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Europhysics Letters (EPL)

Geometry and topological order in the Luttinger liquid state

H. V. Kruis, Ian P. McCulloch, Zohar Nussinov, Jan Zaanen

Published Feb 1, 2004• 13 citations
Publisher

Abstract

We demonstrate the existence of a form of topological order in the Luttinger liquid state, which appears crucial to the notion of spin-charge separation. This order, which is closely related to the hidden order characterizing the Haldane spin chains, appears to be incompletely specified in the Tomonaga-Luttinger picture from bosonization. Thus, we use numerical calculations to complete the picture.

DOI
10.1209/epl/i2003-10114-3

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We demonstrate the existence of a form of topological order in the Luttinger liquid state, which appears crucial to the notion of spin-charge separation. This order, which is closely related to the hidden order characterizing the Haldane spin chains, appears to be incompletely specified in the Tomonaga-Luttinger picture from bosonization. Thus, we use numerical calculations to complete the picture.
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.
The Haldane phase is a celebrated example of a magnetic topological phase but rarely reported in materials. The authors show here an NMR study on the spin-1 chain magnet NiC2{}_{2}2​O4{}_{4}4​\ifmmode\cdot\else\textperiodcentered\fi{}2NH3{}_{3}3​, which determines its ground state to be a Haldane phase at zero field. An applied magnetic field induces a magnetic ordering at low temperatures and a Tomonaga-Luttinger liquid phase above the ordering temperature with attractive fermionic interactions.
We investigate the T=0T=0T=0 phase diagram of a variant of the one-dimensional extended Hubbard model where particles interact via a finite-range soft-shoulder potential. Using Density Matrix Renormalization Group (DMRG) simulations, we evidence the appearance of Cluster Luttinger Liquid (CLL) phases, similarly to what first predicted in a hard-core bosonic chain [M. Mattioli, M. Dalmonte, W. Lechner, and G. Pupillo, Phys. Rev. Lett. 111, 165302]. As the interaction strength parameters change, we find different types of clusters, that encode the order of the ground state in a semi-classical approximation and give rise to different types of CLLs. Interestingly, we find that the conventional Tomonaga Luttinger Liquid (TLL) is separated by a critical line with a central charge c=5/2c=5/2c=5/2, along which the two (spin and charge) bosonic degrees of freedom (corresponding to c=1c=1c=1 each) combine in a supersymmetric way with an emergent fermionic excitation (c=1/2c=1/2c=1/2). We also demonstrate that there are no significant spin correlations.
We study the entanglement spectrum (ES) and entropy between two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. This problem gives access to the entanglement properties of various interesting systems, such as spin ladders as well as two-dimensional topological phases. By expanding interchain interactions to quadratic order in bosonic fields, we are able to calculate the ES for both gapped and gapless systems using only methods for free theories. In certain gapless phases of coupled nonchiral TLLs, we interestingly find an ES with a dispersion relation proportional to the square root of the subsystem momentum, which we relate to a long-range interaction in the entanglement Hamiltonian. We numerically demonstrate the emergence of this unusual dispersion in a model of hard-core bosons on a ladder. In gapped phases of coupled nonchiral TLLs, which are relevant to spin ladders and topological insulators, we show that the ES consists of linearly dispersing modes, which resembles the spectrum of a single-chain TLL but is characterized by a modified TLL parameter. Based on a calculation for coupled chiral TLLs, we are also able to provide a very simple proof for the correspondence between the ES and the edge-state spectrum in quantum Hall systems consistent with previous numerical and analytical studies.
We consider the problem of two-coupled Luttinger liquids both at half filling and at low doping levels, to investigate the problem of competing orders in quasi-one-dimensional strongly correlated systems. We use bosonization and renormalization group equations to investigate the phase diagrams, to determine the allowed phases, and to establish approximate boundaries among them. Because of the chiral translation and reflection symmetries in the charge mode away from half filling, orders of charge-density wave (CDW) and spin Peierls (SP), diagonal current (DC), and d-density wave (DDW) form two doublets and thus can be at most quasi-long-range ordered. At half filling, Umklapp terms break this symmetry down to a discrete group and thus Ising-type ordered phases appear as a result of spontaneous breaking of the residual symmetries. Quantum disordered Haldane phases are also found, with finite amplitudes of pairing orders and triplet counterparts of CDW, SP, DC, and DDW. Relations with recent numerical results and implications to similar problems in two dimensions are discussed.