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Small-cluster study of interacting fermions on a square lattice with next-nearest-neighbour interactions
Physica A 183 (1992) 209-222 North-Holland
B
L
~
Small-cluster study of interacting fermions on a square lattice with next-nearest-neighbour interactions G e o r g e S. K l i r o s Department of Physics, University of Patras, Patras 26110, Greece Received 12 June 1991 We calculate the ground state energies and wavefunctions for small systems of interacting fermions on a square lattice with cylindrical topology. We examine the role of next-nearestneighbour interactions on the observed ground states: the fractional quantum Hall ground state and the super-conducting ground state. Several different probes of fermion-pairing and off-diagonal long-range order as flux quantization, binding energies, pair-correlation functions and critical external magnetic field are investigated. Antiperiodic boundary conditions imposed in one direction provide a test of our numerical results within current size limitations.
I. Introduction T h e r e is currently considerable interest in the study of strongly correlated fermions on a square lattice in a magnetic field because of its importance in various fascinating topics as the fractional quantum Hall effect ( F Q H E ) [1], the c o m m e n s u r a t e flux states in the t - J model for the high t e m p e r a t u r e superconductors [2], anyon superconductivity [3], etc. T h e single-particle spectrum of tight-binding fermions on a square lattice with periodic boundary conditions in both directions (toroidal geometry) in a magnetic field shows surprising features such as self-similarity, nesting properties and a Landau-level-like structure, which have been studied by several authors [4, 5]. W h e n the magnetic flux per plaquette, 4 , is a rational n u m b e r , p / q , the spectrum has q subbands with well-defined gaps. It has been shown [6] that each subband carries an integral Hall conductance and the integer has a topological origin. O n e of the important consequences of the structure of the spectrum is the stabilization of the Fermi sea by the gaps at every rational value of magnetic flux. If the n u m b e r of electrons per site is fixed, the total energy of fermions exhibits cusplike minima when the Fermi energy jumps 0378-4371/92/$05.00 (~ 1992- Elsevier Science Publishers B.V. All rights reserved
210
G.S. Kliros / Interacting fermions on a square lattice
across a gap as a function of magnetic flux per plaquette, and a sharp global minimum appears exactly at plus or minus one flux quantum per particle [7]. A linear proportionality between the lattice filling and the optimal flux has been shown with high accuracy [8]. This result is closely connected with the mean-field approximation for the t - J Hamiltonian [9]. The main features of the above spectrum survive if we consider periodic conditions in one direction and hard walls in the other (cylindrical geometry). Although this geometry allows continuous variation of the magnetic flux through the plaquettes and is suitable for testing flux quantization, the average degeneracy of the energy levels is reduced and pronounced size effects may appear [10]. Exact diagonalization techniques have proved to be very powerful in studying the difficult problem of interacting particles giving partial but useful answers to interesting questions like the influence of the attractive or repulsive interaction on the nature of the ground state of the system and the effective interaction between two particles. A pioneering small-cluster study of anyons (including as a special case the fermions) has been done by Canright, Girvin, and Brass [11]. They found that as a function of statistics and applied magnetic field the ground state of "free" anyons on a square lattice with cylindrical symmetry could be either super-conducting or quantum-Hall-like. They also found from the ground-state energy versus the threading of the cylinder flux diagram that there exists a paired state for interacting fermions and semions. Also, the validity of the vector-mean-field theory for statistical transmutation has been tested using both exact diagonalization and Monte Carlo methods [12]. Recently, Kliros and d'Ambrumenil [13] calculated the ground-state energies and wavefunctions for small systems of fermions on a square lattice in the presence of an external magnetic field using singly and doubly periodic boundary conditions. They identified fractional quantum Hall states on a lattice from the minima appearing in the ground-state energy when the magnetic flux per plaquette corresponds to filling fractions with odd denominators. They also found that the minima for two systems with the same flux density and lattice, one with cylindrical and the other with toroidal topology, closely resemble each other when the number of particles per plaquette is the same in the two cases. In this paper we present a systematic numerical study of the properties of tight-binding interacting fermions on a square lattice with cylindrical topology including up to next-nearest-neighbour (NNN) interactions. Repulsive interparticle interactions lead to FQH states while attractive interactions lead to electron pairing and existence of an off-diagonal long range order (ODLRO) for a large range of the interaction parameters. We look for independent probes of possible "superconducting" pairing between fermions and examine the effect of NNN interactions on them.
G . S . Kliros / Interacting f e r m i o n s on a square lattice
211
The paper is organized as follows. In the next section we present the model and describe some features of the non-interacting case such as the form of the single-particle spectrum and the existence of an O D L R O in the corresponding free hard-core bosons system. In section 3 we examine the effect of NNN interactions on the FQHE-Iike minima found in ref. [13]. A numerical test of the f e r m i o n - b o s o n mapping is performed as a further check of the occurrence of F Q H states. Several signatures for binding of fermions and O D L R O in the case of attractive interactions and the effect of NNN interactions are discussed in section 4. Section 5 contains a review of our numerical results.
2. Model Hamiltonian and free-particles' energy spectrum We study numerically the ground state energy of small systems of interacting spin-polarized fermions on a square lattice considering nearest-neighbour (NN) hopping and up to next-nearest-neighbour (NNN) interparticle interactions. We apply periodic boundary conditions in the x-direction keeping hard walls in the y-direction so that our system is topologically equivalent to a cylinder. The system is minimally coupled to a magnetic field parameterized by a flux per plaquette qo and a " t e s t " flux @t is considered as in ref. [11]. The fluxes are in units of the magnetic flux quantum 40 -= hc/e. We work with the following Hamiltonian on a two-dimensional square lattice: Y( = - t ~ ei°qc~cj q- h.c. + U 1 ~] nin ] + U 2 Z ninj, (1) (ij)
(ij)
[ij] t
where c i are the usual fermion operators on the lattice, n i = ci ci is the particle n u m b e r operator, and (ij) and [ij] stand for NN and NNN bonds of the particular lattice, respectively. The parameters U 1 and U 2 give the strength of NN and NNN interaction, respectively. The phases 0,j defined by ]
Oij=-e
(A + A t ) ' d l ,
(2)
i
where A and A t a r e vector potentials that correspond to flux through the plaquettes and " t e s t " flux, respectively, are distributed along the links of the lattice so that the total flux through each plaquette is the same modulo an integer. We have diagonalized the Hamiltonian (1) numerically using the Lanczos technique [14]. In this paper we present results for a 4 × 4 lattice. Less than 50 Lanczos steps were necessary to obtain the ground state with enough accuracy
212
G.S. Kliros / Interacting fermions on a square lattice
(1 part i n ~ 1 0 4 ) . Independent runs have been carried out starting from a trial state corresponding to different configurations in order to be sure we have obtained the true ground state. The single-particle energy spectrum of eq. (1) for the non-interacting case (U~ = U2 = 0) as a function of magnetic flux per plaquette @ is shown in fig. 1. The ground-state energy for N non-interacting particles is just the sum of the N lowest single-particle (kinetic) energies which can be calculated exactly. Due to the existence of well-defined gaps, the ground-state energy as a function of magnetic flux per plaquette @ exhibits deep local minima at certain filling fractions as has already been observed for the case of doubly periodic boundary conditions [7]. Considering q0 = 0, we can also obtain the behaviour 6
L?~':Z/. i
•-o °
°° ,°
.,°
o-°-:~
° ."
Ooo °° °
..~°o.
%
•...oo"
,-*.~
~
".°. oO.
-6 0.0
0.2
0.4
0.6
0.8
1.0
e Fig. 1. Single-particle spectrum o f fermions on a 4 x 4 lattice with cylindrical symmetry in the presence o f a magnetic field parameterized by a flux per plaquette ~.
