2.4 A QMC study of the Hubbard model doped with nonmagnetic impurities
2.4.4 Two-leg ladder
Next we examine the effects of nonmagnetic impurity doping in a two-leg Hubbard ladder of size 20×2. The interaction potential is taken to be U = 6t. The impurity site configurations are chosen in such a way that the ladder remains always connected.
We calculate the magnetic susceptibility defined as, χ(q)= 1
N X
i j
e−iq(ri−rj) Z β
0
dτD
(ni↑(τ)−ni↓(τ))(nj↑(0)−nj↓(0))E
(2.53)
as a function of temperature, T for the pure and doped ladder. Here β is the inverse tem- perature and τ is the imaginary time. The results for the uniform magnetic susceptibility, χ0(T ) (= χ(q = 0,T )) where T = 1/β, are shown in Fig. 2.6. For the pure case, suscepti-
0.5 1 1.5
0 0.5 1 1.5 2
χ0(T)
T
pure ladder 5 impurities
〈n〉 = 1.0
F2.6: Uniform susceptibility, χ(q = 0) versus temperature, T for the pure and doped ladder at half filling.
bility increases initially with decreasing temperature, reaches a maximum and then decreases sharply at further lower temperatures. However when the ladder is doped with nonmagnetic impurities,χ0(T ) shows a cusp-like feature followed by a dramatic increase at lower tempera- tures. Similar behavior of magnetic susceptibility was observed experimentally in Ref. [181].
The results for the pure case can be understood as follows. At higher temperatures spins are almost free due to weak magnetic correlations and hence susceptibility shows the expected Curie-Weiss like increase with decreasing temperature. As the temperature is further low- ered, spin fluctuations are reduced and spins start to form singlet pairs which results in the decrease of susceptibility below a certain critical temperature. When the ladder is doped with nonmagnetic impurities, free local moments are induced around impurity sites[186] giving rise to the rapid increase in susceptibility at low temperatures. The occurrence of the cusp has the following explanation. There exists two characteristic temperature scales in the system.
One, below which susceptibility starts to decrease, marks the onset of spin singlet formation.
The other, slightly lower, signals the onset of localization of induced free magnetic moment as indicated by the sharp increase in susceptibility. The observed cusp is a result of these two phenomena occurring at two different temperatures.
To examine the enhancement of magnetic correlations by nonmagnetic impurities, we cal-
at and away from half filling. Fig.2.7shows the results for spin correlations along a leg for the pure ladder and the ladder doped with one nonmagnetic impurity, at half filling. It is observed
-0.4 -0.2 0 0.2 0.4
2 4 6 8 10
c(i, j)
rj (ri = 1) β = 2 β = 5 β = 10
〈n〉 = 1.0
pure ladder
(a)
-0.4 -0.2 0 0.2 0.4
2 4 6 8 10
c(i, j)
rj (ri = 1) β = 2 β = 5 β = 10
〈n〉 = 1.0
1 impurity at r1 = 0
(b)
F2.7: Spin spin correlations, c(i,j) between sites ri(=1) and rj(=2,. . . ,10) belonging to the same leg for (a) the pure ladder and (b) the ladder doped with one nonmagnetic impurity placed at site ri =0. The sites in the leg are numbered as 0,1,2, . . .respectively. Band filling,
hni=1.0. The values ofβ(in units of t) are shown in the figures.
that in the presence of the impurity, the antiferromagnetic (AF) correlations are enhanced, a feature also noted in studies on Heisenberg ladders[185, 187]. This effect is more prominent at lower temperatures. Forβ = 2 shown in figure, no noticeable impurity effect is observed.
For the two other values of β, viz. 5 and 10, it is interesting to note that in addition to the nearest neighbor, the long distance spin correlations are also enhanced by the impurity. We carried out similar calculations for a 10×10 square lattice and found that the enhancement at larger distances for the ladder is greater than that in the square lattice. Away from half filling, at electron densitieshni = 0.95 and 0.90, the effect of an impurity on spin correlations for β < 5 is found to be weaker. Simulation at still lower temperatures is prevented by the nega- tive sign problem. Further, we calculate the spin structure factor S (q)= 1/NP
i jeiq.(ri−rj)c(i, j) at q = (π, π) for various band fillings with different number of impurities, to study the effect of finite impurity concentration on the magnetic order. The results are shown in Fig.2.8. At half filling, S (π, π) increases for small impurity concentration indicating strengthening of AF order in the ladder, which is in agreement with experimental results[181]. For larger impurity concentrations, long range spin correlations are heavily destroyed by the presence of impuri- ties and thus S (π, π) decreases. Away from half filling, the enhancement of spin correlations by impurity is anyway small for the range of temperatures considered here. Hence S (π, π)
decreases slowly with increasing impurity concentration due to the dilution of magnetic sites.
2 2.5 3 3.5 4
0 2 4 6
S(π,π)
No. of impurities
〈n〉 = 1.00
〈n〉 = 0.95
〈n〉 = 0.90 β = 5
F2.8: S (π, π) versus number of nonmagnetic impurities at various band fillings.
We also examine the effects of nonmagnetic impurities on the spin wave excitation modes in the ladder. We calculate the spin wave velocity, vswas a function of impurity concentration in the following way. The magnon dispersion, ω(q) was calculated for the two dimensional Hubbard model using an approximate relation given by,ω(q)= 2S (q)/χ(q)[188]. We use the same relation to calculate magnon dispersion for the ladder. Then the spin wave velocity for long wavelength (q → 0) spin wave modes is obtained using vsw(q) = ω(q)/q. In Fig. 2.9, we show the results for vsw(q) for a few small values of q calculated for different number of impurities forβ= 8. The plot clearly suggests that spin wave velocity gets suppressed due to softening of the spin wave excitations.
Finally, to verify whether nonmagnetic impurity affects hole mobility as suggested by experiments[178], we calculate the kinetic energy,hc†iσcjσiof the electrons around an impurity site for the doped ladder and compare it with that for the pure ladder, slightly away from half filling. The results of our calculation do not show any evidence of any effect of the impurities on the mobility of the electrons neighboring it. To quote some numbers, the value ofhc†iσcjσi for two nearest neighbor sites i, j for the pure ladder is 0.165 for β = 5,hni = 0.95. For the doped case also, the value for any pair around an impurity site is found to be the same within statistical error bars.
0.6 0.8 1 1.2
(π/10) (3π/10) (5π/10)
vsw(q)
qx (qy = 0) imp = 0 imp = 1 imp = 5 β = 8
〈n〉 = 1.0
F 2.9: Long wavelength spin wave velocity, vsw(q) for the pure and impurity doped ladder.β=8 andhni=1.0.