Chapter 7 Thermodynamics versus local density fluctuations in the two-dimensional Hubbard model
(a) U/t=1.6(2)
0.0 0.5 1.0 1.5
0.00 0.25 0.50 0.75 1.00
Structure factor S(q=0)
(b) U/t=8.2(5)
0.0 0.5 1.0 1.5
0.00 0.25 0.50 0.75 1.00
(c) U/t=12.0(7)
0.0 0.5 1.0 1.5
0.00 0.25 0.50 0.75 1.00
0.5 1.5 2.5 3.5
kBT (t)
(d)
0.0 0.5 1.0 1.5
Filling n
i 0.00
0.25 0.50 0.75 1.00
On-Site fluctuations n i2 /n i (e)
0.0 0.5 1.0 1.5
Filling n
i 0.00
0.25 0.50 0.75 1.00
(f)
0.0 0.5 1.0 1.5
Filling n
i 0.00
0.25 0.50 0.75 1.00
Figure 7.3: Comparison of the thermodynamic and local density fluctuations in the two-dimensional Hubbard model. Left column (a,d):U/t=1.6(2), middle column (b,e):U/t =8.2(5), right column (c,f):U/t=12.0(7). Top row: static structure factor S(k =0)measuring density fluctuations in thermodynamically large volumes;
bottom row: density fluctuations at a single site of the optical lattice. Dashed line: on-site fluctuations in the non-interacting limit. Solid line: on-site fluctuations in the infinite-interactions limit. Temperature is encoded as colour (see legend in (c)).
which is indicated by the dashed line in Figure7.3. The measured local fluctuations for U/t = 1.6 (Fig. 7.3(d)) reproduce the ideal Fermi gas prediction very well, with an additional small suppression of the fluctuations which we attribute to the finite interaction strength. In the limit of infinite repulsive interactions (U= +∞) andni ≤ 1, either zero or one fermion occupy one lattice site and hence
δn2i/ni =1−ni. (7.16)
Forni > 1 we find in the same limit
δn2i/ni =3−ni−2/ni (7.17)
The on-site fluctuations in the strongly-interacting case are insensitive to temperature unlesskBT ∼U for which thermally induced double-occupancies contribute to the fluctuations. In general, the on-site density fluctuationsδn2i/nfor a given filling and temperature are higher than the fluctuationsδN2/Nof the same data set in the thermodynamic limit. This shows that the thermodynamic fluctuations contain a non-local contribution from density-density correlations on different length scales. In order to understand the difference between local and thermodynamic fluctuations, we transcribe the fluctuation-dissipation
96
theorem7.13to a lattice with discrete sites labelled by the indicesi j
κ= 1
a2kBT
"
δn2i +Õ
j,i
hnˆinˆji − hnˆii hnˆji
#
. (7.18)
Here, we have separated the local fluctuation δn2i from the non-local density correlations. On the microscopic level, the violation of the local fluctuation-dissipation theorem [141,142] is rooted in the spatial correlations of the density fluctuations and hence is governed by the nature of the underlying quantum state. In a perfectly localised state, such as a band insulator at zero temperature, the off-site correlations are zero since the correlation function factorises hnˆinˆji = hnˆii hnˆji. In contrast, for a delocalised state at zero temperature, such as a Fermi gas in a partially filled Bloch band, the compressibility cannot be described by local fluctuations alone since the off-site correlations are not negligible. At finite temperature, however, the thermal correlation length limits the range of density correlations and for a classical gas the correlations are restricted to on-site fluctuations. In order to highlight this effect, Figure7.4shows the non-local density correlation
δnn.l.2 =a2κkBT−δn2i (7.19)
as a function of temperature and lattice filling. For low filling,ni . 0.4, the results for all interaction strengths agree very well with the non-interacting Fermi gas in a two-dimensional square lattice with nearest-neighbour tunnelling (solid line). This shows that at low filling the atoms delocalise in the lattice irrespective of the explored interaction strength and that for low temperatures the delocalisation gives rise to non-local density correlations which significantly affect the compressibility. At ni ≥ 0.5 we observe the onset of interaction effects in the deviations from the ideal Fermi gas theory. While the weakly-interacting data (blue) remain closer to the ideal Fermi gas prediction, the strongly-interacting data exhibit a suppression of long-range density correlations. Finally, at half-filling, we observe a clear distinction between the two cases. For example, theU/t=12 data show an almost complete suppression of off-site density correlations, while theU/t=1.6 data are still close to the free Fermi gas expectation.
The former is a signature for the atoms having formed a localised Mott insulator, the thermodynamics of which is entirely described by local quantities in the density sector. Finally, Figure 7.5shows the residual non-local density fluctuationsδn2n.l. as a function of the atom number per sitenfor the lowest temperatures achieved at each interaction strength. The data highlight that, within the experimental resolution, at low filling the off-site density correlations are essentially independent of interaction strength and equal to the ideal Fermi gas prediction. Upon approaching half-filling, off-site correlations are highly suppressed for the strongly-interacting gas and signal the onset of a Mott insulator with localised spins.
