Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2102.05145 · Nature Communications

Uncomputably complex renormalisation group flows

James D. Watson, Emilio Onorati, Toby S. Cubitt

Published Dec 9, 2022• 11 citations
arXivHTMLPDF

Abstract

Renormalisation group methods are among the most important techniques for analysing the physics of many-body systems: by iterating a renormalisation group map, which coarse-grains the description of a system and generates a flow in the parameter space, physical properties of interest can be extracted. However, recent work has shown that important physical features, such as the spectral gap and phase diagram, may be impossible to determine, even in principle. Following these insights, we construct a rigorous renormalisation group map for the original undecidable many-body system that appeared in the literature, which reveals a renormalisation group flow so complex that it cannot be predicted. We prove that each step of this map is computable, and that it converges to the correct fixed points, yet the resulting flow is uncomputable. This extreme form of unpredictability for renormalisation group flows had not been shown before and goes beyond the chaotic behaviour seen previously.

DOI
10.1038/s41467-022-35179-4

Related papers

Uncomputability of phase diagrams

Johannes Bausch, Toby S. Cubitt, James D. Watson

The phase diagram of a material is of central importance in describing the properties and behaviour of a condensed matter system. In this work, we prove that the task of determining the phase diagram of a many-body Hamiltonian is in general uncomputable, by explicitly constructing a continuous one-parameter family of Hamiltonians H(φ), where [Formula: see text], for which this is the case. The H(φ) are translationally-invariant, with nearest-neighbour couplings on a 2D spin lattice. As well as implying uncomputablity of phase diagrams, our result also proves that undecidability can hold for a set of positive measure of a Hamiltonian's parameter space, whereas previous results only implied undecidability on a zero measure set. This brings the spectral gap undecidability results a step closer to standard condensed matter problems, where one typically studies phase diagrams of many-body models as a function of one or more continuously varying real parameters, such as magnetic field strength or pressure.

arXiv:1910.01631•Nature Communications•Jan 19, 2021•20 citations

Scaling Analysis and Renormalisation Group for General (Quantum) Many Body Systems in the Critical Regime

Manfred Requardt

With the help of a smooth scaling and coarse-graining approach of observables, developed recently by us in the context of so-called fluctuation operators (inspired by prior work of Verbeure et al) we perform a rigorous renormalisation group analysis of the critical regime. The approach is quite general, encompassing classical, quantum, discrete and continuous systems, the main thrust going to quantum many body systems. Our central topic is the analysis of the emergent properties of critical systems on the intermediate scales and in the scaling limit. To mention some particularly interesting points, we show that systems typically loose part of their quantum character in the scaling limit (vanishing of commutators) and we rigorously prove, with the help of the KMS-condition, the emergence of the phenomenon of critical slowing down together with the necessity of renormalising the time variable. These general features are then illustrated with the help of an instructive class of models and are related to the singular structure of quasi particle excitation modes for vanishing energy-momentum.

arXiv:math-ph/0303064•Mar 27, 2003•0 citations

A Renormalisation group for TCSA

Giovanni Feverati, Graham, Kevin, Paul A. Pearce +2 more

We discuss the errors introduced by level truncation in the study of boundary renormalisation group flows by the Truncated Conformal Space Approach. We show that the TCSA results can have the qualitative form of a sequence of RG flows between different conformal boundary conditions. In the case of a perturbation by the field phi(13), we propose a renormalisation group equation for the coupling constant which predicts a fixed point at a finite value of the TCSA coupling constant and we compare the predictions with data obtained using TBA equations.

arXiv:hep-th/0612203•Dec 19, 2006•25 citations

Scaling Limit and Renormalisation Group in the Critical Point Analysis of General (Quantum) Many Body Systems

Manfred Requardt

We employ the machinery of smooth scaling and coarse-graining of observables, developed recently by us in the context of so-called fluctuation operators (inspired by prior work of Verbeure et al) to make a rigorous renormalisation group analysis of the critical regime. The approach appears to be quite general, encompassing classical, quantum, discrete and continuous systems. One of our central topics is the analysis of the famous `scaling hypothesis', that is, we make a general investigation under what conditions on the l-point correlation functions a scale invariant (non-trivial) limit theory can be actually attained. Furthermore, we study in a rigorous manner questions like the quantum character of the system in the scaling limit, the phenomenon of critical slowing down etc.

arXiv:math-ph/0205011•May 8, 2002•1 citation

Physics-informed renormalisation group flows

Friederike Ihssen, Pawlowski, Jan M.

