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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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Landau level in particular cases

If we consider an array of identical uncoupled spinless non-interacting one-dimensional wires, as shown in Fig.(a) with a single-particle electronic dispersion E(k), which we can take to be parabolic ...
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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 \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_\...
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Relation between bosonisation and mirror symmetry in 3d

In 3d bosonisation, there is a process, also known as flux attachment, that changes the spin-statistics of a particle. In the simplest case with only one flavour, this comes in the form of coupling ...
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Sign on Particle-Vortex Duality from 3d Bosonization

This is most likely a question with a very simple answer but well, on the paper "Particle-Vortex Duality from 3d Bosonization" by Karch and Tong, on page 11 it is claimed that the first three terms on ...
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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 ...
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Expectation value of Hamiltonian of Heisenberg spin chain

The following is Hamiltonian of Heisenberg spin half chain mapped onto Hard core bosons by Holestein-Primakoff transformation. $$ H = t\sum_{j}b_{j}^{\dagger}b_{j}+h.c + V\sum_{j}n_{j}n_{j+1}+\sum_{j}...
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Bosonization and Commutation Relation

I'm playing a bit with bosonization $ψ→:e^{-φ}:$ and $ψ^*→:e^{φ}:$ in the sense that $$ \Bigg\langle 0_\mathrm{F} \Bigg|∏_{i=1}^nψ(z_i)ψ^*(w_i)\Bigg|0_\mathrm{F}\Bigg\rangle = \Bigg\langle 0_\mathrm{...
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How the understand the idea of spatial dependent Fermi wave vector?

Recently, I have been reading the book by Naoto Nagaosa on Quantum field theory in Strongly Correlated Electronic Systems, but I got a problem in Chapter 3.2. When he discuss the idea of Bosonization ...
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Relation between bosonization and conformal field theory

Recently I have been studying bosonization for 1-dimensional system. There are often some claims of bosonization being related to conformal field theory. I know that one could map 1+1D quantum field ...
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Jordan-Wigner transformation v.s. Bosonization

Jordan-Wigner transformation is a powerful tool, mapping between models with spin-1/2 degrees of freedom and spinless fermions. The key idea is that there is a simple mapping between the Hilbert space ...
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Majorana Fermions representation of Klein factors

In Abelian Bosonization Klein Factors are introduced as raising and lowering operators which connect Hilbert spaces with different numbers of particles. Also for more than one species of bosons, the ...
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Uniform and $2k_F$ density operators in 1+1D bosonization

Consider the standard operator identities in 1+1D bosonization $$\psi^\dagger_{L,R}\sim e^{i(\phi\pm\theta)},\\\rho(x)\equiv\psi^\dagger_L\psi_L+\psi^\dagger_R\psi_R=\frac{1}{\pi}\partial_x\theta.$$ ...
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chiral anomaly and translation symmetry in 1+1D

A Luttinger liquid at low energies can be captured by Dirac fermions in 1+1D, where the two component fermion field is given by $$\Psi(x)=\left(\begin{array}{c}\psi_R(x)\\\psi_L(x)\end{array}\right),$$...
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Mathematical content of Thirring/Sine-Gordon duality

I'm a mathematician who is intrigued by the duality between the Thirring and Sine-Gordon models as established by Sidney Coleman. Can someone explain the mathematical content of this duality to me? (...
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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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$\phi^4$ theory kinks as fermions?

In 1+1 dimensions there is duality between models of fermions and bosons called bosonization (or fermionization). For instance the sine-Gordon theory $$\mathcal{L}= \frac{1}{2}\partial_\mu \phi \...
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Can the terms in the microscopic model with nonzero conformal spin generate some new term(s) under RG (renormalization group) flow?

As in the book Bosonization and Strongly Correlated Systems at page 66, it says that "We see that the original perturbation with nonzero conformal spin generates the perturbation with zero conformal ...
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Interchaging boson and fermion on an infinite 1 dimensional line

In 1+1 dim bosonization, one introduce the Klein factors, which are Hermitian and satisfies Clifford algebra. (1) In the case of 1 dim space is a 1D ring ($S^1$ circle), then one have left-right ...
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Bosonization on the lattice fermion - a rigorous mapping

An inquiry: usually the bosonization is done on the field theory side. The mapping between the fermion operator to the boson operator is done for the field theory operators. As far as we know for the ...