# 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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### 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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### 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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### 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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### 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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### 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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### "$\theta$-$\phi$ duality" and $T$-duality

When bosonizing an interacting spinless Luttinger liquid, the action can be written as S=\frac{K}{2\pi}\int dx d\tau\ (\partial_\mu\phi)^2 = \frac{1}{2\pi K}\int dx d\tau\ (\partial_\...
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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 ...
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### Question on functional bosonization

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 ...
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### 1D Bosonisation, equating number of states in Hilbert space

I was reading von Delft and Schoeller's review (Bosonization for Beginners) and in their Appendix B they prove that the number of states in the fermionic representation's Hilbert space is the same as ...