The physics of strongly correlated systems offers some of the most intriguing physics challenges such as competing orders or the emergence of dynamical composite degrees of freedom. Often, the resolution of these physics challenges is computationally hard, but can be simplified enormously by a formulation in terms of the dynamical degrees of freedom and within an expansion about the physical ground state. Importantly, such a formulation does not only reduce or minimise the computational challenges, it also facilitates the access to the physics mechanisms at play. The tasks of finding the dynamical degrees of freedom and the physical ground state can be systematically addressed within the functional renormalisation group approach with flowing fields which accommodates both, emergent composites as well as the physical ground state. In the present work we use this approach to set up physics-informed renormalisation group flows (PIRG flows): Scale-dependent coordinate transformations in field space induce emergent composites, and the respective flows for the effective action generate a large set of target actions, formulated in these emergent composite fields. This novel perspective on RG flows bears a great potential both for conceptual as well as computational applications: to begin with, PIRG flows allow for a systematic search of the dynamical degrees of freedom and the respective ground state that leads to the most rapid convergence of expansion schemes, thus minimising the computational effort. Secondly, the resolution of the remaining computational tasks within a given expansion scheme can be further reduced by optimising the physics content within a given approximation. Thirdly, the maximal variability of PIRG flows can be used to reduce the analytic and numerical effort of solving the flows within a given approximation.

arXiv:2409.13679•Sep 20, 2024•2 citations

High-precision thermodynamic and critical properties from tensor renormalization-group flows

Michael Hinczewski, A. Nihat Berker

The recently developed tensor renormalization-group (TRG) method provides a highly precise technique for deriving thermodynamic and critical properties of lattice Hamiltonians. The TRG is a local coarse-graining transformation, with the elements of the tensor at each lattice site playing the part of the interactions that undergo the renormalization-group flows. These tensor flows are directly related to the phase diagram structure of the infinite system, with each phase flowing to a distinct surface of fixed points. Fixed-point analysis and summation along the flows give the critical exponents, as well as thermodynamic functions along the entire temperature range. Thus, for the ferromagnetic triangular lattice Ising model, the free energy is calculated to better than 10(-5) along the entire temperature range. Unlike previous position-space renormalization-group methods, the truncation (of the tensor index range D) in this general method converges under straightforward and systematic improvements. Our best results are easily obtained with D=24, corresponding to 4624-dimensional renormalization-group flows.