3 >_ 2 n" I u.J Z 0 l.U -I -2 -3 -6
0.0
0.2
0.4
0.6
®,
0.8
1.0
Fig. 2. Single-particle energies of non-interacting fermions on a 4 x 4 lattice with cylindrical symmetry versus "test" magnetic flux @, through the bore of the cylinder.
213
G.S. Kliros / Interacting fermions on a square lattice
o f the single-particle s p e c t r u m as the " t e s t " flux @t varies. T h e spectrum shows several level crossings and is invariant u n d e r changing q~, by an integral n u m b e r of flux q u a n t a (fig. 2). F u r t h e r m o r e , as test of o u r numerical calculations, we investigated the g r o u n d - s t a t e e n e r g y o f free h a r d - c o r e ( H C ) b o s o n s in zero external m a g n e t i c field (@ = 0). H C b o s o n s are superfluid and show an O D L R O in the o n e - b o d y density matrix. In fig. 3 we show the g r o u n d state e n e r g y of H C bosons on the 4 x 4 lattice with cylindrical s y m m e t r y as a function of flux @t threading the cylinder for different n u m b e r s o f b o s o n s N. It is periodic with a p e r i o d - o n e flux q u a n t u m , which is a characteristic p r o p e r t y of a charge e superfluid. In fig. 4 we have d r a w n the p e a k s of the e n e r g y barrier E ( @ ) - E(0) as a function o f 2.8
8
N=? 2.4 2.0 1.6 1.2 0.8 0.4
0.0 0.0 0.1 0.2 0.3 0.4 0.5 0,6 0,? 0.8 0.9 1,0 MAGNETIC FLUX @t
Fig. 3. Ground-state energy as a function of the "test" magnetic flux @, for non-interacting bosons on a 4 × 4 lattice with cylindrical topology. 2.75 '~2.50 L.I.I
• °'°
~
""~"°
h
"'~
2.25 /
/
z /
2. O0
\
l /
I . 75
%,,•
/
I . 50 3
|
i
,
,
,
,
4
5
6
7
8
9
i
I0
|
II
12
13
N Fig. 4. Scaling behaviour of the ground-state energy maxima with particle number for bosons in the 4 x 4 lattice. Note the deviation from linear behaviour near half-filling.
214
G.S. Kliros / Interacting fermions on a square lattice
n u m b e r of particles N. The observed scaling behaviour is a clear evidence of O D L R O in the system. Of course, in the thermodynamic limit the energy barrier between two adjacent minima of E(q~) is infinite. The deviation from scaling behaviour near half filling is attributed to lattice effects and it seems to be independent of the imposed boundary conditions since it has also been found in calculations on a sphere [15].
3. Repulsive interactions: fractional quantum Hall states The most widely accepted theory of the F Q H E is based on Laughlin's wavefunction [16] for filling factors 1/q and q an odd integer and associated hierarchical models [17] for the other fractions with odd denominators. The "binding" of q flux quanta to each particle gives rise to q zeros in the wavefunction as a function of the relative coordinate of any pair of particles. This reduces the probability that any two particles approach each other closely and the interaction energy for any short-range repulsive interaction between particles is therefore lowered [18]. The importance of zeros in the wavefunction as a function of the relative coordinate of any pair of particles to the short-range behaviour is less obvious for particles on a lattice, whereas all the "coarse grained" properties identified as important in the Laughlin theory should be well reproduced. We should notice that on a lattice analytic wavefunctions in the presence of a magnetic field do not exist. The restriction to the lowest Landau level, which requires the wavefunction to be an entire function of the particles' complex coordinates, does not apply on the lattice and one expects lines of zeros rather than "point" zeros [12]. In a recent small-cluster study of interacting fermions on a square lattice with NN interactions [13], F Q H states were identified from the deep minima developed in the ground-state energy as a function of the filling factor determined by the ratio of the number of particles per plaquette divided by the n u m b e r of flux quanta per plaquette. More specifically, one can define the quantity fint = E g s -
(~gs[~kinl~gs),
(3)
where Eg s is the ground state energy and ~kin is the hopping part of the Hamiltonian. The variation of Eg s due to the single-particle band structure has been removed from Ein t making the variation of the contributions from the interaction energy more apparent. It has been found [13] that, considering only NN interactions, Ei. t exhibits
G.S. Kliros / Interacting fermions on a square lattice
215
d e e p m i n i m a at filling factors with o d d d e n o m i n a t o r s for a large range of the interaction strength U 1 > - 3 t . T h e s e m i n i m a give a clear evidence that F Q H states o c c u r in small systems on a lattice. A n interesting question addressed h e r e c o n c e r n s the role of N N N interactions with respect to the o b s e r v e d F Q H E on a lattice. We have t h e r e f o r e calculated the quantity Ein t by exact diagonalization of Y( (eq. (1)) for repulsive N N and N N N interactions (U~, U 2 > 0) on a 4 x 4 lattice. W e f o u n d that for U 2 < 0.2U1, the m i n i m a in Ein t a p p e a r at the same filling fractions as without N N N interactions. W h e n U 2 is increased a b o v e 0 . 2 U 1 the m i n i m a m o v e t o w a r d s the filling factor v = 1 / 2 and for U 2 - 0 . 5 U ~ the F Q H like m i n i m a have b e e n destroyed. Fig. 5 shows Ein t versus filling factor for a 3 / 4 x 4 lattice (three fermions on a 4 × 4 lattice) for UI = 5t and U 2 = 0.1U~ = 0.5t. F Q H states are identified at 1, = 4 / 7 and v = 4 / 9 . T h e r e are also local m i n i m a at v = 3 / l l and ~ , = 3 / 1 3 . W h e n the N N N interaction strength is increased to U 2 = 0 . 5 U 1 = 2 . 5 t , the sharp F Q H E m i n i m a have disappeared and m i n i m a are o b s e r v e d at the special fillings ~, = 1 / 2 and v = 1 / 4 as is shown in fig. 6. We can say that in the a b o v e regime of competitive interactions the n a t u r e o f the g r o u n d state has b e e n changed. T h e new m i n i m a might be a reminiscent o f the lattice effect showing a qualitative difference of o u r app r o a c h f r o m a c o n t i n u u m calculation [19]. So far we have discussed only one p r o b e of F Q H states on a lattice; the m i n i m a a p p e a r e d in the " e f f e c t i v e " interaction Ei, t as we sweep c o n t i n u o u s l y the m a g n e t i c flux and hence the filling factor v. A n o t h e r p r o b e of the F Q H E g r o u n d states is the overlap of the f e r m i o n g r o u n d state wavefunction with that o f free b o s o n s in zero magnetic field. L a r g e overlap implies a singular O D L R O
1.0
_/
•+-, 0 . 8
t.UO. 6
I I
I I
4,j
0.4
I I
4,/, I I
0.2
0.0 0.2
0.4
0.6
0.8
1.O
V
Fig. 5. Ground-state energy on a 3/4 x 4 lattice with cylindrical topology as a function of filling factor v. U~ = 5t and U2 = 0.5t.
216
G.S. Kliros / Interacting fermions on a square lattice
1.2
e
~.