Chapter 7 Thermodynamics versus local density fluctuations in the two-dimensional Hubbard model
(a)
ni=0.10(10)
-0.2 -0.1 0.0
a2 k BT - n i2
U/t=1.6(2) U/t=6.1(4) U/t=8.2(5) U/t=10.3(6) U/t=12.0(7)
(b)
ni=0.25(10)
kBT (t) -0.2
-0.1 0.0
a2 k BT - n i2
(c)
ni=0.50(10)
kBT (t) -0.2
-0.1 0.0
a2 k BT - n i2
(d)
ni=0.75(10)
kBT (t) -0.2
-0.1 0.0
a2 k BT - n i2
(e)
ni=1.00(10)
0.5 1.0 1.5 2.0 2.5 3.0 3.5
kBT (t) -0.2
-0.1 0.0
a2 k BT - n i2
Figure 7.4: Non-local density-density correlations for different lattice site occupationsni. The solid line is the theoretical expectation of the ideal Fermi gas in a two-dimensional lattice. Colour code: purple:U/t =1.6(2), light blue:U/t=6.1(4), greenU/t=8.2(5), yellow:U/t=10.3(6), red:U/t=12.0(7).
98
0.00 0.25 0.50 0.75 1.00 1.25 1.50 -0.3
-0.2 -0.1 0. 0
a2 k BT -n i2
U/t=1.6(2) U/t=6.1(4) U/t=8.2(5) U/t=10.3(6) U/t=12.0(7)
n
Figure 7.5: Non-local density-density correlations for the lowest temperatures achieved in the experiment. Colour code: purple: U/t=1.6(2),kBT/t =0.75(3), light blue: U/t =6.1(4),kBT/t =0.64(1), green: U/t =8.2(5), kBT/t =0.65(3), yellow:U/t=10.3(6),kBT/t =0.93(8), red:U/t =12.0(7),kBT/t=0.65(2). The solid line shows the prediction of the ideal Fermi gas on a square lattice forkBT/t=0.65.
Observation of antiferromagnetic spin correlations in the two-dimensional Hubbard model
This chapter presents results published in "Antiferromagnetic correlations in two-dimensional fermionic Mott-Insulating and metallic phases" by J. H. Drewes, L. A. Miller, E. Cocchi, C. F. Chan, N. Wurz, M.
Gall, D. Pertot, F. Brennecke and M. Köhl (compare Ref. [143]) and closely follows the description therein.
This chapter presents a study of spin correlations in the 2D Hubbard model. Section8.1motivates our work. Section8.2provides a summary of the measurement protocol and thermometry. Section8.3 presents the method used to infer the amount of spin correlations from the experimental data. Finally, Sections8.4and8.5presents the uniform spin structure factor as a function of interaction, temperature and doping.
8.1 Motivation and previous work
Quantum magnetism arises from repulsive short-range interactions at low temperatures. The occurrence of magnetic long range order however, depends crucially on the interplay of interactions, lattice geometry, dimensionality and doping. Even though the consequences of this interplay are not yet fully understood, quantum magnetism is believed to be connected to a range of complex phenomena in the solid state, most prominently, in the context of high-Tc superconductivity [2,13]. Ultracold atomic Fermi gases in optical lattices are an ideal platform for studying quantum magnetism in order to reveal the microscopic origin of the magnetic susceptibility in the Hubbard model. Previous experiments have detected evidence for antiferromagnetic (AFM) correlations in measurements averaging over inhomogeneous systems [25, 128]. In parallel with the work presented here, the spin correlation function has been probed with site resolved resolution using quantum gas microscopes [36,38,144]. More recently, a system with quasi long-range order could be realised in which the correlation length extended over the whole sample in a system consisting of a small number of lattice sites [39].
Here, we detect the emergence of AFM correlations through the uniform magnetic susceptibility and study its dependence on temperature, interaction strength and filling. To this end, we employ high-resolution absorption imaging (cf. Chapter 5) to record the density distributions of both spin components in a single measurement and determine the spin structure factor as well as the magnetic susceptibility. Since the spin correlations are expected to be strongest at half filling and sensitively depend on the filling, we employ a spatially-resolved detection scheme and, due to the spatially varying trapping potential, determine the spin structure factor as a function of doping. We compare our results
Chapter 8 Observation of antiferromagnetic spin correlations in the two-dimensional Hubbard model
Singles (a)
-100 -50 0 50 100 x (a) -100
-50 0 50 100
y (a)
Singles
-100 -50 0 50 100 x (a)
0 0.5
Sq=0
(b)
-100 -50 0 50 100 x (a) -100
-50 0 50 100
y (a)
0.0 0.3
(c)
-12 -8 -4 0
U/2 (t) 0.0
0.1 0.2
Spin correlations
Sq=0
m2 m2NLCE
Figure 8.1: Spin-resolved singles density distribution and uniform magnetic structure factor. (a) Average distributions of spin-up and spin-down atoms on singly occupied lattice sites. The data shown are averaged over 36 experimental realisations for an on-site interactionU/t =8.2(5). (a) Map of the uniform magnetic structure factorSq=0mag. (c) Equipotential average of the uniform magnetic structure factor (circles) and the local momentC0,0(diamonds). For the data shown, we extract a temperaturekBT/t=0.63+0.09−0.02from fits of the averaged spin distributions to NLCE data (solid line).
to state-of-the-art numerical calculations and find agreement where results are available, which is, in particular, at half filling.