arXiv:0709.2803•Physical Review E•Jan 7, 2008•14 citations
Renormalisation group methods are among the most important techniques for analysing the physics of many-body systems: by iterating a renormalisation group map, which coarse-grains the description of a system and generates a flow in the parameter space, physical properties of interest can be extracted. However, recent work has shown that important physical features, such as the spectral gap and phase diagram, may be impossible to determine, even in principle. Following these insights, we construct a rigorous renormalisation group map for the original undecidable many-body system that appeared in the literature, which reveals a renormalisation group flow so complex that it cannot be predicted. We prove that each step of this map is computable, and that it converges to the correct fixed points, yet the resulting flow is uncomputable. This extreme form of unpredictability for renormalisation group flows had not been shown before and goes beyond the chaotic behaviour seen previously.
The phase diagram of a material is of central importance in describing the properties and behaviour of a condensed matter system. In this work, we prove that the task of determining the phase diagram of a many-body Hamiltonian is in general uncomputable, by explicitly constructing a continuous one-parameter family of Hamiltonians H(φ), where [Formula: see text], for which this is the case. The H(φ) are translationally-invariant, with nearest-neighbour couplings on a 2D spin lattice. As well as implying uncomputablity of phase diagrams, our result also proves that undecidability can hold for a set of positive measure of a Hamiltonian's parameter space, whereas previous results only implied undecidability on a zero measure set. This brings the spectral gap undecidability results a step closer to standard condensed matter problems, where one typically studies phase diagrams of many-body models as a function of one or more continuously varying real parameters, such as magnetic field strength or pressure.
With the help of a smooth scaling and coarse-graining approach of observables, developed recently by us in the context of so-called fluctuation operators (inspired by prior work of Verbeure et al) we perform a rigorous renormalisation group analysis of the critical regime. The approach is quite general, encompassing classical, quantum, discrete and continuous systems, the main thrust going to quantum many body systems. Our central topic is the analysis of the emergent properties of critical systems on the intermediate scales and in the scaling limit. To mention some particularly interesting points, we show that systems typically loose part of their quantum character in the scaling limit (vanishing of commutators) and we rigorously prove, with the help of the KMS-condition, the emergence of the phenomenon of critical slowing down together with the necessity of renormalising the time variable. These general features are then illustrated with the help of an instructive class of models and are related to the singular structure of quasi particle excitation modes for vanishing energy-momentum.
We discuss the errors introduced by level truncation in the study of boundary renormalisation group flows by the Truncated Conformal Space Approach. We show that the TCSA results can have the qualitative form of a sequence of RG flows between different conformal boundary conditions. In the case of a perturbation by the field phi(13), we propose a renormalisation group equation for the coupling constant which predicts a fixed point at a finite value of the TCSA coupling constant and we compare the predictions with data obtained using TBA equations.
We employ the machinery of smooth scaling and coarse-graining of observables, developed recently by us in the context of so-called fluctuation operators (inspired by prior work of Verbeure et al) to make a rigorous renormalisation group analysis of the critical regime. The approach appears to be quite general, encompassing classical, quantum, discrete and continuous systems. One of our central topics is the analysis of the famous `scaling hypothesis', that is, we make a general investigation under what conditions on the l-point correlation functions a scale invariant (non-trivial) limit theory can be actually attained. Furthermore, we study in a rigorous manner questions like the quantum character of the system in the scaling limit, the phenomenon of critical slowing down etc.
The physics of strongly correlated systems offers some of the most intriguing physics challenges such as competing orders or the emergence of dynamical composite degrees of freedom. Often, the resolution of these physics challenges is computationally hard, but can be simplified enormously by a formulation in terms of the dynamical degrees of freedom and within an expansion about the physical ground state. Importantly, such a formulation does not only reduce or minimise the computational challenges, it also facilitates the access to the physics mechanisms at play. The tasks of finding the dynamical degrees of freedom and the physical ground state can be systematically addressed within the functional renormalisation group approach with flowing fields which accommodates both, emergent composites as well as the physical ground state. In the present work we use this approach to set up physics-informed renormalisation group flows (PIRG flows): Scale-dependent coordinate transformations in field space induce emergent composites, and the respective flows for the effective action generate a large set of target actions, formulated in these emergent composite fields. This novel perspective on RG flows bears a great potential both for conceptual as well as computational applications: to begin with, PIRG flows allow for a systematic search of the dynamical degrees of freedom and the respective ground state that leads to the most rapid convergence of expansion schemes, thus minimising the computational effort. Secondly, the resolution of the remaining computational tasks within a given expansion scheme can be further reduced by optimising the physics content within a given approximation. Thirdly, the maximal variability of PIRG flows can be used to reduce the analytic and numerical effort of solving the flows within a given approximation.
The recently developed tensor renormalization-group (TRG) method provides a highly precise technique for deriving thermodynamic and critical properties of lattice Hamiltonians. The TRG is a local coarse-graining transformation, with the elements of the tensor at each lattice site playing the part of the interactions that undergo the renormalization-group flows. These tensor flows are directly related to the phase diagram structure of the infinite system, with each phase flowing to a distinct surface of fixed points. Fixed-point analysis and summation along the flows give the critical exponents, as well as thermodynamic functions along the entire temperature range. Thus, for the ferromagnetic triangular lattice Ising model, the free energy is calculated to better than 10(-5) along the entire temperature range. Unlike previous position-space renormalization-group methods, the truncation (of the tensor index range D) in this general method converges under straightforward and systematic improvements. Our best results are easily obtained with D=24, corresponding to 4624-dimensional renormalization-group flows.