,,
~0.8 i,m 0.6
0.4 0.2 0.2
0.4
0.6
0.8
1.0
V
Fig. 6. Ground-state energy on a 3/4 × 4 lattice as a function of filling factor ~,, U1= 5t and U2 = 0.5U1. for the corresponding ground state [20]. This is well connected with the b o s o n - f e r m i o n mapping: fermions can be treated as bosons with attached "fictitious" flux tubes. Since at v - - 1 / q there are q flux quanta available to each electron, we can m a k e a mean-field approximation substituting the flux tubes by an average "statistical" flux giving a statistical angle 0 = q~r [21]. Then the statistical flux cancels the external magnetic flux leaving bosons in zero magnetic field which condense and show long-range order [22]. Investigating the minima of the quantity Ein t v e r s u s the external magnetic field on a 3/4 x 5 system, we are able to identify the lattice analog of v = 1/3 and v = 1/5 F Q H ground states. We found significant overlaps ( - 0 . 9 2 ) between these fermion ground states and the corresponding hard-core boson ground state in zero magnetic field but this probe of F Q H states seems to be lattice-size dependent.
4. Attractive interactions: superconducting pairing and ODLRO
The purpose of this section is a systematic study of the model Hamiltonian 9( (eq. (1)) for the case of attractive interactions and especially the investigation of the role of N N N interactions. Several signatures of a superfluid ground state consisting of pairs of particles on our finite lattice are discussed. Canright et al. [11] studied small systems with only NN interactions. They argued, based on the ground-state energy versus test-flux diagram, that fermions on a square lattice with cylindrical geometry show pairing in some p a r a m e t e r range for N N interparticle interaction.In the present study, we have diagonalized Hamiltonian eq. (1) considering • = 0, U 1, U e < 0, for different
G . S . Kliros / Interacting f e r m i o n s on a square lattice
217
values of the test-flux ~t- We present results for the system 10/4 × 4, which can also be viewed as having six holes. As a first probe of superconducting pairing one can consider the periodicity of the ground-state energy Eg S sweeping the test-flux q0 through the bore of the cylinder and the flux quantization defined by the occurrence of equally spaced minima in the structure of Egs versus q~t. We found that well-defined minima in Eg s versus ~t survive for the values of NNN interaction parameter in the range of interest 0 < U 2 <~ U~/2. Fig. 7 shows periodicity of Egs vs. ~t with period • o/2, which is the flux quantum of a paired state in a BCS superconductor. The observed flux quantization implies that there exists a coherent, paired state with O D L R O . As is well known [23], mixed or other variable boundary conditions are generally useful in checking the importance of finite-size effects. To ensure that the above flux quantization curves are not a finite-size result, we also performed calculations with antiperiodic boundary conditions. We imposed antiperiodic boundary conditions in the x-direction replacing t by - t on certain bonds along the x-direction and changing all hopping integrals by a proper phase factor in order to preserve translational invariance along this direction. A typical curve of our calculations for U 1 = - 3 . 0 t and U: = 0.2U 1 = - 0 . 6 t is shown in fig. 8. We observe that both periodicity in the ground-state energy versus flux and well-defined minima at multiples of q00/2 persist. Another question to be addressed is whether the observed quantization "survives" in the presence of an external magnetic field. We have therefore diagonalized Hamiltonian eq. (1) for • ~ 0. We found that flux quantization persists only in very small external magnetic fields. As is shown in fig. 9, an -39.8 e) -40.0 L~J -40.2
-40.4
-40.6
0.0
0.2
0.4
0.6
0.8
1.0
e, Fig. 7. G r o u n d - s t a t e e n e r g y of f e r m i o n s o n t h e 4 x 4 lattice v e r s u s " t e s t " m a g n e t i c flux ~t for U 1 = - 2 . 5 t a n d t h r e e d i f f e r e n t v a l u e s of N N N i n t e r a c t i o n s : (a) U z = 0.0, (b) U z = 0 . 2 U 1 a n d (c) U 2 = 0 . 4 U 1. T w o o f the c u r v e s (b a n d c) are shifted in e n e r g y so as to lie in the s a m e d i a g r a m .
218
G . S . Kliros / Interacting f e r m i o n s on a square lattice -40. I -40.2 i.~ - 4 0 . 3 -40.4
-40.5 -40.6 0.0
0.2
0.4
0.6
0.8
1.0
Ot Fig. 8. G r o u n d - s t a t e e n e r g y v e r s u s " t e s t " flux • t for f e r m i o n s on a 4 × 4 lattice with a n t i p e r i o d i c c o n d i t i o n s i m p o s e d o n the x - d i r e c t i o n . E q u a l l y s p a c e d m i n i m a o c c u r for U L = - 3 t a n d U 2 = 0 . 2 U , .
-45.9 -46.0 -46. I -46.2 -46.3 -46.4 -46.5 0.0
i
i
i
0.2
0. 6,
0.6
i
®,
0.8
1.0
Fig. 9. G r o u n d - s t a t e e n e r g y v e r s u s " t e s t " flux O, for f e r m i o n s o n a 4 x 4 lattice in the p r e s e n c e of a n e x t e r n a l m a g n e t i c flux • = 00/2. UI = - 2 . 5 t and U 2 = 0 . 2 U , .
external magnetic flux • = 00/2 is sufficient to destroy the " p r o p e r " minima in the structure of the ground-state energy. This observation is a further indication of O D L R O . In the following, we deal with a second signature of superconducting pairing, the binding energy of two fermions defined by EB, 2 = [ E g ~ ( N - 2 ) - E g ~ ( N ) ] - 2 [ E g s ( N -
i) - E~(N)I,
(4)
where Egs(N ) is the ground state energy of N fermions. We can easily see that EB,z represents the effective interaction between
G . S . Kliros / Interacting f e r m i o n s o n a square lattice
219
particles: If we imagine the infinite system consisting of a collection of finite clusters, eq. (4) gives the difference b e t w e e n the state with two particles a d d e d in the same cluster and the o n e in which fermions are a d d e d to different clusters. W h e n EB, 2 is negative, the lower-energy state is that in which the two extra f e r m i o n s w e r e a d d e d to the same cluster and, hence, we can interpret EB, 2 as an attractive interaction b e t w e e n them. We must, h o w e v e r , distinguish " s u p e r c o n d u c t i n g " pairing from phase separation or clustering o f particles. We have therefore also calculated the binding e n e r g y of four fermions defined by EB, 4 -~ [Egs(N - 4) - Egs(N)] - 2[Egs(N - 2) -
Eg,(N)I
(5)
as well as the binding e n e r g y o f three fermions with respect to splitting into a pair and a single-particle, defined by EB, 3 = [Egs(N - 3) - Egs(N)] - [Egs(N - 2) - Egs(N)]
(6)
- [ E g s ( N - 1) - Eg~(N)].
