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).

Filter by
Sorted by
Tagged with
0
votes
0answers
14 views

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 ...
0
votes
1answer
42 views

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 ...
2
votes
1answer
69 views

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 ...
0
votes
0answers
37 views

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 $\sum_k(C^{\dagger}_{k+q_1+q_2}C_{k}-C^{\dagger}_{k+q_1}C_{k-q_2})$...
0
votes
0answers
20 views

“Physical” field fermionic operator

I'm studying an article on bosonization and came across this: \begin{equation} c_j=\sum_{k\in BZ}\frac{e^{ikj}}{\sqrt{L}}c_k \end{equation} First note that the continuum limit ($a\rightarrow0$, where ...
3
votes
1answer
48 views

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 =&...
0
votes
0answers
42 views

Bosonization subtleties

I have a doubt on the bosonization prescription commonly used to work out effective low energy theories from lattice Hamiltonians. I am trying to work out the bosonization representation of the ...
5
votes
1answer
56 views

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 ...
6
votes
0answers
68 views

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 $\...
1
vote
0answers
47 views

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'}(...
0
votes
1answer
54 views

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_{...
1
vote
0answers
17 views

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 ...
1
vote
0answers
67 views

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 ...
2
votes
2answers
74 views

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....
6
votes
0answers
92 views

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 ...
6
votes
2answers
144 views

Is bosonization possible in any number of dimensions?

Is bosonization applicable to an arbitrary number of spacetime dimensions?
0
votes
1answer
53 views

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 (...
2
votes
0answers
54 views

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{\...
0
votes
1answer
57 views

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 ...
2
votes
1answer
53 views

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\...
1
vote
0answers
19 views

$\cos(\sqrt{8} \phi_\sigma)$ 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)^...
2
votes
2answers
91 views

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 ...
5
votes
1answer
134 views

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 $e^{i\sqrt{\pi}\left(\phi\pm\bar\phi\right)}$. ...
1
vote
0answers
25 views

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)\...
3
votes
0answers
67 views

(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 $$O_{SU}(r)=\Psi^\dagger(r)\Psi^\dagger(r+a)$$ ...
1
vote
1answer
41 views

(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 ...
2
votes
0answers
45 views

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 $AdS_3/CFT_2$? ...
2
votes
1answer
75 views

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 $...
3
votes
0answers
153 views

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 $\psi_r (x)$ ...
2
votes
0answers
50 views

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 ...
1
vote
0answers
41 views

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 ...
2
votes
1answer
136 views

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 |...
3
votes
1answer
463 views

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, ...
1
vote
0answers
33 views

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 ...
3
votes
2answers
260 views

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 ...
3
votes
2answers
536 views

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 ...
9
votes
2answers
435 views

“$\theta$-$\phi$ 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_\...
0
votes
0answers
108 views

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 ...
6
votes
1answer
426 views

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 ...
1
vote
0answers
63 views

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 ...
0
votes
1answer
366 views

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}...
6
votes
1answer
290 views

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{...
5
votes
0answers
126 views

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 ...
5
votes
1answer
494 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 ...
13
votes
2answers
2k views

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 ...
3
votes
0answers
281 views

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 ...
1
vote
0answers
132 views

Uniform and $2k_F$ density operators in 1+1D bosonization

Consider the standard operator identities in 1+1D bosonization $$\psi^\dagger_{L,R}\sim e^{i(\phi\pm\theta)},\\\rho(x)\equiv\psi^\dagger_L\psi_L+\psi^\dagger_R\psi_R=\frac{1}{\pi}\partial_x\theta.$$ ...
3
votes
1answer
260 views

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 $$\Psi(x)=\left(\begin{array}{c}\psi_R(x)\\\psi_L(x)\end{array}\right),$$...
9
votes
1answer
354 views

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? (...
2
votes
1answer
178 views

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 ...