June 7, 2007

The purpose of reporting physics in one-dimension (1D) is twofold: First of all, in 1D the mathematical models are simpler than their higher dimensional counterparts, and more often than not one can has access to exact solutions. Even in 1D the physics is very rich and complicated! Secondly, nowadays there are a number of real systems, such as carbon nanotube, quantum wire, organic superconductors, etc., that should be described by 1D models. Thus 1D physics is no longer artificial but of great concern to both theorists and experimentalists. The content here is mainly based on Ref.1, and briefer introduction to this subject can be found in ordinary many-body textbooks[2,3].

J. M. Luttinger in 1960s discussed an exactly
solvable model of a 1D fermionic system[4]. For this reason 1D
interacting fermions now bear the name *Luttinger liquid*. Because
of the 1D nature, an electron cannot propogate without pushing its
neighbors. No individual motion is possible. Unlike the Fermi liquid,
where the quasiparticle is well defined, the low-energy excitation of
interacting fermions in 1D are collective. Recall that when collective
motion (such as plasma oscillation) occurs, the equation of motion for
the density fluctuation resembles that of a harmonic oscillator, which
obeys Bose-Einstein statistics. This suggests that we must treat the
problem with bosonic fields. This procedure is known as
*Bosonization*.

In this report I shall introduce the technique of Bosonization, which is one of the most important tools for 1D systems, and then discuss some interesting features of Luttinger liquid. In addition, I will address the possibility of 1D superconductivity (since we have learned superconductivity for almost a whole quarter). A few theoretical papers in the 1960s concluded that there should be no superconductive phase in 1D [5,6], yet experiments told us that we do have organic superconductors which can be regarded as 1D systems. Their conclusion is incorrect, and I will try to explain what's the problem with their arguments.

For a 1D electron gas, the excitaion energy of a particle-hole pair is proportional to q, the momentum trasnfer, in the vicinity of the fermi level:

The dispersion relation can be linearized as long as
q is small compared to the fermi momentum k_{F}. If we
use the linear dispersion relation, we are forced to consider the right-
and left- going fermions seperately (see Fig. 1). Therefore, the
Hamiltonian of the kinetic energy part becomes (for simplicity, we
consider spinless fermions first):

where R and L represent right- and left-going particles, respectively.

We now have the kinetic energy part and need the interaction part (for spinless fermions). A typical interaction is:

where L is the length for the system of interest.

We can decompose the interaction into three different
types as shown in Fig. 2. For identical spinless fermions, g_{2}
and g_{1} processes are the same. The use of notation g to
distinguish different processes is historical and called "g-ology".

Historically, the first attempt to solve the full Hamiltonian was made using the fermionic language, which is complicated because the appearance of 4-fermionic operators. However, as mentioned at the beginning, density flucuations (bosonic fields) are a natural basis to use because of the collective tendencies in 1D:

The density operator is bosonic, as follows from the
anticommutation relations of fermion operators. Let us then write the
density operator as some linear combination of b_{q} and
b_{q}^{+}, where b_{q} and
b_{q}^{+} are boson annihilation and creaion operators,
respectively. Written in the boson basis, the interacting Hamiltonian
then becomes:

This Hamiltonian is quadratic and thus easier to diagonize.

It can be shown that the mapping from all fermionic operators into the bosonic basis is complete. One can then look up the "Bosonization dictionary" to rewrite Hamiltonian in the boson basis. The kinetic energy of the resultant Hamiltonian is

where the parameters inside the integrand are defined as

Note that since we have introduced an infinite number of occupied states, the Dirac sea, we have to be careful in defining the density operator to avoid infinities [1].

If we calculate the g_{4} process, we will
find its contibution to be

which is proportionl to the free Hamiltonian.
Therefore, the g_{4} process only serves to renormalize the
velocity of excitations, which becomes

Similarly, if we include the calculation for the
g_{2} process, the total Hamiltonian then becomes

which is *still quadratic*. The parameters
u and K are defined as

Therefore, dealing with the 1D interacting problem in the language of Bosonization is no more complicated than solving the free Hamiltonian.