B o t h EB, 3 and EB, 4 w o u l d be negative if phase separation occurs. Superconducting pairing is t h e r e f o r e indicated by E B , 2 being negative and EB, 3 and EB, 4 being positive. O u r results concerning EB, 2 and EB, 4 for the 10/4 × 4 cluster with and w i t h o u t N N N interactions are shown in table I. It is seen that f e r m i o n pairing a p p e a r s only with a well-defined " w i n d o w " of N N interaction p a r a m e ter and this " w i n d o w " is b r o a d e n e d in the presence of N N N interaction. Also, N N N interaction decreases EB, 4 for all the values of N N interaction. In the a b o v e range, all the values of EB, 3 have b e e n f o u n d to be positive. A n o t h e r i m p o r t a n t quantity concerning two-particle interactions is the pair-
Table I Pair-correlation functions on a 4 x 4 lattice with cylindrical topology for several values of NN interaction U1. The strength of NNN interactions has been taken U, = 0.2U1. U~/t
R = 1
R = V~
R =2
R = V-5
R = 2V'2
0.0 -0.5 - 1.0 - 1.5 -2.0 2.5 -3.0 -3.5 -4.0
0.6039 0.6377 0.6848 0.7558 0.8467 0.8560 0.8612 0.9139 0.9188
0.5749 0.5998 0.6355 0.6864 0.7227 0.7290 0.7332 0.7459 0.7472
0.3890 0.3851 0.3807 0.3750 0.3969 0.3968 0.3963 0.3872 0.3852
0.6703 0.6776 0.6816 0.6726 0.6666 0.6644 0.6628 0.63ll 0.6294
0.1890 0.1864 0.1803 0.1643 0.1521 0.1494 0.1475 0.1293 0.1281
-
220
G.S. Kliros / Interacting fermions on a square lattice
c o r r e l a t i o n f u n c t i o n d e f i n e d by [ri-rll=R
C(R)
= N~
nin~ ~gs
s
,
(7)
tl
N s b e i n g t h e n u m b e r o f l a t t i c e sites, w h i c h r e p r e s e n t s t h e p r o b a b i l i t y t h a t t h e r e is a f e r m i o n at R g i v e n t h e r e is o n e at t h e o r i g i n . R e s u l t s f o r t h e 4 x 4 l a t t i c e w i t h U~ = - 2 . 5 t
a n d U 2 = 0.2U~ a r e g i v e n b y t a b l e II. T h e p a r t i c l e s t e n d t o g e t
closer together
as t h e a b s o l u t e v a l u e o f U ~ / t i n c r e a s e s a n d t h e r e is a c r i t i c a l
v a l u e U~ - - 1 . 5 t
b e l o w w h i c h f e r m i o n s a r e p r e d o m i n a n t l y in N N sites. Fig. 10
g i v e s t h e N N a n d N N N p a i r - c o r r e l a t i o n f u n c t i o n s f o r U~ = - 2 . 5 t different values of NNN interaction parameter
and for three
f o r c o m p a r i s o n . It is s e e n t h a t
Table II Binding energies of two fermions EB.2 and four fermions EB, 4 for the 10/4 × 4 cluster. Ut/t
E,.2(U~- = 0.0)
E..4(U 2 :
0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5 -4.0
0.2370 0.2893 0.4579 0.8297 0.0348 -0.0384 -0.2486 -0.3555 1.4557
1.9996 1.4420 0.8850 0.2240 0.4092 0.4565 2.4953 4.1097 7.6152
0.0)
EB,2(U2 : 0.2U~ )
E..~(U z = 0.2U~)
0.2370 0.3346 0.5952 0.8721 -0.2603 -0.0726 -0.1136 -0.3623 -0.7882
1.9996 1.3707 0.6935 0.2875 0.3392 0.6790 2.4187 3.9(134 5.1980
O. 95 0.90
O. 85
0.80 0
0.F5
O. 70
,. ,° ..~,-
/
0.65 0.60
O. 55
~-'" .g" ,t~.= .
. 0.0 0.5
.
.
.
.
.
.
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 -Ult
Fig. 10. NN (R = 1, solid line) and NNN (R = X/2, dashed line) correlation function C(R) in the ground state of fermions on a 4 × 4 lattice as a function of N N interaction parameter U~ for three different values of NNN interaction parameter: U2 =0.0 (triangles), U2 =0.2Ut (diamonds), U2 = 0.4U~ (polygons).
G.S. Kliros / Interacting fermions on a square lattice
221
NNN interactions increase the probability for two fermions to be at nearest or next-nearest sites, especially at U1 = U~ ~ - 1 . 5 t .
5. Conclusions
We have calculated by exact diagonalization the ground-state energies and wave-functions for small systems of spin-polarized fermions on a square lattice with cylindrical topology assuming both repulsive and attractive nearestneighbour (NN) and next-nearest-neighbour (NNN) interactions. Fractional quantum Hall (FQH) states have been identified from the deep minima developed in the ground-state energy when the external magnetic flux corresponds to filling fractions with odd denominators. These minima are destroyed when strong enough (greater than 1 / 5 of the value of the NNinteraction parameter) NNN interactions are imposed. Therefore, competing interparticle interactions U 1 and U2 do not preserve the nature of the ground state. A further evidence that FQHE states occur on a lattice is the large overlap between the ground state at certain odd-denominator filling factors with the free hard-core bosons' ground state in zero magnetic field. This also indicates the existence of a singular ODLRO. It is our aim to examine the O D L R O in the FQHE on a lattice by means of the nonsingular boson-fermion mapping recently introduced by Xie et al. [24] as well as the quantum Hall states of anyons on a lattice with different periodic conditions [25]. Assuming attractive NN and NNN interactions between spin-polarized fermions, we have seen several different signatures of superfluidity as flux quantization, binding energies, pair-correlation functions and the effect of external magnetic field on the pairing features. Flux quantization has been observed for a large range of NNN interactions and survives when antiperiodic conditions in one direction are imposed. This indicates that the flux quantization is not an artifact due to finite size system effects. The pairing features persist in a small external magnetic field but are destroyed when the external flux exceeds a critical value. We have computed the binding energy of two and four fermions as well as that of three fermions with respect to splitting into a pair and a single particle. We found that NNN interactions enlarge the range of effective pairing. This is in good agreement with the results from the calculation of pair-correlation functions. We have seen that NNN interactions increase the probability for two fermions to be at nearest and next-nearest sites. The scaling behaviour of the above properties of interacting fermions on a lattice with particle number, lattice size and aspect ratio is an open problem
222
G.S. Kliros / Interacting ferrnions on a square lattice
d u e to the p r o h i b i t i v e difficulty of e x t e n d i n g the d i a g o n a l i z a t i o n a p p r o a c h to sufficiently larger systems.
Acknowledgements I a m i n d e b t e d to Prof. P.N. B u t c h e r for his hospitality a n d e n c o u r a g e m e n t d u r i n g m y stay at the D e p a r t m e n t of Physics, U n i v e r s i t y of W a r w i c k , U . K . , a n d to Dr. N. d ' A m b r u m e n i l for helpful discussions. I a m grateful to A . Y a n n a c o p o u l o s for v a l u a b l e c o m m e n t s r e g a r d i n g the m a n u s c r i p t .
