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arXiv:hep-th/0603154 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Bosonization of nonrelativistic fermions on a circle: Tomonaga’s problem revisited

Avinash Dhar, Gautam Mandal

Published Nov 6, 2006• 10 citations
arXivHTMLPDF

Abstract

We use the recently developed tools for an exact bosonization of a finite number $N$ of nonrelativistic fermions to discuss the classic Tomonaga problem. In the case of noninteracting fermions, the bosonized Hamiltonian naturally splits into an $\mathrm{O}(N)$ piece and an O(1) piece. We show that in the large-$N$ and low-energy limit, the $\mathrm{O}(N)$ piece in the Hamiltonian describes a massless relativistic boson, while the O(1) piece gives rise to cubic self-interactions of the boson. At finite $N$ and high energies, the low-energy effective description breaks down and the exact bosonized Hamiltonian must be used. We also comment on the connection between the Tomonaga problem and pure Yang-Mills theory on a cylinder. In the dual context of baby universes and multiple black holes in string theory, we point out that the $\mathrm{O}(N)$ piece in our bosonized Hamiltonian provides a simple understanding of the origin of two different kinds of nonperturbative $\mathrm{O}({e}^{\ensuremath{-}N})$ corrections to the black hole partition function.

DOI
10.1103/physrevd.74.105006

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We use the recently developed tools for an exact bosonization of a finite number NNN of nonrelativistic fermions to discuss the classic Tomonaga problem. In the case of noninteracting fermions, the bosonized Hamiltonian naturally splits into an O(N)\mathrm{O}(N)O(N) piece and an O(1) piece. We show that in the large-NNN and low-energy limit, the O(N)\mathrm{O}(N)O(N) piece in the Hamiltonian describes a massless relativistic boson, while the O(1) piece gives rise to cubic self-interactions of the boson. At finite NNN and high energies, the low-energy effective description breaks down and the exact bosonized Hamiltonian must be used. We also comment on the connection between the Tomonaga problem and pure Yang-Mills theory on a cylinder. In the dual context of baby universes and multiple black holes in string theory, we point out that the O(N)\mathrm{O}(N)O(N) piece in our bosonized Hamiltonian provides a simple understanding of the origin of two different kinds of nonperturbative O(e\ensuremath−N)\mathrm{O}({e}^{\ensuremath{-}N})O(e\ensuremath−N) corrections to the black hole partition function.
A bstract We describe a new exact lattice bosonization of matrix quantum mechanics (equivalently of non-relativistic fermions) that is valid for arbitrary rank N of the matrix. It is based on the exact operator bosonization of non-relativistic fermions introduced earlier in [1]. The trace identities, which characterize finite rank matrices, are automatically incorporated in the bosonic theory. The finite number N of fermions is reflected in the finite number N of bosonic annihilation-creation operators, and equivalently to the finite number N of lattice points. The fermion Hamiltonian is exactly mappable to a bosonic Hamiltonian. At large N , the latter becomes local and corresponds to the lattice version of a relativistic boson Hamiltonian, with a lattice spacing of order 1/ N . The finite lattice spacing leads to a finite entanglement entropy (EE) of the bosonic theory, which reproduces the finite EE of the fermionic theory. Such a description is not available in the standard bosonization in terms of fermion density fluctuations on the Fermi surface, which does not have a built-in short distance cut-off (see, however, a recent description of the finiteness of the fermionic EE in a collective field theory formalism [2]). The bosonic lattice constructed in our work is equipped with a geometry that is determined by the matrix potential or equivalently by the shape of the Fermi surface. Our bosonization also works in the double scaled c = 1 matrix model; in particular the bosonic EE again turns out to be finite, with the short distance cut-off turning into g s l s , and reproduces the matrix result. Once again, this is to be contrasted with the conventional dual 2D string theory, where the bosonic EE is naturally identified with that of the “tachyon”, the massless string mode, where one may imagine the short-distance cut-off to be the string length l s . This appears to indicate our bosonization as a different dual description to the c = 1 matrix model appropriate for “local physics” like quantum entanglement, by contrast with the conventional duality to the eigenvalue density which works well for asymptotic observables like S-matrices. We briefly discuss possible relation of our bosonization to D0 branes.
The theoretical description of interacting fermions in one spatial dimension is simplified by the fact that the low energy spectrum of noniteracting fermions is identical to the one of a harmonic chain. This fermion-boson transmutation allows to describe interacting fermions as a system of coupled oscillators. Tomonaga's model of interacting nonrelativistic fermions on a ring is presented and first discussed in low order perturbation theory. After introducing the concept of two independent species of right and left moving fermions the exact solution of the Tomonaga-Luttinger model is discussed in detail. The momentum distribution and spectral functions are calculated using the method of the bosonization of the field operator. The general Luttinger liquid phenomenology is shortly discussed.
We derive the exact form of the bosonized Hamiltonian for a many-body fermion system in one spatial dimension with arbitrary dispersion relations, using the droplet bosonization method. For a single-particle Hamiltonian polynomial in the momentum, the bosonized Hamiltonian is a polynomial of one degree higher in the bosonic ``boundary'' field and includes subleading lower-order and derivative terms. This generalizes the known results for massless relativistic and nonrelativistic fermions (quadratic and cubic bosonic Hamiltonians, respectively). We also consider two-body interactions and demonstrate that they lead to interesting collective behavior and phase transitions in the Fermi sea.
