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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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 $\{ ...
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Why does normal-ordering ensure finiteness?

I will be using Jan von Delft's rigorous construction of bosonization/refermionization as an example, but I will try to explain my question in more general terms. Consider an (countably) infinite-dim (...
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SUSY vs bosonization and fermionization

(New to the concepts.) From what was known SUSY described a theory consisted of both a boson and a fermion pair as a symmetric counter part. Bosonization and Fermionization on the other hand described ...
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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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Peierls term in bosonization

Let's say we want to bosonize the contact interaction term: $:[\Psi^{\dagger}(x)\Psi(x)]^2:$, where $::$ denotes normal ordering. If one first writes $\Psi(x)={\rm e}^{-i k_F x}\psi_L(x) +{\rm e}^{i ...
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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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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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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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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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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 $\...
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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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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: \begin{equation} b_{k}=\sum_{l} u_{...
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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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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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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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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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Is two dimension equal to three for bosonization?

I have been reading about bosonization lately and really appreciated Luttinger liquid bosonization in 1 dimension. Also, I got interested in higher dimensional bosonization but I only find Haldane's (...
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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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Interacting electrons in 1D

In Senechal’s review on Bosonization https://arxiv.org/pdf/cond-mat/9908262.pdf he starts with a microscopic Hamiltonian $$H_{\mathrm{F}}=\sum_{k} \varepsilon(k) c^{\dagger}(k) c(k)$$ before stating ...
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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 $2\...
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$\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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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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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 $e^{i\sqrt{\pi}\left(\phi\pm\bar\phi\right)}$. ...
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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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(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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(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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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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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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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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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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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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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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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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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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