Exact Results of the 1D $1/r^2$ Supersymmetric t-J Model without Translational Invariance

13 Pages·0.12 MB·English

by  C. Gruber

IPT-EPFL, 1993
Exact Results of the 1D 1/r2 Supersymmetric t-J Model
without Translational Invariance
C. Gruber and D. F. Wang
4 Institut de Physique Th´eorique
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9 E´cole Polytechnique F´ed´erale de Lausanne
1

n
PHB-Ecublens, CH-1015 Lausanne-Switzerland.
a
J

7

3
v In this work, we continue the study of the supersymmetric t-J model with 1/r2
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0
hopping and exchange without translational invariance. A set of Jastrow eigen-
0
1
0 functions are obtained for the system, with eigenenergies explicitly calculated. The
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ground state of the t-J model is included in this set of wavefunctions. The spectrum
/
t
a
m of this t-J model consists of equal-distant energy levels which are highly degenerate.
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o PACS number: 71.30.+h, 05.30.-d, 74.65+n, 75.10.Jm
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:
v
i
X
r
a
Typeset Using REVTEX
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In recent years, there have been considerable interests in study of low dimensional elec-
tronic models of strong correlation, due to the possibility that the normal state of the two
dimensional novel superconductivity [1] may share some interesting feature of 1D interact-
ing electron system (non-Fermi liquid behavior). The one-band two dimensional Hubbard
model reduces to the t-J model in the large on-site energy limit. The Hubbard model and
the t-J model have been under intense study through various approaches. For these strongly
correlated electron models, few exact results may be obtained in two dimensions. The high
dimensional versions are much harder to study than their one dimensional ones. In one
dimension, however, the Bethe-ansatz technique may allow us to exactly solve Hamiltonians
in some special cases, such as the Lieb-Wu solution [2] and the ordinary t-J model at its
supersymmetric point [3]. In particular, the 1D long range exactly solvable electronic mod-
els have attracted a lot of attention, since they display interesting physics with solutions of
simple mathematical structure [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21].
Recently, we have been able to explicitly construct all the constants of motion for the
translational invariant long range supersymmetric t-J model, by mapping the system to a
mixture of fermions and bosons, with the super-algebra representation for the electron fields
[12]. Moreover, we have introduced a one dimensional supersymmetric t-J model with 1/r2
hopping and exchange without translational invariance. This system has also been shown
by us to be completely integrable, with infinite number of conserved quantities explicitly
constructed [12]. In this work, we continue the study of this t-J model. A set of Jastrow
eigenfunctions, as well as their eigenenergies, are obtained explicitly. The ground state of
the t-J model is included in this set of wavefunctions. We also briefly discuss the structure
of the full spectrum for the system.
The Hamiltonian for the one-dimensional t-J model is given by [12]
N
H = (1/2)P − t (c† c )+ J [P −(1−n )(1−n )] P , (1)
tJ G ij iσ jσ ij ij i j  G
1≤i6=j≤Lσ=1 1≤i6=j≤L
X X X
 
