Optical Signatures of Antiferromagnetic Ordering of Fermionic Atoms in an Optical Lattice

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Optical Signatures of Antiferromagnetic Ordering of Fermionic Atoms in an Optical Lattice

Francisco Cordobes Aguilar, Andrew F. Ho, and Janne Ruostekoski

Phys. Rev. X 4, 031036 – Published 2 September 2014

Abstract

We show how off-resonant light scattering can provide quantitative information on antiferromagnetic ordering of a two-species fermionic atomic gas in a tightly-confined two-dimensional optical lattice. We analyze the emerging magnetic ordering of atoms in the mean-field and in random phase approximations and show how the many-body static and dynamic correlations, evaluated in the standard Feynman-Dyson perturbation series, can be detected in the scattered light signal. The staggered magnetization reveals itself in the magnetic Bragg peaks of the individual spin components. These magnetic peaks, however, can be considerably suppressed in the absence of a true long-range antiferromagnetic order. The light scattered outside the diffraction orders can be collected by a lens with highly improved signal-to-shot-noise ratio when the diffraction maxima are blocked. The collective and single-particle excitations are identified in the spectrum of the scattered light. We find that the spin-conserving and spin-exchanging atomic transitions convey information on density, longitudinal spin, and transverse spin correlations. The different correlations and scattering processes exhibit characteristic angular distribution profiles for the scattered light, and e.g., the diagnostic signal of transverse spin correlations could be separated from the optical response by the scattering direction, frequency, or polarization. We also analyze the detection accuracy by estimating the number of required measurements, constrained by the heating rate that is determined by inelastic light-scattering events. The imaging technique could be extended to the two-species fermionic states in other regions of the phase diagram where the ground-state properties are still not fully understood.

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Received 29 November 2013

DOI:https://doi.org/10.1103/PhysRevX.4.031036

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Francisco Cordobes Aguilar^1,2, Andrew F. Ho^1,*, and Janne Ruostekoski^2,†

¹Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom
²Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom

^*Corresponding author. andrew.ho@rhul.ac.uk
^†Corresponding author. J.Ruostekoski@soton.ac.uk

Popular Summary

Atoms can be used as emulators of strongly interacting phenomena from other areas of physics that are too complex even for numerical studies. Solid-state crystal structures can be simulated in a controlled, defect-free laboratory setting by confining cold atoms in a periodic optical lattice. Quantum magnetism in solids is thought to play a central role in high-temperature superconductivity, and simulating magnetism in cold-atom systems is fundamentally related to the development of accurate diagnostic methods. We propose an optical imaging technique to measure magnetic correlations of atoms in a cold-atom emulator.

Scattered light can be separated into elastic and inelastic components, where the former arises when an atom transitions back to its original state and the latter occurs when an atom transitions between two states. Similarly to the x-ray scattering from crystalline solids that reveals the spatial lattice structure in constructive interference peaks, the light intensity peaks from the elastically scattered atoms reveal the underlining magnetic ordering of the atoms. On the other hand, the inelastic component can be used to map the quantum correlations of atomic spins onto the fluctuations of light scattered off of atoms, which we undertake; the strength of the peaks and fluctuations of the scattered background light around the peaks can convey information about specific spin correlations or tell us whether or not the correlations have a long-range nature. Recent experiments have shown antiferromagnetic ordering in ultracold gases; we consider a tightly-confined two-dimensional lattice and show that antiferromagnetic order and spin excitations can be inferred from the Bragg peaks and spectral features of the scattered light where energy resolution is available.

Our imaging method provides several advantages over commonly used single-site atom microscopy: There is no need to scan atoms individually in large lattices; the fluctuations of the scattered light can also probe collective magnetic many-body excitations (magnons), and the scattered light from spin-changing transitions provides access to a larger variety of atomic correlation phenomena.

