Projects in Holger Frahm's Research Group

FOR 2316: Correlations in Integrable Quantum Many-Body Systems

The goal of the research unit is the development of many-body standard reference systems with known static and dynamical correlation functions at arbitrary temperature, on the lattice and in the continuum, very much like Bosonization provides standard reference systems near \(T = 0\), namely conformal field theories with central charges \(c = 1\) and known static low-temperature properties.

Correlations in Integrable Quantum Many-Body Systems

Participating Instutions

Project: Spin chains and vertex models based on superalgebras

Spin chains and two-dimensional vertex models with an underlying superalgebra structure appear naturally in certain statistical physics models, e.g. intersecting loops, and disordered electron systems. In several examples for such models the low energy effective theory describing their critical behaviour has displayed rather unusual properties: the lack of unitarity in these systems allows for continua of critical exponents leading to a fine structure with strong subleading corrections to scaling in the finite size spectrum. This is a signature of non-compact degrees of freedom emerging in the continuum limit of these models. In this project we plan to study the properties of these systems in the context of integrable superspin chains. In particular we want to identify the corresponding conformal field theories and characterize the continuous part of their spectrum. In addition, the effect of boundary conditions on the critical properties will be addressed. To deal with the strong finite-size effects present in these systems we shall develop new analytical methods for the analysis of the spectral problem.

Publications

Frahm, H., Gehrmann, S., & Kotousov, G. A. (2023). Scaling limit of the staggered six-vertex model with \(U_q(\mathfrak{sl}(2))\) invariant boundary conditions. https://arxiv.org/abs/2312.11238
Frahm, H., Gehrmann, S., Nepomechie, R. I., & Retore, A. L. (2023). The D(2)3spin chain and its finite-size spectrum. Journal of High Energy Physics, 95(11), [095]. https://doi.org/10.1007/JHEP11(2023)095
Frahm, H., & Martins, M. J. (2023). Uq[OSp(3|2)] quantum chains with quantum group invariant boundaries. Nuclear Physics B, 995, [116329]. https://doi.org/10.48550/arXiv.2307.09412, https://doi.org/10.1016/j.nuclphysb.2023.116329
Kotousov, G. A., & Lukyanov, S. L. (2023). On the scaling behaviour of an integrable spin chain with Zr​ symmetry. https://arxiv.org/abs/2305.03620
Frahm, H., & Gehrmann, S. (2023). Integrable boundary conditions for staggered vertex models. Journal of Physics A: Mathematical and Theoretical, 56(2), [025001]. https://doi.org/10.48550/arXiv.2209.06182, https://doi.org/10.1088/1751-8121/acb29f
Frahm, H., & Martins, M. J. (2022). \(OSp(n|2m)\) quantum chains with free boundaries. Nuclear Physics B, 980, [115799]. https://doi.org/10.1016/j.nuclphysb.2022.115799
Frahm, H., & Gehrmann, S. (2022). Finite size spectrum of the staggered six-vertex model with \(U_q(sl(2))\)-invariant boundary conditions. Journal of High Energy Physics, 2022(1), [70]. https://doi.org/10.1007/JHEP01(2022)070
Hobuß, K. (2019). Spin chains and vertex models based on superalgebras. [Doctoral thesis, Leibniz University Hannover]. Leibniz Universität Hannover. https://doi.org/10.15488/8827
Frahm, H., Hobuß, K., & Martins, M. J. (2019). On the critical behaviour of the integrable q-deformed OSp(3|2) superspin chain. Nuclear Physics B, 946, [114697]. https://doi.org/10.1016/j.nuclphysb.2019.114697, https://doi.org/10.15488/10411
Frahm, H., & Martins, M. J. (2018). The fine structure of the finite-size effects for the spectrum of the OSp(n|2m) spin chain. Nuclear Physics B, 930, 545-562. https://doi.org/10.1016/j.nuclphysb.2018.03.016, https://doi.org/10.15488/3390
Frahm, H., & Hobuß, K. (2017). Spectral flow for an integrable staggered superspin chain. Journal of Physics A: Mathematical and Theoretical, 50(29), [294002]. https://doi.org/10.1088/1751-8121/aa77e7

Datasets

Frahm, H. & Martins, M. J. (2023). Dataset: Finite size data for \(U_q[OSp(3|2)]\) quantum chains with quantum group invariant boundaries. DOI: 10.25835/ypipefbz

Frahm, H.,Hobuß, K., & Martins, M. J. (2019). Dataset: Finite size data for the q-deformed OSp(3|2) superspin chain. https://doi.org/10.25835/0064330

Project: Non-Abelian anyons

Quasi particles in topological quantum liquids such as the fractional Quantum Hall states and certain two-dimensional frustrated magnets display unconventional quantum statistics. The conserved topological charge of these non-Abelian anyons is protected and has spawned interest for such systems in the context of quantum computation. In this project we plan to study the properties of interacting many-anyon systems whose construction is based on the mathematical structures describing the fundamental operations of fusion and braiding. Upon fine-tuning of the interactions these models can be embedded into a family of commuting operators. We shall develop functional methods to exploit local identities present in these integrable models for the solution of their spectral problem. Our investigation of integrable anyon chains will be complemented by studies of non-integrable deformations thereof to gain understanding into the emergence of unconventional boundary degrees of freedom and their realization as topological quantum impurities in electronic systems.

Publications

Westerfeld, D. (2022). Functional methods for correlation functions of integrable face and anyon models. [Doctoral thesis, Leibniz University Hannover]. Leibniz Universität Hannover. https://doi.org/10.15488/11992
Ardonne, E., Finch, P. E., & Titsworth, M. (2021). Classification of Metaplectic Fusion Categories. Symmetry, 13(11), [2102]. https://doi.org/10.3390/sym13112102
Frahm, H., & Westerfeld, D. (2021). Density matrices in integrable face models. SciPost Physics, 11(3), [057]. https://doi.org/10.21468/SciPostPhys.11.3.057
Borcherding, D., & Frahm, H. (2019). Condensates of interacting non-Abelian SO(5)Nf anyons. Journal of High Energy Physics, 2019(10), [54]. https://doi.org/10.1007/JHEP10(2019)054
Borcherding, D., & Frahm, H. (2018). Condensation of non-Abelian SU(3) Nf anyons in a one-dimensional fermion model. Journal of Physics A: Mathematical and Theoretical, 51(49), [495002]. https://doi.org/10.48550/arXiv.1808.05808, https://doi.org/10.1088/1751-8121/aaea9b
Borcherding, D., & Frahm, H. (2018). Signatures of non-Abelian anyons in the thermodynamics of an interacting fermion model. Journal of Physics A: Mathematical and Theoretical, 51(19), [195001]. https://doi.org/10.48550/arXiv.1706.09822, https://doi.org/10.1088/1751-8121/aaba1e
Finch, P. E., Flohr, M., & Frahm, H. (2018). Zn clock models and chains of so(n)2 non-Abelian anyons: Symmetries, integrable points and low energy properties. Journal of Statistical Mechanics: Theory and Experiment, 2018(2), [023103]. https://doi.org/10.1088/1742-5468/aaa788