Algorithmic differentiation and the calculation of forces by quantum Monte Carlo

We describe an efficient algorithm to compute forces in quantum Monte Carlo using adjoint algorithmic differentiation. This allows us to apply the space warp coordinate transformation in differential form, and compute all the 3M force components of a system with M atoms with a computational effort comparable with the one to obtain the total energy. Few examples illustrating the method for an electronic system containing several water molecules are presented. With the present technique, the calculation of finite-temperature thermodynamic properties of materials with quantum Monte Carlo will be feasible in the near future.Comment: 32 pages, 4 figure, to appear in The Journal of Chemical Physic

Ising transition in the two-dimensional quantum J1−J2J_1-J_2 Heisenberg model

We study the thermodynamics of the spin-$S$ two-dimensional quantum Heisenberg antiferromagnet on the square lattice with nearest ($J_1$) and next-nearest ($J_2$) neighbor couplings in its collinear phase ($J_2/J_1>0.5$), using the pure-quantum self-consistent harmonic approximation. Our results show the persistence of a finite-temperature Ising phase transition for every value of the spin, provided that the ratio $J_2/J_1$ is greater than a critical value corresponding to the onset of collinear long-range order at zero temperature. We also calculate the spin- and temperature-dependence of the collinear susceptibility and correlation length, and we discuss our results in light of the experiments on Li$_2$VOSiO$_4$ and related compounds.Comment: 4 page, 4 figure

Spin-lattice coupling in frustrated antiferromagnets

We review the mechanism of spin-lattice coupling in relieving the geometrical frustration of pyrochlore antiferromagnets, in particular spinel oxides. The tetrahedral unit, which is the building block of the pyrochlore lattice, undergoes a spin-driven Jahn-Teller instability when lattice degrees of freedom are coupled to the antiferromagnetism. By restricting our considerations to distortions which preserve the translational symmetries of the lattice, we present a general theory of the collective spin-Jahn-Teller effect in the pyrochlore lattice. One of the predicted lattice distortions breaks the inversion symmetry and gives rise to a chiral pyrochlore lattice, in which frustrated bonds form helices with a definite handedness. The chirality is transferred to the spin system through spin-orbit coupling, resulting in a long-period spiral state, as observed in spinel CdCr2O4. We discuss explicit models of spin-lattice coupling using local phonon modes, and their applications in other frustrated magnets.Comment: 23 pages, 6 figures. Lecture notes for Trieste Summer School, August 2007. To appear as a chapter in "Highly Frustrated Magnetism", Eds. C. Lacroix, P. Mendels, F. Mil

Real-Time Risk Management: An AAD-PDE Approach

We apply adjoint algorithmic differentiation (AAD) to the risk management of securities when their price dynamics are given by partial differential equations (PDE). We show how AAD can be applied to forward and backward PDEs in a straightforward manner. In the context of one-factor models for interest rates or default intensities, we show how price sensitivities are computed reliably and orders of magnitude faster than with a standard finite-difference approach. This significantly increased efficiency is obtained by combining (i) the adjoint forward PDE for calibrating model parameters, (ii) the adjoint backward PDE for derivatives pricing, and (iii) the implicit function theorem to avoid iterating the calibration procedure

Thermodynamics of the quantum easy-plane antiferromagnet on the triangular lattice

The classical XXZ triangular-lattice antiferromagnet (TAF) shows both an Ising and a BKT transition, related to the chirality and the in-plane spin components, respectively. In this paper the quantum effects on the thermodynamic quantities are evaluated by means of the pure-quantum self-consistent harmonic approximation (PQSCHA), that allows one to deal with any spin value through classical MC simulations. We report the internal energy, the specific heat, and the in-plane correlation length of the quantum XX0 TAF, for S=1/2, 1, 5/2. The quantum transition temperatures turn out to be smaller the smaller the spin, and agree with the few available theoretical and numerical estimates.Comment: 4 pages,3 postscript figure

Spiral order by disorder and lattice nematic order in a frustrated Heisenberg antiferromagnet on the honeycomb lattice