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Vol. 4, Iss. 3 — July - September 2014

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Figure 1
Illustration of Brillouin zone indicated by the outer (red) square and RBZ by the inner (blue) square in an $8 \times 8$ lattice. The filled squares represent filled momentum states in the Fermi sea at half-filling.
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Figure 2
Diagram showing the $(U, T)$ dependence of the staggered magnetization, $m$ , obtained by solving Eq. (29) in a $40 \times 40$ lattice.
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Figure 3
Schematic phase diagram of the 2D Hubbard model at half-filling (based on Fig. 2 from Ref. [60]). $T_{C, MFT}$ is the critical temperature computed from the gap equation, Eq. (29). $T_{X}$ is the temperature at which the crossover from short-range to (quasi-)long-range order happens.
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Figure 4
Transverse spin susceptibility for $U = 5 J$ and $q = (π / a, 0)$ computed with a finite convergence factor $δ = 0.1 J$ in a $20 \times 20$ lattice. (a) We show for the MFT case, the individual momentum terms $χ_{(0)}^{+ - (k)} (q, q; ω)$ of Eq. (42), $χ_{(0)}^{+ -} (q, q; ω) = \sum_{k} χ_{(0)}^{+ - (k)} (q, q; ω)$ . Each term contributes with a Lorentzian-shaped peak at $ω = E_{k} - E_{k + q}$ . (b) Imaginary (solid line) and real (dashed line) parts of the denominator in the RPA case $χ_{RPA}^{+ -} (q, q; ω)$ [Eq. (47)]. The inset in (b) zooms in close to the origin and shows the zero crossing in the real part.
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Figure 5
Imaginary part of the MFT and RPA density, longitudinal spin, and transverse spin susceptibilities for a lattice size of $40 \times 40$ . Other parameters are as in Fig. 4. (a) Imaginary part of the transverse spin susceptibility for the MFT case $χ_{(0)}^{+ -} (q, q; ω)$ of Eq. (42) (dashed line) and for the RPA case $χ_{RPA}^{+ -} (q, q; ω)$ of Eq. (47) (solid line). (b) RPA density susceptibility $χ_{RPA}^{ρ ρ} (q, q; ω)$ of Eq. (44) (dashed line) and RPA longitudinal spin susceptibility $χ_{RPA}^{ρ ρ} (q, q; ω)$ of Eq. (45) (dotted line). Note that in the MFT case, $χ_{(0)}^{ρ ρ} (q, q; ω)$ (solid line) coincides with the MFT $χ_{(0)}^{z z} (q, q; ω)$ ; see Eqs. (39) and (40). However, the RPA renormalizes these two susceptibilities differently.
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Figure 6
Schematic illustration of the light-scattering setup. The atoms are confined in the optical lattice close to the $x y$ plane. The incident light field with the wave vector $k_{1}$ propagates perpendicular to the lattice in the positive $z$ direction. The wave vector of the scattered light is denoted by $k_{2}$ , with the scattering direction determined by the coordinates $θ$ and $ϕ$ .
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Figure 7
Schematic illustration of the atomic-level structure. The atoms are illuminated by an incident light with the $σ^{-}$ polarization, exciting atoms $| ↑ ⟩ \to | 2 ⟩$ and $| ↓ ⟩ \to | 1 ⟩$ . The state $| 1 ⟩$ decays to $| ↓ ⟩$ , while the state $| 2 ⟩$ can decay either to $| ↑ ⟩$ or to $| ↓ ⟩$ .
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Figure 8
Angular distribution of the elastic component of the scattered light intensity along the direction $ϕ = π / 4$ . The calculations use the AFM-order parameter $m$ computed with RPA corrections. Here, the number of sites is $40 \times 40$ and the lattice depth $s = 25$ . (a) $I_{e} (Δ k)$ for different values of $m$ , when both species are detected. Different curves represent $U = 7.3 J$ and $m_{RPA} = 0.3$ (solid line), $U = 3.9 J$ and $m_{RPA} = 0.25$ (dashed line), and $U = 2.0 J$ and $m_{RPA} = 0.15$ (short dashed line). In (b), we compare the results for the total density with the single-species detection ( $m_{RPA} = 0.3$ ). Curves from top to bottom: $I_{e}^{↓} (Δ k)$ (dashed), $I_{e}^{↑} (Δ k)$ (dash-dotted), and $I_{e} (Δ k)$ (solid). The magnetic Bragg peak is observable since $κ = 1.5 > \sqrt{2}$ . Note that the highest peak in (a) corresponds to the smallest one in (b).
