arXiv:cond-mat/9508008 · Journal of Physics A Mathematical and General

Critical properties of the Calogero - Sutherland model with boundaries

Takashi Yamamoto, Norio Kawakami, Sung-Kil Yang

Published Jan 21, 1996• 9 citations

Abstract

Critical properties of the Calogero - Sutherland model of -type ( -CS model) are studied. Using the asymptotic Bethe ansatz spectrum of the -CS model, we calculate finite-size corrections in the energy spectrum. Since the -CS model does not possess translational invariance, the finite-size spectrum acquires the contributions coming from `boundaries'. We show that the low-energy critical behaviour of the model is described by c = 1 boundary conformal field theory. Thus the universality class of the model is identified as a chiral Tomonaga - Luttinger liquid.

DOI: 10.1088/0305-4470/29/2/012

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arXiv:cond-mat/9508008 · Journal of Physics A Mathematical and General

Critical properties of the Calogero - Sutherland model with boundaries

Takashi Yamamoto, Norio Kawakami, Sung-Kil Yang

Published Jan 21, 1996• 9 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/0305-4470/29/2/012

Related papers

Critical Properties of the Calogero-Sutherland Model with Boundaries

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Critical properties of the Calogero-Sutherland model of BCN-type (BCN-CS model) are studied. Using the asymptotic Bethe-ansatz spectrum of the BCN-CS model, we calculate finite-size corrections in the energy spectrum. Since the BCN-CS model does not possess translational invariance, the finite-size spectrum acquires the contributions coming from “boundaries”. We show that the low-energy critical behavior of the model is described by c = 1 boundary conformal field theory. Thus the universality class of the model is identified as a chiral Tomonaga-Luttinger liquid. 1 I.Critical properties of the Calogero-Sutherland model of BCN-type (BCN-CS model) are studied. Using the asymptotic Bethe-ansatz spectrum of the BCN-CS model, we calculate finite-size corrections in the energy spectrum. Since the BCN-CS model does not possess translational invariance, the finite-size spectrum acquires the contributions coming from “boundaries”. We show that the low-energy critical behavior of the model is described by c = 1 boundary conformal field theory. Thus the universality class of the model is identified as a chiral Tomonaga-Luttinger liquid. 1 I.

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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 $(\mathrm{TMTSF}{)}_{2}X,$ where TMTSF is tetramethyltetraselenafulvalene, and $X={\mathrm{ClO}}_{9},$ ${\mathrm{PF}}_{6},$ $\mathrm{Re}{\mathrm{O}}_{4},$ 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.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 $(\mathrm{TMTSF}{)}_{2}X,$ where TMTSF is tetramethyltetraselenafulvalene, and $X={\mathrm{ClO}}_{9},$ ${\mathrm{PF}}_{6},$ $\mathrm{Re}{\mathrm{O}}_{4},$ 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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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 $c=1$ 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.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 $c=1$ 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.

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Applying boundary conformal field theory we study the low-energy critical behavior of the Calogero-Sutherland model of

BC_N

-type. The universality class of the model is found to be a {\it chiral} Tomonaga-Luttinger liquid. Various correlation exponents depending on the interaction strength are obtained.

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.

(\mathrm{TMTSF}{)}_{2}X,

where TMTSF is tetramethyltetraselenafulvalene, and

X={\mathrm{ClO}}_{9},

{\mathrm{PF}}_{6},

\mathrm{Re}{\mathrm{O}}_{4},

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.

c=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.