where we take the hopping matrix and the spin exchange interaction to be t /2 = J =
ij ij
1/(r −r )2. L is the number of sites on the chain. In this model, the positions of the sites
i j
2
{r }aregivenbytherootsoftheHermitepolynomialH (x), andthespincomponentσ takes
i L
values from1 toN. The operatorc† is thethecreation operatorforanelectron atsite iwith
iσ
spin σ, c is the corresponding annihilation operator. Their anti-commutation relations are
iσ
given by {c ,c† } = δ δ ,{c ,c } = 0,{c† ,c† } = 0. The Gutzwiller projection
iσi jσj + ij σiσj iσi jσj + iσi jσj +
operator P projects out all the double or multiple occupancies, P = L P (i), and
G G i=1 G
P (i) = δ + δ , with n = N c† c . The operator P = N NQ c† c c† c
G 0,ni 1,ni i σ=1 iσ iσ ij σ=1 σ′=1 iσ iσ′ jσ′ jσ
P P P
exchanges the spins of the electrons at sites i and j, if both sites are occupied, and is zero
otherwise. At half-filling, our t-J model becomes the long range spin model, introduced first
by Polychronakos on such a non-translational-invariant lattice [14].
In terms of the b and f fields, the eigenequation of the t-J model can be written as [12]
− (q −q )−2M φ({q},{σ}) = Eφ({q},{σ}), (2)
i j ij
1≤i<j≤L
X
where φ({q},{σ}) = φ(q σ ,q σ ,···,q σ |q ,q ,···,q ) is the amplitude for the
1 1 2 2 Ne Ne Ne+1 Ne+2 L
f fermions to be at q ,q ,···,q , while the spinless b bosons are at q ,q ,···,q .
1 2 Ne Ne+1 Ne+2 L
Here, {σ} = (σ ,σ ,···,σ ) and {q} = (q ,q ,···,q ) = (x ,x ,···,x ,y ,y ,···,y ).
1 2 Ne 1 2 L 1 2 Ne 1 2 Q
The wavefunction φ is symmetric in the b boson positions {y}, while antisymmetric when
exchanging x σ and x σ . The operator M exchanges only the position variables q and
i i j j ij i
q , that is, M φ({q},{σ}) = φ({q′},{σ}), with {q} = (q ,q ,···,q ,···,q ,···,q ) and
j ij 1 2 i j L
{q′} = (q ,q ,···,q ,···,q ,···,q ). In this approach, the f fermions and the b bosons
1 2 j i L
occupy the whole chain, i.e. {q} and {q′} are permutations of the sites {r ,r ,···,r }, and
1 2 L
we work in the Hilbert space where at each site there is exactly one particle. Q, the number
of the b bosons, is also the number of holes in the original problem; N , the number of the
e
f fermions, is also the number of the c electrons on the lattice. N˜ , with α = 1,2,···N,
α
the number of the f fermions with spin component α, is also the number of the c electrons
with spin component α on the chain.
The Hamiltonian in the first quantization, as given by Eq. (2), is
H = − (q −q )−2M . (3)
i j ij
1≤i<j≤L
X
3
It commutes with the permutation operator T = PσM exchanging the f fermion spin and
ij ij ij
position simultaneously. Let us work in the Hilbert space where the number of fermions of
each flavor is fixed, i.e. N˜ , σ = 1,2,...,N, is fixed. Consider the following wavefunction
σ
in Jastrow product form,
φ(x σ ,x σ ,···,x σ |y ,y ,···,y ) = (x −x )δσiσjeiπ2sgn(σi−σj), (4)
1 1 2 2 Ne Ne 1 2 Q i j
i<j
Y
where {x} and {y} span the whole lattice. We would like to show that this wavefunction is
an eigenstate of the system. The Hamiltonian in Eq. (2) can be broken up into three parts,
the first part H exchanges the f fermions, the second H exchanges the b bosons, and the
1 2
third H exchanges the bosons and the fermions:
3
H = (−1) (q −q )−2M
1 i j ij
1≤iX<j≤Ne
H = (−1) (q −q )−2M
2 α β αβ
1+Ne≤Xα<β≤L
H = (−1) (q −q )−2M . (5)
3 α j αj
Ne+X1≤α≤L1≤Xj≤Ne
We then calculate the effects of these three parts when acting the Jastrow wavefunction
given by Eq. (4). The contribution from H is immediate:
2
H φ = − (y −y )−2φ. (6)
2 α β
α<β
X
The contributions from H and H are harder to deal with since many particle terms are
1 3
involved. Using a similar trick introduced in Refs. [8,19], we have
y −x
H φ = − (x −y )−2 ( α j)δσiσjφ
3 i α
x −x
Xi Xα jY(6=i) i j
y −x
= − (x −y )−2 (1+δ α i)φ
i α σiσjx −x
Xi Xα jY(6=i) i j
δ
= − (x −y )−2 + σiσj φ−Rest, (7)
i α
 (y −x )(x −x )
Xi Xα jX(6=i) α i i j
 
where
N Ne 1
Rest = [ (y −x )r−3 φ], (8)
α i
(x −x )
Xα σX=1rX=3 J|XJ⊂|=℘σr Xi∈J xj∈YXJ/xi i j
4
with ℘σ = {k ∈ {1,2,···,N };σ = σ},X = {x ;j ∈ J}. Then using the fact that for any
e k J j
set X = (x ,x ,···,x ), we have the following identity ( see Appendix A )
1 2 n
n 1
xt = 0, (9)
i (x −x )
Xi=1 xj∈YX/xi i j
for all t = 0,1,2,···,n−2, we conclude that Rest = 0.
The contribution from H is calculated in a similar manner:
1
x −x x −x
H1φ = − (xi −xj)−2(1−2δσiσj) (xk −xj)δσiσk(xk−xi)δσjσk ·φ
Xi<j k(Y6=i,j) k i k j
x −x x −x
= δ (x −x )−2φ− (x −x )−2(1−δ ) (1+δ i j −δ i j )
σiσj i j i j σiσj σiσkx −x σjσkx −x
Xi<j Xi<j k(Y6=i,j) k i k j
= δ (x −x )−2φ− (x −x )−2φ−
σiσj i j i j
i6=j i<j
X X
x −x x −x
− (x −x )−2(1−δ ) δ i j · i j (−δ )φ. (10)
i j σiσj σiσkx −x x −x σjσk
Xi<j K1⊂{1,X2,···,Ne}/ij kY∈K1 k i kY∈K2 k j
K2⊂{1,2,···,Ne}/ij
K1∪K26=0
Using the sum rule Eq. (9), and the fact that (x − x )−1(x − x )−1δ δ = 0,
i6=j6=k k i k j σiσj σiσk
P
the last term in the above equation becomes
−φ (1−δ ) (x −x )−1(x −x )−1δ −(x −x )−1(x −x )−1δ
σiσj i j k i σiσk i j k j σjσk
Xi<j k(X6=i,j)h i
= −φ (x −x )−1(x −x )−1δ . (11)
i j k i σiσk
Xi6=jk(X6=i,j)
In the end, we have
Hφ = − (q −q )−2 φ+ (x −x )−1δ (x −x )−1 + (x −y )−1 φ.
 i j  i j σiσj  i k i α 
1≤iX<j≤L Xi6=j kX(6=i) Xα
   