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Figure 9
Typical configurations for the Ising model at two different temperatures. We show the cases far above ( $K = - 0.3$ , on the left) and slightly above ( $K = - 0.43$ , on the right) the transition temperature. The corresponding correlation lengths are $ξ_{Ising} (- 0.3) \approx 3 a$ and $ξ_{Ising} (- 0.43) \approx 40 a$ .
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Figure 10
Angular distribution of the elastic component of the scattered light intensity in the presence of short-range order [Eq. (107)] along the $ϕ = π / 4$ direction. The lattice depth is taken to be $s = 25$ , $κ = 1.5$ [Eq. (59)], and $m_{RPA} \approx 0.3$ . We only show the magnetic Bragg peaks for the case where only the down species is imaged. In (a)–(c), the solid lines represent the ensemble averages of the intensity over 100 stochastic realizations at different temperatures: (a) $K = - 0.3$ , (b) $K = - 0.42$ , and (c) $K = - 0.43$ . The corresponding fluctuations (sample standard deviation) are given by the dashed lines. The different average intensities of (a)–(c) are shown together in (d). Typical individual lattice configurations are shown in Fig. 9.
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Figure 11
Angular distribution of the inelastically scattered light intensity for different values of the interaction strength $U$ at $T = 0$ along the direction $ϕ = π / 4$ . The calculations are based on MFT. Here, the number of sites is $40 \times 40$ . The ratio between the wave number of the probe light to the effective wave number of the optical lattice light $κ = 1.05$ [Eq. (59)]. The lattice height is $s = 25$ . Intensity contributions from (a) the density and longitudinal spin components, (b) the transverse spin component, and (c) total scattered light intensity. The scattered intensity decreases with increasing magnetization because of the changes in the density and longitudinal spin susceptibility.
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Figure 12
Comparison of the angular distribution of the inelastic scattered light intensity based on MFT (solid lines) and RPA (dashed lines) at $T = 0$ along the direction $ϕ = π / 4$ , with $κ = 1.5$ . The rest of the parameters are as in Fig. 11. The MFT and the RPA results notably differ, owing to the collective modes that significantly modify the transverse spin component. The magnitude of the order parameter [Eq. (15)] is $m = 0.19$ ( $m_{RPA} ≃ 0.15$ ) for $U = 2 J$ and $m = 0.4$ ( $m_{RPA} ≃ 0.28$ ) for $U = 5.3 J$ .
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Figure 13
Angular distribution of the inelastically scattered light intensity for different values of the interaction strength $U$ at $T = 0$ along the direction $ϕ = π / 4$ . The calculations are based on RPA. We use the same parameters as in Fig. 11. Intensity contributions from (a) the density and longitudinal spin components, (b) the transverse spin component, and (c) total scattered light intensity. The scattered intensity increases with increasing magnetization near the perpendicular direction $θ \sim π / 2$ since, because of the collective modes, the transverse spin component dominates the scattered light.
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Figure 14
Angular distribution of the inelastically scattered light intensity at different temperatures along the direction $ϕ = π / 4$ . The calculations are based on MFT. Here, the on-site interaction is fixed at $U = 5.3 J$ . This means that lower $T$ corresponds to higher values of $m$ . We use the same parameters as in Fig. 11. The intensity contributions from (a) the density and longitudinal spin components, (b) the transverse spin component, and (c) total scattered light intensity. An increase in temperature enhances scattering in the near-forward direction.
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Figure 15
