Transfer Matrices and Excitations with Matrix Product States

We investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low energy excitations using the formalism of tensor network states. In particular, we show that the Matrix Product State Transfer Matrix (MPS-TM) - a central object in the computation of static correlation functions - provides important information about the location and magnitude of the minima of the low energy dispersion relation(s) and present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TM's eigenspectrum and give several arguments for the close relation between the structure of the low energy spectrum of the system and the form of static correlation functions. Finally, we discuss how the MPS-TM connects to the exact Quantum Transfer Matrix (QTM) of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of MPS, which allows to reinterpret variational MPS techniques (such as the Density Matrix Renormalization Group) as an application of Wilson's Numerical Renormalization Group along the virtual (imaginary time) dimension of the system.Comment: 39 pages (+8 pages appendix), 14 figure

Symmetry Breaking and the Geometry of Reduced Density Matrices

The concept of symmetry breaking and the emergence of corresponding local order parameters constitute the pillars of modern day many body physics. The theory of quantum entanglement is currently leading to a paradigm shift in understanding quantum correlations in many body systems and in this work we show how symmetry breaking can be understood from this wavefunction centered point of view. We demonstrate that the existence of symmetry breaking is a consequence of the geometric structure of the convex set of reduced density matrices of all possible many body wavefunctions. The surfaces of those convex bodies exhibit non-analytic behavior in the form of ruled surfaces, which turn out to be the defining signatures for the emergence of symmetry breaking and of an associated order parameter. We illustrate this by plotting the convex sets arising in the context of three paradigmatic examples of many body systems exhibiting symmetry breaking: the quantum Ising model in transverse magnetic field, exhibiting a second order quantum phase transition; the classical Ising model at finite temperature in two dimensions, which orders below a critical temperature $T_c$; and a system of free bosons at finite temperature in three dimensions, exhibiting the phenomenon of Bose-Einstein condensation together with an associated order parameter $\langle\psi\rangle$. Remarkably, these convex sets look all very much alike. We believe that this wavefunction based way of looking at phase transitions demystifies the emergence of order parameters and provides a unique novel tool for studying exotic quantum phenomena.Comment: 5 pages, 3 figures, Appendix with 2 pages, 3 figure

Introducing the Complexity of Educational Diversification

Peer reviewe

A joint physics and radiobiology DREAM team vision - Towards better response prediction models to advance radiotherapy.

Radiotherapy developed empirically through experience balancing tumour control and normal tissue toxicities. Early simple mathematical models formalized this practical knowledge and enabled effective cancer treatment to date. Remarkable advances in technology, computing, and experimental biology now create opportunities to incorporate this knowledge into enhanced computational models. The ESTRO DREAM (Dose Response, Experiment, Analysis, Modelling) workshop brought together experts across disciplines to pursue the vision of personalized radiotherapy for optimal outcomes through advanced modelling. The ultimate vision is leveraging quantitative models dynamically during therapy to ultimately achieve truly adaptive and biologically guided radiotherapy at the population as well as individual patient-based levels. This requires the generation of models that inform response-based adaptations, individually optimized delivery and enable biological monitoring to provide decision support to clinicians. The goal is expanding to models that can drive the realization of personalized therapy for optimal outcomes. This position paper provides their propositions that describe how innovations in biology, physics, mathematics, and data science including AI could inform models and improve predictions. It consolidates the DREAM team's consensus on scientific priorities and organizational requirements. Scientifically, it stresses the need for rigorous, multifaceted model development, comprehensive validation and clinical applicability and significance. Organizationally, it reinforces the prerequisites of interdisciplinary research and collaboration between physicians, medical physicists, radiobiologists, and computational scientists throughout model development. Solely by a shared understanding of clinical needs, biological mechanisms, and computational methods, more informed models can be created. Future research environment and support must facilitate this integrative method of operation across multiple disciplines

A joint physics and radiobiology DREAM team vision - towards better response prediction models to advance radiotherapy

Radiotherapy developed empirically through experience balancing tumour control and normal tissue toxicities. Early simple mathematical models formalized this practical knowledge and enabled effective cancer treatment to date. Remarkable advances in technology, computing, and experimental biology now create opportunities to incorporate this knowledge into enhanced computational models. The ESTRO DREAM (Dose Response, Experiment, Analysis, Modelling) workshop brought together experts across disciplines to pursue the vision of personalized radiotherapy for optimal outcomes through advanced modelling. The ultimate vision is leveraging quantitative models dynamically during therapy to ultimately achieve truly adaptive and biologically guided radiotherapy at the population as well as individual patient-based levels. This requires the generation of models that inform response-based adaptations, individually optimized delivery and enable biological monitoring to provide decision support to clinicians. The goal is expanding to models that can drive the realization of personalized therapy for optimal outcomes. This position paper provides their propositions that describe how innovations in biology, physics, mathematics, and data science including AI could inform models and improve predictions. It consolidates the DREAM team's consensus on scientific priorities and organizational requirements. Scientifically, it stresses the need for rigorous, multifaceted model development, comprehensive validation and clinical applicability and significance. Organizationally, it reinforces the prerequisites of interdisciplinary research and collaboration between physicians, medical physicists, radiobiologists, and computational scientists throughout model development. Solely by a shared understanding of clinical needs, biological mechanisms, and computational methods, more informed models can be created. Future research environment and support must facilitate this integrative method of operation across multiple disciplines. [Abstract copyright: Copyright © 2024 The Author(s). Published by Elsevier B.V. All rights reserved.

A joint physics and radiobiology DREAM team vision - towards better response prediction models to advance radiotherapy

Radiotherapy developed empirically through experience balancing tumour control and normal tissue toxicities. Early simple mathematical models formalized this practical knowledge and enabled effective cancer treatment to date. Remarkable advances in technology, computing, and experimental biology now create opportunities to incorporate this knowledge into enhanced computational models. The ESTRO DREAM (Dose Response, Experiment, Analysis, Modelling) workshop brought together experts across disciplines to pursue the vision of personalized radiotherapy for optimal outcomes through advanced modelling. The ultimate vision is leveraging quantitative models dynamically during therapy to ultimately achieve truly adaptive and biologically guided radiotherapy at the population as well as individual patient-based levels. This requires the generation of models that inform response-based adaptations, individually optimized delivery and enable biological monitoring to provide decision support to clinicians. The goal is expanding to models that can drive the realization of personalized therapy for optimal outcomes. This position paper provides their propositions that describe how innovations in biology, physics, mathematics, and data science including AI could inform models and improve predictions. It consolidates the DREAM team’s consensus on scientific priorities and organizational requirements. Scientifically, it stresses the need for rigorous, multifaceted model development, comprehensive validation and clinical applicability and significance. Organizationally, it reinforces the prerequisites of interdisciplinary research and collaboration between physicians, medical physicists, radiobiologists, and computational scientists throughout model development. Solely by a shared understanding of clinical needs, biological mechanisms, and computational methods, more informed models can be created. Future research environment and support must facilitate this integrative method of operation across multiple disciplines