# Questions tagged [bosonization]

Bosonization is a mathematical procedure mapping a system of interacting fermions in 1+1 dimensions to a system of massless, bosons (excitations).

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### Point splitting in bosonization

I was following two lecture notes on bosonization: https://arxiv.org/abs/cond-mat/9805275 and https://stanford.idm.oclc.org/login?url=https://www.worldscientific.com/doi/10.1142/9789814447027_0006 I ...
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### Connection of Sugawara construction to the regular energy-momentum tensor

Update: As pointed out by @ConnorBehan this problem is related to the rearrangement lemma in the 'Yellow Book'. In fact this problem is already mentioned page 649 in the book in the discussion about ...
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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 ...
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### Derivation of anomalous commutators of currents in Fradkin's book

I am trying to understand the derivation of the anomalous commutators of the left- (and right)-moving currents in Fradkin's book (see e.g. here). I am not sure I understand how (6.71) leads to (6.72). ...
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### Anomalous commutators in bosonization

I am trying to learn about abelian bosonization, and I'm finding there to be very many subtle issues. One of these issues lies in the anomalous commutator, on which the constructive approach to ...
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### Interacting fermions in one dimension: Connection between the Fourier transform of interaction $V(q)$ to the $g_2, g_4$ constants?

In the book Condensed Matter Field Theory by Altland and Simons, a treatment of fermions in one dimension interacting via density-density interaction is presented. The authors first consider a general ...
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### Compressiblity for Tomonaga-Luttinger liquid

I'm trying to understand how Giamarchi (in his book "Quantum Physics in One Dimension", section 2.2.1) evaluates the compressibility in the Tomonaga-Luttinger model, that is a linear ...
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### Excitations in Luttinger liquids

It's not clear to me what are the elementary excitations of Luttinger liquids. Quoting from Giamarchi's book Quantum Physics in One Dimension: In one dimension, [...], an electron that tries to ...
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### Bosonization and peculiarities of 1-D systems of interacting fermions

I'm studying bosonization and from what I've understood the main reasons why it's useful are that: For models such as the Hubbard model the Bethe Ansatz, though it allows to evaluate eigenvalues and ...
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### A question about normal ordering in bosonization

I am learning abelian bosonization, and I need to find the commutator of two density operatorS. In this process, I need to calculate $\sum_k(C^{\dagger}_{k+q_1+q_2}C_{k}-C^{\dagger}_{k+q_1}C_{k-q_2})$...
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### Luttinger Liquid Parameter Physical Meaning - Attractive or Repulsive

For 1+1D systems at long wavelength, it is known that the (Tomonaga)-Luttinger Liquid can be rewritten in terms of bosonic parameters, where the Hamiltonian densities can be written \begin{align*} H =&...
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### Bosonisation of two non-interacting Fermions

Assume we have 2 sets of non-interacting fermions which I show by $\psi^{\pm}$ and $\chi^{\pm}$ where we have $\left< \psi^{+}(z) \psi^{-}(0) \right>=\frac{1}{z}$ and similar for $\chi$. Now we ...
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### Can a constant term be added to the new operators in the Bogoliubov transform?

The Bogoliubov transformation picks a set of boson operators $\{a_{k},a^{\dagger}_{k}\}$ and transforms them into a new set of boson operators generally written as: b_{k}=\sum_{l} u_{...
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### When are we required to use the Wess-Zumino term?

I was recently reading about non-Abelian bosonization, and I had a question concerning the Wess-Zumino term. In particular, I have been reading this short introduction by Ivan Karmazin, which states ...
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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, ...
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### Co-propagating fermions with tunneling

When we have coupled fermions with an opposite chirality, the existence of the tunneling term will effectively act as a mass term and opens up the gap. When we bosonize the theory this mass term ...
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### Bosonization for unequal left/right Fermi velocities

The standard exposition of bosonization/Luttinger liquid theory in textbooks treats the case that left and right channels share the same absolute value of Fermi velocity. Is it possible to relax this ...
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### What is the meaning of propagator in the context of lattice theory?

Say in $1+1D$ free fermion theory, it is easy to calculate the propagator in the (effective) field theory to be $$\langle \psi^\dagger(z)\psi(z')\rangle = \frac{1}{2\pi}\frac{1}{z-z'}$$ (in the ...
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### Anticommutation relation for the exponential field of the bosonic field

In 1+1 dimensions, the massless KG equation has the general solution $$\phi(x,t)=\int_{-\infty}^{\infty}dp/(4\pi E_p)[a_pe^{i(px-E_pt)}+a^{\dagger}_pe^{-i(px-E_pt)}]$$ where $E_p^2=p^2$. The operator ...
Let me be sort of specific. We consider a Weyl particles on $S^{1}$, with the following Hamiltonian $$\mathcal{H}=v\int_{0}^{2\pi}\psi^{\dagger}(x)(-i\partial_{x})\psi(x)dx$$ Such particles have the ...