Fig. 3:The distribution function in momentum
space. |

Now we readily see one interesting feature of
Luttinger liquid: the velocity of the excititaion u is being
renormalized with the presence of the interactions g_{4} and
g_{2}. In general, for fermions with spin, the interactions
involving charge and spin are different, so two distinctive excitation
velolcities appear. This is known as the *spin charge
seperation.*

Another interesting feature of Luttinger liquid is
the occupation number in momentum space. This was first discussed by
Luttinger[4] and later refined by Mattis and Lieb[6]. As can be seen in
Fig. 3, instead of a discontinuity at k_{F} -the signature of
quasiparticle excitation in Fermi liquid - one finds in 1D an essential
singularity at k_{F} (the point with infinite slope).

In the Fermi liquid theory, one decomposes the
spectral function A into a sharp excitation with peak value Z and a
smooth background. The distribution in Fig. 3 then means that no
individual fermionic excitation can survive in 1D, for this corresponds
to Z=0 in Fermi liquid theory. Note that while the shape of the
distribution function may be largely distorted, the singularity is still
at k_{F}, thanks to the conservation law known as Luttinger
theorem: *Although the shape of the Fermi surface can be affected by
interactions, the "volume" enclosed by the Fermi surface is an
invariant*. In fact, some of the most convincing observations of
Luttinger liquid behavior lie on the measurements in the tunneling
density of states. Also, as pointed out by Luttinger, though the
distribuion function in momentum space is altered because of
interactions, some behaviors near the fermi level (for example, the Kohn
effect, a singularity in (q-2k_{F}) remain unchanged.

In statistical mechanics it is known from studies of the 1D Ising model that no magnetic ordering exists at finite temperature. (The exception is when there are long-range interactions, which can change the correlation functions qualitatively.) More generally, no phase exhibiting long-range order (LRO) can exist in 1D if only short-range forces exist. Since both the BCS Hamiltonian or the actual physical system involve forces only of finite range, it is then argued that there should be no superconducting phase in 1D. Even if one takes into account modification to BCS by including the compressional mode, it has then be shown by R. Ferrell[5] that these compressional modes, which is more dominant in 1D, prevent the establishment of LRO, which is required for superconductive phenomena[8]. The calculation involves the known theorem that there is no long-range lattice order for 1D geometry[7]. D. Mattis and E. Lieb also showed that in 1D no phase transition at any finite temperature is found based on their model, which proved more rigorously Ferrell's argument[6].

Their arguments appear to be convincing, yet in reality we do have 1D superconductivity. The flaw of the above argument lies in its assumption that a nonvanishing value of the correlation function G(x-x') for large (x-x') is required by superconducting phenomenon. In other words, LRO isn't required for superconductivity. There is still a critial temperature below which one can tell the difference between an exponential decay and a power law decay for the correlation function. Therefore, there is still a (superconductive) phase even though there is no LRO in the correlation fucntion. However, for a 1D system, if you are sure that there are no long-range interactions and you see a phase with LRO, you probably made a mistake in your experiment (for example, you didn't wait long enough for the system to come to equilibrium).

© 2007 Cheng-Chien Chen. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] Thierry Giamarchi, *Quantum Physics in One
Dimension* (Oxford, 2003).

[2] H. Bruus and K. Flensberg, *Many-Body Quantum
Theory in Condensed Matter Physics*, (Oxford, 2004).

[3] P. Phillips, *Advanced Solid State
Physics* (Westview Press, 2003).

[4] J. M. Lutiinger, J. Math Phys. **4**, 1154
(1963).

[5] R. A. Ferrell, Phys. Rev. Lett. **13**, 330
(1964).

[6] D. Mattis and E. Lieb, J. Math Phys. **6**, 304
(1965).

[7] L. D. Landau and E. M. Lifshitz, *Statistical
Physics*, 3rd Ed. Part 1 (Elsevier, 1980).

[8] C. N. Yang, Rev. Mod. Phys. **34**, 694
(1962).