(12)
Using the properties of the roots of the Hermite polynomial
r = (r −r )−1, (r −r )−2 = L(L−1)/4, (13)
i i j i j
jX(6=i) 1≤iX<j≤L
we thus conclude that the wavefunction φ is an eigenstate with eigenvalue
N
E = −L(L−1)/4+(1/2) (N˜ −1)N˜ . (14)
σ σ
σ=1
X
5
Although it is expected that this wavefunction is the lowest energy state in the subspace
of fixed N˜ ,N˜ ,···,N˜ , we were not able to prove it. However, in the case of SU(2), the
1 2 N
small lattice diagonalization up to 8 sites confirms this conjecture. Moreover, the discussion
below will also confirm this idea for the general case. For fixed number N of the electron
e
number, the minimum of the energy is obtained when |N˜ −N˜ | is as small as possible for
σ σ′
each pair σ 6= σ′.
In the SU(2) case, the above result becomes
E = (−1)L(L−1)/4+(1/2)N˜ (N˜ −1)+(1/2)N˜ (N˜ −1), (15)
↑ ↑ ↓ ↓
where N˜ and N˜ are the numbers of the up-spin electrons and the down-spin electrons
↑ ↓
respectively. For fixed number of electrons on the chain, i.e. for fixed N , the minimum of
e
the energy given in Eq. (15) is obtained when S = 0 for even N , or when S = ±1/2 for
z e z
odd N . Therefore, the ground state is a spin singlet ( respectively spin 1/2 ) state for even
e
( respectively odd) number of electrons on the chain. In particular, for an even number of
electrons on the chain, the ground state energy is
E = (−1/4)L(L−1)+N2/4−N /2, (16)
G e e
while for an odd number of electrons it is
N −1
E = (−1/4)L(L−1)+( e )2. (17)
G
2
The charge susceptibility of the ground state χ is given by χ−1 = ∂2E /∂N2 = 1/2,
c c G e
independent of the electron concentration. Very unexpectedly, the charge susceptibility
is also finite at half-filling N = L, in spite of the existence of a metal-insulator phase
e
transition at half-filling for this system. This is in contrast to the case of the periodic 1/r2
supersymmetric t-J model, where the charge susceptibility is divergent at half-filling, at
which the metal-insulator phase transition occurs.
To study the spectrum of the system away fromhalf-filling, we follow the idea introduced
in Ref. [14]. Let us define the operators
6
π = i (q −q )−1M = π†,
j j k jk j
kX(6=j)
a† = π +iq ,
j j j
a = π −iq , (18)
j j j
which satisfy the following commutation relations:
[π ,π ] = 0
j k
[q ,H] = iπ
j j
[π ,H] = −2i (q −q )−3. (19)
j j k
kX(6=j)
Then using the property of the roots of the Hermite polynomial we have
[π ,H] = −iq , (20)
j j
and
[a†,H] = −a†
j j
[a ,H] = a . (21)
j j
Therefore the operators A†(ν) = a†S(ν),i = 1,2,···,N , where ν = 0,±,z for the SU(2)
i i i e
(0)
case with S = 1, will act as raising operators, while their hermitian conjugate A (ν) will
i i
act as lowering operators. It thus follows that the wavefunction
Ne
φ{n},{ν} = (A†i(νi))nPiφ (22)
P i=1
XY
with {n} = (n ,n ,···,n ),n ≥ 0,{ν} = (ν ,ν ,···,ν ), is either an eigenstate with
1 2 Ne i 1 2 Ne
energy
Ne
E = E + n (23)
{n} i
i=1
X
or zero.
Moreover, it is shown in the Appendix B that the operators Ne A (ν ) with ν = 0 or z,
i=1 i i i
and a annihilate the wavefunction φ, and also Ne A (±)φ =P0, confirming the conjecture
α i=1 i G
P
7
that φ is the lowest energy state in the subspace. We then arrive at the conjecture that the
excitation spectrum of the system is of the form
E(s) = E +s;
s ∈ (0,1,2,···,s ), (24)
max
i. e., thespectrum of thist-Jmodelconsists of equal-spaced energy levels. Since the modelis
on a finite chain, s is finite. In the special case of SU(2), the small lattice diagonalization
max
up to 8 sites suggests that the highest energy level is given by
E (Q) = L(L−1)/4−Q(Q−1)/2, (25)
max
i.e., for an even number of electrons,
E = E +(1/4)N (4L−3N ), (26)
max G e e
where Q is the number of holes on the chain, N = L−Q is the number of electrons. For
