Geometry and the hidden order of Luttinger liquids: The universality of squeezed space

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 ${Z}_{2}$ 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 ${Z}_{2}$ 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.