We use our recently developed functional bosonization approach to bosonize interacting fermions in arbitrary dimension d beyond the Gaussian approximation. Even in d=1 the finite curvature of the energy dispersion at the Fermi surface gives rise to interactions between the bosons. In higher dimensions scattering processes describing momentum transfer between different patches on the Fermi surface (around-the-corner processes) are an additional source for corrections to the Gaussian approximation. We derive an explicit expression for the leading correction to the bosonized Hamiltonian and the irreducible self-energy of the bosonic propagator that takes the finite curvature as well as around-the-corner proceses into account. In the special case that around-the-corner scattering is negligible, we show that the self-energy correction to the Gaussian propagator is negligible if the dimensionless quantities (qc{\mathit{q}}_{\mathit{c}}qc​/kF{\mathit{k}}_{\mathit{F}}kF​)d{)}^{\mathit{d}})dF0{\mathit{F}}_{0}F0​ [1+F0{\mathit{F}}_{0}F0​${]}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\mu}/${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\Vert}\ensuremath{\partial}${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$/\ensuremath{\partial}\ensuremath{\mu}\ensuremath{\Vert} are small compared with unity for all patches \ensuremath{\alpha}. Here qc{\mathit{q}}_{\mathit{c}}qc​ is the cutoff of the interaction in wave-vector space, kF{\mathit{k}}_{\mathit{F}}kF​ is the Fermi wave vector, \ensuremath{\mu} is the chemical potential, F0{\mathit{F}}_{0}F0​ is the usual dimensionless Landau interaction parameter, and ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$ is the local density of states associated with patch \ensuremath{\alpha}. We also show that the well-known cancellation between vertex and self-energy corrections in one-dimensional systems, which is responsible for the fact that the random-phase approximation (RPA) for the density-density correlation function is exact in d=1, exists also in d>1, provided (1) the interaction cutoff qc{\mathit{q}}_{\mathit{c}}qc​ is small compared with kF{\mathit{k}}_{\mathit{F}}kF​, and (2) the energy dispersion is locally linearized at the Fermi surface. Finally, we suggest a systematic method to calculate corrections to the RPA, which is based on the perturbative calculation of the irreducible bosonic self-energy arising from the non-Gaussian terms of the bosonized Hamiltonian.
This tutorial gives an elementary and self-contained review of abelian bosonization in 1 dimension in a system of finite size LLL, following and simplifying Haldane's constructive approach. As a non-trivial application, we rigorously resolve (following Furusaki) a recent controversy regarding the tunneling density of states, ρdos(ω)ρ_{dos} (ω)ρdos​(ω), at the site of an impurity in a Tomonaga-Luttinger liquid: we use finite-size refermionization to show exactly that for g=1/2 its asymptotic low-energy behavior is ρdos(ω)∼ωρ_{dos}(ω) \sim ωρdos​(ω)∼ω. This agrees with the results of Fabrizio & Gogolin and of Furusaki, but not with those of Oreg and Finkel'stein (probably because we capture effects not included in their mean-field treatment of the Coulomb gas that they obtained by an exact mapping; their treatment of anti-commutation relations in this mapping is correct, however, contrary to recent suggestions in the literature). --- The tutorial is addressed to readers unfamiliar with bosonization, or for those interested in seeing ``all the details'' explicitly; it requires knowledge of second quantization only, not of field theory. At the same time, we hope that experts too might find useful our explicit treatment of certain subtleties -- these include the proper treatment of the so-called Klein factors that act as fermion-number ladder operators (and also ensure the anti-commutation of different species of fermion fields), the retention of terms of order 1/L, and a novel, rigorous formulation of finite-size refermionization of both e−iΦ(x)e^{-i Φ(x)}e−iΦ(x) and the boson field Φ(x)Φ(x)Φ(x) itself.
For pure fermionic systems, and in the absence of symmetry breaking (e.g. magnetism, superconductivity, …), two types of metallic phases are well established: (i) the “normal” Fermi liquid phase in d = 3 presented in Chapters 12 and 19 via the response and the Green functions techniques; and (ii) the “anomalous” Luttinger liquid phase in d = 1 originally formulated with the bosonization technique by which the fermionic operators are represented in terms of “bosonic” density operators (Luther and Peschel (1974); Haldane (1981)) with the exact solution (Mattis and Lieb (1965)) of the Tomonaga–Luttinger model (Tomonaga (1950); Luttinger (1963)) discussed inmany reviews and books (see for instance the book by Giamarchi (2004)). Here, to stress similarities and differences between the Fermi liquid and the Luttinger liquid, we will follow the alternative route of response and Green functions techniques (Dzyaloshinskii and Larkin (1973), Di Castro and Metzner (1991), Metzner et al. (1998)) and add a few comments on the renormalization group approach (Metzner and Di Castro (1993)). As we have seen in Chapters 12 and 19, the Fermi liquid is described asymptotically by the free low-lying single-particle excitations, i.e., the quasiparticles with a zero-kelvin discontinuous occupation number in momentum space n k (see Eq. (18.54)) at a well defined Fermi surface (k = k F ): where here k F> and k F< indicate the limit k → k F from above and below. This discontinuity still marks the Fermi surface in the interacting system. The finite reduction of the single-particle spectral weight z kF with respect to the Fermi gas (z kF = 1) is given by the finite (i.e. non-critical) “wavefunction renormalization” (see Eq. (18.94)). The presence of the discontinuity at the Fermi surface, together with the Pauli principle, compel the inverse quasiparticle lifetime to be τ −1 ≈ max( T 2 , ϵ 2 ), where _ is the deviation of the energy of the quasiparticle from the Fermi energy. The energy uncertainty ħτ −1 due to the finite lifetime of a quasiparticle near the Fermi surface is small compared to its energy ϵ , and the quasiparticle concept is well defined. All the momentum transferring scattering processes become asymptotically ineffective, as derived phenomenologically in Chapter 12 and confirmed microscopically in Chapter 19.