time
6.3 Observation of pair formation
0.25 0.5 0.75 1.0 0.2
0.6 1 1.4 1.8
S(q=0)
a
0.25 0.5 0.75 1.0
−2 0 2 4 6 8 10
∫(g(2)(r)−1)d2r
b
0.25 0.5 0.75 1.0
n
0 2 4 6 8 10 12
∫(g(2) ↑↓(r)−1)d2r
c
−2
−4
−6
−8
−10
U/t
Figure 6.4:Density correlations. aDensity structure factor at zero momentumS(q=0)for various interaction strengths and filling. b∫
(g(2)(r) −1)d2r versus interaction strengthsU/t. The black solid line shows the non-interacting expectation at the lowest temperature in the data sets. Data points above the horizontal dashed line signal that the unequal-sping(↑↓2)(r)outweighs its equal-spin counterpart, and vice versa. cInteracting contribution∫
(g(↑↓2)(r) −1)d2ris obtained by subtracting the equal-spin contribution∫
(g↑↑(2)(r) −1)d2r. Data are forU/t =−1.83,−3.90,−6.09,−7.62 and−9.61, and the corresponding temperatureskBT/tof the data set are 1.26(4), 1.31(5), 1.73(8), 2.05(11)and 2.16(8)respectively.
dominates over the Pauli blocking.2
In order to quantitatively compare the contributions from the Pauli principle and the attractive interaction, we note that the equal-spin contribution∫ h
g(↑↑2)(r) −1i
d2r can be calculated using the tight-binding dispersion relation and temperature, see solid line in Fig.6.4b. Strictly speaking, this calculation is exact only for the non-interacting case because interactions in principle modify the dispersion from a simple sinusoidal energy band. However, in the low-filling regime, this estimate remains a faithful approximation since most of the occupied part of the energy band remains harmonic. Subtracting the equal-spin contribution from∫ h
g(2)(r) −1i
d2r, we obtain the interacting
2While it is true that the tightly-bound local dimers would become composite bosons in the large-Ulimit, our measurement scheme does not detect the pair-pair correlation. The bunching as reflected in theg(2)(r), therefore, is not a sign for the bosonisation of fermions.
∫ h
g(↑↓2)(r) −1i
d2r, as shown in Fig.6.4c. We observe that the interacting contribution maintains a similar dependence asS(q =0). For either decreasing filling or increasing interaction strength, the interacting pair correlation increases. This highlights the parameter space at which the interaction effect ong2↑↓(r)is most prominent.
6.3.3 Pair correlation length
To gain further insight into the signature of local pairing, we turn to estimate a length scale up to which the pair correlation extends. We start by noticing that the interacting pair correlation amplitude atr =0 is
g↑↓(2)(0)=4 hn↑n↓i
hn↑+n↓i2 = D
(S↑+D)2. (6.10)
This implies that the amplitude can be directly obtained from our local density measurement of singly- and doubly-occupied site occupations. Although the analytical form of the unequal-spin pair correlation function is not known, an exponential decaye− |r|/ξis expected to be a good approximation, see inset of Fig.6.5a. Combining the knowledge of the integral∫ h
g(↑↓2)(r) −1i
d2rand the amplitude atr =0, we then infer the characteristic length scaleξas
2πξ2≈
∫ h
g↑↓(2)(r) −1i d2r h
g(↑↓2)(0) −1i . (6.11)
In Fig.6.5a, we plot the estimated pair correlation lengthξ as a function of fillingn. For low filling n. 0.5, we observe a correlation lengthξas large as 0.92(4)afor the lowest interaction strength atU/t = −1.83, wherea =532 nm denotes the in-plane lattice spacing. Although the attractive interaction is purely on-site, we observe that its effect extends beyond local site, similar to the Pauli blocking leading to beyond-local density suppression.
Filling dependence The decreasing trend ofξas filling increases could be attributed to two reasons.
