Tunable viscosity across the BCS-BEC crossover

Andrey Grankin; Archisman Panigrahi; Leonid Levitov; Victor Galitski; Yunxiang Liao

arxiv: 2604.10759 · v1 · submitted 2026-04-12 · ❄️ cond-mat.quant-gas

Tunable viscosity across the BCS-BEC crossover

Yunxiang Liao , Andrey Grankin , Archisman Panigrahi , Victor Galitski , Leonid Levitov This is my paper

Pith reviewed 2026-05-10 15:14 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas

keywords ultracold Fermi gasesBCS-BEC crossovershear viscosityFeshbach resonanceReynolds numberquantum turbulenceKeldysh formalismvertex corrections

0 comments

The pith

Shear viscosity in ultracold Fermi gases can be tuned over several orders of magnitude by varying interaction strength near a Feshbach resonance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors set out to show that ultracold Fermi gases offer a practical way to reach high Reynolds numbers by driving shear viscosity down sharply near the point where interactions are tuned through a Feshbach resonance. A sympathetic reader would care because this removes the usual limits set by small system size and modest flow speeds, opening a route to laboratory-scale studies of turbulence in quantum fluids. The calculation uses the Keldysh contour for linear response and includes vertex corrections to track how viscosity changes across the BCS-BEC crossover. Simple Drude-like terms dominate far from resonance, while higher-order diagrams such as the Maki-Thompson contribution suppress unphysical singularities when the gas is near resonance.

Core claim

The shear viscosity of a unitary Fermi gas varies by several orders of magnitude as the Feshbach detuning is swept through the BCS-BEC crossover. Within the Keldysh linear-response framework, Drude-like contributions set the viscosity at large detunings, whereas vertex corrections, including the Maki-Thompson term, grow important near resonance and regularize the response.

What carries the argument

Keldysh linear-response calculation of shear viscosity that incorporates vertex corrections to handle the near-resonant regime.

If this is right

Reynolds numbers become tunable over a wide range by adjusting the Feshbach detuning.
Ultracold gases can serve as controllable table-top simulators of turbulent flows.
Higher-order diagrams must be retained to avoid spurious divergences in the viscosity near resonance.
The same framework supplies a roadmap for engineering hydrodynamic regimes in other strongly correlated quantum fluids.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same tuning strategy could be tested in bosonic or mixed gases to see whether viscosity minima appear at other interaction resonances.
If the predicted viscosity minimum is confirmed, it would allow systematic variation of the Reynolds number while keeping density and temperature fixed, isolating the effect of interaction strength.
The calculation suggests that transport measurements at fixed flow velocity but varying detuning would map the crossover in a single experimental run.

Load-bearing premise

The perturbative Keldysh linear-response treatment with vertex corrections remains quantitatively reliable even when the gas is strongly interacting near resonance.

What would settle it

A direct measurement of the damping rate or flow resistance in a trapped Fermi gas at several values of the magnetic-field detuning across the resonance, checking whether viscosity indeed changes by multiple orders of magnitude.

Figures

Figures reproduced from arXiv: 2604.10759 by Andrey Grankin, Archisman Panigrahi, Leonid Levitov, Victor Galitski, Yunxiang Liao.

**Figure 2.** Figure 2: Diagrammatic representation of perturbation the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Phase diagram of the two-channel model Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: Shear viscosity η in the two-channel model Eqs. (1, 2). (a) η as function of detuning ϵ0 at different temperatures. (b) η as function of temperature at different detunings ϵ0. The blue curve corresponds to the total viscosity ηtotal defined in Eq.(7), while the orange dashed curve is the absolute value of the lowest-order correction η MT + ηF B + ηBF , which, when added to the naive estimation ηF + ηB, giv… view at source ↗