Comparison of the angular distribution of the different components of the scattered light intensity at $T = 0$ along the direction $ϕ = π / 4$ . Here, the on-site interaction strength $U = 1.76 J$ and the order parameter $m = 0.16$ ( $m_{RPA} ≃ 0.13$ ). We use the same parameters as in Fig. 11. We show the MFT elastic component (solid line), the RPA inelastic intraband component (dashed line), and the inelastic interband component (dotted line). We schematically represent the position of a block and a $NA = 0.8$ lens by vertical lines.
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Figure 16
Schematic illustration of the experimental configurations to detect AFM ordering of fermionic atoms in an optical lattice. The atoms are confined in the 2D optical lattice close to the $z = 0$ plane, and the incident light propagates towards the positive $z$ direction. In (a), the elastically scattered light corresponding to the emerging additional Bragg peak generated by the AFM ordering is collected by a small lens. In (b), the setup is closely related to that of Ref. [26]. The two lenses have focal lengths $f_{1}$ and $f_{2}$ . The light scattered in the near-forward direction is first collected by lens 1. In the focal plane, the scattered light is selectively stopped by a block in order to suppress the intensity of the elastically scattered light at the detector. The shape and the size of the block can be optimized for different measurements of collective excitations or of temperature. In (c), the scattered light is collected near the perpendicular scattering direction of $θ = π / 2$ , to measure transverse spin correlations.
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Figure 17
Measurement accuracy at $κ = 1.5$ when light along the direction of the emerging magnetic Bragg peak is detected. The calculations are based on RPA susceptibilities at $T = 0$ with an RPA corrected order parameter $m_{RPA}$ for the elastic component. We show the collected photon rates vs the RPA corrected AFM-order parameter $m_{RPA}$ (left column) and the number of experimental realizations $τ$ to achieve a relative accuracy $Δ m / m_{ref}$ (right column). In all the plots (a)–(d), dashed curves correspond to the configuration of a lens with $NA = 0.4$ pointing in the perpendicular direction [Fig. 16], and solid curves are for a small lens with $NA = 0.2$ pointing in the direction of the emerging magnetic Bragg peak [Fig. 16]. Note that the lens in the perpendicular direction also collects the signal from two magnetic Bragg peaks. Plots (a) and (b) show the case when both spin species are detected. Plots (c) and (d) show the case when only the $| ↓ ⟩$ atoms are detected. Far fewer experimental realizations are needed for a given accuracy when only one species is detected. In plots (b) and (d), we show the $τ$ values for two different reference states: $m_{ref} ≃ 0.12$ in black (top two curves) and $m_{ref} ≃ 0.19$ in green (bottom two curves).
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Figure 18
$T = 0$ collected photon rates vs the RPA corrected AFM-order parameter $m_{RPA}$ with a lens of $NA = 0.8$ in the forward direction (top row) and the corresponding estimated number of experimental realizations to achieve a relative accuracy $Δ m / m_{ref}$ (bottom row). The calculations are computed with RPA susceptibilities at $T = 0$ and with an RPA corrected order parameter $m_{RPA}$ for the elastic component. In both (a) and (b), we show a fixed value of $κ = 1.05$ and compare two different lattice depths $s = 7.8$ (dotted line) and $s = 25$ (solid line). In (b), the top two curves (in black) are for $m_{ref} ≃ 0.12$ , and the bottom two curves (in green) are for $m_{ref} ≃ 0.19$ . The top two curves are essentially on top of one another, similarly for the bottom two curves. In both (c) and (d), we show a fixed lattice depth $s = 25$ and compare three different values of the parameter $κ = 0.66$ (dotted line), $κ = 1.05$ (solid line), and $κ = 1.5$ (dashed line). Note that varying $κ$ changes the width of the elastic diffraction peak. Thus, to block out the main elastic diffraction peaks, a different block width is required for each $κ$ value (see Table 1). In (d), the top three curves (in black) are for $m_{ref} ≃ 0.12$ , and the bottom three curves (in green) are for $m_{ref} ≃ 0.19$ . The top three curves are close together, and even more so for the bottom three curves.
