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Exact solutions of the two-hole and hole-magnon bound states have been found in a ferromagnetic system with hole hopping involved. A phenomenological Hamiltonian that contains spin-spin interaction and hole-hopping terms without doubly occupied sites is considered. Two-hole and hole-magnon bound-state conditions, binding energies, and wave functions are shown for one-dimensional and two-dimensional lattices. They illustrate the interplay of charge and spin excitations. The physical picture of bound states is discussed.
©1990 The American Physical Society.
PACS: 75.10.Jm 74.20.-z
One interesting aspect of high T_{c_}superconductors (HTS's) is the possibility that superconductivity in these materials may be primarily of an electronic origin rather than due to the electron phonon interaction. A common point of departure is to assume that the parent compounds of the HTS's are Mott insulators and can be described, at least in a first approximation, by the s-band Hubbard model in the strong coupling limit.^{1}In this simple picture, the Cu-O planes of the HTS's are modeled by a two dimensional square lattice with each site representing a CuO_{2_}unit cell. Electrons can hop between nearest neighbors, and for two electrons on the same site there is strong Coulomb repulsion modeled by the Hubbard U term. In the strong coupling limit the Coulomb repulsion prevents double occupancy of a lattice site and an effective Hamiltonian can be derived in which there is nearest neighbor antiferromagnetic exchange and a hole hopping term (t-J model) (Ref. 2)
H = - t sum from {langle i,j rangle sigma} (1-n_{{i,-}sigma} )c_{{i}sigma}^{dag}c_{{j}sigma} (1-n_{{j-}sigma} ) - J sum from {langle i,j rangle} S vec_{i}cdot S vec_{j} , L (1)
where J=-4t^{2}/ U<0. The sums are over all lattice sites i and j and over the two spin values symbolized by sigma = +- 1. S vec_{i_}is the spin operator for an electron at site i, c_{{i}sigma}^{dag_}is the fermion creation operator for site i and spin value sigma, and n_{{i}sigma}_ is the fermion occupation number operator. The components of S vec_{i_}can be written in terms of fermion operators as
S_{i^{+}=}c_{i1^{dag}c}sub {i 1 bar} , (2)
S_{i^{-}=}c_{{i}1 bar}^{dag}c_{i1} , (3)
S_{i^{z}=}cf12 ( n_{i1}- n_{{i}1 bar} ) . (4)
We are considering the case where the electrons are not allowed to double occupy lattice sites and have used the notation 1 bar == -1. In the half filled band case this model is equivalent to the spin cf12 antiferromagnetic Heisenberg model, which from simulation studies^{3}and calculations done for the related nonlinear sigma model^{4}is known to describe well the antiferromagnetic correlations in La_{2}CuO_{4}.^{5_}
Numerical diagonalization studies of the strong coupled Hubbard model on small lattices suggest that hole pairing can occur for some range of values of the interaction parameter t/U .^{6}More recently analytical and numerical techniques have been employed to study the possibility of such pairing in this model and its possible causes.
It may be interesting to consider the analytic continuation of the t-J model (1) in the positive J plane. We should mention that this model cannot be derived from the strong coupling Hubbard model and it is not perhaps directly relevant to the HT superconductors. However, the fact that it has a trivial ground state when holes are absent makes its study simple and can illustrate the interplay of charge and spin excitations. Investigations of a ferromagnetic system with hole hopping involved would be of interest, especially since a number of results regarding two hole and hole magnon bound states can be obtained exactly.
In this paper we will calculate eigenstates of H given by equation (1), but with J>0, exactly for two cases: (1) two holes in a ferromagnetically aligned lattice and (2) one hole and one spin deviation in a ferromagnetically aligned lattice. We find that a two hole bound state always exists for total momentum K vec at the Brillouin zone boundary in both one and two dimensions, but the range of K vec values over which the bound state exists depends on the value of J/t. If J/t is greater than a certain critical value then bound states exist over the whole range of K vec values, otherwise there exists some region surrounding K vec =0 for which no bound state exists. For the case of one hole and one magnon in a ferromagnetically ordered lattice, we find that in one dimension a bound state always exists at K vec =0 for any value of J/t greater than zero while in two dimensions there is a finite range of values, J_{l}/t <= J/t <= J_{u}/t_, for which a bound state can exist.
