Improved Hilbert space exploration algorithms for finite temperature calculations

Computing correlation functions in strongly-interacting quantum systems is one of the most important challenges of modern condensed matter theory, due to their importance in the description of many physical observables. Simultaneously, this challenge is one of the most difficult to address, due to the inapplicability of traditional perturbative methods or the few-body limitations of numerical approaches. For special cases, where the model is integrable, methods based on the Bethe Ansatz have succeeded in computing the spectrum and given us analytical expressions for the matrix elements of physically important operators. However, leveraging these results to compute correlation functions generally requires the numerical evaluation of summations over eigenstates. To perform these summations efficiently, Hilbert space exploration algorithms have been developed which has resulted most notably in the ABACUS library. While this performs quite well for correlations on ground states or low-entropy states, the case of high entropy states (most importantly at finite temperatures or after a quantum quench) is more difficult, and leaves room for improvement. In this work, we develop a new Hilbert space exploration algorithm for the Lieb-Liniger model, specially tailored to optimize the computational order on finite-entropy states for correlations of density-related operators.Comment: 32 pages, 9 figure

Statistics of matrix elements of local operators in integrable models

We study the statistics of matrix elements of local operators in the basis of energy eigenstates in a paradigmatic integrable many-particle quantum theory, the Lieb-Liniger model of bosons with repulsive delta-function interaction. Using methods of quantum integrability we determine the scaling of matrix elements with system size. As a consequence of the extensive number of conservation laws the structure of matrix elements is fundamentally different from, and much more intricate than, the predictions of the eigenstate thermalization hypothesis for generic models. We uncover an interesting connection between this structure for local operators in interacting integrable models, and the one for local operators that are not local with respect to the elementary excitations in free theories. We find that typical off-diagonal matrix elements $\langle\boldsymbol{\mu}|O|\boldsymbol{\lambda}\rangle$ in the same macro-state scale as $\exp(-c^{ O}L\ln(L)-LM^{O}_{\boldsymbol{\mu},\boldsymbol{\lambda}})$ where the probability distribution function for $M^{O}_{\boldsymbol{\mu},\boldsymbol{\lambda}}$ are well described by Fr\'echet distributions and $c^{O}$ depends only on macro-state information. In contrast, typical off-diagonal matrix elements between two different macro-states scale as $\exp(-d^{ O}L^2)$, where $d^{O}$ depends only on macro-state information. Diagonal matrix elements depend only on macro-state information up to finite-size corrections.Comment: 30 pages, 40 figure

Learning to imitate facial expressions through sound

The question of how young infants learn to imitate others’ facial expressions has been central in developmental psychology for decades. Facial imitation has been argued to constitute a particularly challenging learning task for infants because facial expressions are perceptually opaque: infants cannot see changes in their own facial configuration when they execute a motor program, so how do they learn to match these gestures with those of their interacting partners? Here we argue that this apparent paradox mainly appears if one focuses only on the visual modality, as most existing work in this field has done so far. When considering other modalities, in particular the auditory modality, many facial expressions are not actually perceptually opaque. In fact, every orolabial expression that is accompanied by vocalisations has specific acoustic consequences, which means that it is relatively transparent in the auditory modality. Here, we describe how this relative perceptual transparency can allow infants to accrue experience relevant for orolabial, facial imitation every time they vocalise. We then detail two specific mechanisms that could support facial imitation learning through the auditory modality. First, we review evidence showing that experiencing correlated proprioceptive and auditory feedback when they vocalise – even when they are alone – enables infants to build audio-motor maps that could later support facial imitation of orolabial actions. Second, we show how these maps could also be used by infants to support imitation even for silent, orolabial facial expressions at a later stage. By considering non-visual perceptual domains, this paper expands our understanding of the ontogeny of facial imitation and offers new directions for future investigations

New approximations for the cone of copositive matrices and its dual

We provide convergent hierarchies for the cone C of copositive matrices and its dual, the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner) approximations for C (resp. for its dual), thus complementing previous inner (resp. outer) approximations for C (for the dual). In particular, both inner and outer approximations have a very simple interpretation. Finally, extension to K-copositivity and K-complete positivity for a closed convex cone K, is straightforward.Comment: 8

Some geochemical constraints upon models for the crystallization of the upper critical zone-main zone interval, northwestern Bushveld complex

Ratios between elements Mg, Fe, Co, Cr, Ni, V, and Sc are consistently different in mafic rocks of the upper critical zone, and those above the Bastard unit. Within the 300 m section above the Merensky Reef, 87Sr/86Sr ratios increase from c.0.7063 to c.0.7087, irrespective of rock type. Decoupling of Mg/(Mg + Fe2+) ratios and the Ca contents of plagioclase, and wide variations in the proportions of anorthosite within the Bastard, Merensky, and Merensky Footwall units, are inconsistent with anorthosite formation by simple fractional crystallization of magma batches of limited volume

Integer Polynomial Optimization in Fixed Dimension

We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an integer polynomial over the lattice points of a convex polytope, we show an algorithm to compute lower and upper bounds for the optimal value. For polynomials that are non-negative over the polytope, these sequences of bounds lead to a fully polynomial-time approximation scheme for the optimization problem.Comment: In this revised version we include a stronger complexity bound on our algorithm. Our algorithm is in fact an FPTAS (fully polynomial-time approximation scheme) to maximize a non-negative integer polynomial over the lattice points of a polytop

Effect of Extending the Original Eligibility Criteria for the CROSS Neoadjuvant Chemoradiotherapy on Toxicity and Survival in Esophageal Cancer

Patients with curable esophageal cancer (EC) who proceed beyond the original Chemoradiotherapy for Oesophageal Cancer Followed by Surgery Study (CROSS) eligibility criteria are also treated with neoadjuvant chemoradiotherapy (nCRT). This study assessed the effect that extending the CROSS eligibility criteria for nCRT has on treatment-related toxicity and overall survival (OS) in EC. The study enrolled 161 patients with locally advanced EC (T1N1-3/T2-4aN0-3/M0) treated with the CROSS schedule followed by esophagectomy. Group 1 consisted of 89 patients who met the CROSS criteria, and group 2 consisted of 72 patients who met the extended eligibility criteria, i.e. a tumor length greater than 8 cm (n = 24), more than 10% weight loss (n = 35), more than 2-4 cm extension in the stomach (n = 21), celiac lymph node metastasis (n = 13), and/or age over 75 years (n = 2). The study assessed the differences in nCRT-associated toxicity [National Cancer Institute's Common Terminology Criteria for Adverse Events (CTCAE) grade3] and 90-day postoperative mortality. Moreover, the prognostic value for OS was assessed with multivariate Cox regression analysis. No difference was found in nCRT-associated toxicity (P = 0.117), postoperative complications (P = 0.783), and 90-day mortality (P = 0.492). The OS differed significantly (P = 0.004), with a median of 37.3 months [95% confidence interval (CI), 10.4-64.2 months] for group 1 and 17.2 months (95% CI 13.8-20.7 months) for group 2. Pathologic N stage (P = 0.023), pathologic T stage (P = 0.043), and group 2 (P = 0.008) were independent prognostic factors for OS. Extension of the CROSS study eligibility criteria for nCRT did not affect nCRT-associated toxicity, postoperative complications, and postoperative mortality, but was prognostic for OS