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Figure 19
Finite-temperature plots of (a) collected photon rates vs the MFT AFM-order parameter $m$ with a lens of $NA = 0.8$ in the forward direction, and (b) the corresponding estimated number of experimental realizations to achieve a relative accuracy $Δ m / m_{ref}$ . Results are obtained with MFT at finite $T$ with fixed $U = 5.3 J$ , and with $s = 25$ and $κ = 1.05$ . The cross block width is 0.048 rad. At fixed $U$ , increasing $T$ leads to decreasing $m$ , and in (b), we show from top to bottom (corresponding to lowering $T$ ), $m_{ref} = 0.13$ , 0.18, 0.22, 0.26, 0.31, 0.35, 0.4. In (b), the data points are joined with straight lines.
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Figure 20
Collected photon rates vs the AFM-order parameter $m$ with a lens pointing in the perpendicular direction (left column) and the corresponding estimated number of experimental realizations $τ$ to achieve a relative accuracy $Δ m / m_{ref}$ (right column). The calculations are computed with RPA susceptibilities at $T = 0$ and with a RPA corrected order parameter $m_{RPA}$ for the elastic component. We compare the case when [(a) and (b)] all the density and spin components are collected by a lens with $NA = 0.4$ , to the case when [(c) and (d)] only the transverse spin component of the scattered light is collected by a lens with $NA = 0.5$ . In both (a) and (b), solid lines are for $κ = 1.05$ and dashed lines for $κ = 1.5$ . In (b), the top two curves (in black) are for $m_{ref} ≃ 0.12$ and the bottom two curves (in green) are for $m_{ref} ≃ 0.19$ . Note that the bottom black dashed curve almost overlaps the top green solid curve. In both (c) and (d), dotted lines are for $κ = 0.66$ , solid lines are for $κ = 1.05$ , and dashed lines are for $κ = 1.5$ . In (d), the top three curves (in black) are for $m_{ref} ≃ 0.12$ and the bottom three curves (in green) are for $m_{ref} ≃ 0.19$ .
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Figure 21
Angular distribution of the inelastically scattered light intensity along the direction $ϕ = π / 4$ when the scattered light is projected along specific polarization directions. We show different values of the interaction strength $U$ at $T = 0$ . The calculations are based on RPA, $κ = 1.5$ , and the other parameters are as in Fig. 11. (a) The total scattered light intensity; (b) the total scattered intensity from the spin-conserving transition; (c) the total scattered intensity from the spin-exchanging transition. In (d) and (e), the light has been projected along the direction of the scattered light from the spin-conserving transition. (d) Projected component of the scattered intensity from the spin-conserving transition, which in this case is equal to (b); (e) projected component of the scattered intensity from the spin-exchanging transition. In (f) and (g), the light has been projected along the direction of the scattered light from the spin-exchanging transition. (f) Projected component of the scattered intensity from the spin-conserving transition; (g) projected component of the scattered intensity from the spin-exchanging transition, which in this case is equal to (c).
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Figure 22