This study forms a complement to the problem of two magnon states in a ferromagnetically ordered lattice where in one- and two dimensional lattices bound states exist over the whole range of two magnon total momentum merging with the continuum at K vec =0, and in three dimensions bound states exist for some range of K vec values but not in a finite region surrounding K vec = 0 .^{7}
In the next section we will describe how to solve the Schroumldinger equation in the two hole subspace. The equation determining the bound state eigenenergy as a function of total momentum K vec of the two hole pair will be derived. Section III contains a discussion of the hole magnon states for one and two dimensional lattices. Equations determining energy eigenvalues for the bound states are derived and some results are presented.
The atomic representation introduced by Hubbard can be a useful formalism for treating more complicated correlated electron models.^{8}We will introduce it here as an illustration of its use in a simple context. Each site can be in one of three states: |i , 1 rangle has one up spin electron at site i, |i, 1 bar rangle has one down spin electron and |i, 0 rangle is an empty site. Any single site operator can be expressed in this representation from the resolution of the identity
1 hat_{i}= |i, 1 rangle langle i,1|+|i, 0 rangle langle i,0|+|i, 1 bar rangle langle i, 1 bar | . (5)
Define the atomic representation operators U_{rs^{i_}as}
U_{rs^{i}=|}i,r rangle langle i,s | . (6)
These operators satisfy the multiplication rule
U_{pq^{i}U}sub rs^{i}= delta_{qr}U_{ps^{i }(7)
}and the commutation relation
[U_{pq^{i},}U_{rs^{j}]}sub -+ = delta_{ij}( delta_{qr}U_{ps^{i}-+}delta_{ps}U_{rq^{i})} , (8)
where the upper sign is taken when at least one of the operators has boson character, and the lower sign is taken only if both operators have fermion character. For the model considered here U_{10^{i}_,}U_{{1}bar 0}^{i_,}U_{01^{i_,}and}U_{{0}1 bar}^{i_}are the operators encountered with fermionic character. The operators present in H can be expressed in terms of the operators U_{rs^{i_}
}S_{i^{+}=}U_{{1}1 bar}^{i} , (9)
S_{i^{-}=}U_{{1}bar 1}^{i} , (10)
S_{i^{z}=}cf12 (U_{11^{i}-}U_{{1}bar 1 bar}^{i}) , (11)
c_{is^{+}(1-}n_{{i}s bar} )= U_{s0^{i} , }(12)
(1-n_{{i}s bar} ) c_{is}= U_{0s^{i} . }(13)
The Hamiltonian H can be rewritten
H = mark - sum from (i,j)s t_{ij}U_{s0^{i}U}sub 0s^{j }L
lineup - sum from (i,j) J_{ij}[ U_{{1}1 bar}^{i}U_{{1}bar 1}^{j}+ case 1 over 4 ( U_{11^{i}-}U_{{1}bar 1 bar}^{i})( U_{11^{j}-}U_{{1}bar 1 bar}^{j})] . (14)
In this section we are interested in eigenstates of H in subspaces with one or two holes introduced on a lattice where each site contains an up spin. Define scrF_{1_}and scrF_{2_}to be the one hole and two hole subspaces. A basis for scrF_{1_}is formed from the states
| r vec_{i}rangle = U_{01^{i}|}F rangle , (15)
where r vec_{i_}refers to a lattice site and |F rangle is the state with all sites containing one spin up electron. A spanning set for scrF_{2_}is formed by
| r vec_{i}, r vec_{j}rangle = U_{01^{i}U}sub 01^{j}| F rangle . (16)
We will be working with the Fourier transform of these states to momentum space
| k vec rangle = 1 over {sqrt N} sum from {r vec_{1}}e^{{i}k vec_{1}cdot r vec_{1}}| r vec_{1}rangle (17)
and
| k vec_{1}, k vec_{2}rangle = 1 over N sum from {r vec_{1}r vec_{2}}e^{{i}k vec_{1}cdot r vec_{1}}e^{{i}k vec_{2}cdot r vec_{2}}| r vec_{1}, r vec_{2}rangle . (18)
We note that the transverse exchange terms in H have no effect on states in scrF_{1_}and scrF_{2_}so that they can be dropped for the remainder of this section. The states | k vec rangle are eigenstates of H with energy eigenvalues
curlep ( k vec ) =- Jz over 4 (N-2) +t z gamma_{k}vec , (19)
gamma_{k}vec = 1 over z sum from delta e^{{i}k vec cdot delta} , (20)
and z is the number of nearest neighbors delta.
