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
6
votes
2answers
80 views

Is bosonization possible in any number of dimensions?

Is bosonization applicable to an arbitrary number of spacetime dimensions?
0
votes
1answer
36 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
33 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
42 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
46 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
13 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
75 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 ...
3
votes
1answer
77 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 $e^{i\sqrt{\pi}\left(\phi\pm\bar\phi\right)}$. ...
0
votes
0answers
23 views

Two inconsistent expression of density operator $\rho(x)$ in Giamarchi's book

I am currently reading Giamarchi's "Quantum Physics in One Dimension". From (2.28) one easily obtain $$ \rho(x)=\rho_R(x)+\rho_L(x)=-\frac{1}{\pi}\nabla\phi(x).$$ (Actually (2.55) exactly states this ...
1
vote
0answers
19 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
59 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
28 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 ...
1
vote
0answers
33 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
55 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 $...
0
votes
0answers
30 views

Density fluctuation and derivative of the field operator $\partial_x \phi(x)$

In a literature about bosonization, one argues that $\partial_x \phi(x)$ represents the density fluctuation. Why can we think that the derivative is about the density fluctuation? My guess: For small ...
3
votes
0answers
145 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
36 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
33 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
83 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
286 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
29 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
197 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
390 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 ...
8
votes
2answers
298 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
69 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 ...
5
votes
1answer
362 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
61 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
311 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}...
5
votes
1answer
253 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
112 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 ...
4
votes
1answer
414 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 ...
11
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
264 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
128 views

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

Consider the standard operator identities in 1+1D bosonization $$\psi^\dagger_{L,R}\sim e^{i(\phi\pm\theta)},\\\rho(x)\equiv\psi^\dagger_L\psi_L+\psi^\dagger_R\psi_R=\frac{1}{\pi}\partial_x\theta.$$ ...
2
votes
1answer
230 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 $$\Psi(x)=\left(\begin{array}{c}\psi_R(x)\\\psi_L(x)\end{array}\right),$$...
8
votes
1answer
258 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
154 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 ...
26
votes
2answers
743 views

$\phi^4$ 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 \...
2
votes
0answers
83 views

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 ...
1
vote
1answer
183 views

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 ...
4
votes
0answers
276 views

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 ...