Lattice CP Research Articles

Loop-TNR analysis of CP(1) model with theta term

The phase structure of the two dimensional lattice CP(1) model in the presence of the θ term is analyzed by tensor network methods. The tensor renormalization group, which is a standard renormalization method of tensor networks, is used for the regions θ = 0 and θ ≠ 0. Loop-TNR, which is more suitable for the analysis of near criticality, is also implemented for the region θ = 0. The application of Loop-TNR for the region θ ≠ 0 is left for future work.

Tensor renormalization group analysis ofCP(N−1)model

We apply the higher-order tensor renormalization group to the lattice CP($N-1$) model in two dimensions. A tensor network representation of the CP($N-1$) model in the presence of the $\theta$ term is derived. We confirm that the numerical results of the CP(1) model without the $\theta$ term using this method are consistent with that of the O(3) model which is analyzed by the same method in the region $\beta \gg 1$ and that obtained by the Monte Carlo simulation in a wider range of $\beta$. The numerical computation including the $\theta$ term is left for future challenges.

Simulating the all-order strong coupling expansion IV: [formula omitted] as a loop model

We exactly reformulate the lattice CP ( N − 1 ) spin model on a D-dimensional torus as a loop model whose configurations correspond to the complete set of strong coupling graphs of the original system. A Monte Carlo algorithm is described and tested that samples the loop model with its configurations stored and manipulated as a linked list. Complete absence of critical slowing down and correspondingly small errors are found at D = 2 for several observables including the mass gap. Using two different standard lattice actions universality is demonstrated in a finite size scaling study. The topological charge is identified in the loop model but not yet investigated numerically.

Sublattices of the Lattices of Strong P-Congruences on P-Inversive Semigroups

In this paper, we describe strong P-congruences and sublattice-structure of the strong P-congruence lattice CP(S) of a P-inversive semigroup S(P). It is proved that the set of all strong P-congruences CP(S) on S(P) is a complete lattice. A close link is discovered between the class of P-inversive semigroups and the well-known class of regular ⋆-semigroups. Further, we introduce concepts of strong normal partition/equivalence, C-trace/kernel and discuss some sublattices of CP(S). It is proved that the set of strong P-congruences, which have C-traces (C-kernels) equal to a given strong normal equivalence of P (C-kernel), is a complete sublattice of CP(S). It is also proved that the sublattices determined by C-trace-equaling relation θ and C-kernel-equaling relation κ, respectively, are complete sublattices of CP(S) and the greatest elements of these sublattices are given.

A transition path for the pressure-induced wurtzite- to NaCl-type transformation described inPna21

Quite recently, two further mechanisms for the pressure-induced transition from the wurtzite to the NaCl type were proposed [Shimojo et al. (2004). Phys. Rev. B, 70, 184111-1-6] but no symmetry information was given. It will be shown that a slight modification of one of the assumed transition pathways allows a crystallographic description on the basis of a deformation of a heterogeneous 4-connected sphere packing in Pna2(1). All investigations were done with the help of the corresponding homogeneous packing in Pnma where the transition may be described as a deformation of a lonsdaleite configuration into a cubic primitive lattice cP. During the transformation, all sphere contacts are maintained. The new transition model is compared with the well known Cmc2(1) mechanism. Further related mechanisms can also be derived.

Lattice matrix elements and CP violation inB andK physics: Status and outlook

Status of lattice calculations of hadron matrix elements along with CP violation inB and inK systems is reviewed. Lattice has provided useful input which, in conjunction with experimental data, leads to the conclusion that CP-odd phase in the CKM matrix plays the dominant role in the observed asymmetry inB → ψK s. It is now quite likely that any beyond the SM, CP-odd, phase will cause only small deviations in B-physics. Search for the effects of the new phase(s) will consequently require very large data samples as well as very precise theoretical predictions. Clean determination ofall the angles of the unitarity triangle therefore becomes essential. In this regardB → KD0 processes play a unique role. RegardingK-decays, remarkable progress made by theory with regard to maintenance of chiral symmetry on the lattice is briefly discussed. First application already provide quantitative information onB K and the ΔI = 1/2 rule. In the lattice calculation, the enhancement in Re A0 appears to arise solely from tree operators, esp. Q2; penguin contribution toRe A0 appears to be very small. However, improved calculations are necessary for ε’/ε as the contributions of QCD penguins and electroweak penguins largely seem to cancel. There are good reasons, though, to believe that these cancellations will not survive improvements that are now underway. Importance of determining the unitarity triangle purely fromK-decays is also emphasized.