e
N = 1 or N = L ( half -filling ), this formular gives right results; moreover, at half-filling,
e e
this corresponds to all spins polarized in one direction.
The feature of the t-J model spectrum consisting of equal-distant energy levels may also
be seen by taking the strong interaction limit of the Sutherland-Calogero-Morse quantum
system for a mixture of fermions and bosons,
L L
H = (−1/2) ∂2/∂q2 + l2q2/2+ l(l−M )/(q −q )2, (27)
i i ij i j
i=1 i=1 i<j
X X X
where there are N fermions with spins and Q spinless bosons, M permutes the positions
e ij
of the particles i and j only. The mixture gas has equal-distant energy levels described in
terms of effective harmonic oscillators. In the strong interaction limit, the elastic modes
decouple from the internal degrees of freedom. Since elastic modes also consists of equal-
distant energy levels, we thus are led to the conclusion that the spectrum of the internal
dynamics, which isthat ofour t-Jmodel, also consists ofequal-distant energy levels. Further
work is necessary for a fully complete proof that the t-J model full spectrum takes the form
Eq. (24).
8
Finally, we would like to point out that the states of the t-J model in the whole Hilbert
space are grouped into a structure of “spin supermultiplets”, as indicated by the small
lattice diagonalization, similar to that of the periodic 1/r2 supersymmetric t-J model. Such
pattern of the Hilbert space is related to the symmetries associated with the Hamiltonian.
It is highly worth while to identify them more explicitly, and we would like to study these
aspects in further work.
In summary, a set of Jastrow eigenfunctions have been found for the t-J model, with the
eigenenergies explicitly calculated. The expected groundstate ofthe t-Jmodelis included in
this set of wavefunctions. The full spectrum of the t-J model is found to have equal-distant
energy levels which are highly degenerate. It would be very interesting to understand the
underlying symmetry principles that give rise to such simple Hilbert space structure. It
remains to study various correlation functions, as well as the thermodynamics, for this
strongly correlated electron system. It would also be very interesting to study the effective
field theory for the low-lying excitations for this t-J model.
We wish to thank Dr. James T. Liu and Dr. Nicolas Macris for conversations. In
particular, we are very grateful to Dr. James T. Liu for his substantial numerical support.
We also would like to thank the World Laboratory Foundation for the financial support.
9
APPENDIX A:
In this appendix, we provide a brief proof for the sum rule Eq. (9) for reader’s con-
venience. The same argument can also be found in previous works Refs. [8,19]. Let
X = (x ,x ,···,x );X = X/x , we wish to show
1 2 n i i
n 1
xt = 0,∀t = 0,1,2,···,(n−2). (A1)
i x −x
Xi=1 xjY∈Xi i j
Indeed, the Vandermonde determinant V(X) has the property:
n V(Xx)
(x−x ) = , (A2)
j
V(X)
j=1
Y
where X = (x ,x ,···,x ),Xx = (x ,x ,···,x ,x). Therefore we obtain
1 2 n 1 2 n
n 1 n V(X )
xt = xt i =
i x −x iV(X x )
Xi=1 xjY∈Xi i j Xi=1 i i
xt xt ··· xt
1 2 n
 
1 1 ··· 1
n V(X ) (−1)n−1  
= (−1)i−nxt i = det x x ··· x  = 0. (A3)
i V(X) V(X)  1 2 n 
Xi=1  .. .. .. 
 . . ··· . 
 
 
xn−2 xn−2 ··· xn−2
 1 2 n 
 
This thus proves the sum rule Eq. (9).
APPENDIX A:
In this appendix, we shall show that the lowering operators A (z) and a give zero
i α
when acting on the Jastrow wavefunction φ given by Eq. (4). This will yield a partial
confirmation that the wavefunction φ is the lowest energy state in the subspace where the
number of particles of each spin component is fixed. For the b boson degrees of freedom, we
have following property
a φ = 0,i ∈ (N +1,N +2,···,L), (A1)
i e e
10