First, for dilute filling, particles are described by delocalised wave-packets. With an increasing filling, however, the reduced inter-particle spacing limits the available range for pairing. Second, such localisation of particles means that the continuous integral of Eq. (6.11) starts to deviate from the discretised sum in a lattice. If the interacting contribution is dominated by the local termg↑↓(0), the continuous approximation of Eq. (6.11) would result inξ/a≈p
1/2π≈0.4, which is in agreement with our observation in Fig.6.5a.
Interaction dependence Upon changingU/t, we observe that the correlation lengthξshrinks as interaction strength increases, as shown in Fig.6.5b. This signals the formation of tightly-bound pairs and this pairing behaviour is most prominent below quarter filling (n. 0.5). Above quarter filling, we do not observe a discernible trend ofξas a function ofU/t, as indicated by the lowest data points in Fig.6.5b. Although we do not expect any macroscopic long-range coherence in our
system, we observe the precursor at which attractive interaction induces the formation of local pairs.
And the dependence in density and interaction are in qualitative agreement with the low-temperature phase diagram, as shown in Fig.6.1.
Temperature dependence Last but not least, we investigate the temperature dependence of the correlation lengths. For temperatures below the critical temperature and low density, the pairing would be described by a BCS-BEC type behaviour. Although the temperatures reached in our experiments remain in the normal phase, we observe a resemblance in the behaviour of the pair correlation length. As shown in Fig.6.5c, we plotξas a function of temperaturekBT/tatn=0.2.
The correlation length rises as temperature decreases for the weakly attractive case. In the strongly attractive case, we observe a much less significant trend in temperature dependence. Since the pairing occurs at an energy scale of the interactionU, it is also informative to recast the temperature with respect to|U|. In Fig.6.5(d), we compare the temperature with respect to the interaction energy by plotting the correlation lengths as a function ofkBT/|U|. Despite a lower achievedkBT/|U|for large interaction strengths, the correlation length remains small due to the energetically favourable dimer
0.25 0.5 0.75 1.0
n 0.3
0.4 0.5 0.6 0.7 0.8 0.9 1.0
ξ/a
a
2 4 6 8 10
U/t 0.3
0.4 0.5 0.6 0.7 0.8 0.9 1.0
ξ/a
b
1.0 1.5 2.0 2.5 3.0 3.5
kBT/t 0.3
0.4 0.5 0.6 0.7 0.8 0.9
ξ/a
c
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 kBT/|U|
0.3 0.4 0.5 0.6 0.7 0.8 0.9
ξ/a
d
−2
−4
−6
−8
−10
U/t
g (r)
4<n n >
<n + n >
(2) 2
0 r
1 ξ
Figure 6.5:Inferred pair correlation length. aCorrelation lengthξversus fillingn. As the interaction strength increases, we observe that the correlation length shrinks for low fillingn.0.5. For increasing filling, we observe thatξsettles at approximately the limit of the continuous approximation. The inset exemplifies the estimation of the length scale according to Eq. (6.11), which is effectively dividing the area bound by the g(↑↓2)(r)and the uncorrelated value (integral) by height obtained by local pair correlationsg(↑↓2)(r)(amplitude). b Correlation lengthξfor specificnversusU/t.n=0.1, 0.15, 0.2, 0.3 and 0.5 correspond to square, up-triangle, diamond, down-triangle and pentagon markers respectively. Temperatures of data points shown inaandbare same as those in Fig.6.4. cCorrelation lengthξversus temperaturekBT/t atn=0.2. For low filling, we observe in general a decreasing trend as temperature rises. While already atn=0.5, no significant temperature dependence is observed. The solid lines are linear fits to the data points. dCorrelation lengthξ versus temperaturekBT/|U|atn=0.2. When rescaled with respect to the interaction strength, the temperatures reached in the weakly-interacting case are higher than the ones in the strongly interacting case.
state. For weak interactions, the pair correlation length rises at much higherkBT/|U|, signalling the tendency to delocalise and form longer-range pairs.