The two hole eigenstates can be labeled by the total momentum K vec = k vec_{1}+ k vec_{2_}of the pair since it is a conserved quantity. The relative momentum q vec = cf12 ( k vec_{1}- k vec_{2})_ will also be introduced.
We will find that coefficients f_{k}vec ( q vec_) can be found such that the state | PSI_{K}vec rangle_ in scrF_{2_}is an eigenstate of H where
| PSI_{K}vec rangle = sum from q vec f_{K}( q vec ) | k vec_{1}, k vec_{2}rangle . (21)
Projecting the time independent Schr o umldinger equation with the bra langle lcurl K vec / 2 + q vec prime rcurl , lcurl K vec / 2 -q vec prime rcurl | we obtain
E[f_{K}vec ( q vec ) - f_{K}vec ( - q vec )] L
= mark [ E_{0}+ t ( K vec over 2 + q vec right ) +t ( K vec over 2 - q vec right ) [f_{K}vec ( q vec ) - f_{K}vec ( - q vec )]
lineup - 1 over 2N sum from {q vec prime} J ( q vec prime ) [f_{{K}vec} ( q vec + q vec prime ) - f_{{K}vec} ( - q vec + q vec prime ) ] , (22)
where E_{0}=- (Jz/4) N+Jz_, J( q vec )=Jz gamma_{q_}and t( q vec ) = tz gamma_{{q}vec}_. Since we are interested in lattices with an inversion symmetry, gamma_{{q}vec}_ has the property gamma_{{-}q vec} = gamma_{{q}vec}_. Therefore
sum from {q vec prime} J ( q vec prime ) f_{{K}vec} ( - q vec + q vec prime ) = sum from {q vec prime} J ( q vec prime ) f_{{K}vec} (- q vec - q vec prime ) , (23)
since the sum is over the entire Brillouin zone. With k vec prime defined by k vec prime = q vec + q vec prime, equation (22) can be written as
[ E-E_{0}-t ( {K vec} over 2 + q vec right ) - t ( K vec over 2 - q vec right ) right ] g_{{K}vec} ( q vec ) L
=- 1 over 2N sum from {k vec prime} J ( k vec prime - q vec ) g_{{K}vec} ( k vec prime ) , R
where (25) g_{{K}vec} ( q vec ) = f_{{K}vec} ( q vec ) - f_{{K}vec} ( - q vec ) . (24)
Equation (24) must be solved for g_{{K}vec} ( q vec )_ and E. We note that the free hole states
| PSI_{{K}vec} rangle = C | k vec_{1}, k vec_{2}rangle (26)
are eigenstates to order 1/N with energy
scrE_{{K}vec} ( q vec ) = E_{0}+ t ( K vec over 2 + q vec right ) + t ( K vec over 2 - q vec right ) . (27)
Therefore each value of total momentum K vec has an associated band of solutions that extends from the minimum to the maximum value of scrE_{{K}vec} ( q vec_). Coherence between the free hole states caused by the interaction can produce bound state solutions with E- scrE_{K}vec^{min}< 0_.
To find the bound state solutions Eq. (24) is written
g_{K}vec ( q vec ) =- J over 2N sum from {k vec prime , delta vec} exp (-iq vec cdot delta vec ) exp ( i k vec prime cdot delta vec ) {g_{K}vec ( k vec prime )} over {E- scrE_{K}vec ( q vec )} .
