Specific heat of a one-dimensional interacting Fermi system: Role of anomalies

We develop a new method that allows us to map models of interacting fermions onto bosonic models describing collective excitations in an arbitrary dimension. This mapping becomes exact in the thermodynamic limit in the presence of a bath. The boson models can be written either in the form of a model of non-interacting bosons in a fluctuating auxiliary field or in the form of a superfield theory of interacting bosons. We show how one can study the latter version using perturbation theory. Using the developed diagrammatic technique we compared the first two orders of perturbation theory with the corresponding results for the original fermion model and found a perfect agreement. As concerns the former representation, we suggest a scheme that may be suitable for Monte Carlo simulations and demonstrate that it is free of the fermionic sign problem. We discuss in details the properties of the bosonic representation and argue that there should not be any obstacles preventing from an efficient computation.

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“…(2.9), in the form 19) and the HamiltonianĤ 0 now given bŷ 20) with the modified chemical potential µ ′ from Eq. (2.14).…”

Section: Fermionic Model and Bosonizationmentioning

confidence: 99%

Describing systems of interacting fermions by boson models: Mapping in arbitrary dimension and applications

2010

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“…19 Another example of such an anomaly is the specific heat of 1D fermions. 10 Connection to bosonization. The fermion language is not a preferred one: one can equally well obtain the same results in the boson language.…”

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confidence: 99%

Dynamical spin structure factor of one-dimensional interacting fermions

Zyuzin

Maslov

2015

Phys. Rev. B

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We revisit the dynamic spin susceptibility, χ(q, ω), of one-dimensional interacting fermions. To second order in the interaction, backscattering results in a logarithmic correction to χ(q, ω) at q ≪ kF , even if the single-particle spectrum is linearized near the Fermi points. Consequently, the dynamic spin structure factor, Imχ(q, ω), is non-zero at frequencies above the single-particle continuum. In the boson language, this effect results from the marginally irrelevant backscattering operator of the sine-Gordon model. Away from the threshold, the high-frequency tail of Imχ(q, ω) due to backscattering is larger than that due to finite mass by a factor of kF /q. We derive the renormalization group equations for the coupling constants of the g-ology model at finite ω and q and find the corresponding expression for χ(q, ω), valid to all orders in the interaction but not in the immediate vicinity of the continuum boundary, where the finite-mass effects become dominant.Introduction Bosonization is the most common way to describe one-dimensional (1D) interacting fermions. 1 If the lattice effects are not essential, an exact correspondence between the fermion and fermion-hole (boson) operators maps the charge sector of the system onto a gas of free bosons [the Tomonaga-Luttinger liquid (TLL)]. The spin sector, however, is not free but maps onto the sine-Gordon model. The non-Gaussian (cosine) term of this model results from backscattering of fermions with opposite spins. If the interaction between the original fermions is repulsive, the backscattering term represents a marginally irrelevant operator and is renormalized down to zero at the fixed point, where the spin sector also becomes free. At intermediate energy scales, such marginally irrelevant operators lead to logarithmic renormalizations of the observables. 2 Since the original paper by Dzyaloshinskii and Larkin (DL), 3 it has been known that the backscattering operator gives rise to the logarithmic temperature (or external magnetic field) corrections to the static spin susceptibility. In Refs. 4 and 5, it was shown that the static spin susceptibility also depends logarithmically on the external momentum q at small q. In addition, both the spin-and charge susceptibilities at 2k F acquire multiplicative logarithmic renormalizations. 1,6 In this work, we focus on dynamics of the longwavelength part of the spin response. First, we need to outline the differences between the charge and spin sectors. As charge bosons are free at all energies, the dynamical charge structure factor (the imaginary part of the charge susceptibility at finite frequency, ω, and momentum, q) is a delta function centered at the boson dispersion, ω = v c q, which is represented by a straight line in Fig. 1A. This result differs from that for free fermions only in that the Fermi velocity, v F , is replaced by the renormalized charge velocity, v c . A non-zero width of the charge structure factor appears only if one goes beyond the TLL paradigm by taking into account finite curvature (inverse ...

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“…This time, the integral comes from energies |ω m | ∼ |ε k | ∼ W , i.e., from high energies in our nomenclature. The fact that the same result can be obtained in two ways implies that the N F term in the polarization operator for free fermions is an "anomaly", 15,16 which can be viewed as either as a low-or high-energy contribution, depending on the regularization procedure. This feature is a consequence of the double pole in the integrand of Eq.…”

Section: Bosonic Propagator In Rpa and In Fl Theorymentioning

confidence: 74%

“…For example, diagram c in Fig. 2 in this case contain a product of two blocks (16) where k = k + q. Because the interaction is static, integrals over ω m and ω m in Eq.…”

Section: A Formulation Of the Problem And Backgroundmentioning

confidence: 99%

Gradient terms in quantum-critical theories of itinerant fermions

et al. 2017

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We investigate the origin and renormalization of the gradient (Q 2 ) term in the propagator of soft bosonic fluctuations in theories of itinerant fermions near a quantum critical point (QCP) with Q = 0. A common belief is that (i) the Q 2 term comes from fermions with high energies (roughly of order of the bandwidth) and, as such, should be included into the bare bosonic propagator of the effective low-energy model, and (ii) fluctuations within the low-energy model generate Landau damping of soft bosons, but affect the Q 2 term only weakly. We argue that the situation is in fact more complex. First, we found that the high-and low-energy contributions to the Q 2 term are of the same order. Second, we computed the high-energy contributions to the Q 2 term in two microscopic models (a Fermi gas with Coulomb interaction and the Hubbard model) and found that in all cases these contributions are numerically much smaller than the low-energy ones, blue especially in 2D. This last result is relevant for the behavior of observables at low energies, because the low-energy part of the Q 2 term is expected to flow when the effective mass diverges near QCP. If this term is the dominant one, its flow has to be computed self-consistently, which gives rise to a novel quantum-critical behavior. Following up on these results, we discuss two possible ways of formulating the theory of a QCP with Q = 0.

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Specific heat of a one-dimensional interacting Fermi system: Role of anomalies

Cited by 9 publications

References 36 publications

Describing systems of interacting fermions by boson models: Mapping in arbitrary dimension and applications

Describing systems of interacting fermions by boson models: Mapping in arbitrary dimension and applications

Dynamical spin structure factor of one-dimensional interacting fermions

Gradient terms in quantum-critical theories of itinerant fermions

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