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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$\phi^4$ 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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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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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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"$\theta$-$\phi$ 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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1answer
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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 and supersymmetry

In 2D (time + space) there is no notion of statistic. So particles can be described in terms of bosonic and fermionic fields. Well-known example is Thirring/Sine-Gordon duality. There are also some ...
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Is bosonization possible in any number of dimensions?

Is bosonization applicable to an arbitrary number of spacetime dimensions?
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Bosonization and gauge symmetry

The bosonization map relates the fermionic current $\bar{\psi}\gamma\psi$ to the bosonic current $\partial\phi$, and also the components of $\psi$ to $e^{i\sqrt{\pi}\left(\phi\pm\bar\phi\right)}$. ...
6
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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 ...
6
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1answer
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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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Jordan-Wigner transformation on a circle and spin structures?

Is there an analog of the Jordan-Wigner transformation between fermion algebra on a circle and a Pauli algebra? For example, the continuum analog of bosonization of "compact boson $\...
5
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1answer
556 views

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 ...
5
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1answer
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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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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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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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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 ...
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2answers
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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 ...
3
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1answer
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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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1answer
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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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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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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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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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(Giamarchi) Physical meaning of $\Psi^\dagger(r)\Psi^\dagger(r+a)$

I am currently reading Giamarchi's "Quantum Physics in One Dimension". The below is the part of the book: Question: 1. Eq. (2.72) of the book defines $$O_{SU}(r)=\Psi^\dagger(r)\Psi^\dagger(r+a)$$ ...
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Fermion creation operator in boson basis

I've been reading Giamarchi, Quantum Physics in One Dimension, Chapter 2 on 1d bosonization, and in appendix B.1, he derives equation B.2, which represents the fermion creation operator $\psi_r (x)$ ...
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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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1answer
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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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2answers
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1+1D Bosonization and Zero Modes

I have been reading Senechal's lecture notes on bosonization, and I appreciate the care that he takes in dealing with the zero modes of the massless boson. However, when it comes to applications - e.g....
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2answers
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Computing correlation function $\langle e^{i\beta \phi(x)}e^{-i\beta\phi(0)}\rangle$ for massless scalar field $\phi$

I am currently reading Shankar's "Bosonization: How to make it work for you in condensed matter" (http://inspirehep.net/record/408901/). In page 9, I am stuck with computing the correlation function ...
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1answer
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Two definitions for normal ordering of $c_{k+q}^\dagger c_k$

Consider the fermionic operator $c_k, c^\dagger_k$, and where $k$ is discrete and unbounded. (Note: This situation frequently arises in bosonization.) Let the vacuum $|0\rangle$ be the state with all $...
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1answer
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Why does equal commutator relation imply equal operator?

In 1d bosonization, Giamarchi (Quantum Physics in One Dimension) Chap 2, shows that fermionic Hamiltonian $$H_f=\sum_k k(R_k^* R_k -L_k L_k)$$ is equal to the bosonic representation $$H_b = \sum_k |...
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1answer
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Simple question on computing commutation relation

In bosonization, one faces with the following commutator: $$[\phi(x_1), \theta(x_2)]=\sum_{q\neq 0} \frac{\pi}{Lq} e^{iq(x_2-x_1)-\alpha |q|}\tag{1}$$ where $q$ is an non-zero integer multiple of $2\...
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1answer
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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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Bosonization for the Tomonaga-Luttinger liquid

I'm having a hard time understanding bosonization applied to the Tomonaga-Luttinger liquid. My difficulties appear when contrasting some references which approach the problem in different ways I can't ...
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Bosonization in 0+1 dimensions?

We know examples of bosonization in 1+1 dimensional spacetimes, as well as some examples in 2+1 and 3+1 space-time dimensions are known in literature. Does this also work in quantum mechanical models ...
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Bosonization left and right moving fields in 1D

In’s Senechal’s Bosonization review (https://arxiv.org/pdf/cond-mat/9908262.pdf)for the free boson he defines separate left and right moving parts for the field $\phi$ As $$\phi(x,t)=\phi(x-vt)+\bar{\...
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Do large $N$ free fermion or WZW theories have a holographic dual in $AdS_3/CFT_2$?

I was wondering if for $N$ free Dirac fermions (or equivalently by bosonization, $N$ free bosons or an $SU(N)_1$ WZW theory plus an extra boson) have a holographic dual description via $AdS_3/CFT_2$? ...
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Is there an explicit mapping between N free bosonic fields and the $SU(N)_1$ WZW model + free boson?

Witten's nonabelian bosonization tells us that $N$ free Dirac fields can by written in terms of an $SU(N)_1$ WZW model and one free boson. But bosonization also tells us that we could just as well ...
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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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1answer
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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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1answer
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(Giamarchi) Meaning of slowly varying field in bosonization

I am currently reading Giamarchi's Quantum Physics in One Dimension. Eq. (2.30) of the book says $$ \psi_r(x)=\frac{U_r}{\sqrt{2\pi\alpha}}e^{irk_Fx}e^{-i(r\phi(x)-\theta(x))} $$ where $U_r$ is the ...
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Can we use Bosonization to study theories without $U(1)$ symmetry?

When studying lattice models using bosonization, we expect the total charge is conserved so that the elementary excitation is particle-hole-like bosonic degrees of freedom. How about models without $U(...
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Mathematical problem in 1D bosonization

I am reading the following article on bosonization : https://arxiv.org/abs/cond-mat/9805275 and I encountered the following set of equalities. $$\begin{align} [\phi_\eta (x),\partial_{x'}\phi_{\eta'}(...
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Features of plasmon and surface plasmon polariton

What is the difference between surface plasmon polariton and plasmon in the Hamiltonian? So let's say that I can diagonalize the Hamiltonian of the system I am studying no matter how complicated that ...
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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)^...
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Simple question on Giamarchi's book

I have trouble from Eq.(2.62) to Eq.(2.63) in Giamarchi's "Quantum Physics in One Dimension". The book says as follows, but I think that some terms are missing. By straight computation of $\rho(r)\...
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Explicit form of Klein factors in Giamarchi

In Giamarchi, Quantum Physics in One Dimension, Appendix B, I don't understand how he did his last step in equation B.8, as shown below. If anyone has gone over the derivation, I would really ...
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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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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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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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1answer
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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 ...