This equation can be solved in the following way. First we take (29) G_{K}vec ( delta ) = 1 over N sum from {k vec} exp ( i k vec prime cdot delta vec ) g_{K}vec k vec prime ) . (28)
Then we have
G_{{K}vec} ( delta vec_{i}) = sum from delta C_{{delta}sub i , delta} G_{{K}vec} ( delta ) , (30)
where
C_{{delta}sub i , delta} =- J over 2N sum from q vec {exp [i q vec cdot ( delta vec_{i}- delta vec )] } over {E- scrE_{K}vec ( q vec )} . (31)
To find the bound state energies the matrix equation
| open1 bhat - C | =0 (32)
needs to be solved. In fact this equation can be simplified by considering the symmetry of the lattice. For a hypercubic d-dimensional lattice the solutions of Eq. (32) become
-1= J over N sum from q vec { sin^{2}( q_{i})} over {E- scrE_{K}vec ( q vec )} (33)
with q_{i}= q_{1},..., q_{d_}and taking the lattice constant as a unit of distance. Thus for a chain and a two dimensional square lattice the bound state energies can be found by solving Eq. (33).
Some features of two hole bound states in a square lattice will be described. Figure 1 shows the two hole bound state energies along selected paths in the first Brillouin zone for selected values of J/t. For J/t >= 11.008 there exists a bound state for all values for total momentum. This indicates an instability in the ground state of the free hole system. Two distinct solutions exist along the paths GAMMA M and MX, whereas the two solutions are degenerate along GAMMA X due to the symmetry K_{x}= K_{y_.}For 4 <= J/t <= 11.008 the bound state solutions merge with the bottom of the two hole continuum along the GAMMA M and GAMMA X paths at K vec_{m_.}Figure 2 shows the K vec_{m_}for which this merging occurs along these paths. K vec_{m_}is obtained by substituting the energy of the bottom of the continuum for a particular value of K vec into Eq. (33) and calculating J/t. For example along GAMMA X we obtain
K_{mx}= K_{my}= 2 ^ arccos(J eta / 4t ) , (34)
eta = 1 over pi int_{0^{pi}dq}sub x {sin^{2}q_{x}}over {[(2+ cos q_{x})^{2}-1]^{1/2}}= 0.363^380^23 .
Note that two solutions exist for all values of K along MX for this range of J/t values. For J/t <4 only one solution exists along GAMMA M. Along MX one solution merges with the bottom of the continuum at (36) K_{my}= 2 ^ arccos(J/4t) . (35)
The other solution along MX exists for all values of K_{y_}as long as J/t>0.
It is interesting to look at the hole positions in position space to see whether the bound state is localized or not. By projecting PSI_{K}vec_ onto the state | r vec_{1}, r vec_{2}rangle_ with one hole located at site r vec_{1_}and another hole located at site r vec_{2_}we obtain
langle r vec_{1}, r vec_{2}| PSI_{K}vec rangle = ie^{{i}K vec cdot R vec} 1 over N sum from q g_{K}vec ( q vec ) sin ( q vec cdot r vec ) , (37)
where
r vec = r vec_{1}- r vec_{2}= n_{x}x bhat + n_{y}y bhat , R vec = { r vec_{1}+ r vec_{2}}over 2 (38)
and n_{x}, n_{y}= lcurl 0, +- 1 , +- 2 , .^.^. rcurl_ (the lattice constant is taken to be one).
This two hole wave function can be calculated analytically. Consider the one dimensional case first. The solution of Eq. (33) is
curlep = J over 2 [ 1+ ( 4t over J cos K/2 right )^{2}right ] (39)
with the condition (4t ^ cos K/2)^{2}/ J^{2}< 1 where curlep = E_{0}- E_. Substitution of the energy eigenvalue back into the equation for g_{K}(q)_ yields
g_{K}(q) = {2^ sin q} over {1+ ( 4t over J cos K over 2 right )^{2}+ 8t over J cos K over 2 cos q} A_{K} , (40)
where A_{K_}is a normalization factor.
