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Questions tagged [tomonaga-luttinger-liquid]

Tomonaga-Luttinger liquid, more often referred to as simply a Luttinger liquid, is a theoretical model describing interacting electrons (or other fermions) in a one-dimensional conductor (e.g. quantum wires such as carbon nanotubes). Such a model is necessary as the commonly used Fermi liquid model breaks down for one dimension.

Tomonaga-Luttinger liquid, more often referred to as simply a Luttinger liquid, is a theoretical model describing interacting electrons (or other fermions) in a one-dimensional conductor (e.g. quantum wires such as carbon nanotubes). Such a model is necessary as the commonly used Fermi liquid model breaks down for one dimension.

The Tomonaga-Luttinger liquid was first proposed by Tomonaga in 1950. The model showed that under certain constraints, second-order interactions between electrons could be modelled as bosonic interactions. In 1963, Luttinger reformulated the theory in terms of Bloch sound waves and showed that the constraints proposed by Tomonaga were unnecessary in order to treat the second-order perturbations as bosons. But his solution of the model was incorrect, the correct one was given by Mattis and Lieb 1965.

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Compact or non-compact boson from bosonization?

In some discussions of bosonization, it is stressed that the duality between free bosons and free fermions requires the use of a compact boson. For example, in a review article by Senechal, the following statement is made: In order for bosonization…
Zack
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"$\theta$-$\phi$ duality" and $T$-duality -- is the free fermion theory self-dual?

When bosonizing an interacting spinless Luttinger liquid, the action can be written as \begin{equation} S=\frac{K}{2\pi}\int dx d\tau\ (\partial_\mu\phi)^2 = \frac{1}{2\pi K}\int dx d\tau\ (\partial_\mu\theta)^2, \end{equation} where…
pathintegral
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Can Fermi liquid be obtained by a canonical transformation?

The basic assumption of the Ferm-liquid theory is the one-to-one correspondence between the states of an interacting Fermi gas to those of a gas of non-interacting quasiparticles. The question is then, whether one can perform a canonical…
Roger V.
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The correct definition of Klein Factor

Klein factors are the operators which make sure that the anticommutation between the different species is correct during the bosonization procedure. According to this famous review by Jan Von Delft, they are the operators responsible for raising or…
Boa_Constrictor
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Nature of the ferromagnetic transition in the 1D quantum XXZ model

Consider the 1D quantum XXZ model defined by the Hamiltonian: $$ \begin{aligned} H &= ∑_i S^x_{i}S^x_{i+1} + S^y_{i}S^y_{i+1} + Δ_1 S^z_{i}S^z_{i+1} \\ &= ∑_i \frac{1}{2}(S^+_{i}S^-_{i+1}+\mathrm{H.c.}) + Δ_1 S^z_{i}S^z_{i+1} \end{aligned} $$ Or in…
Jacob
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Renormalization-group approach to half-filling charge density wave

In Shankar's noted review paper on the renormalization group (RG) approach to many-body physics, Sec. IV deals with RG in a 1D lattice nearest-neighbour (quartically) interacting model, which leads to the conclusion of marginal quartic interaction…
xiaohuamao
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How do you bosonise the spin-$1/2$ operator $S_z$?

Consider a 1D spin-$1/2$ chain. After a Jordan-Wigner transformation, the spin-$1/2$ operator $S^z_i$ takes the form $$ S^z_i = c^\dagger_i c_i - \frac{1}{2} \equiv \rho_i - \frac{1}{2}$$ where $\{ c_i \}$ are a set of fermionic modes obeying $\{…
Hermitian_hermit
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What's the meaning of these pseudo-momentum and pseudo-position operators?

Consider the following hamiltonian, describing a system of independent bosons: $$ \hat H = \sum_{q \neq 0} \hbar c_q |q|\hat \beta^\dagger_q \hat \beta_q \tag{1}$$ where $\hat \beta_q$ and $\hat \beta^\dagger_q$ are the destruction and creation…
DoeJohn
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What's the relation between SDW/CDW and spinon/holon in the one dimensional repulsive Hubbard model?

As is well known, spin-charge separation occurs in the one dimensional repulsive Hubbard model. This phenomenon can be well understood by the Luttinger liquid theory, where spin density wave(SDW) and charge density wave(CDW) are described by…
Waltergu
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The validity of infinite many Conformal Field Theories on the Fermi surface

The naive $2$-dimensional Fermi sea in $k$-space (with a convex structure and positive Gaussian curvature, some nice properties, etc) in $2+1$-dimensional spacetime may be viewed as an infinite collection of the $1$-dimensional in k-space chiral…
wonderich
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Critical Behaviour of transverse variance

When dealing with phase transitions, the behavior of physical quantities can be described by the theory of critical exponents. For example, let's consider a one-dimensional spin $1/2$ $XY$ Hamiltonian with a field $\Omega$ applied along $x$ $$ H =…
Filippo Caleca
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Is the Luttinger liquid a limit of the Kitaev chain model?

From what I understand, they are both models of electrons on a nanowire, but the Hamiltonian is different, just in the pairing term. When I look up the connection between them, there's surprisingly little literature on the topic. So I wonder if it's…
Juan
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Interaction Term in Tomonaga-Luttinger Model

I am studying Tomonaga-Luttinger Model from Altland and Simon's textbook called Condensed Matter Field Theory. From the derivation, I am stuck with showing that the contribution to the interaction term comes only from $(k, k', q) = (\pm k_F, \pm…
Neophyte
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$\cos(\sqrt{8} \phi_\sigma)$ term when bosonizing the Luttinger Hamiltonian

I am currently reading "Fermi liquids and Luttinger liquids" by Schulz (https://arxiv.org/abs/cond-mat/9807366). In page 27 it says the following: My question is about how $$\frac{g_1}{(2\pi\alpha)^2} \int dx \cos(\sqrt 8 \phi_\sigma)$$ arises in…
Laplacian
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Gapping out edge modes by backscattering

I was reading this paper by Michael Levin about protected edge states without symmetry. In the introduction, he makes the argument that backscattering terms or other perturbations gap out left and right moving modes in equal numbers. How can we…
Boa_Constrictor
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