous magnetization for both types of lattice sites.

The Critical Behavior of the Ising Model on Union Jack Lattice11The project supported by Science Foundation of Shanghai Jiaotong University

In this paper, we have investigated the critical behavior of the ferromagnetic Ising model on union jack lattice. The model is equivalent to the eight-vertex model which can be solved as a free fermion model with the free fermion approximation. The critical exponents have been obtained as and .

Mixed-spin Ising model on the union jack lattice

We show that the mixed-spin Ising model on the union jack lattice with S= 1/2 and S=1 spin operators placed on different sublattices is equivalent to the eight-vertex model. On a two-dimensional manifold in a parameter space, the model corresponds to the symmetric eight-vertex model and thus is solvable. We locate exactly a coexistence surface between two ordered phases of the mixed-spin model. This surface is bounded by a bicritical line with continuously changing exponents.

The Critical Points of a Generalized Mixed Spin Model with Free Fermion Approximation

A generalized mixed spin model with anisotropic interactions is considered on the union jack lattice. We find that the model is equivalent to a free fermion model. The critical temperature is obtained. In special cases, we get the same results as exact solutions.

Ising model with a new class of four-spin interactions

Abstract The Ising model on a Union-Jack lattice, described by a Hamiltonian with second-neighbor pair-pair, four-spin, infinite-range interactions is considered. The model is solved exactly and the results are compared with MFA predictions. Within the exact treatment two new classes of phase transitions are obtained. The first one includes transitions from a disordered to a metastable, ordered and then to a stable and ordered phase with decreasing temperature. The metastable phase does not appear if the temperature is increased. The second one contains transitions between ordered and partialy ordered, partialy frustrated phases.

FREE-FERMION APPROXIMATE SOLUSION OF THE ISING MODEL ON UNION JACK LATTICE

In this paper,the Ising model with two-body anisotropic interactions and three-body interactions on union Jack lattice is solved by free-fermion approximation. We have obtained its free energy, spontaneous magnetisation and the equation of critical points. When the interaction coefficients are simplified as in a square lattice Ising model, the same solusion as Onsager's is obtained.

CRITICAL TEMPERATURE CURVES FOR THE GENERALIZED MIXED SPIN MODEL

In this paper, we propose a generalized mixed spin model on the union jack lattice. The mean field. free-fermion approximations and equiscale transformation in the same universality class method are applied to this model. The critical temperature curves obtained by different methods agree with each other quite well. We also compare the three different methods. In some special cases, the same critical points as in exactly solved models are obtained.

Spontaneous magnetizations of the Ising model on the union Jack lattice

We demonstrate that the (Union-Jack lattice) Ising model maps onto an isotropic free-fermion eight vertex model. Using this result we show the claim of Lin and Lee, that the spontaneous magnetization given by Vaks, Larkin and Ovchinnikov is in error, to be incorrect. By the use of correlation identities for the checkerboard Ising model, we derive the mean spontaneous magnetization on the Union-Jack lattice, which agrees with low-temperature series expansion.

Spontaneous magnetization of the Ising model on a union Jack lattice

A formula for the spontaneous magnetization of the Ising model on a Union Jack lattice is proposed. Our result agrees with the exact low-temperature series expansion up to the 24th order.