HAL Id: jpa-00247048
https://hal.archives-ouvertes.fr/jpa-00247048
Submitted on 1 Jan 1995
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
liquid: I. Landau fixed point and integrability
R. Mélin
To cite this version:
R. Mélin. Level spacing statistics of the bidimensional Fermi liquid: I. Landau fixed point and integrability. Journal de Physique I, EDP Sciences, 1995, 5 (2), pp.159-179. �10.1051/jp1:1995105�.
�jpa-00247048�
Classification Physics Al~stracts
71.70 05.30F
Level spacing statistics of the bidimensional Fermi liquid:
I. Landau fixed point and integrability
R. Mélin
CRTBT-CNRS, BP 166X, 38042 Grenoble Cedex, France
(Received 29 April 1994, revised 28 September1994, accepted 4 November1994)
Abstract. We investigate trie statistical properties of trie excitation spectruIn of one- and two-dimensional mortels for Landau liquids. Trie level spacing statistics are found to be Poisso-
mari for non-zero separations and sufficiently strong interactions. In trie Poissonian regime, trie
level spacing statistics are independent of trie precise form of trie interaction parameters.
l. Introduction
The statistical distribution of energy level spacings lias been recently studied in several strongly
correlated fermion systems [1-4]. One potentially very interesting conclusion is the preseuce of energy level repulsion iii some parameter domain of the two-dimensional t-J model [1], wuicu is one of the most investigated models of strongly correlated fermions. This behàviour is
interesting since most integrable systems exhibit by contrast an uucorrelated distribution ôf energy levels. This property lias first been proved in the context of semi-dassical quantization of dassically integrable systems with a finite number of degrees of freedom [6]. It bas also been
found in a broad Mass of exactly solvable models in one dimension, such as spin 1/2 chains [2]
or the Luttiuger model [4]. In more than one spatial dimension, it is much more diflicult to
construct integrable interacting fermion models. This paper is the second one devoted to the
study of level spacing statistics of integrable quantum liqmds. Non integrability effects iii trie Fermi liquid shall be studied in a forthcoming paper. Trie one-dimensional Luttinger liqmd bas already been analyzed in reference [4], in connection with trie breakdown of trie Fermi liquid behaviour as trie g2 interaction is switched on. Iii this case, trie spectrum was treated in terms of trie bosonic excitations, which generate trie q # 0 part of trie Hilbert space. We now give trie main idea of trie present paper.
A normal Fermi liquid in trie vicinity of its ground state is descnbed by a set of elementary
excitations (trie Landau quasipartides) which are in one to one correspondence with trie Fermi gas excitations. Landau lias shown that trie energy of a configuration is given by
Hiiôn~ii
= ~ e~ôn~ + j ~ J~~,ôn~ôn~> +..
,
ii)
k (kk,)
Q Les Editions de Physique 1995
where only the first two terms are relevant at low temperatures. D is the spatial dimension
(1 or 2). The system is in a cubic box of linear size L. Only the spinless case is studied in this paper, ônk is the difference between the single quasiparticle occupation numbers in the
given configuration, and in the ground state. ônk
" 1 for a quasiparticle, and ônk " -1 for a
quasihole. The renormalized dispersion e(k) is
flk) ~2
= ~, (2)
where m denotes the effective mass. The ground state is obtained by filling ail the states under the Fermi level eF
" k)/2m. We consider the energies given by (1) as the exact eigenvalues of
a toy model Dia Fermi liquid, for which the eigenstates are labelled in the same way as for the Fermi gas. We study the statistics of level spacings for a Due- and twc-dimensional phase space.
However, the one-dimensional case can only be seen as a toy model since the Fermi liquid theory
breaks down in one dimension. The Hamiltonian il) is a quadratic form of trie quasiparticle
and quasihole occupation numbers, and is obviously integrable. The remaining correlations may only originate from a special form of the interaction functions fk,k, which generate the
spectrum. The aim of this article is to check that the interaction between quasiparticles induces
a cross-aven towards a tandem level spacmg statistics, and to study the cross-over as a function
of the strength of the interactions, the system size, the temperature and the dimensionality.