To obtain the wave function we must calculate the integral
scrS_{n}( alpha ) = 1 over pi int_{0^{pi}dq}{sin (q) sin (q n)} over {alpha + cos ( q)} (41)
for n= lcurl +- 1 , +- 2 , +- 3 , .^.^. rcurl. We obtain
scrS_{n}( alpha ) mark = 1 over pi int dq {cos (n-1)q} over {alpha + cos q} - 1 over pi int dq { cos (nq) cos q} over {alpha + cos q}
lineup = alpha C_{n}( alpha ) + C_{n-1}( alpha ) , (42)
where
C_{n}( alpha ) = 1 over pi int_{0^{pi}dq}{cos ( qn)} over {alpha + cos q} . (43)
A recursion formula for C_{n_}can be derived by using the appropriate recursion formula for cos ( qn):
C_{n}( alpha ) =- 2 alpha C_{n-1}( alpha ) - C_{n-2}( alpha ) . (44)
This is a homogeneous linear difference equation with initial conditions
C_{0}( alpha ) = 1 over {( alpha^{2}- 1)^{1/2} }(45)
and
C_{1}( alpha ) =1- alpha C_{0}( alpha ) . (46)
It can be solved by the choice C_{n}( alpha ) = c b_{alpha^{n_.}We}obtain
C_{n}( alpha ) = 1 over {( alpha^{2}-1 )^{1/2}}[- alpha + ( alpha^{2}-1)^{1/2}]^{n} . (47)
Summarizing the results for one dimension gives
langle r_{1}r_{2}| psi_{K}rangle = scrN_{K}e^{{i(K/2)}(r_{1}+ r_{2})} [ {( alpha_{K^{2}-}1 )^{1/2}}- alpha_{K}]^{{|r}sub 1 - r_{2}|} L
with r_{1}!= r_{2_}and (49) alpha_{k}= {1+ ( 4 over J cos K over 2 right )^{2}}over {8 over J cos K over 2} (48)
and scrN_{K_}is a normalization factor.
In two dimensions there are two bound state solutions. For the solution corresponding to q_{i}= q_{x_}in Eq. (33) we obtain
langle r vec_{1}r vec_{2}| psi_{K}vec rangle_{1}= scrN_{K}vec e^{{i}K vec / 2 cdot ( r vec_{1}+ r vec_{2})} I_{n^{x}(}alpha_{K}vec ) , (50)
where
I_{n^{x}(}alpha_{K}vec ) = 1 over pi int_{0^{pi}dq}sub y scrS_{{n}sub x} ( alpha_{K}vec , q_{y}) cos ( q_{y}n_{y}) , L (51)
scrS_{{n}sub x} ( alpha_{K}vec , q_{y}) = alpha_{K}vec ( q_{y}) C_{{n}sub x} (alpha_{K}vec ( q_{y}) )+ C_{{n}sub x -1} (alpha_{K}vec ( q_{y}) ) , L (52)
alpha_{K}vec ( q_{y}) = {curlep + 4 ^ cos ( K_{y}/2) cos q_{y}}over {4^ cos ( K_{x}/ 2)} , L (53)
and C_{{n}sub x} (alpha_{K}vec ( q_{y}) )_is given by Eq. (47). For the other solution a similar result is obtained
langle r vec_{1}r vec_{2}| psi_{K}vec rangle_{2}= scrN_{K}vec e^{{i}K vec / 2 cdot ( r vec_{1}+ r vec_{2})} I_{n^{y}(}alpha_{K}vec ) , (54)
where I_{n^{y}(}alpha_{K}vec )_ is defined from Eqs. (51) and (52) with n_{x}, q_{y_}replaced by n_{y}, q_{x_.}
The two hole wave functions take especially simple forms for K vec = [ pi ,0]. We find that
I_{n^{x}=}c_{1}left "{" J over 8t - [ 1+ ( J over 8t right )^{2}right ]^{1/2}right "}"^{{|n}sub y |} ( delta_{{n}sub x ,1} + delta_{{n}sub x ,-1} ) L (55)
and
I_{n^{y}=}c_{2}(- 4t / J)^{{|n}sub y |} delta_{{n}sub x ,0} , (56)
where n_{y}!= 0_ and c_{1_}and c_{2_}are constants involving J/t. The first solution has one hole at the origin and the other hole on a line parallel to the y axis but displaced by one lattice spacing in either the plus or the minus x direction. The second solution has a hole at the origin and the second hole along the y axis. Both solutions are very localized in the x direction with the amount of y localization depending on J/t. The average hole separation can be calculated from
langle l rangle = {sum_{n}l_{n}( I_{n^{i})^{2}}over}{sum_{n}( I_{n^{i})^{2}} , }(57)
where i = lcurl x,y rcurl and
l_{n}= (1+n_{y^{2})^{1/2} }for i= x ,
l_{n}= n_{y} for i=y . N (58)
Figure 3 shows langle l rangle as a function of J/t for the two cases. It is apparent that both solutions show the hole remaining tightly bound until the binding energy is very small compared to J/t. This occurs at J/t=0 for the first case and J/t =4 for the second case.
