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arXiv:2207.04395 · Physical review. B./Physical review. B

Universal description of dissipative Tomonaga-Luttinger liquids with SU(N) spin symmetry: Exact spectrum and critical exponents

Kazuki Yamamoto, Norio Kawakami

Published Jan 9, 2023• 30 citations
arXivHTMLPDF

Abstract

Universal scaling relations for dissipative Tomonaga-Luttinger (TL) liquids with $\mathrm{SU}(N$) spin symmetry are obtained for both fermions and bosons, by using asymptotic Bethe-ansatz solutions and conformal field theory (CFT) in one-dimensional non-Hermitian quantum many-body systems with $\mathrm{SU}(N$) symmetry. We uncover that the spectrum of dissipative TL liquids with $\mathrm{SU}(N$) spin symmetry is described by the sum of one charge mode characterized by a complex generalization of $c=1$ U(1) Gaussian CFT, and $N\ensuremath{-}1$ spin modes characterized by level-$1\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N$) Kac-Moody algebra with the conformal anomaly $c=N\ensuremath{-}1$, and thereby dissipation only affects the charge mode as a result of spin-charge separation in one-dimensional non-Hermitian quantum systems. The derivation is based on a complex generalization of Haldane's ideal-gas description, which is implemented by the $\mathrm{SU}(N$) Calogero-Sutherland model with inverse-square long-range interactions.

DOI
10.1103/physrevb.107.045110

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We demonstrate the universal properties of dissipative Tomonaga-Luttinger (TL) liquids by calculating correlation functions and performing finite-size scaling analysis of a non-Hermitian XXZ spin chain as a prototypical model in one-dimensional open quantum many-body systems. Our analytic calculation is based on effective field theory with bosonization, finite-size scaling approach in conformal field theory, and the Bethe-ansatz solution. Our numerical analysis is based on the density-matrix renormalization group generalized to non-Hermitian systems (NH-DMRG). We uncover that the model in the massless regime with weak dissipation belongs to the universality class characterized by the complex-valued TL parameter, which is related to the complex generalization of the c=1c=1c=1 conformal field theory. As the dissipation strength increases, the values of the TL parameter obtained by the NH-DMRG begin to deviate from those obtained by the Bethe-ansatz analysis, indicating that the model becomes massive for strong dissipation. Our results can be tested with the two-component Bose-Hubbard system of ultracold atoms subject to two-body loss.
We show that one-dimensional quantum systems with gapless degrees of freedom and open boundary conditions form a universality class of quantum critical behavior, which we propose to call ``bounded Luttinger liquids.'' They share the following properties with ordinary (periodic) Luttinger liquids: the absence of fermionic quasiparticle excitations, charge-spin separation, and anomalous power-law correlations with exponents whose scaling relations are parametrized by a single coupling constant per degree of freedom, ${K}_{\ensuremath{\nu}}.$ The values of ${K}_{\ensuremath{\nu}}$ are independent of boundary conditions, but the representation of the critical exponents in terms of these ${K}_{\ensuremath{\nu}}'\mathrm{s}$ depends on boundary conditions. We illustrate these scaling relations by exploring general rules for boundary critical exponents derived earlier using the Bethe ansatz solution of the one-dimensional Hubbard model together with boundary conformal field theory, and the theory of Luttinger liquids in finite-size systems. We apply this theory to the photoemission properties of the organic conductors (TMTSF)2X,(\mathrm{TMTSF}{)}_{2}X,(TMTSF)2​X, where TMTSF is tetramethyltetraselenafulvalene, and X=ClO9,X={\mathrm{ClO}}_{9},X=ClO9​, PF6,{\mathrm{PF}}_{6},PF6​, ReO4,\mathrm{Re}{\mathrm{O}}_{4},ReO4​, and discuss to what extent the assumption of finite strands with open boundaries at the sample surface can reconcile the experimental results with independent information on the Luttinger-liquid state in these materials.
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Asymptotic behavior of the Tomonaga-Luttinger liquid in the vicinity of the phase-separated region is investigated in the one-dimensional $t\ensuremath{-}J$ model to study the universal property of the c=1c=1c=1 conformal field theory with U(1) symmetry near the $K\ensuremath{\rightarrow}\ensuremath{\infty}$ instability. By analogy to the spinless fermion, we argue that the compressibility behaves as $\ensuremath{\kappa}\ensuremath{\propto}{(J}_{c}\ensuremath{-}{J)}^{\ensuremath{-}1},$ and that the Drude weight is constant and changes to zero discontinuously at the phase boundary. This speculation is confirmed by analyzing the finite-size effect from the result of the exact diagonalization.
The critical properties of Tomonaga-Luttinger liquids including a mobile impurity are studied by means of the Bethe ansatz method, conformal field theory (CFT), and bosonization. We calculate conformal dimensions by performing the CFT analysis of the energy spectrum for an integrable impurity model, and show that this system is classified as c=1c=1c=1 shifted Gaussian CFT. The exact critical exponents of various correlation functions are obtained as functions of the impurity mass and its momentum. Passing to the continuum limit, we then construct an effective field theory in terms of bosonization, and directly calculate some correlation functions. A possible application to the Fermi-edge singularity in quantum wires is also discussed.