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arXiv:0707.1160 · Physical Review Letters

Renormalization Group for Treating 2D Coupled Arrays of Continuum 1D Systems

Robert Konik, Yury Adamov

Published Mar 4, 2009• 25 citations
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Abstract

We study the spectrum of two dimensional coupled arrays of continuum one-dimensional systems by wedding a density matrix renormalization group procedure to a renormalization group improved truncated spectrum approach. To illustrate the methodology, we study the spectrum of large arrays of coupled quantum Ising chains. We demonstrate explicitly that the method can treat the various regimes of chains, in particular, the three dimensional Ising ordering transition the chains undergo as a function of interchain coupling.

DOI
10.1103/physrevlett.102.097203

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We study the spectrum of two dimensional coupled arrays of continuum one-dimensional systems by wedding a density matrix renormalization group procedure to a renormalization group improved truncated spectrum approach. To illustrate the methodology, we study the spectrum of large arrays of coupled quantum Ising chains. We demonstrate explicitly that the method can treat the various regimes of chains, in particular, the three dimensional Ising ordering transition the chains undergo as a function of interchain coupling.
We present a renormalization group (RG) procedure which works naturally on a wide class of interacting one-dimension models based on perturbed (possibly strongly) continuum conformal and integrable models. This procedure integrates Wilson's numerical renormalization group with Zamolodchikov's truncated conformal spectrum approach. The key to the method is that such theories provide a set of completely understood eigenstates for which matrix elements can be exactly computed. In this procedure the RG flow of physical observables can be studied both numerically and analytically. To demonstrate the approach, we study the spectrum of a pair of coupled quantum Ising chains and correlation functions in a single quantum Ising chain in the presence of a magnetic field.
We describe an algorithm for studying the entanglement entropy and spectrum of two-dimensional (2D) systems, as a coupled array of NNN one-dimensional chains in their continuum limit. Using the algorithm to study the quantum Ising model in 2D (both in its disordered phase and near criticality), we confirm the existence of an area law for the entanglement entropy and show that near criticality there is an additive piece scaling as cefflog(N)/6{c}_{\mathrm{eff}}\phantom{\rule{0.16em}{0ex}}\mathrm{log}(N)/6ceff​log(N)/6 with ${c}_{\mathrm{eff}}\ensuremath{\approx}1$. Studying the entanglement spectrum, we show that entanglement gap scaling can be used to detect the critical point of the 2D model. When short-range (area law) entanglement dominates we find (numerically and perturbatively) that this spectrum reflects the energy spectrum of a single quantum Ising chain.
Quasi-one-dimensional (Q1D) systems, i.e., three- and two-dimensional (3D/2D) arrays composed of weakly coupled one-dimensional lattices of interacting quantum particles, exhibit rich and fascinating physics. They are studied across various areas of condensed matter and ultracold atomic lattice-gas physics, and are often marked by dimensional crossover as the coupling between one-dimensional systems is increased or temperature decreased, i.e., the Q1D system goes from appearing largely 1D to largely 3D. Phase transitions occurring along the crossover can strongly enhance this effect. Understanding these crossovers and associated phase transitions can be challenging due to the very different elementary excitations of 1D systems compared to higher-dimensional ones. In the present work, we combine numerical matrix product state (MPS) methods with mean-field (MF) theory to study paradigmatic cases of dimensional crossovers and the associated phase transitions in systems of both hard-core and soft-core lattice bosons, with relevance to both condensed matter physics and ultracold atomic gases. We show that the superfluid-to-insulator transition is a first order one, as opposed to the isotropic cases, and calculate transition temperatures for the superfluid states, finding excellent agreement with analytical theory. At the same time, our MPS+MF\mathrm{MPS}+\mathrm{MF}MPS+MF approach keeps functioning well where the current analytical framework cannot be applied. We further confirm the qualitative and quantitative reliability of our approach by comparison to exact quantum Monte Carlo calculations for the full 3D arrays.
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.