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Journal of Physics A Mathematical and General

The Luttinger mode, Thirring strings, and enhanced internal symmetries

Uwe K. Albertin, Michael G. Cleveland, Tim Pritchett

Published Jun 21, 1990• 0 citations
Publisher

Abstract

The standard solution of the Luttinger model (1963) is compared with recent formulations of the Thirring model used in describing Thirring strings. These models may in turn be described by compactified boson models of particular radii. An interesting discrepancy is noted between the apparent radius of the bosons in the Luttinger model and those in the Thirring model. This difference manifests itself in different conformal weights for the fermion fields. The authors resolve this discrepancy by performing an additional rescaling of the Luttinger model bosons. They then examine the enhanced symmetry points that occur in the boson theory at particular values of the radii. In particular, for the Thirring model with two families of spinors, which corresponds to a conformal theory with central extension c=2, one finds an enhanced SU(3) L *SU(3) R symmetry for certain values of the boson radii. This theory is equivalent to the Luttinger model with spin. By examining the response functions for the Luttinger model it is found that the point of SU(3) L *SU(3) R symmetry lies on the Luttinger model phase transition lines.

DOI
10.1088/0305-4470/23/12/040

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The standard solution of the Luttinger model (1963) is compared with recent formulations of the Thirring model used in describing Thirring strings. These models may in turn be described by compactified boson models of particular radii. An interesting discrepancy is noted between the apparent radius of the bosons in the Luttinger model and those in the Thirring model. This difference manifests itself in different conformal weights for the fermion fields. The authors resolve this discrepancy by performing an additional rescaling of the Luttinger model bosons. They then examine the enhanced symmetry points that occur in the boson theory at particular values of the radii. In particular, for the Thirring model with two families of spinors, which corresponds to a conformal theory with central extension c=2, one finds an enhanced SU(3) L *SU(3) R symmetry for certain values of the boson radii. This theory is equivalent to the Luttinger model with spin. By examining the response functions for the Luttinger model it is found that the point of SU(3) L *SU(3) R symmetry lies on the Luttinger model phase transition lines.
We achieve a bosonization of one-dimensional ideal gas of particles obeying exclusion statistics $\ensuremath{\lambda}$ (so called $\ensuremath{\lambda}$ exclusons) at low temperatures, resulting in a new variant of c=1c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1c=1 conformal field theory with compactified radius $R\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\sqrt{1/\ensuremath{\lambda}}$. These ideal excluson gases exactly reproduce the low- TTT critical properties of Luttinger liquids, so they can be used to characterize the fixed points of the latter. Generalized ideal gases with mutual statistics and nonideal gases with Luttinger-type interactions have also similar low- TTT behavior, controlled by an effective statistics varying in a fixed-point line.
We extend the scope of the Klein bottle entropy, originally introduced by Tu [Phys. Rev. Lett. 119, 261603 (2017)] in the rational conformal field theory (CFT), to the compactified boson CFT, which are relevant to the studies of Luttinger liquids. We first review the Klein bottle entropy in rational CFT and discuss details of how to extract the Klein bottle entropy from lattice models using the example of the transverse field Ising model. We then go beyond the scope of rational CFT and study the Klein bottle entropy lnglnglng in the compactified boson CFT, which turns out to have a straightforward relation to the compactification radius RRR, lng=lnRlng=lnRlng=lnR. This relation indicates a convenient and efficient method to extract the Luttinger parameter from lattice model calculations. Our numerical results on the Klein bottle entropy in the spin-1/21/21/2 XXZ chain show excellent agreement with the CFT predictions, up to some small deviations near the isotropic point, which we attribute to the marginally irrelevant terms. For the S=1S=1S=1 XXZ chain that cannot be exactly solved, our numerical results provide an accurate numerical determination of the Luttinger parameter in this model.
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We present the case where Luttinger liquids are characterized by a form of hidden order which is similar, but distinct in some crucial regards, to the hidden order characterizing spin-1 Heisenberg chains. We construct a string correlator for the Luttinger liquid which is similar to the string correlator constructed by den Nijs and Rommelse for the spin chain. We reanalyze the spin one chain, introducing a precise formulation of the geometrical principle behind the so-called ``squeezed space'' construction, to demonstrate that the physics at long wavelength can be reformulated in terms of a Z2{Z}_{2}Z2​ gauge theory. Peculiarly, the normal spin chain lives at infinite gauge coupling where it is characterized by deconfinement. We identify the microscopic conditions required for confinement thereby identifying a novel phase of the spin chain. We demonstrate that the Luttinger liquid can be approached in the same general framework. The difference from the spin chain is that the gauge sector is critical in the sense that the Luttinger liquid is at the phase boundary where the Z2{Z}_{2}Z2​ local symmetry emerges. In addition, the ``matter'' (spin) sector is also critical. We evaluate the string correlator analytically for the strongly coupled Hubbard model and we further demonstrate that the squeezed space structure is still present even in the noninteracting fermion gas. This adds new insights to the meaning of bosonization. These structures are hard wired in the mathematical structure of bosonization and this becomes obvious by considering string correlators. Numerical results are presented for the string correlator using a non-abelian version of the density matrix renormalization group algorithm, confirming in detail the expectations following from the theory. We conclude with some observations regarding the generalization of bosonization to higher dimensions.
Despite great successes in the study of gapped phases, a comprehensive understanding of the gapless phases and their transitions is still under development.In this paper, we study a general phenomenon in the space of (1 + 1)-dimensional critical phases with fermionic degrees of freedom described by a continuous family of conformal field theories (CFTs), also known as the conformal manifold.Along a one-dimensional locus on the conformal manifold, there can be a transition point, across which the fermionic CFTs on the two sides differ by stacking an invertible fermionic topological order (IFTO), point by point along the locus.At every point on the conformal manifold, the order and disorder operators have power-law two-point functions, but their critical exponents cross over with each other at the transition point, where stacking the IFTO leaves the fermionic CFT unchanged.We call this continuous transition on the fermionic conformal manifold a topological transition.By gauging the fermion parity, the IFTO stacking becomes a Kramers-Wannier duality between the corresponding bosonic CFTs.Both the IFTO stacking and the Kramers-Wannier duality are induced by the electromagnetic duality of the (2 + 1)-dimensional Z 2 topological order.We provide several examples of topological transitions, including the familiar Luttinger model of spinless fermions (i.e., the c = 1 massless Dirac fermion with the Thirring interaction) and a class of c = 2 examples describing U(1) × SU(2)-protected gapless phases.