Geometry and the Hidden Order of Luttinger Liquids: the Universality of Squeezed Space

Abstract

We present the case that 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. From a geometric prespective on the so-called `squeezed space' construction, we 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. We evaluate the string correlator analytically and show that the squeezed space structure is present both for the strongly coupled Hubbard model and the non-interacting fermion gas. 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.

24 pages, 14 eps figures, Revtex4

References in corpus (21)

• One-Dimensional Fermi liquids
• Thermodynamic limit of the density matrix renormalization for the spin-1 Heisenberg chain
• The Quantum-Mechanical Position Operator in Extended Systems
• Z_2 Gauge Theory of Electron Fractionalization in Strongly Correlated Systems
• Electron Localization in the Insulating State
• Order and quantum phase transitions in the cuprate superconductors
• Diverging Entanglement Length in Gapped Quantum Spin Systems
• The Non-Abelian Density Matrix Renormalization Group Algorithm
• Generalized Jordan-Wigner Transformations
• Competing orders in a magnetic field: spin and charge order in the cuprate superconductors
• Phase String Effect in the t-J Model: General Theory
• Exact Correlation Amplitude for the S=1/2 Heisenberg Antiferromagnetic Chain
• Numerical renormalization group study of the correlation functions of the antiferromagnetic spin-$\frac{1}{2}$ Heisenberg chain
• Lattice Twisting Operators and Vertex Operators in Sine-Gordon Theory in One Dimension
• Charged domain walls as quantum strings living on a lattice
• Understanding correlated electron systems by a classification of Mott insulators
• Order-out-of-disorder in a gas of elastic quantum strings in 2+1 dimensions
• Quantizing Charged Magnetic Domain Walls: Strings on a Lattice
• Stripe fractionalization II: the quantum spin nematic and the Abrikosov lattice
• New Types of Off-Diagonal Long Range Order in Spin-Chains
• Dynamical stripe correlations and the spin fluctuations in cuprate superconductors

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