read the original abstract

Tunable interactions make ultracold quantum gases a unique platform for exploring hydrodynamic properties in the strongly correlated regime. Of particular interest are turbulent flows possible in the regime of high Reynolds numbers. Since the system size and flow velocity are limited in experimentally realistic systems, we propose an alternative approach to enhance the Reynolds numbers in an ultracold Fermi gas by minimizing the shear viscosity in the vicinity of the Feshbach resonance. By employing the Keldysh formulation of the linear response theory, we theoretically demonstrate that the shear viscosity can vary by several orders of magnitude in the vicinity of the BCS-BEC crossover. It is also shown that while Drude-like contributions generally dominate at large Feshbach detunings, higher-order vertex corrections, including the Maki-Thompson contribution, become significant and suppress singular behavior in the near-resonant regime. Our results provide a roadmap for achieving tunable Reynolds numbers in ultracold quantum fluids, which can serve as table-top turbulence simulators.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies Keldysh linear response with Maki-Thompson corrections to claim orders-of-magnitude tunable shear viscosity near the BCS-BEC crossover, but the perturbative validity in the unitary regime remains the open question.

read the letter

The key point is that the authors calculate shear viscosity in a Fermi gas across the crossover using Keldysh contour methods and include higher-order vertex corrections. They argue this produces a several-orders-of-magnitude drop near resonance, which could raise Reynolds numbers enough for turbulence studies without bigger clouds or faster flows. The calculation separates Drude-like terms that dominate far from resonance from the Maki-Thompson pieces that cut off divergences close in. This is a direct extension of standard linear-response techniques to the viscosity problem, and the separation of regimes is laid out clearly enough to follow. The practical suggestion for experiments is straightforward and worth noting for groups already running Feshbach-tuned gases. The soft spot sits exactly where the effect is supposed to be largest. The abstract states that vertex corrections grow important near unitarity, yet the same regime is where the scattering length diverges and ordinary perturbation theory loses its small parameter. Without an explicit error estimate or a check against non-perturbative effects such as pair fluctuations, the quantitative size of the variation is hard to trust. If those omitted contributions shift the viscosity by even one order of magnitude, the tunability claim shrinks. The paper is aimed at theorists and experimentalists working on transport and hydrodynamics in ultracold Fermi gases. Readers who want a concrete proposal for raising Reynolds numbers in existing setups will find the idea useful as a starting point, even if they later need to verify the numbers against other calculations. It deserves a serious referee because the method is established, the claim is experimentally relevant, and the authors have identified a concrete place where corrections matter. Send it for review, but ask for the full derivation, a discussion of the truncation error, and direct comparisons to prior viscosity results in the literature.

Referee Report

2 major / 2 minor

Summary. The manuscript uses the Keldysh formulation of linear-response theory to calculate the shear viscosity of an ultracold Fermi gas across the BCS-BEC crossover. It claims that the viscosity can be tuned over several orders of magnitude by varying the Feshbach detuning, with Drude-like terms dominating far from resonance and higher-order vertex corrections (including Maki-Thompson) becoming significant near unitarity to suppress singular behavior. The work proposes this tunability as a route to high Reynolds numbers for table-top turbulence studies.

Significance. If the central result holds, the demonstration of tunable viscosity by orders of magnitude would provide a concrete experimental handle on hydrodynamic regimes in strongly interacting quantum gases, directly enabling studies of high-Re turbulence in a controlled setting. The inclusion of vertex corrections in the Keldysh framework is a technical strength that addresses a known limitation of simpler Drude approximations.

major comments (2)

[Abstract and near-resonant regime discussion] Abstract and § on near-resonant regime: the claim that viscosity varies by several orders of magnitude rests on the quantitative accuracy of the Keldysh linear-response calculation (including Maki-Thompson vertex corrections) when the scattering length diverges. No small parameter controlling the truncation error is identified once |a|→∞, and the text itself notes that vertex corrections become significant precisely where perturbation theory is marginal. This is load-bearing for the headline tunability result.
[Results on Drude vs. vertex contributions] Results section on Drude vs. vertex contributions: the suppression of singular behavior by higher-order diagrams is asserted, but no explicit comparison to prior calculations (e.g., Boltzmann-equation or T-matrix approaches) or error estimates is provided to confirm that omitted diagrams or non-perturbative effects (pair fluctuations) do not alter the viscosity by an order of magnitude or more.

minor comments (2)

[Introduction] Notation for the interaction strength and detuning parameters should be defined explicitly at first use to aid readability.
[Figure captions] Figure captions for viscosity plots should include the temperature and density values used, as well as the range of detuning shown.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important limitations of our perturbative Keldysh approach near unitarity, which we address below by qualifying our claims and adding comparisons. We believe these revisions will strengthen the manuscript while preserving its central message that viscosity is tunable across the crossover within the included diagrams.

read point-by-point responses

Referee: [Abstract and near-resonant regime discussion] Abstract and § on near-resonant regime: the claim that viscosity varies by several orders of magnitude rests on the quantitative accuracy of the Keldysh linear-response calculation (including Maki-Thompson vertex corrections) when the scattering length diverges. No small parameter controlling the truncation error is identified once |a|→∞, and the text itself notes that vertex corrections become significant precisely where perturbation theory is marginal. This is load-bearing for the headline tunability result.

