Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-abelian bosonization to truncated spectrum methods

TL;DR

The paper surveys non-perturbative tools for low-dimensional strongly correlated systems, centering on non-Abelian bosonization and the truncated spectrum approach (TSA). It details the conformal embedding of massless fermions into commuting WZNW sectors, the Kac–Moody current algebra, and operator correspondences that enable tractable analyses of complex symmetries in 1D and related cold-atom systems. It then presents TSA as a robust numerical framework for perturbed integrable or conformal theories, with extensive demonstrations in Ising, TIM, sine-Gordon, and related models, and discusses numerical renormalization-group and RG-based refinements to remove cutoff artifacts. The review culminates with applications to cold-atom physics, high-symmetry spin ladders, and two-dimensional extensions that couple TSA with matrix-product-state methods, highlighting both the capabilities and current limitations of these non-perturbative approaches. Overall, the work provides a detailed toolbox for extracting spectra and correlation functions in strongly correlated, symmetry-rich quantum systems, with clear implications for experiments in ultracold atoms and low-dimensional materials."

Abstract

We review two important non-perturbative approaches for extracting the physics of low-dimensional strongly correlated quantum systems. Firstly, we start by providing a comprehensive review of non-Abelian bosonization. This includes an introduction to the basic elements of conformal field theory as applied to systems with a current algebra, and we orient the reader by presenting a number of applications of non-Abelian bosonization to models with large symmetries. We then tie this technique into recent advances in the ability of cold atomic systems to realize complex symmetries. Secondly, we discuss truncated spectrum methods for the numerical study of systems in one and two dimensions. For one-dimensional systems we provide the reader with considerable insight into the methodology by reviewing canonical applications of the technique to the Ising model (and its variants) and the sine-Gordon model. Following this we review recent work on the development of renormalization groups, both numerical and analytical, that alleviate the effects of truncating the spectrum. Using these technologies, we consider a number of applications to one-dimensional systems: properties of carbon nanotubes, quenches in the Lieb-Liniger model, 1+1D quantum chromodynamics, as well as Landau-Ginzburg theories. In the final part we move our attention to consider truncated spectrum methods applied to two-dimensional systems. This involves combining truncated spectrum methods with matrix product state algorithms. We describe applications of this method to two-dimensional systems of free fermions and the quantum Ising model, including their non-equilibrium dynamics.

Figures (54)

• Figure 1: (a) A schematic depiction of the spectrum of $H_{\rm known}$ in the infinite volume (left) and the finite volume (right). In the infinite volume, there is a continuum of states, whilst in the finite volume the spectrum is discrete (and possibly with finite degeneracy). (b) A cartoon illustration of the TSA procedure; a cutoff energy $E_c$ is introduced and states in the spectrum of $H_{\rm known}$ above this energy are discarded.
• Figure 2: Phase diagram of the one-dimensional quantum Ising model with $h=0$. Order is only possible at $T=0$, with $g=0$ separating an ordered phase ($J<0$ ferromagnet, $J>0$ antiferromagnet, shown as a solid orange bar) from a disordered phase.
• Figure 3: (a) A sketch of two domain walls (dashed lines) in an ordered background of the quantum Ising chain. We see that the spins between the domain walls are overturned relative to the system's overall order. (b) In the presence of a longitudinal magnetic field, the domain walls are linearly confined (energy cost $\Delta E$ grows with domain size $D$).
• Figure 4: Raw TSA data for the lowest lying energy levels of the Hamiltonian in Eq. (\ref{['continuumH']}) for $h=(2m)^{15/8}$, $m=1$, and a cutoff of $N=RE_c/(2\pi)=30$ plotted against the dimensionless system size, $Rh^{8/15}$. The presented data focuses on the zero-momentum (ground state) sector. One sees that the energy levels all roughly have a constant negative slope. The dashed black line that has a positive slope corresponds to a false vacuum state (equal to one of the linear combinations, $|NS\rangle +{\rm sign}(h)|R\rangle$ -- see text).
• Figure 5: The same data as presented in Fig. \ref{['TSAfig1']} with the ground state energy subtracted.