References [1] D.H. Lee and M.P.A. Fisher, Phys. Rev. Lett. 63 (1989) 903. [2] P.W. Anderson, Phys. Scr. T27 (1989). D. Poilblanc, Y. Hasegawa and T.M. Rice, Phys. Rev. B 41 (1990) 1949. [3] Y.H. Chen, F. Wilczek, E. Witten and B.I. Halperin, Int. J. Mod. Phys. B3 (1989) 1001. G.S. Canright, S.M. Girvin and A. Brass, Phys. Rev. Lett. 63 (1989) 2291, 2295. [4] D.R. Hofstadter, Phys. Rev. B 14 (1976) 2239. [5] F.H. Claro and G. Wannier, Phys. Rev. B 19 (1979) 6008. R. Rammal, J. Phys. (Paris) 46r (1985) 1345. Y. Hasegawa, Y. Hatsugai, M. Kohmoto and G. Montambaux, Phys. Rev. B 41 (1990) 9174. [6] D.J. Touless, M. Kohmoto, P. Nightingale and M. den Nijs, Phys. Rev. Lett. 49 (1982) 405. [7] G. Montambaux, Phys. Rev. Lett. 63 (1989) 1657. [8] F. Nori, E. Abrahams and G. Zimanyi, Phys. Rev. B 41 (1990) 7277. [9] P. Lederer, D. Poiblanc and T.M. Rice, Phys. Rev. Lett. 63 (1989) 1519. [10] D. Bormann, T. Schneider and M. Frick, Europhys. Lett. 14 (1991) 101. [11] G.S. Canright, S.M. Girvin and A. Brass, Phys. Rev. Lett. 63 (1989) 2291. [12] C. Gros, S.M. Girvin, G. Canright and M. Johnson, Phys. Rev. B 43 (1991) 5883. [13] G.S. Kliros and N. d'Ambrumenil, J. Phys.: Condens. Matter 3 (1991) 4241. [14] E. Dagotto and A. Moreo, Phys. Rev. D 31 (1985) 865. E.R. Gagliano, E. Dagono, A. Moreo and F.C. Alcazar, Phys. Rev. Lett. B 34 (1987) 1677. [15] W. Wu, C. Kallin and A. Brass, Phys. Rev. B 42 (1990) 2222. [16] R.E. Prange and S.M. Girvin, eds., The Quantum Hall Effect (Springer, New York, 1987) ch. 7. R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. [17] F.D.M. Haldane, Phys. Rev. Lett. 51 (1983) 605. B.I. Halperin Phys. Rev. Lett. 52 (1984) 1583. [18] B.I. Halperin, Helv. Phys. Acta 56 (1983) 75. [19] G.S. Kliros, Nuovo Cimento D 13 (1991) 111. [20] S.M. Girvin and A. MacDonald, Phys. Rev. Lett. 58 (1987) 1252. S.C. Zhang, T. Hansson and S. Kivelson, Phys. Rev. Lett. 62 (1989) 82. N. Read, Phys. Rev. Lett. 62 (1989) 86. [21] F. Wilczek, Phys. Rev. Lett. 49 (1982) 114. [22] C.N. Yang, Rev. Mod. Phys. 34 (1962) 694. [23] R.M. Fye, M.J. Martins and R. Scalettar, Phys. Rev. B 42 (1990) 6809. [24] X.C. Xie, Songe He and S. das Sarma, Phys. Rev. Lett. 66 (1991) 389. [25] M. Ma and F. Zhang, Phys. Rev. Lett. 66 (1991) 1769. G.S. Kliros, Theory of quantum Hall effects on a lattice, Int. School on Low-Dimensional Semiconductor Structures, IC-90-136 (ICTP, Trieste, 1990).
B
L
~
Small-cluster study of interacting fermions on a square lattice with next-nearest-neighbour interactions G e o r g e S. K l i r o s Department of Physics, University of Patras, Patras 26110, Greece Received 12 June 1991 We calculate the ground state energies and wavefunctions for small systems of interacting fermions on a square lattice with cylindrical topology. We examine the role of next-nearestneighbour interactions on the observed ground states: the fractional quantum Hall ground state and the super-conducting ground state. Several different probes of fermion-pairing and off-diagonal long-range order as flux quantization, binding energies, pair-correlation functions and critical external magnetic field are investigated. Antiperiodic boundary conditions imposed in one direction provide a test of our numerical results within current size limitations.
I. Introduction T h e r e is currently considerable interest in the study of strongly correlated fermions on a square lattice in a magnetic field because of its importance in various fascinating topics as the fractional quantum Hall effect ( F Q H E ) [1], the c o m m e n s u r a t e flux states in the t - J model for the high t e m p e r a t u r e superconductors [2], anyon superconductivity [3], etc. T h e single-particle spectrum of tight-binding fermions on a square lattice with periodic boundary conditions in both directions (toroidal geometry) in a magnetic field shows surprising features such as self-similarity, nesting properties and a Landau-level-like structure, which have been studied by several authors [4, 5]. W h e n the magnetic flux per plaquette, 4 , is a rational n u m b e r , p / q , the spectrum has q subbands with well-defined gaps. It has been shown [6] that each subband carries an integral Hall conductance and the integer has a topological origin. O n e of the important consequences of the structure of the spectrum is the stabilization of the Fermi sea by the gaps at every rational value of magnetic flux. If the n u m b e r of electrons per site is fixed, the total energy of fermions exhibits cusplike minima when the Fermi energy jumps 0378-4371/92/$05.00 (~ 1992- Elsevier Science Publishers B.V. All rights reserved
210
G.S. Kliros / Interacting fermions on a square lattice
across a gap as a function of magnetic flux per plaquette, and a sharp global minimum appears exactly at plus or minus one flux quantum per particle [7]. A linear proportionality between the lattice filling and the optimal flux has been shown with high accuracy [8]. This result is closely connected with the mean-field approximation for the t - J Hamiltonian [9]. The main features of the above spectrum survive if we consider periodic conditions in one direction and hard walls in the other (cylindrical geometry). Although this geometry allows continuous variation of the magnetic flux through the plaquettes and is suitable for testing flux quantization, the average degeneracy of the energy levels is reduced and pronounced size effects may appear [10]. Exact diagonalization techniques have proved to be very powerful in studying the difficult problem of interacting particles giving partial but useful answers to interesting questions like the influence of the attractive or repulsive interaction on the nature of the ground state of the system and the effective interaction between two particles. A pioneering small-cluster study of anyons (including as a special case the fermions) has been done by Canright, Girvin, and Brass [11]. They found that as a function of statistics and applied magnetic field the ground state of "free" anyons on a square lattice with cylindrical symmetry could be either super-conducting or quantum-Hall-like. They also found from the ground-state energy versus the threading of the cylinder flux diagram that there exists a paired state for interacting fermions and semions. Also, the validity of the vector-mean-field theory for statistical transmutation has been tested using both exact diagonalization and Monte Carlo methods [12]. Recently, Kliros and d'Ambrumenil [13] calculated the ground-state energies and wavefunctions for small systems of fermions on a square lattice in the presence of an external magnetic field using singly and doubly periodic boundary conditions. They identified fractional quantum Hall states on a lattice from the minima appearing in the ground-state energy when the magnetic flux per plaquette corresponds to filling fractions with odd denominators. They also found that the minima for two systems with the same flux density and lattice, one with cylindrical and the other with toroidal topology, closely resemble each other when the number of particles per plaquette is the same in the two cases. In this paper we present a systematic numerical study of the properties of tight-binding interacting fermions on a square lattice with cylindrical topology including up to next-nearest-neighbour (NNN) interactions. Repulsive interparticle interactions lead to FQH states while attractive interactions lead to electron pairing and existence of an off-diagonal long range order (ODLRO) for a large range of the interaction parameters. We look for independent probes of possible "superconducting" pairing between fermions and examine the effect of NNN interactions on them.
G . S . Kliros / Interacting f e r m i o n s on a square lattice
211
The paper is organized as follows. In the next section we present the model and describe some features of the non-interacting case such as the form of the single-particle spectrum and the existence of an O D L R O in the corresponding free hard-core bosons system. In section 3 we examine the effect of NNN interactions on the FQHE-Iike minima found in ref. [13]. A numerical test of the f e r m i o n - b o s o n mapping is performed as a further check of the occurrence of F Q H states. Several signatures for binding of fermions and O D L R O in the case of attractive interactions and the effect of NNN interactions are discussed in section 4. Section 5 contains a review of our numerical results.