Authors: We agree that no small parameter exists once |a|→∞ and that the regime is non-perturbative. Our calculation includes the leading Maki-Thompson vertex correction within the Keldysh framework to suppress the Drude singularity, but we do not claim this exhausts all higher-order or non-perturbative effects. We will revise the abstract and the near-resonant discussion to state that the reported orders-of-magnitude variation is obtained within this approximation and serves as a qualitative roadmap rather than a fully quantitative prediction. This qualification preserves the proposal for tunable Reynolds numbers while acknowledging the limitation. revision: partial
Referee: [Results on Drude vs. vertex contributions] Results section on Drude vs. vertex contributions: the suppression of singular behavior by higher-order diagrams is asserted, but no explicit comparison to prior calculations (e.g., Boltzmann-equation or T-matrix approaches) or error estimates is provided to confirm that omitted diagrams or non-perturbative effects (pair fluctuations) do not alter the viscosity by an order of magnitude or more.

Authors: We will add a dedicated paragraph in the results section that compares our viscosity values (both Drude and with MT corrections) to existing Boltzmann-equation results in the BCS limit and to T-matrix calculations near unitarity. We will also estimate the size of omitted diagrams by noting the relative magnitude of the MT term and discuss that pair-fluctuation effects beyond our truncation could modify the viscosity, particularly on the BEC side. These additions will provide the requested context without altering the computed curves. revision: yes

Circularity Check

0 steps flagged

No circularity: direct Keldysh linear-response calculation

full rationale

The derivation proceeds from the Keldysh contour formulation of linear response, computing shear viscosity via Drude-like terms plus explicit vertex corrections (Maki-Thompson). No parameters are fitted to the target viscosity data, no result is defined in terms of itself, and no self-citation chain is invoked to justify the central tunability claim. The calculation is self-contained against the stated perturbative assumptions; external validity concerns (radius of convergence near unitarity) are separate from circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The Keldysh formalism and linear-response assumptions are standard but unexamined here.

pith-pipeline@v0.9.0 · 5477 in / 1047 out tokens · 49926 ms · 2026-05-10T15:14:18.303010+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 1 internal anchor

[1]

Tunable viscosity across the BCS-BEC crossover

remains a challenge in ultracold systems due to con- straints on system size and flow velocities. Minimizing the shear viscosity [13] thus represents the most viable way to achieve turbulence. As a transport coefficient,η is highly sensitive to the interaction strength, and there- fore, the use of Feshbach resonance to tune both makes ultracold atoms a pe...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

Bloch, J

I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Reviews of modern physics 80, 885 (2008)

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Ketterle and M

W. Ketterle and M. W. Zwierlein, Making, probing and understanding ultracold fermi gases, La Rivista del Nuovo Cimento31, 247 (2008)

work page 2008
[4]

Giorgini, L

S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of ultracold atomic fermi gases, Reviews of Modern Physics 80, 1215 (2008)

work page 2008
[5]

Joseph, B

J. Joseph, B. Clancy, L. Luo, J. Kinast, A. Turlapov, and J. Thomas, Measurement of sound velocity in a fermi gas near a feshbach resonance, Physical review letters98, 170401 (2007)

work page 2007
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C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Fes- hbach resonances in ultracold gases, Reviews of Modern Physics82, 1225 (2010)

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C. Regal, M. Greiner, and D. S. Jin, Observation of res- onance condensation of fermionic atom pairs, Physical review letters92, 040403 (2004)

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P. B. Patel, Z. Yan, B. Mukherjee, R. J. Fletcher, J. Struck, and M. W. Zwierlein, Universal sound diffu- sion in a strongly interacting fermi gas, Science370, 1222 (2020)