In the last section we explored how the ferromagnetic background can affect holes moving in the lattice. The moving holes should also affect the magnetic states of the system. We will explore here the possibility that a hole and a magnon interact to form a bound states. The Hamiltonian H will be diagonalized in a subspace scrF_{11_}with one hole and one spin deviation placed on a lattice with all up spins. A spanning set for scrF_{11_}is formed from the states
| r vec_{i}, r vec_{j}rangle = U_{01^{i}U}sub {1 bar 1}^{j}| F rangle (59)
An eigenstate phi_{K}vec_ of H can be constructed in the subspace scrF_{11_}
| phi_{K}vec rangle = sum from q vec f_{K}vec ( q vec ) | k vec _{1}, k vec_{2}rangle . (60)
We can obtain the eigenvalue equation for H:
[ E-E_{0}- t ( K vec over 2 + q vec right ) +J ( K vec over 2 - q vec right ) right ] f_{K}vec ( q vec ) = 1 over N sum from {q vec prime} [ t( q vec prime + q vec ) - J ( q vec prime - q vec right ) + J ( K vec over 2 - q vec right ) -t ( K vec over 2 + q vec right ) right ] f_{K}vec ( q vec prime ) , (61)
where
E_{0}=- NJz over 4 + case 3 over 2 Jz . (62)
Similar to the two hole case, Eq. (61) yields a band of free hole magnon states for each K vec with E ranging from the minimum to the maximum value of scrE_{K}vec ( q vec_) where
scrE_{K}vec ( q vec ) = E_{0}+ t ( K vec over 2 + q vec right ) - J ( K vec over 2 - q vec right ) . (63)
To obtain the hole magnon bound state solutions for two dimensional square lattices we must solve the determinant equation
| open1 bhat -C| =0 (64)
with C given by
( matrix { ccol { a alpha_{2000}above a alpha_{1100}above a alpha_{1010}above a alpha_{1001}above a alpha_{1000}}ccol { - b alpha_{1100}above - b alpha_{0200}above - b alpha_{0110}above -b alpha_{0101}above - b alpha_{0100}}ccol { a alpha_{1010}above a alpha_{0110}above a alpha_{0020}above a alpha_{0011}above a alpha_{0110}}ccol { - b alpha_{1001}above -b alpha_{0101}above - b alpha_{0011}above -b alpha_{0002}above -b alpha_{0001}}ccol { c_{1}above c_{2}above c_{3}above c_{4}above c_{5}}}right ) . (65)
We have used the definitions
alpha_{ijkl}= 1 over N sum from q vec {( cos q_{x})^{i}( sin q_{x})^{j}( cos q_{y})^{k}( sin q_{y})^{l}}over {E - scrE_{K}vec ( q vec )} , (66)
c_{r}= mark - a^ cos (K_{x}/2) alpha_{r1}+ b^ sin ( K_{x}/2) alpha_{r2} ,
lineup - a ^ cos ( K_{y}/2) alpha_{r3}+ b ^ sin ( K_{y}/ 2) alpha_{r4} , (67)
where alpha_{ri_,}i=1,2,3,4, are the integrals alpha_{ijkl_}involved in the first four entries of the rth row in C vec. Also, a=2(t-J) and b=2(t+J).
The largest binding energy occurs at K vec =[0,0] and solutions only exist for 0.466^ 942 <= J/t <= 2.141^593. At K vec = [ pi ,0] a bound state can also exist, but it occurs for a smaller J/t interval: 0.710^ 952 <= J/t <= 1.406^565. At K vec = [ pi , pi ] the bound state is always coincident with the bottom of the hole magnon continuum. The energy is the lowest at K vec = [ pi , pi ].