Domain wall fermion andCPsymmetry breaking

We examine the CP properties of chiral gauge theory defined by a formulation of the domain wall fermion, where the light field variables $q$ and $\bar q$ together with Pauli-Villars fields $Q$ and $\bar Q$ are utilized. It is shown that this domain wall representation in the infinite flavor limit $N=\infty$ is valid only in the topologically trivial sector, and that the conflict among lattice chiral symmetry, strict locality and CP symmetry still persists for finite lattice spacing $a$. The CP transformation generally sends one representation of lattice chiral gauge theory into another representation of lattice chiral gauge theory, resulting in the inevitable change of propagators. A modified form of lattice CP transformation motivated by the domain wall fermion, which keeps the chiral action in terms of the Ginsparg-Wilson fermion invariant, is analyzed in detail; this provides an alternative way to understand the breaking of CP symmetry at least in the topologically trivial sector. We note that the conflict with CP symmetry could be regarded as a topological obstruction. We also discuss the issues related to the definition of Majorana fermions in connection with the supersymmetric Wess-Zumino model on the lattice.

The NaCl- to CsCl-type phase transition discussed on the basis of the cP to cI deformation with the symmetry Cmcm 4(c) m2m

A structure forming a cubic primitive lattice cP may be deformed into a structure forming a cubic body-centred lattice cI in the space group Cmcm at position 4(c) m2m 0,y,1/4. If in related structures the sites are alternately occupied by unlike atoms, the NaCl and the CsCl types occur, respectively. The corresponding phase transition can be described as a deformation of a heterogeneous sphere packing in the subgroup Pmmn (a,-c,b) of Cmcm. All sphere configurations with symmetry Cmcm 4(c) m2m were derived. On the basis of this information, further possibilities for phase transitions that also correspond to sphere-packing deformations were found with this symmetry. Two of them possibly may take place in metals. The first one leads from a primitive hexagonal lattice to a hexagonal close packing, the other from a cubic body-centred lattice also to a hexagonal close packing.

Monte Carlo simulations of the CP 3 model and U(1) gauge theory in the presence of a θ term

A θ term, which couples to topological charge, is added to the two-dimensional lattice CP 3 model and U(1) gauge theory. Monte Carlo simulations are performed and compared to strong-coupling character expansions. In certain instances, a flattering behavior occurs in the free-energy at sufficiently larger θ, but the effect is an artifact of the simulation methods.

Monte Carlo studies of two-dimensional systems with aθterm

A $\theta$ term, which couples to topological charge, is added to the two-dimensional lattice CP^3 model and U(1) gauge theory. Monte Carlo simulations are performed and compared to strong-coupling character expansions. In certain instances, a flattening behavior occurs in the free-energy at sufficiently large $\theta$, but the effect is an artifact of the simulation methods.

Scaling topological charge in the CP 3 model using a fixed point action

We define a fixed point action in two-dimensional lattice CP N−1 models. The fixed point action is a classical perfect lattice action, which is expected to show strongly reduced cut-off effects in numerical simulations. Furthermore, the action has scale invariant instanton solutions, which enables us to define a topological charge without topological defects. We present results for the scaling of the topological suceptibility from a Monte Carlo simulation in the CP 3 model.

Monte Carlo simulation of lattice CP N−1 models at large N

In order to check the validity and the range of applicability of the 1 N expansion, we performed numerical simulations of the 2D lattice CP N−1 models at large N, in particular we considered the CP 20 and the CP 40 models. Quantita agreement with the large- N predictions is found for the dimensionless quantities χt ξ G 2and σξ G 2, where ξ G , χt and σ are respectively the correlation length defined by the second moment of the correlation function, the topological susceptibility and the string tension. On the other hand, quantities involving the mass gap are still far from the large- N results showing a very slow approach to the asymptotic regime.