2. Model Hamiltonian and free-particles' energy spectrum We study numerically the ground state energy of small systems of interacting spin-polarized fermions on a square lattice considering nearest-neighbour (NN) hopping and up to next-nearest-neighbour (NNN) interparticle interactions. We apply periodic boundary conditions in the x-direction keeping hard walls in the y-direction so that our system is topologically equivalent to a cylinder. The system is minimally coupled to a magnetic field parameterized by a flux per plaquette qo and a " t e s t " flux @t is considered as in ref. [11]. The fluxes are in units of the magnetic flux quantum 40 -= hc/e. We work with the following Hamiltonian on a two-dimensional square lattice: Y( = - t ~ ei°qc~cj q- h.c. + U 1 ~] nin ] + U 2 Z ninj, (1) (ij)
(ij)
[ij] t
where c i are the usual fermion operators on the lattice, n i = ci ci is the particle n u m b e r operator, and (ij) and [ij] stand for NN and NNN bonds of the particular lattice, respectively. The parameters U 1 and U 2 give the strength of NN and NNN interaction, respectively. The phases 0,j defined by ]
Oij=-e
(A + A t ) ' d l ,
(2)
i
where A and A t a r e vector potentials that correspond to flux through the plaquettes and " t e s t " flux, respectively, are distributed along the links of the lattice so that the total flux through each plaquette is the same modulo an integer. We have diagonalized the Hamiltonian (1) numerically using the Lanczos technique [14]. In this paper we present results for a 4 × 4 lattice. Less than 50 Lanczos steps were necessary to obtain the ground state with enough accuracy
212
G.S. Kliros / Interacting fermions on a square lattice
(1 part i n ~ 1 0 4 ) . Independent runs have been carried out starting from a trial state corresponding to different configurations in order to be sure we have obtained the true ground state. The single-particle energy spectrum of eq. (1) for the non-interacting case (U~ = U2 = 0) as a function of magnetic flux per plaquette @ is shown in fig. 1. The ground-state energy for N non-interacting particles is just the sum of the N lowest single-particle (kinetic) energies which can be calculated exactly. Due to the existence of well-defined gaps, the ground-state energy as a function of magnetic flux per plaquette @ exhibits deep local minima at certain filling fractions as has already been observed for the case of doubly periodic boundary conditions [7]. Considering q0 = 0, we can also obtain the behaviour 6
L?~':Z/. i
•-o °
°° ,°
.,°
o-°-:~
° ."
Ooo °° °
..~°o.
%
•...oo"
,-*.~
~
".°. oO.
-6 0.0
0.2
0.4
0.6
0.8
1.0
e Fig. 1. Single-particle spectrum o f fermions on a 4 x 4 lattice with cylindrical symmetry in the presence o f a magnetic field parameterized by a flux per plaquette ~.
3 >_ 2 n" I u.J Z 0 l.U -I -2 -3 -6
0.0
0.2
0.4
0.6
®,
0.8
1.0
Fig. 2. Single-particle energies of non-interacting fermions on a 4 x 4 lattice with cylindrical symmetry versus "test" magnetic flux @, through the bore of the cylinder.
213
G.S. Kliros / Interacting fermions on a square lattice
o f the single-particle s p e c t r u m as the " t e s t " flux @t varies. T h e spectrum shows several level crossings and is invariant u n d e r changing q~, by an integral n u m b e r of flux q u a n t a (fig. 2). F u r t h e r m o r e , as test of o u r numerical calculations, we investigated the g r o u n d - s t a t e e n e r g y o f free h a r d - c o r e ( H C ) b o s o n s in zero external m a g n e t i c field (@ = 0). H C b o s o n s are superfluid and show an O D L R O in the o n e - b o d y density matrix. In fig. 3 we show the g r o u n d state e n e r g y of H C bosons on the 4 x 4 lattice with cylindrical s y m m e t r y as a function of flux @t threading the cylinder for different n u m b e r s o f b o s o n s N. It is periodic with a p e r i o d - o n e flux q u a n t u m , which is a characteristic p r o p e r t y of a charge e superfluid. In fig. 4 we have d r a w n the p e a k s of the e n e r g y barrier E ( @ ) - E(0) as a function o f 2.8
8
N=? 2.4 2.0 1.6 1.2 0.8 0.4
0.0 0.0 0.1 0.2 0.3 0.4 0.5 0,6 0,? 0.8 0.9 1,0 MAGNETIC FLUX @t
Fig. 3. Ground-state energy as a function of the "test" magnetic flux @, for non-interacting bosons on a 4 × 4 lattice with cylindrical topology. 2.75 '~2.50 L.I.I
• °'°
~
""~"°
h
"'~
2.25 /
/
z /
2. O0
\
l /
I . 75
%,,•
/
I . 50 3
|
i
,
,
,
,
4
5
6
7
8
9
i
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|
II
12
13
N Fig. 4. Scaling behaviour of the ground-state energy maxima with particle number for bosons in the 4 x 4 lattice. Note the deviation from linear behaviour near half-filling.
214
G.S. Kliros / Interacting fermions on a square lattice
n u m b e r of particles N. The observed scaling behaviour is a clear evidence of O D L R O in the system. Of course, in the thermodynamic limit the energy barrier between two adjacent minima of E(q~) is infinite. The deviation from scaling behaviour near half filling is attributed to lattice effects and it seems to be independent of the imposed boundary conditions since it has also been found in calculations on a sphere [15].
3. Repulsive interactions: fractional quantum Hall states The most widely accepted theory of the F Q H E is based on Laughlin's wavefunction [16] for filling factors 1/q and q an odd integer and associated hierarchical models [17] for the other fractions with odd denominators. The "binding" of q flux quanta to each particle gives rise to q zeros in the wavefunction as a function of the relative coordinate of any pair of particles. This reduces the probability that any two particles approach each other closely and the interaction energy for any short-range repulsive interaction between particles is therefore lowered [18]. The importance of zeros in the wavefunction as a function of the relative coordinate of any pair of particles to the short-range behaviour is less obvious for particles on a lattice, whereas all the "coarse grained" properties identified as important in the Laughlin theory should be well reproduced. We should notice that on a lattice analytic wavefunctions in the presence of a magnetic field do not exist. The restriction to the lowest Landau level, which requires the wavefunction to be an entire function of the particles' complex coordinates, does not apply on the lattice and one expects lines of zeros rather than "point" zeros [12]. In a recent small-cluster study of interacting fermions on a square lattice with NN interactions [13], F Q H states were identified from the deep minima developed in the ground-state energy as a function of the filling factor determined by the ratio of the number of particles per plaquette divided by the n u m b e r of flux quanta per plaquette. More specifically, one can define the quantity fint = E g s -
(~gs[~kinl~gs),
(3)
where Eg s is the ground state energy and ~kin is the hopping part of the Hamiltonian. The variation of Eg s due to the single-particle band structure has been removed from Ein t making the variation of the contributions from the interaction energy more apparent. It has been found [13] that, considering only NN interactions, Ei. t exhibits
G.S. Kliros / Interacting fermions on a square lattice
215
d e e p m i n i m a at filling factors with o d d d e n o m i n a t o r s for a large range of the interaction strength U 1 > - 3 t . T h e s e m i n i m a give a clear evidence that F Q H states o c c u r in small systems on a lattice. A n interesting question addressed h e r e c o n c e r n s the role of N N N interactions with respect to the o b s e r v e d F Q H E on a lattice. We have t h e r e f o r e calculated the quantity Ein t by exact diagonalization of Y( (eq. (1)) for repulsive N N and N N N interactions (U~, U 2 > 0) on a 4 x 4 lattice. W e f o u n d that for U 2 < 0.2U1, the m i n i m a in Ein t a p p e a r at the same filling fractions as without N N N interactions. W h e n U 2 is increased a b o v e 0 . 2 U 1 the m i n i m a m o v e t o w a r d s the filling factor v = 1 / 2 and for U 2 - 0 . 5 U ~ the F Q H like m i n i m a have b e e n destroyed. Fig. 5 shows Ein t versus filling factor for a 3 / 4 x 4 lattice (three fermions on a 4 × 4 lattice) for UI = 5t and U 2 = 0.1U~ = 0.5t. F Q H states are identified at 1, = 4 / 7 and v = 4 / 9 . T h e r e are also local m i n i m a at v = 3 / l l and ~ , = 3 / 1 3 . W h e n the N N N interaction strength is increased to U 2 = 0 . 5 U 1 = 2 . 5 t , the sharp F Q H E m i n i m a have disappeared and m i n i m a are o b s e r v e d at the special fillings ~, = 1 / 2 and v = 1 / 4 as is shown in fig. 6. We can say that in the a b o v e regime of competitive interactions the n a t u r e o f the g r o u n d state has b e e n changed. T h e new m i n i m a might be a reminiscent o f the lattice effect showing a qualitative difference of o u r app r o a c h f r o m a c o n t i n u u m calculation [19]. So far we have discussed only one p r o b e of F Q H states on a lattice; the m i n i m a a p p e a r e d in the " e f f e c t i v e " interaction Ei, t as we sweep c o n t i n u o u s l y the m a g n e t i c flux and hence the filling factor v. A n o t h e r p r o b e of the F Q H E g r o u n d states is the overlap of the f e r m i o n g r o u n d state wavefunction with that o f free b o s o n s in zero magnetic field. L a r g e overlap implies a singular O D L R O
1.0
_/
•+-, 0 . 8
t.UO. 6
I I
I I
4,j
0.4
I I
4,/, I I
0.2
0.0 0.2
0.4
0.6
0.8
1.O
V
Fig. 5. Ground-state energy on a 3/4 x 4 lattice with cylindrical topology as a function of filling factor v. U~ = 5t and U2 = 0.5t.