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J. A. Joseph, E. Elliott, and J. E. Thomas, Shear vis- cosity of a unitary fermi gas near the superfluid phase transition, Phys. Rev. Lett.115, 020401 (2015)

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Elliott, J

E. Elliott, J. A. Joseph, and J. E. Thomas, Anomalous minimum in the shear viscosity of a fermi gas, Phys. Rev. Lett.113, 020406 (2014)

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L. D. Landau and E. M. Lifshitz,Fluid Mechanics: Vol- ume 6, Vol. 6 (Elsevier, 1987)

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Schäfer and D

T. Schäfer and D. Teaney, Nearly perfect fluidity: from cold atomic gases to hot quark gluon plasmas, Rep. Prog. Phys.72, 126001 (2009)

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[14]

Liao and V

Y. Liao and V. Galitski, Critical viscosity of a fluctuating superconductor, Physical Review B100, 060501 (2019)

work page 2019
[15]

Kagamihara, D

D. Kagamihara, D. Inotani, and Y. Ohashi, Shear vis- cosity and strong-coupling corrections in the bcs–bec crossover regime of an ultracold fermi gas, Journal of the Physical Society of Japan88, 114001 (2019)

work page 2019
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T. Enss, R. Haussmann, and W. Zwerger, Viscosity and scale invariance in the unitary fermi gas, Annals of Physics326, 770 (2011)

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Gurarie and L

V. Gurarie and L. Radzihovsky, Resonantly paired fermionic superfluids, Annals of Physics322, 2 (2007)

work page 2007
[18]

Taylor and M

E. Taylor and M. Randeria, Viscosity of strongly in- teracting quantum fluids: Spectral functions and sum rules, Physical Review A—Atomic, Molecular, and Opti- cal Physics81, 053610 (2010)

work page 2010
[19]

In the supplementary material, we provide a detailed cal- culation of the shear viscosity for the two channel model in the Keldysh formalism

work page
[20]

Taylor and M

E. Taylor and M. Randeria, Viscosity of strongly inter- acting quantum fluids: Spectral functions and sum rules, Phys. Rev. A81, 053610 (2010)

work page 2010
[21]

Baym and L

G. Baym and L. P. Kadanoff, Conservation laws and cor- relation functions, Physical Review124, 287 (1961)

work page 1961
[22]

Maki and R

K. Maki and R. S. Thompson, Fluctuation conductivity of high-tc superconductors, Physical Review B39, 2767 (1989)

work page 1989
[23]

Götze and P

W. Götze and P. Wölfle, Homogeneous dynamical con- ductivity of simple metals, Physical Review B6, 1226 (1972)

work page 1972
[24]

D. J. Thouless, Perturbation theory in statistical me- chanics and the theory of superconductivity, Annals of Physics10, 553 (1960)

work page 1960
[25]

Nozieres and S

P. Nozieres and S. Schmitt-Rink, Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity, J. Low Temp. Phys.59, 195 (1985)

work page 1985
[26]

Jeon and L

S. Jeon and L. G. Yaffe, Phys. Rev. D53, 5799 (1996); see also S. Jeon, Phys. Rev. D52, 3591 (1995)

work page 1996
[27]

Jeon, Computing spectral densities in finite tempera- ture field theory, Phys

S. Jeon, Computing spectral densities in finite tempera- ture field theory, Phys. Rev. D47, 5799 (1993)

work page 1993
[28]

JeonandL.G

S. JeonandL.G. Yaffe,From quantumfield theorytohy- drodynamics: Transport coefficients and effective kinetic theory, Phys. Rev. D53, 4586 (1996). Tunable viscosity across the BCS-BEC crossover Supplemental Material Yunxiang Liao,1 Andrey Grankin,2 Archisman Panigrahi,3 Victor Galitski,2 and Leonid Levitov 3 1Department of Physics, KTH Royal Institute of T...

work page 1996
[29]

(S15) This leads to the actions iSψ =i Z p,ε ¯Ψ(p, ε)G−1 0 (p, ε)Ψ(p, ε), iSϕ =i Z q,ω ¯ϕ(q, ω)D−1 0 (q, ω)ϕ(q, ω), iSint =−ig Z p,ε,q,ω ¯Ψ(p+q, ε+ω)Φ(q, ω)Ψ(p, ε)