In a first step, we analyse the level spacing statistics of one-dimensional toy models for Lan- dau Fermi liquids. This gives us the possibility to compare the behaviour Dia one-dimensional liquid of fermionic Landau quasiparticules and the behaviour of a Luttinger liquid [4] which has bosonic excitations. The technical difference between the present paper and the study of
Luttinger liquids is that the one-dimensional Fermi system is treated here directly in terms of fermions, whereas we used bosonic excitations m [4]. In the case of a Fermi surface consisting
of two points, one is left with a collection of infinite size Hilbert space sectors, labelled by
the total momentum. By contrast, the sectors of given momentum have a finite size for a one branch model. The momentum of the single partiale states are restricted ta the range kF -1 ta kF +1, 1 being a cut-off on the momenta. The level spacing statistics are analyzed in this regularized Hilbert space HA. Within this approach, we analyze the effects of a quadratic
correction ta the linear Luttinger liquid dispersion relation on the level spacing statistics. The statistics are found ta be smgular, and depend on the magnetic flux through the ring. As far as the one-dimensional Landau liquid is concerned, the quasipartide interaction function is taken
as
fk,k> " Vfl)lk k'l), 13)
where R is the range of the interactions, V their intensity, and f a decreasing function. If the intensity is given, a cross-over is found as the range mcreases, ta a level spacing distribution
slightly different from the expected Poisson distribution. The atm of the one-dimensional toy model is ta specify the cross-over scales to the Poisson distribution level spacing statistics. The level spacing statistics of the integrable 1/r2 Haldane-Shastry spin chain are analyzed as well.
The Hamiltoman has a Landau form m terms of the semiomc excitations [5].
As far as two-dimensional Landau hquids are concerned, the quasipartide interaction func- tion fk,k, has been assumed ta be of the form
fkk, " A(1 + B cas Ù(k,k')), (4)
where Ù(k, k') denotes the angle between k and k'. The statistics are found ta be Poissonian for sulliciently large parameters A and B.
2. Level Spacing Statistics of Fermions on a Non-Linear Dispersion Relation 2.1. INTRODUCTION. We begin with the study of one-dimensional models. As mentioned in the introduction, the study of the one-dimensional Luttinger liquid fixed point has already been done in [4]. It should be interesting ta compare the case of a one-dimensionnai Hamiltonian of the Landau form, with interacting fermionic quasiparticules. This shall be done in the
next section. Up ta now, we consider only free fermions, and study the effects of a non-linear
dispersion relation on the level spacing statistics. The standard treatment of the excitations of a Fermi gas in the vicinity of its Fermi surface involves the linearization of the dispersion
relation in the vicinity of the Fermi surface. In this case, the excitation spectrum is made up of equidistant degenerate levels and thus leads to singular level spacing statistics. We address the
question of what are the level statistics if one incorporates non-linear terms in the dispersion
relation. In the case of the parabolic free electron dispersion relation, one obtains a second order deviation to the linearized dispersion relations, which should modify the high energy
excitation spectrum. If the deviation to the dispersion relation is a polynomial with a low degree, the energy spectrum depends only on a small number of independent parameters. For instance, in the case Dia parabolic dispersion relation, the excitation spectrum depends only
on two parameters, the mass and the position of the Fermi level. One must moreover add the constraint that the mean value of the level spacing is normalized to unity. Since only a
few parameters enter its composition, the spectrum is correlated in a non-universal way, which depends on the detailed value of the parameters. If the degree of the polynomial defining
the deviation to the dispersion relation mcreases, one expects that the level spacing statistics should transit to a universal Poisson regime. We did net test this assertion in the present paper, but an example is provided in [4], where we studied the level spacing statistics of a model of two coupled Luttiger liquids [11]. The dispersion relation has the form
f-iQ)
= )i~p ~«)Q i)i~p ~«)Q)~ + 4ti, là)
where ltp andlta are the charge and spin velocities, respectively, ti is the transverse hopping.
One can develop the square root of là) in powers of q(, where the length scale ( is defined by ( = (ltp -lta /4ti, and obtain the deviation to the linear dispersion relation (ltp -lta )q/2 2ti
as a serres m powers of q(. In the regime q( » 1, the number of terms required to describe
the dispersion relation as a series in q( is high. By contrast to the case of a dispersion relation depending only on a small number of pararneters, the transition to a universal level spacing
statistics thus requires a higher number of independent parameters. If they are numerous enough, the level spacing statistics are net influenced by trie detailed value of the parameters, but fait in a universal regime, described by trie Poissonian case of random matrix theory.
