[Paper Review] The patching of critical points using quantum group

Sher Alam, M. O. Rahman+3 more|arXiv (Cornell University)|Apr 20, 2000

Physics of Superconductivity and Magnetism5 references3 citations

TL;DR

This paper proposes a quantum group-based gauge theory framework to model critical points in strongly correlated electron systems like cuprates by patching fixed points from different Wess-Zumino-Witten (WZW) models. By identifying critical points via quantum symmetry groups such as $\mathrm{SO}_q(N)$, constructing effective field theories, and defining a partition function over these points, the approach yields a non-perturbative formulation of condensed matter Hamiltonians, linking Hubbard models, gauge theories, and string theory concepts.

ABSTRACT

Following our recent conjecture to model the phenomenona of antiferromagnetism and superconductivity by quantum symmetry groups, we discuss in the present note how to construct a workable scenario using this symmetry. In particular we propose to patch the relevant critical points. This means we identify fixed points, corresponding to various $k$ or $q$ [since the two are related] and make expansion around these points, to control these expansion we can impose gauge structure and we thus arrive at quantum group based gauge theory or collection of classical gauge theories which represent the condensed matter system such as cuprates. This is different than ordinary gauge theories in several ways, for in ordinary field theory one has well-defined critical point here the critical points are not unique or simple. In short the real transition than in condensed matter system is represented by collection or {\it ensemble average} of the several chosen critical points which come from ordinary field theory [gauge theory]. This idea may reveal the connection between Hubbard model and gauge theory and string theory. In short it can lead to the non-perturbative formualtion of Hubbard and other condensed matter Hamiltonians.

Motivation & Objective

• To develop a non-perturbative framework for describing high-temperature superconductivity and antiferromagnetism in strongly correlated systems.
• To unify fixed points from different WZW models (e.g., $k=1$, $k=2$) into a single effective theory via quantum group symmetry.
• To construct a gauge-theoretic description of condensed matter systems by patching critical points corresponding to distinct $k$-levels of $\mathrm{SU}(2)_k$ or $\mathrm{SO}_q(N)$ quantum groups.
• To establish a connection between the Hubbard model, quantum group gauge theories, and string theory through critical point patching.
• To provide a smeared or ensemble-based effective field theory that captures the physics of doping crossover in 1D to 2D systems.

Proposed method

• Label critical points using quantum symmetry groups such as $\mathrm{SO}_q(N)$ or $\mathrm{SU}_q(N)$, corresponding to specific $k$-levels in Wess-Zumino-Witten models.
• Identify low-energy spectra of these critical points, including $c=3/2$ (three Majorana fermions) for $k=2$ and $c=1$ for $k=1$.
• Construct a partition function that weights contributions from multiple critical points, forming a representative effective or 'smeared' gauge structure.
• Use the $q$-deformation of Lie groups to unify classical symmetries and topological features, enabling a non-perturbative formulation of the system.
• Apply this framework to the doped spin-1 chain, where finite doping interpolates between $k=2$ (undoped) and $k=1$ (fully doped) fixed points.
• Derive an effective field theory with mixed central charges ($c=3/2 + 1/2$) and different velocities, reflecting the emergence of a fourth Majorana fermion in the doped regime.

Experimental results

Research questions

• RQ1 How can quantum group symmetries be used to describe critical points in doped Mott insulators like cuprates?
• RQ2 What is the role of $k$-level Wess-Zumino-Witten models in characterizing fixed points of strongly correlated systems?
• RQ3 How does patching multiple critical points—each with distinct $k$-levels—yield a unified effective field theory for doped spin chains?
• RQ4 Can a partition function over quantum group-based critical points lead to a non-perturbative formulation of the Hubbard Hamiltonian?
• RQ5 What is the connection between the resulting quantum group gauge theory and both gauge theories and string theory in condensed matter systems?

Key findings

• The undoped spin-1 chain corresponds to the $\mathrm{SU}(2)_{k=2}$ WZW model with central charge $c=3/2$, equivalent to three massless Majorana fermions.
• The fully doped limit ($x=1$) corresponds to $\mathrm{SU}(2)_{k=1}$, with $c=1$, representing a single free bosonic mode in the charge sector.
• Finite doping leads to a spin sector with a direct sum of $c=3/2$ and $c=1/2$ models, indicating the emergence of a fourth Majorana fermion, consistent with two-channel Kondo physics.
• The patching of critical points via quantum group symmetry allows for a non-perturbative effective field theory that interpolates between distinct fixed points.
• The proposed framework realizes a quantum group-based gauge theory that unifies classical gauge structures and topological features, offering a new route to non-perturbative condensed matter Hamiltonians.
• The method provides a conceptual bridge between the Hubbard model, gauge theories, and string theory through the critical point patching mechanism.