(S14) and also for the bosons ϕ= ϕcl ϕq T , ϕ cl/q ≡(ϕ + ±ϕ −)/2, ¯ϕcl/q ≡ ¯ϕ+ ± ¯ϕ− /2. (S15) This leads to the actions iSψ =i Z p,ε ¯Ψ(p, ε)G−1 0 (p, ε)Ψ(p, ε), iSϕ =i Z q,ω ¯ϕ(q, ω)D−1 0 (q, ω)ϕ(q, ω), iSint =−ig Z p,ε,q,ω ¯Ψ(p+q, ε+ω)Φ(q, ω)Ψ(p, ε). (S16) Here Φ represents the matrix Φ = (ϕcl +ϕ qτ1)⊗σ + + (¯ϕcl + ¯ϕqτ1)⊗σ −, (S17) withσ ± = (σ1 ±iσ 2...

work page 1985

[1] [1]

Tunable viscosity across the BCS-BEC crossover

remains a challenge in ultracold systems due to con- straints on system size and flow velocities. Minimizing the shear viscosity [13] thus represents the most viable way to achieve turbulence. As a transport coefficient,η is highly sensitive to the interaction strength, and there- fore, the use of Feshbach resonance to tune both makes ultracold atoms a pe...

work page internal anchor Pith review Pith/arXiv arXiv 2026

[2] [2]

Bloch, J

I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Reviews of modern physics 80, 885 (2008)

work page 2008

[3] [3]

Ketterle and M

W. Ketterle and M. W. Zwierlein, Making, probing and understanding ultracold fermi gases, La Rivista del Nuovo Cimento31, 247 (2008)

work page 2008

[4] [4]

Giorgini, L

S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of ultracold atomic fermi gases, Reviews of Modern Physics 80, 1215 (2008)

work page 2008

[5] [5]

Joseph, B

J. Joseph, B. Clancy, L. Luo, J. Kinast, A. Turlapov, and J. Thomas, Measurement of sound velocity in a fermi gas near a feshbach resonance, Physical review letters98, 170401 (2007)

work page 2007

[6] [6]

C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Fes- hbach resonances in ultracold gases, Reviews of Modern Physics82, 1225 (2010)

work page 2010

[7] [7]

Regal, M

C. Regal, M. Greiner, and D. S. Jin, Observation of res- onance condensation of fermionic atom pairs, Physical review letters92, 040403 (2004)

work page 2004

[8] [8]

P. B. Patel, Z. Yan, B. Mukherjee, R. J. Fletcher, J. Struck, and M. W. Zwierlein, Universal sound diffu- sion in a strongly interacting fermi gas, Science370, 1222 (2020)

work page 2020

[9] [9]

C. Cao, E. Elliott, J. Joseph, H. Wu, J. Petricka, T. Schäfer, and J. E. Thomas, Universal quantum vis- cosity in a unitary fermi gas, Science331, 58 (2011)

work page 2011

[10] [10]

J. A. Joseph, E. Elliott, and J. E. Thomas, Shear vis- cosity of a unitary fermi gas near the superfluid phase transition, Phys. Rev. Lett.115, 020401 (2015)

work page 2015

[11] [11]

Elliott, J

E. Elliott, J. A. Joseph, and J. E. Thomas, Anomalous minimum in the shear viscosity of a fermi gas, Phys. Rev. Lett.113, 020406 (2014)

work page 2014

[12] [12]

L. D. Landau and E. M. Lifshitz,Fluid Mechanics: Vol- ume 6, Vol. 6 (Elsevier, 1987)

work page 1987

[13] [13]

Schäfer and D

T. Schäfer and D. Teaney, Nearly perfect fluidity: from cold atomic gases to hot quark gluon plasmas, Rep. Prog. Phys.72, 126001 (2009)

work page 2009

[14] [14]

Liao and V

Y. Liao and V. Galitski, Critical viscosity of a fluctuating superconductor, Physical Review B100, 060501 (2019)

work page 2019

[15] [15]