2.2. REGULARISATION OF THE HILBERT SPACE. Trie dispersion relation is right-left sym-
metric e(k)
= fi-k). The boundary conditions are periodic. The Hamiltoman reads
Hi "
~j ufakAkônk + ~j ôe(Ak)ônk. (6)
k k
The first term is the Luttinger gas Hamiltonian, with VF
" de(kF)/dk the Fermi velocity. k is
a multiple of 21rIL, and runs over the momentum of the excitations, ak " +1, -1
= R, L and
Ak = k akkf. The second term contains non-linear deviations to the linearized dispersion
branches. The spectrum of the Luttinger gas is made up of equidistant degenerate levels, with separations equai ta 2~uF/L, ieading to a singular level spacing statistics. We analyze the
statistics as the deviation from the Luttinger model dispersion relation ôe(Ak) increases. The non-linear terms in the energy are expected to randomize the Luttinger gas spectrum, and the randomization is expected to inciease with the energy. The dimension of a sector of the two-branch model with a given momentum is infinite m the absence of a momentum cut-off,
and is obtained as a sum over the infinity of direct products of the right and left sectors with moménta kR> kL such as k
= kR + kL. The Hilbert space is regularized in the followmg way:
only one branch is taken into account, and a momentum cut-off1is introduced, restricting
the particle momenta between 21ruFIL and À, and the hales momenta from -À + 21ruFIL to 0.
In the sonne way as for the twc-branch model, sectors with different momenta are superposed,
leading to a subspace HA containing (2À)!/(À!)2 states. The spectrum shall be denoted Sp>.
In the absence of a deviation ôe(Ak), the spectrum shall be denoted Sp(, and is in the energy range 0 to À~LUF/21r. Sp( has the property that Sp( C Sp(> if1 < 1'. Sp( is exact only for energies lower than VF1, but states are missing for energies greater than VF1. The partide-hale
symmetry transforms each level ES into a level l~LuF/21r ES which belongs to the spectrum,
so that the density of states p>je) reaches its maximum value for the energy l~LuF/41r if lL/21r
is even, and
~)VFII~))~ + l) Ii)
if lL/21r is odd. The density of states p( is plotted in Figure 1, and is numerically found to be Gaussian around its maximum value. The density of states pie) in the limit 1 - +oo is
computed recursively from the bosomc basis [4] and is plotted in Figure 2. The asymptotic
@fi
4
ÀL/2~
= 12
ÀL/2~
= 13
3 _5 ÀL/2~ = 14
3
2. 5
2
1. 5
1
o. 5
0
0 o-1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fig. 1. Density of states of trie truncated spectrum Sp(. The density of states p>(E) was computed
for AL/2x
= 12,13,14. ~/log(pmax/p>(E)) is plotted as a function of 2~E/VFÀ~L. Trie density of states bas a Gaussian shape around its maximum value.
j)p>(EjdE
4e+06
3. 5e+06
/, ÀL/2~ = là
,' ?
3e+06 1"
/ ÀL/2~ =14
/
2. 5e+06 / ;" ÀL/2x
= 13
j~ ,./ ,'
2e+06
jf~"""
,/
,." AL/2x
= 12
1, 5e+06 ,/ ;.'
,"
,,' ,,,"
,," _:"
le+06 ~~,?'- ~" ,,
~'j_ ~jÎS'~~
500000
""'
_,,
-''''"''
-'''''
0 e
40 45 50 55 60 65 70 75 80
Fig. 2. Integrated density of states of trie truncated spectra conlpared ta integrated density of
states in the absence of
a cut-off. Trie integrated density of states corresponding ta the spectra Sp( is plotted for ÀL/2x = 12,13,14, là, as well as trie integrated density of states p°(E) calculated fronlthe boson basis. Trie energy variable is EL/2xvF.
form of the density of states is [4]
so that the density of states p°(e) of the restricted spectrum Sp( converges non-uniformly to
p°je). The level spacing statistics were computed only in the low energy part of the spectrum
Sp(, for the first slices of about one million levels, as shown m Figure 2.
The cut-off1 may be seen as a temperature in the following way. The probability of thermally activated excitations is given by the Fermi distribution function. The energy scale associated to the vanishing of the Fermi distribution function in the vicinity of the Fermi surface is kBT,
with kB the Boltzman constant. The cut-off1 may thus be seen as an effective temperature T
through the relation T = VF1/kB.
2.3. RANDOM CORRECTIONS TO THE LINEAR DISPERSION RELATION. We first analyze the
generic case of a random correction to the linear dispersion relation. One is left with only one parameter: the amplitude of the random deviations compared to the interlevel spacing of the
Luttinger gas, equal to 21ruFIL. However, each level is decoupled from each other since there is no deterministic dispersion relation. The level spacmg statistics are thus expected to transit to a Poissonian level spacing statistics as the amplitude of trie random corrections increases.
The individual fermionic energy levels are chosen to be e(Ak)
= VF(k kF) + AOR(Ak), where
R(Ak) is chosen randomly between 0 and 1, with a constant density, and Ao controls the
amplitude of the random deviations. The statistics are studied in the complete Sp> spectrum.