216
G.S. Kliros / Interacting fermions on a square lattice
1.2
e
~.
,,
~0.8 i,m 0.6
0.4 0.2 0.2
0.4
0.6
0.8
1.0
V
Fig. 6. Ground-state energy on a 3/4 × 4 lattice as a function of filling factor ~,, U1= 5t and U2 = 0.5U1. for the corresponding ground state [20]. This is well connected with the b o s o n - f e r m i o n mapping: fermions can be treated as bosons with attached "fictitious" flux tubes. Since at v - - 1 / q there are q flux quanta available to each electron, we can m a k e a mean-field approximation substituting the flux tubes by an average "statistical" flux giving a statistical angle 0 = q~r [21]. Then the statistical flux cancels the external magnetic flux leaving bosons in zero magnetic field which condense and show long-range order [22]. Investigating the minima of the quantity Ein t v e r s u s the external magnetic field on a 3/4 x 5 system, we are able to identify the lattice analog of v = 1/3 and v = 1/5 F Q H ground states. We found significant overlaps ( - 0 . 9 2 ) between these fermion ground states and the corresponding hard-core boson ground state in zero magnetic field but this probe of F Q H states seems to be lattice-size dependent.
4. Attractive interactions: superconducting pairing and ODLRO
The purpose of this section is a systematic study of the model Hamiltonian 9( (eq. (1)) for the case of attractive interactions and especially the investigation of the role of N N N interactions. Several signatures of a superfluid ground state consisting of pairs of particles on our finite lattice are discussed. Canright et al. [11] studied small systems with only NN interactions. They argued, based on the ground-state energy versus test-flux diagram, that fermions on a square lattice with cylindrical geometry show pairing in some p a r a m e t e r range for N N interparticle interaction.In the present study, we have diagonalized Hamiltonian eq. (1) considering • = 0, U 1, U e < 0, for different
G . S . Kliros / Interacting f e r m i o n s on a square lattice
217
values of the test-flux ~t- We present results for the system 10/4 × 4, which can also be viewed as having six holes. As a first probe of superconducting pairing one can consider the periodicity of the ground-state energy Eg S sweeping the test-flux q0 through the bore of the cylinder and the flux quantization defined by the occurrence of equally spaced minima in the structure of Egs versus q~t. We found that well-defined minima in Eg s versus ~t survive for the values of NNN interaction parameter in the range of interest 0 < U 2 <~ U~/2. Fig. 7 shows periodicity of Egs vs. ~t with period • o/2, which is the flux quantum of a paired state in a BCS superconductor. The observed flux quantization implies that there exists a coherent, paired state with O D L R O . As is well known [23], mixed or other variable boundary conditions are generally useful in checking the importance of finite-size effects. To ensure that the above flux quantization curves are not a finite-size result, we also performed calculations with antiperiodic boundary conditions. We imposed antiperiodic boundary conditions in the x-direction replacing t by - t on certain bonds along the x-direction and changing all hopping integrals by a proper phase factor in order to preserve translational invariance along this direction. A typical curve of our calculations for U 1 = - 3 . 0 t and U: = 0.2U 1 = - 0 . 6 t is shown in fig. 8. We observe that both periodicity in the ground-state energy versus flux and well-defined minima at multiples of q00/2 persist. Another question to be addressed is whether the observed quantization "survives" in the presence of an external magnetic field. We have therefore diagonalized Hamiltonian eq. (1) for • ~ 0. We found that flux quantization persists only in very small external magnetic fields. As is shown in fig. 9, an -39.8 e) -40.0 L~J -40.2
-40.4
-40.6
0.0
0.2
0.4
0.6
0.8
1.0
e, Fig. 7. G r o u n d - s t a t e e n e r g y of f e r m i o n s o n t h e 4 x 4 lattice v e r s u s " t e s t " m a g n e t i c flux ~t for U 1 = - 2 . 5 t a n d t h r e e d i f f e r e n t v a l u e s of N N N i n t e r a c t i o n s : (a) U z = 0.0, (b) U z = 0 . 2 U 1 a n d (c) U 2 = 0 . 4 U 1. T w o o f the c u r v e s (b a n d c) are shifted in e n e r g y so as to lie in the s a m e d i a g r a m .
218
G . S . Kliros / Interacting f e r m i o n s on a square lattice -40. I -40.2 i.~ - 4 0 . 3 -40.4
-40.5 -40.6 0.0
0.2
0.4
0.6
0.8
1.0
Ot Fig. 8. G r o u n d - s t a t e e n e r g y v e r s u s " t e s t " flux • t for f e r m i o n s on a 4 × 4 lattice with a n t i p e r i o d i c c o n d i t i o n s i m p o s e d o n the x - d i r e c t i o n . E q u a l l y s p a c e d m i n i m a o c c u r for U L = - 3 t a n d U 2 = 0 . 2 U , .
-45.9 -46.0 -46. I -46.2 -46.3 -46.4 -46.5 0.0
i
i
i
0.2
0. 6,
0.6
i
®,
0.8
1.0
Fig. 9. G r o u n d - s t a t e e n e r g y v e r s u s " t e s t " flux O, for f e r m i o n s o n a 4 x 4 lattice in the p r e s e n c e of a n e x t e r n a l m a g n e t i c flux • = 00/2. UI = - 2 . 5 t and U 2 = 0 . 2 U , .
external magnetic flux • = 00/2 is sufficient to destroy the " p r o p e r " minima in the structure of the ground-state energy. This observation is a further indication of O D L R O . In the following, we deal with a second signature of superconducting pairing, the binding energy of two fermions defined by EB, 2 = [ E g ~ ( N - 2 ) - E g ~ ( N ) ] - 2 [ E g s ( N -
i) - E~(N)I,
(4)
where Egs(N ) is the ground state energy of N fermions. We can easily see that EB,z represents the effective interaction between
G . S . Kliros / Interacting f e r m i o n s o n a square lattice
219
particles: If we imagine the infinite system consisting of a collection of finite clusters, eq. (4) gives the difference b e t w e e n the state with two particles a d d e d in the same cluster and the o n e in which fermions are a d d e d to different clusters. W h e n EB, 2 is negative, the lower-energy state is that in which the two extra f e r m i o n s w e r e a d d e d to the same cluster and, hence, we can interpret EB, 2 as an attractive interaction b e t w e e n them. We must, h o w e v e r , distinguish " s u p e r c o n d u c t i n g " pairing from phase separation or clustering o f particles. We have therefore also calculated the binding e n e r g y of four fermions defined by EB, 4 -~ [Egs(N - 4) - Egs(N)] - 2[Egs(N - 2) -
Eg,(N)I
(5)
as well as the binding e n e r g y o f three fermions with respect to splitting into a pair and a single-particle, defined by EB, 3 = [Egs(N - 3) - Egs(N)] - [Egs(N - 2) - Egs(N)]
(6)
- [ E g s ( N - 1) - Eg~(N)].