It is worthwhile to compare the two hole bound state situation with this hole magnon bound state in the square lattice. A two hole bound state always exists for any J/t >0, but the hole magnon bound states exist only for a restricted range of J/t values. This result is physically reasonable since when t is small compared to J the hole remains fairly localized: It is harder for it to hop through the lattice at least compared to the ease with which a magnon propagates. Binding of this slowly moving hole with the quickly moving magnon becomes problematic. When t is large compared to J the hole propagates easily compared to the magnon so that again binding becomes problematic.
The largest binding energy occurs for J/t=1. We will present calculations of the energy along the high symmetry path GAMMA MX GAMMA of the first Brillouin zone for this case. Since a =0 there is considerable simplification in the equations.
For K_{y}=0_, along GAMMA M two separable equations for E can be derived. The first is
1+4 alpha_{0002}=0 (68)
and gives a finite binding energy at K vec = [0,0] and at K vec = [ pi ,0]. The other equation is
1-4^ sin ( K_{x}/2) alpha_{0100}+ 4 alpha_{0200}=0 (69)
and gives a solution that merges with the continuum at K vec = [ pi ,0]. One bound state exists along MX. Along the path GAMMA X the two solutions are degenerate. Figure 4 shows the K vec dependence of the bound state energies along the three paths considered. It is interesting to note that for all K vec values [except at K vec = ( pi , pi )] the bound state of a hole with a flipped spin lies below the state in which the hole moves with the same momentum in a ferromagnetically polarized environment.
The wave functions of hole magnon bound states can be found with the same procedure as for the two hole bound states. For the case of J=t and total momentum K vec =0 the wave function has the simple form
| psi_{{K}vec =0} rangle = 1 over 4N sum from r lcurl mark | r vec , r vec + x bhat rangle - | r vec , r vec - x bhat rangle
lineup + | r vec , r vec + y bhat rangle - | r vec , r vec - y bhat rangle rcurl , (70)
where
| r vec , r vec prime rangle = U_{01}( r vec ) U_{{1}bar 1} ( r vec prime )| F rangle . (71)
It is obvious that the bound state has the hole and flipped spin localized next to each other.
In the case of K vec = [ pi , 0], i.e., at the zone boundary point M, the hole magnon bound state wave function becomes
| psi_{{K}sub x = pi , K_{y}=0} rangle L
= C sum from { R vec , n_{x}, n_{y}= +- 1} mark e^{{i}K vec cdot R vec} (-1)^{{(n}sub x + n_{y})/2}
lineup times ( {sqrt 5 -1} over 2 right )^{{|n}sub x |} left | R vec + r vec over 2 , R vec - r vec over 2 rangle2 .
(72) It has a form similar to the wave function for one of the two hole bound states at point M. The average distance between the hole and flipped spin is 1 in the units of the lattice constant.
We have studied the interplay of charge and spin excitations in a quantum ferromagnet model. Exact solutions of two hole and hole magnon bound states, resembling the two magnon bound states of a ferromagnet,^{8}can be obtained for a general lattice.
In this paper we have shown the conditions for obtaining two hole and hole magnon bound states, binding energies and wave functions for one- and two dimensional lattices. The physical picture of bound states can be illustrated by using their wave functions. For example, by studying the two hole wave function in real space we find that pairing is predominantly nearest neighbor except in a very narrow region of J/t close to the critical value. We have also found that if the effective spin spin interaction J is sufficiently strong compared with the hole hopping integral t, the ground state of the system is not just ferromagnetically aligned spins with free hole hopping; the energy of the system with the two hole bound state is lower.
The other multihole bound states have to be studied to determine the ground state with finite density of holes. The kind of ground state that this system has depends on the competition between the energy gain due to bound holes breaking fewer exchange bonds than separate holes and the kinetic energy gain which separates holes have from being more mobile. We expect that as J/t is increased from zero the system goes through a regime where the two hole bound state is lowest in energy, then a regime is obtained where the three hole bound state is lowest, and so on until finally a phase separation occurs. The two hole bound states investigated here will be relevant in that window of J/t where they are the true ground state. Finally, we note that the bound states should influence the low temperature thermodynamics of such systems.
This work was supported in part by the DARPA sponsored Florida Initiative in Advanced Microelectronics and Materials under Contract No. MDA972 88-J-1006 and by the Center for Materials Research and Technology of The Florida State University.
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