Exploring the magnetic properties of the largest single molecule magnets

The giant {Mn₇₀} and {Mn₈₄} wheels are the largest nuclearity single-molecule magnets synthesized to date, and understanding their magnetic properties poses a challenge to theory. Starting from first-principles calculations, we explore the magnetic properties and excitations in these wheels using effective spin Hamiltonians. We find that the unusual geometry of the superexchange pathways leads to weakly coupled {Mn₇} subunits carrying an effective S = 2 spin. The spectrum exhibits a hierarchy of energy scales and massive degeneracies, with the lowest-energy excitations arising from Heisenberg-ring-like excitations of the {Mn₇} subunits around the wheel. We further describe how weak longer-range couplings can select the precise spin ground-state of the Mn wheels out of the nearly degenerate ground-state band

On the generalization of the exponential basis for tensor network representations of long-range interactions in two and three dimensions

In one dimension (1D), a general decaying long-range interaction can be fit to a sum of exponential interactions e^(−λrij) with varying exponents λ, each of which can be represented by a simple matrix product operator with bond dimension D=3. Using this technique, efficient and accurate simulations of 1D quantum systems with long-range interactions can be performed using matrix product states. However, the extension of this construction to higher dimensions is not obvious. We report how to generalize the exponential basis to two and three dimensions by defining the basis functions as the Green's functions of the discretized Helmholtz equation for different Helmholtz parameters λ, a construction which is valid for lattices of any spatial dimension. Compact tensor network representations can then be found for the discretized Green's functions, by expressing them as correlation functions of auxiliary fermionic fields with nearest-neighbor interactions via Grassmann Gaussian integration. Interestingly, this analytic construction in three dimensions yields a D=4 tensor network representation of correlation functions which (asymptotically) decay as the inverse distance (r^(−1)_(ij)), thus generating the (screened) Coulomb potential on a cubic lattice. These techniques will be useful in tensor network simulations of realistic materials

Efficient representation of long-range interactions in tensor network algorithms

We describe a practical and efficient approach to represent physically realistic long-range interactions in two-dimensional tensor network algorithms via projected entangled-pair operators (PEPOs). We express the long-range interaction as a linear combination of correlation functions of an auxiliary system with only nearest-neighbor interactions. To obtain a smooth and radially isotropic interaction across all length scales, we map the physical lattice to an auxiliary lattice of expanded size. Our construction yields a long-range PEPO as a sum of ancillary PEPOs, each of small, constant bond dimension. This representation enables efficient numerical simulations with long-range interactions using projected entangled pair states.Comment: Main Document: 9 pages, 7 figures. Moved supplementary material into main text. Added more discussion of computational cost. Fixed minor errors in Figs 2c and 3

Conversion of projected entangled pair states into a canonical form

We propose an algorithm to convert a projected entangled pair state (PEPS) into a canonical form, analogous to the well-known canonical form of a matrix product state. Our approach is based on a variational gauging ansatz for the QR tensor decomposition of PEPS columns into a matrix product operator and a finite depth circuit of unitaries and isometries. We describe a practical initialization scheme that leads to rapid convergence in the QR optimization. We explore the performance and stability of the variational gauging algorithm in norm calculations for the transverse-field Ising and Heisenberg models on a square lattice. We also demonstrate energy optimization within the PEPS canonical form for the transverse-field Ising and Heisenberg models. We expect this canonical form to open up improved analytical and numerical approaches for PEPS.Comment: 8 pages, 6 Figure

Recommendation to Use Exact P-values in Biomarker Discovery Research

Background: In biomarker discovery studies, markers are ranked for validation using P-values. Standard P-value calculations use normal approximations that may not be valid for small P-values and small sample sizes common in discovery research. Methods: We compared exact P-values, valid by definition, with normal and logit-normal approximations in a simulated study of 40 cases and 160 controls. The key measure of biomarker performance was sensitivity at 90% specificity. Data for 3000 uninformative markers and 30 true markers were generated randomly, with 10 replications of the simulation. We also analyzed real data on 2371 antibody array markers measured in plasma from 121 cases with ER/PR positive breast cancer and 121 controls. Results: Using the same discovery criterion, the valid exact P-values lead to discovery of 24 true and 82 false biomarkers while approximate P-values yielded 15 true and 15 false biomarkers (normal approximation) and 20 true and 86 false biomarkers (logit-normal approximation). Moreover, the estimated numbers of true markers among those discovered were substantially incorrect for approximate P-values: normal estimated 0 true markers discovered but found 15; logit-normal estimated 42 but found 20. The exact method estimated 22, close to the actual number of 24 true discoveries. With real data, exact and approximate P-values ranked candidate breast cancer biomarkers very differently. Conclusions: Exact P-values should be used because they are universally valid. Approximate P-values can lead to inappropriate biomarker selection rules and incorrect conclusions. Impact: Rigorous data analysis methodology in discovery research may improve the yield of biomarkers that validate clinically

Exploring the magnetic properties of the largest single molecule magnets

The giant {Mn₇₀} and {Mn₈₄} wheels are the largest nuclearity single-molecule magnets synthesized to date, and understanding their magnetic properties poses a challenge to theory. Starting from first-principles calculations, we explore the magnetic properties and excitations in these wheels using effective spin Hamiltonians. We find that the unusual geometry of the superexchange pathways leads to weakly coupled {Mn₇} subunits carrying an effective S = 2 spin. The spectrum exhibits a hierarchy of energy scales and massive degeneracies, with the lowest-energy excitations arising from Heisenberg-ring-like excitations of the {Mn₇} subunits around the wheel. We further describe how weak longer-range couplings can select the precise spin ground-state of the Mn wheels out of the nearly degenerate ground-state band