Motivated by recent experiments on Bi$_3$Mn$_4$O$_{12}$(NO$_3$), we study a frustrated $J_1$-$J_2$ Heisenberg model on the two dimensional (2D) honeycomb lattice. The classical $J_1$-$J_2$ Heisenberg model on the two dimensional (2D) honeycomb lattice has N\'eel order for $J_2 J_1/6$, it exhibits a one-parameter family of degenerate incommensurate spin spiral ground states where the spiral wave vector can point in any direction. Spin wave fluctuations at leading order lift this accidental degeneracy in favor of specific wave vectors, leading to spiral order by disorder. For spin $S=1/2$, quantum fluctuations are, however, likely to be strong enough to melt the spiral order parameter over a wide range of $J_2/J_1$. Over a part of this range, we argue that the resulting state is a valence bond solid (VBS) with staggered dimer order - this VBS is a nematic which breaks lattice rotational symmetry. Our arguments are supported by comparing the spin wave energy with the energy of the dimer solid obtained using a bond operator formalism. Turning to the effect of thermal fluctuations on the spiral ordered state, any nonzero temperature destroys the magnetic order, but the discrete rotational symmetry of the lattice remains broken resulting in a thermal analogue of the nematic VBS. We present arguments, supported by classical Monte Carlo simulations, that this nematic transforms into the high temperature symmetric paramagnet via a thermal phase transition which is in the universality class of the classical 3-state Potts (clock) model in 2D. We discuss the possible relevance of our results for honeycomb magnets, such as Bi$_3$M$_4$O$_{12}$(NO$_3$) (with M=Mn,V,Cr), and bilayer triangular lattice magnets.Comment: Slightly revise

AAD and least-square Monte Carlo: fast Bermudan-style options and XVA Greeks

We show how Adjoint Algorithmic Differentiation (AAD) can be used to calculate price sensitivities in regression-based Monte Carlo methods reliably and orders of magnitude faster than with standard finite-difference approaches. We present the AAD version of the celebrated least-square algorithms of Tsitsiklis and Van Roy (2001) and Longstaff and Schwartz (2001). By discussing in detail examples of practical relevance, we demonstrate how accounting for the contributions associated with the regression functions is crucial to obtain accurate estimates of the Greeks, especially in XVA applications

Inhomogeneity Induces Resonance Coherence Peaks in Superconducting BSCCO

In this paper we analyze, using scanning tunneling spectroscopy, the density of electronic states in nearly optimally doped BSCCO in zero field. Focusing on the superconducting gap, we find patches of what appear to be two different phases in a background of some average gap, one with a relatively small gap and sharp large coherence peaks and one characterized by a large gap with broad weak coherence peaks. We compare these spectra with calculations of the local density of states for a simple phenomenological model in which a 2 xi_0 * 2 xi_0 patch with an enhanced or supressed d-wave gap amplitude is embedded in a region with a uniform average d-wave gap.Comment: 4 pages, 3 figure

Treatment responses to antiangiogenetic therapy and chemotherapy in nonsecreting paraganglioma (PGL4) of urinary bladder with SDHB mutation: a case report

Paraganglioma (PGL) is a rare neuroendocrine tumor. Currently, the malignancy is defined as the presence of metastatic spread at presentation or during follow-up. Several gene mutations are listed in the pathogenesis of PGL, among which succinate dehydrogenase (SDHX), particularly the SDHB isoform, is the main gene involved in malignancy. A 55-year-old male without evidence of catecholamine secretion had surgery for PGL of the urinary bladder. After 1 year, he showed a relapse of disease and demonstrated malignant PGL without evidence of catecholamine secretion with a germline heterozygous mutation of succinate dehydrogenase B (SDHB). After failure of a second surgery for relapse, he started medical treatment with sunitinib daily but discontinued due to serious side effects. Cyclophosphamide, vincristine, and dacarbazine (CVD) chemotherapeutic regimen stopped the disease progression for 7 months. Conclusion: Malignant PGL is a very rare tumor, and SDHB mutations must be always considered in molecular diagnosis because they represent a critical event in the progression of the oncological disease. Currently, there are few therapeutic protocols, and it is often difficult, as this case demonstrates, to decide on a treatment option according to a reasoned set of choices. Abbreviations: CVD = cyclophosphamide, vincristine and dacarbazine, HIF-1a = hypoxia inducible factor 1 alpha, PGL = paraganglioma, SDH = succinate dehydrogenase, VEGF = vasoendothelial growth factor

Violation of the Luttinger sum rule within the Hubbard model on a triangular lattice

The frequency-moment expansion method is developed to analyze the validity of the Luttinger sum rule within the Mott-Hubbard insulator, as represented by the generalized Hubbard model at half filling and large $U$. For the particular case of the Hubbard model with nearest-neighbor hopping on a triangular lattice lacking the particle-hole symmetry results reveal substantial violation of the sum rule.Comment: 4 pages, 2 figure