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

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 approach 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.PACS numbers: 05.10.Cc, 75.10.JmThe density matrix renormalization group (DMRG) [1] is one of the primary theoretical tools for the quantitative description of low dimensional lattice models. For a wide range of one dimensional (1D) lattice models, DMRG can characterize the model's spectrum and correlation functions [2]. While there have been notable recent advancements [3,4], its use on 2D lattice models is more circumscribed [5].There exist several strategies to apply DMRG to 2D models. In the first, the 2D lattice is reduced to a 1D lattice with long range interactions [6]. A second approach sees short chains treated as individual lattice sites, allowing the 1D DMRG algorithm to be applied to a model with short ranged interactions directly [7]. In a more sophisticated variant of this methodology, the DMRG is applied in a two stage process [3]. The 2D system is first divided into a set of coupled 1D chains and the DMRG is used to determine a low energy reduction thereof. For the second stage, the reduced chains, coupled together and treated as individual lattice sites in a 1D lattice, are analyzed again using DMRG. In a final approach, the 1D matrix product states underlying the DMRG algorithm [8] are replaced by a higher dimensional generalization, projected entangled pair states [9].In this letter we present a distinct approach to applying DMRG to 2D models. This approach trades upon a description of a 2D system as a mixture of continuum and discrete degrees of freedom. In particular, we approach 2D systems as coupled arrays of continuum 1D chains with truncated Hilbert spaces. This methodology offers several distinct advantages. It allows us to treat any 2D strongly correlated model provided it can be conceived as composed of continuum 1D subunits. Furthermore, the approach affords superior finite size scaling. As a function of the length, R, of the composite 1D systems, finite size corrections behave exponentially. This implies that we can access, at the very least, the infinite volume limit in the dimension parallel to the chains. Finally, the truncation of the underlying 1D Hilbert space dramatically lessens the numerical burden of the DMRG algorithm, while providing a natural means to perform a Wilsonian renormalization group (RG) improvement of any resulting answer [10].The specific type of system that we propose to study takes the form,(1)The 1D continuum subunits of the array are governed by H 1D i which we insist must be either gapless (and so governed by a conformal field theory) or gapped...

Dynamical quantum phase transitions (Review Article)

“…For example [68], studied quantum quenches in the two-dimensional Ising model using the matrix product state approach [69] via the numerical algorithm similar to the density matrix renormalization group approach [70]. The two-dimensional Ising model was presented as a coupled set of one-dimensional Ising chains in the continuum limit.…”

Near-Integrability of Yang-Mills Theories

“…This problem is still under investigation. In the case of the Z 2 gauge theory, dual to the three-dimensional Ising model, the same type of cross-over has been overcome using the densitymatrix renormalization group [19] (in fact, the formulation of the three-dimensional Ising model as coupled two-dimensional Ising models is very similar to the formulation of the 2 + 1-dimensional gauge theory as coupled 1 + 1-dimensional sigma models discussed here). Realistic results for the correlation-length critical exponent ν were obtained this way.…”

“…As mentioned already, this has been accomplished for the Z 2 case [19]. The problem should perhaps be easier for SU(N) theories, as the critical point is the same for both the isotropic and anisotropic theories; it is simply at g 0 = g 0 = 0.…”

Summing planar diagrams by an integrable bootstrap

“…The general solution of (3.17) and (3.18) is 19) where the functions g 1 and g 3 are periodic in θ 1 :…”

“…A similar crossover is an obstacle to using the form factors of the two-dimensional Ising spin field to calculate critical exponents of the three-dimensional Ising model. Konik and Adamov were able to overcome this dimensional crossover for the Ising case with a density-matrix real-space renormalization group [19]. The triviality of the S-matrix as N → ∞ may help defeat the crossover for SU(N) gauge theories.…”