Comparison between the RPA susceptibilities, angle-resolved spectrum, and the spectrum collected by a lens over a range of scattering angles. The ratio between the wave number of the probe light to the effective wave number of the optical lattice light is $κ = 1.5$ [Eq. (59)] and $U = 5 J$ . Each line shown has been normalized to its maximum value for comparison. We show $s_{0.5}^{L} (ω)$ [solid (black) line], $s_{0.1}^{L} (ω)$ [dotted (blue) line], $S (Δ k, ω)$ [dashed (green) line], and $χ (Δ k, Δ k; ω)$ [dot-dashed (red) line]. Panel (a) shows the configuration with a lens in the forward direction [Fig. 16]. The angle-resolved quantities, the spectrum $S (Δ k, ω)$ , and total susceptibility $χ (Δ k, Δ k; ω)$ are computed at a wave vector close to the axis of the lens ( $θ \sim 0$ , $ϕ \sim 0$ ). Panel (b) shows the case for a lens in the perpendicular direction [Fig. 16]. $S (Δ k, ω)$ and $χ (Δ k, Δ k; ω)$ are computed at a wave vector $Δ k$ close to the axis of the lens at ( $θ \sim π / 2$ , $ϕ \sim π / 4$ ). Panel (b) shows the collective mode at low energies, and the inset shows the single-particle excitations at energies above the gap.
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Figure 23
RPA spectrum collected by a lens for different values of the on-site interaction strength $U$ at $T = 0$ . Here, the number of sites is $40 \times 40$ . The lattice height is $s = 25$ and $κ = 1.5$ . Light is collected with a lens of $NA = 0.5$ pointing in (a) the forward direction [Fig. 16] and (b) the perpendicular direction [Fig. 16]. From left to right, the black (solid) line is $m_{RPA} ≃ 0.19$ ( $U \approx 2.6 J$ ), the blue (dashed) line is $m_{RPA} ≃ 0.25$ ( $U \approx 4.0 J$ ), and the green (dot-dashed) line is $m_{RPA} ≃ 0.30$ ( $U \approx 8.3 J$ ). The inset in (b) shows the renormalized single-particle excitations starting at $ω \approx U / J$ . All the curves have been normalized to the maximum of the $m_{RPA} ≃ 0.30$ case.
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Figure 24
RPA scattered spectrum at zero temperature along $ϕ = π / 4$ for $U = 8.3 J$ ( $m_{RPA} ≃ 0.30$ ). This figure only shows the low-energy part of the spectrum that includes the collective modes. Single-particle excitations occur at much higher energies and are not shown. The gray scale on the left gives the magnitude of the spectrum plotted. Panel (a) shows the case $κ = 1.05$ , and (b) shows the case $κ = 1.5$ . $ω_{Δ k}$ (solid line), given by Eq. (49), is the spin-wave dispersion relation of the Heisenberg model. The green (steplike) line connects the maximum values of the collective mode peak of the RPA spectrum for each $θ$ point evaluated.
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Figure 25
Diagrammatic representation of the interaction vertices, spin densities, and Green’s function needed for the RPA calculation of the susceptibilities. Panels (a) and (b) show the interaction vertices [Eq. (a1)], (c) shows the density operator [Eq. (32)], (d) shows the spin projection on the $z$ direction [Eq. (30)], (e) shows the spin-raising operator [Eq. (31)], and (f) shows the MFT Green’s function [Eq. (a3)].
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Figure 26
Diagrammatic representation of the MFT transverse spin susceptibility matrix; see Eq. (37). The sums over internal momentum $k$ and frequency $ν$ have not been written explicitly.
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Figure 27
Diagrammatic representation of the MFT density-density susceptibility matrix; see Eq. (37). The sums over internal momentum $k$ , frequency $ν$ , and spin have not been written explicitly.
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Figure 28
Diagrammatic representation of the RPA series for the density susceptibility. The first-order term is the MFT bubble. The sums over internal momenta $k$ , $k^{'}$ , $k^{''}$ , and frequencies $ν$ , $ν^{'}$ , $ν^{''}$ have not been written explicitly. $\bar{g}$ stands for the opposite spin of $g$ .
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Figure 29
Diagrammatic representation of the alternative form of the interaction vertices suited to computing the transverse spin susceptibility for (a) Eq. (a11) and (b) Eq. (a12). In (c), we show the RPA series for the transverse spin susceptibility. The sums over internal momenta $k$ , $k^{'}$ , $k^{''}$ and frequencies $ν$ , $ν^{'}$ , $ν^{''}$ have not been written explicitly.
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