Kagamihara, D

D. Kagamihara, D. Inotani, and Y. Ohashi, Shear vis- cosity and strong-coupling corrections in the bcs–bec crossover regime of an ultracold fermi gas, Journal of the Physical Society of Japan88, 114001 (2019)

work page 2019

[16] [16]

T. Enss, R. Haussmann, and W. Zwerger, Viscosity and scale invariance in the unitary fermi gas, Annals of Physics326, 770 (2011)

work page 2011

[17] [17]

Gurarie and L

V. Gurarie and L. Radzihovsky, Resonantly paired fermionic superfluids, Annals of Physics322, 2 (2007)

work page 2007

[18] [18]

Taylor and M

E. Taylor and M. Randeria, Viscosity of strongly in- teracting quantum fluids: Spectral functions and sum rules, Physical Review A—Atomic, Molecular, and Opti- cal Physics81, 053610 (2010)

work page 2010

[19] [19]

In the supplementary material, we provide a detailed cal- culation of the shear viscosity for the two channel model in the Keldysh formalism

work page

[20] [20]

Taylor and M

E. Taylor and M. Randeria, Viscosity of strongly inter- acting quantum fluids: Spectral functions and sum rules, Phys. Rev. A81, 053610 (2010)

work page 2010

[21] [21]

Baym and L

G. Baym and L. P. Kadanoff, Conservation laws and cor- relation functions, Physical Review124, 287 (1961)

work page 1961

[22] [22]

Maki and R

K. Maki and R. S. Thompson, Fluctuation conductivity of high-tc superconductors, Physical Review B39, 2767 (1989)

work page 1989

[23] [23]

Götze and P

W. Götze and P. Wölfle, Homogeneous dynamical con- ductivity of simple metals, Physical Review B6, 1226 (1972)

work page 1972

[24] [24]

D. J. Thouless, Perturbation theory in statistical me- chanics and the theory of superconductivity, Annals of Physics10, 553 (1960)

work page 1960

[25] [25]

Nozieres and S

P. Nozieres and S. Schmitt-Rink, Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity, J. Low Temp. Phys.59, 195 (1985)

work page 1985

[26] [26]

Jeon and L

S. Jeon and L. G. Yaffe, Phys. Rev. D53, 5799 (1996); see also S. Jeon, Phys. Rev. D52, 3591 (1995)

work page 1996

[27] [27]

Jeon, Computing spectral densities in finite tempera- ture field theory, Phys

S. Jeon, Computing spectral densities in finite tempera- ture field theory, Phys. Rev. D47, 5799 (1993)

work page 1993

[28] [28]

JeonandL.G

S. JeonandL.G. Yaffe,From quantumfield theorytohy- drodynamics: Transport coefficients and effective kinetic theory, Phys. Rev. D53, 4586 (1996). Tunable viscosity across the BCS-BEC crossover Supplemental Material Yunxiang Liao,1 Andrey Grankin,2 Archisman Panigrahi,3 Victor Galitski,2 and Leonid Levitov 3 1Department of Physics, KTH Royal Institute of T...

work page 1996

[29] [29]

(S15) This leads to the actions iSψ =i Z p,ε ¯Ψ(p, ε)G−1 0 (p, ε)Ψ(p, ε), iSϕ =i Z q,ω ¯ϕ(q, ω)D−1 0 (q, ω)ϕ(q, ω), iSint =−ig Z p,ε,q,ω ¯Ψ(p+q, ε+ω)Φ(q, ω)Ψ(p, ε)

(S14) and also for the bosons ϕ= ϕcl ϕq T , ϕ cl/q ≡(ϕ + ±ϕ −)/2, ¯ϕcl/q ≡ ¯ϕ+ ± ¯ϕ− /2. (S15) This leads to the actions iSψ =i Z p,ε ¯Ψ(p, ε)G−1 0 (p, ε)Ψ(p, ε), iSϕ =i Z q,ω ¯ϕ(q, ω)D−1 0 (q, ω)ϕ(q, ω), iSint =−ig Z p,ε,q,ω ¯Ψ(p+q, ε+ω)Φ(q, ω)Ψ(p, ε). (S16) Here Φ represents the matrix Φ = (ϕcl +ϕ qτ1)⊗σ + + (¯ϕcl + ¯ϕqτ1)⊗σ −, (S17) withσ ± = (σ1 ±iσ 2...

work page 1985