B o t h EB, 3 and EB, 4 w o u l d be negative if phase separation occurs. Superconducting pairing is t h e r e f o r e indicated by E B , 2 being negative and EB, 3 and EB, 4 being positive. O u r results concerning EB, 2 and EB, 4 for the 10/4 × 4 cluster with and w i t h o u t N N N interactions are shown in table I. It is seen that f e r m i o n pairing a p p e a r s only with a well-defined " w i n d o w " of N N interaction p a r a m e ter and this " w i n d o w " is b r o a d e n e d in the presence of N N N interaction. Also, N N N interaction decreases EB, 4 for all the values of N N interaction. In the a b o v e range, all the values of EB, 3 have b e e n f o u n d to be positive. A n o t h e r i m p o r t a n t quantity concerning two-particle interactions is the pair-
Table I Pair-correlation functions on a 4 x 4 lattice with cylindrical topology for several values of NN interaction U1. The strength of NNN interactions has been taken U, = 0.2U1. U~/t
R = 1
R = V~
R =2
R = V-5
R = 2V'2
0.0 -0.5 - 1.0 - 1.5 -2.0 2.5 -3.0 -3.5 -4.0
0.6039 0.6377 0.6848 0.7558 0.8467 0.8560 0.8612 0.9139 0.9188
0.5749 0.5998 0.6355 0.6864 0.7227 0.7290 0.7332 0.7459 0.7472
0.3890 0.3851 0.3807 0.3750 0.3969 0.3968 0.3963 0.3872 0.3852
0.6703 0.6776 0.6816 0.6726 0.6666 0.6644 0.6628 0.63ll 0.6294
0.1890 0.1864 0.1803 0.1643 0.1521 0.1494 0.1475 0.1293 0.1281
-
220
G.S. Kliros / Interacting fermions on a square lattice
c o r r e l a t i o n f u n c t i o n d e f i n e d by [ri-rll=R
C(R)
= N~
nin~ ~gs
s
,
(7)
tl
N s b e i n g t h e n u m b e r o f l a t t i c e sites, w h i c h r e p r e s e n t s t h e p r o b a b i l i t y t h a t t h e r e is a f e r m i o n at R g i v e n t h e r e is o n e at t h e o r i g i n . R e s u l t s f o r t h e 4 x 4 l a t t i c e w i t h U~ = - 2 . 5 t
a n d U 2 = 0.2U~ a r e g i v e n b y t a b l e II. T h e p a r t i c l e s t e n d t o g e t
closer together
as t h e a b s o l u t e v a l u e o f U ~ / t i n c r e a s e s a n d t h e r e is a c r i t i c a l
v a l u e U~ - - 1 . 5 t
b e l o w w h i c h f e r m i o n s a r e p r e d o m i n a n t l y in N N sites. Fig. 10
g i v e s t h e N N a n d N N N p a i r - c o r r e l a t i o n f u n c t i o n s f o r U~ = - 2 . 5 t different values of NNN interaction parameter
and for three
f o r c o m p a r i s o n . It is s e e n t h a t
Table II Binding energies of two fermions EB.2 and four fermions EB, 4 for the 10/4 × 4 cluster. Ut/t
E,.2(U~- = 0.0)
E..4(U 2 :
0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5 -4.0
0.2370 0.2893 0.4579 0.8297 0.0348 -0.0384 -0.2486 -0.3555 1.4557
1.9996 1.4420 0.8850 0.2240 0.4092 0.4565 2.4953 4.1097 7.6152
0.0)
EB,2(U2 : 0.2U~ )
E..~(U z = 0.2U~)
0.2370 0.3346 0.5952 0.8721 -0.2603 -0.0726 -0.1136 -0.3623 -0.7882
1.9996 1.3707 0.6935 0.2875 0.3392 0.6790 2.4187 3.9(134 5.1980
O. 95 0.90
O. 85
0.80 0
0.F5
O. 70
,. ,° ..~,-
/
0.65 0.60
O. 55
~-'" .g" ,t~.= .
. 0.0 0.5
.
.
.
.
.
.
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 -Ult
Fig. 10. NN (R = 1, solid line) and NNN (R = X/2, dashed line) correlation function C(R) in the ground state of fermions on a 4 × 4 lattice as a function of N N interaction parameter U~ for three different values of NNN interaction parameter: U2 =0.0 (triangles), U2 =0.2Ut (diamonds), U2 = 0.4U~ (polygons).
G.S. Kliros / Interacting fermions on a square lattice
221
NNN interactions increase the probability for two fermions to be at nearest or next-nearest sites, especially at U1 = U~ ~ - 1 . 5 t .
5. Conclusions
We have calculated by exact diagonalization the ground-state energies and wave-functions for small systems of spin-polarized fermions on a square lattice with cylindrical topology assuming both repulsive and attractive nearestneighbour (NN) and next-nearest-neighbour (NNN) interactions. Fractional quantum Hall (FQH) states have been identified from the deep minima developed in the ground-state energy when the external magnetic flux corresponds to filling fractions with odd denominators. These minima are destroyed when strong enough (greater than 1 / 5 of the value of the NNinteraction parameter) NNN interactions are imposed. Therefore, competing interparticle interactions U 1 and U2 do not preserve the nature of the ground state. A further evidence that FQHE states occur on a lattice is the large overlap between the ground state at certain odd-denominator filling factors with the free hard-core bosons' ground state in zero magnetic field. This also indicates the existence of a singular ODLRO. It is our aim to examine the O D L R O in the FQHE on a lattice by means of the nonsingular boson-fermion mapping recently introduced by Xie et al. [24] as well as the quantum Hall states of anyons on a lattice with different periodic conditions [25]. Assuming attractive NN and NNN interactions between spin-polarized fermions, we have seen several different signatures of superfluidity as flux quantization, binding energies, pair-correlation functions and the effect of external magnetic field on the pairing features. Flux quantization has been observed for a large range of NNN interactions and survives when antiperiodic conditions in one direction are imposed. This indicates that the flux quantization is not an artifact due to finite size system effects. The pairing features persist in a small external magnetic field but are destroyed when the external flux exceeds a critical value. We have computed the binding energy of two and four fermions as well as that of three fermions with respect to splitting into a pair and a single particle. We found that NNN interactions enlarge the range of effective pairing. This is in good agreement with the results from the calculation of pair-correlation functions. We have seen that NNN interactions increase the probability for two fermions to be at nearest and next-nearest sites. The scaling behaviour of the above properties of interacting fermions on a lattice with particle number, lattice size and aspect ratio is an open problem
222
G.S. Kliros / Interacting ferrnions on a square lattice
d u e to the p r o h i b i t i v e difficulty of e x t e n d i n g the d i a g o n a l i z a t i o n a p p r o a c h to sufficiently larger systems.
Acknowledgements I a m i n d e b t e d to Prof. P.N. B u t c h e r for his hospitality a n d e n c o u r a g e m e n t d u r i n g m y stay at the D e p a r t m e n t of Physics, U n i v e r s i t y of W a r w i c k , U . K . , a n d to Dr. N. d ' A m b r u m e n i l for helpful discussions. I a m grateful to A . Y a n n a c o p o u l o s for v a l u a b l e c o m m e n t s r e g a r d i n g the m a n u s c r i p t .
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