F(s)
4
Ao 0 05
Ao ù-1
3.5 Ao=0.25
PIS)
= exp --s
3
2.5
2
1.5 r
1
o.5
~ s
0 0.5 1 1.5 2 2.5 3 3.5 4
Fig. 3. Level spacing statistics for linear randon1corrections ta trie Luttinger gas dispersion
relation. Trie length L of trie ring is taken equal ta 2x, and trie Fermi velocity equals 1. Trie cut-off
on trie nlomentum of trie fermions is chosen equal to = 11, which generates a Hilbert space of size
705432. Trie statistics are plotted for vanous values of trie amplitude Ao of trie deviation ta trie linear dispersion relation, corresponding ta Ao
= o.05, o.1, o.25. As expected, trie statistics converge towards
a Poisson distribution. Trie cross-over is controlled by trie value Ai = 0.091.
If Ao is non-zero, levels origmating from the degeneracy k are distributed in the energy interval
[ufk, ufk + AOÀL/21r]. The level spacing statistics shall change over to a Poisson distribution
as trie width of the interval goes to 21ruF/L, narnely if Ao > Ai, with
~~ Î ~Î~~'
~~~
The scale Ai is numerically found to control trie cross-over ÉD a Poissonian level spacing dis- tribution, as shown m Figure 3.
2.4. PARABOLIC DISPERSION RELATION
2.4.1. Cross-Over Scales. We now discuss trie level spacing statistics corresponding ÉD trie
dispersion relation (6), ôe(Ak) being a quadratic deviation to trie linear dispersion relation
ôe(Ak)
=
~~~~~. (10)
Non-universal effects are expected ÉD emerge. In order to build trie spectrum, we use trie
regularisation procedure which is described in trie previous section. Due to trie invariance
e(k + Ak)
= e(k) + uFAk of trie hnearized dispersion relation, trie excitation spectrum is
parametrized by trie single pararneter VF- In the presence of quadratic corrections, one has ta additionally specify the Fermi level kF. In this section, the mdependent quantities which
parametrize the parabohc branches are the mass m and the Fermi velocity VF- This choice makes it possible ta switch on the curvature continuously with different choices of the mass m,
while keeping the Fermi velocity constant. The spectrum Sp( in the case of an infinite mass has been discussed in the previous section. Let E° be the energy of a given degeneracy in
Sp>, in the absence of a quadratic term. The effect of the quadratic deviation is to split the
degeneracy. Trie resulting level spacing statistics are however net Poissonian. Trie quadratic
energy term is expected to play a rote only at high enough energies. Moreover, trie lower bound
on trie momenta bas to be positive, for trie excitations to belong to the same right-moving
branch. This condition reads: kF > 1- 21r/L, which restricts the study to masses greater than m)
= il 21r/L)/uF. In order to estimate the energy cross-over, we first calculate the energy scale W(E°) associated with the splitting of a given degeneracy E°, for a finite mass.
The configurations with the maximum and minimum deviations shall be respectively noted
((+),n, ai and ii-), n,À). The excited state ((+),n, ai contains n hales, with wave vectors from kF down to kF 21r(n -1) IL, and n partides in trie highest possible excited particle states, with wave vectors from kF +1- 21r(n 1)IL up to kF +1. Trie contribution of trie
quadratic order terni to trie energy of trie excited state ((+), n,1) reads
AE(+)
=
)(1- )(n -1)). (Il)
The excited state ii-), n,1) contains n hales with wave vectors frein kF -1+ 21r/L up to
kF -1+ 21rnIL, and n particles with wave vectors frein kF + 21rIL up to kF + mn/L, with the
following contribution of the quadratic order terni to trie energy
AE(
=
-)(l )(n +1)). (12)
The levels with an energy E$ in the absence of a quadratic correction are spread on an energy interval of width W)"~
W)"~ = AE(+~ AE(
=
~~ il )n). (13)
If lL/21r is even, the maximum of the width W)"~ equals l~L/81rm, and is reached for a number of partide-hale excitations n equal to n = lL/41r. If lL/21r is odd, the maximum width is reached for
n =
~~
+ 1), (14)
2 21r and equals
à~Î~~~Î~~ ~~' ~~~~
The maximum in the density of states p((e) is found to correspond to the maximum in the
dispersion of the levels coming from a given degeneracy. Equation (3) is exact only for the
degeneracies of the form E(
= VF an. We use it to find an approximation of the expression for trie width W(E°) for an arbitrary energy, net necessarily of trie form E(
= ufIn. We get
w>lE°)
=
fil> ) $). l16)
The influence of trie quadratic term on trie level spacing statistics shall be significant provided WA(E°) > 21ruFIL, which leads to trie expression of trie cross-aven energy
E(~ =
~~~~~
(1- ~).
(17)
41r
Note that this energy is proportional to 21rmu)/ÀL if muf < À~L~.