Topological susceptibility and string tension in CP N−1 models

We determined the topological susceptibility and the string tension in the lattice CP N−1 models for a wide range of values of N, in particular for N = 4, 10, 21, 41. Quantitative agreement with the large- N predictions is found for the CP 20 and the CP 40 models.

Does the topological charge scale in the lattice CP 3-model?

Using a multigrid Monte Carlo algorithm with piecewise-linear interpolation operator for the two dimensional lattice CP 3 model with the standard quadratic action, we compare our results for the correlation length and the topological susceptibility with recent overrelaxation data. We do not confirm the scaling for the universal topological susceptibility found for smaller correlation length in the overrelaxation simulation. We find clear evidence that a single state saturates the 2-point correlation function at large distances.

Topological susceptibility and string tension in the lattice CPN-1 models.

In the lattice CP(N) models we studied the problems related to the measure of the topological susceptibility and the string tension . We perfomed numerical simulations at N=4 and N=10. In order to test the universality, we adopted two different lattice formulations. Scaling and universality tests led to the conclusion that at N=10 the geometrical approach gives a good definition of lattice topological susceptibility. On the other hand, N=4 proved not to be large enough to suppress the unphysical configurations, called dislocations, contributing to the topological susceptibility. We obtained other determinations of the topological susceptibility by the field theoretical method, wich relies on a local definition of the lattice topological charge density, and the cooling method. They gave quite consistent results, showing scaling and universality. The large-N expansion predicts an exponential area law behavior for sufficiently large Wilson loops, which implies confinement, due to the dynamical matter fields and absence of the screening phenomenon. We determined the string tension, without finding evidence of screening effects.

The topological susceptibility of the lattice CP n−1 model on the torus and the sphere

The topological vacuum structure of the two-dimensional CP n−1 model for n = 3, 5, 7 is studied on the lattice. In particular we investigate the small-volume limit on the torus as well as on the sphere and compare with continuum results. For n ⩾ 5, where lattice artifacts should be suppressed, the topological susceptibility shows unexpectedly strong deviations from asymptotic scaling. On the other hand there is an indication of a convergence to values obtained analytically within the limit n → ∞.

Finite size effects and scaling in lattice CP N−1

We present model predictions for the spectrum of CP N−1 in a periodic box and use them to interpret the strong finite size effects observed in lattice simulations at medium values of N. The asymptotic scaling behaviour of alternative lattice actions is discussed along with some aspects of multigrid algorithm efficiency.

The spectrum and potential in lattice CP N

The two-dimensional CP N model is studied on the lattice. Cluster, local Metropolis and hybrid update schemes are used to study the low-lying spectrum and potential of the model for low and moderate values of N. A linearly confining potential is confirmed and accurate measurements of the adjoint bound state mass are obtained. The expected scaling is observed for this mass and for the topological susceptibility. Different forms of the action are shown to give similar results thus demonstrating universality. Little evidence is found for the expected singlet and higher bound states so the 1/ N predictions for the spectrum can not be verified in detail. We independently confirm recent findings that critical slowing is not eliminated by the simplest cluster algorithms but find that a useful reduction in real autocorrelation times can be obtained using an efficiently vectorised hybrid of local and cluster update algorithms.

Topology and universality in the lattice CP 2 model

The topological susceptibility χ t and the correlation length ξ in the CP 2 model on a two-dimensional triangular lattice are calculated by the Monte Carlo simulation method. Using variant lattice actions, we verify the universality of the dimensionless quantity χ t ξ 2 ∼- 0.09 in the scaling region, thus indicating the presence of physical topological effects in the scaling limit. As a by-product, we obtain a combinatorial formula for the topological charge.

Strong coupling versus large N in σ-models

We show that the lattice CP ( N−1) model does not have a phase transition at N = ∞, of the type proposed for SU( N) gauge theories, despite having instantons. It does, however, have non-commuting large- N and strong coupling expansions. This rather unique situation is associated with the screening properties of the model.