$\int_{-\infty}^{\infty}f(t)dt$ (70)

with a sum

$a\sum_{n=-\infty}^{n=+\infty}\left(f(na)+(-1)^{n}g(na)\right)$ (71)

that includes a term oscillating in time through the factor $(-1)^{n}$. $f$
and $g$ are smooth functions, continuous at $t=0$, that vanish in a well-
behaved way as $t\to\infty$.

For the sum over $f$ Eq. (71) is using the standard trapezoidal rule, which
has $a^{2}$ errors. For the sum over $g$ we combine three adjacent terms,
reducing the sum to even values of $n$ only,

$\displaystyle
a\sum_{n=-\infty,n\,{\text{even}}}^{n=+\infty,n\,{\text{even}}}\frac{g(na)-2g(na+a)+g(na+2a)}{2}$
$\displaystyle=a\sum_{n=-\infty,n\,{\text{even}}}^{n=+\infty,n\,{\text{even}}}\left[\frac{a^{2}}{2}{g^{\prime\prime}(na+a)}+\ldots\right]$

Splitting the sum into two pieces, for positive and negative $n$, we have for
positive $n$

$\displaystyle
a\sum_{n=0,2,4}^{n=+\infty}\left[\frac{a^{2}}{2}{g^{\prime\prime}(na+a)}+\ldots\right]$
$\displaystyle=$ $\displaystyle\frac{a^{2}}{4}\int dt\,g^{\prime\prime}(t)$
(73) $\displaystyle=$ $\displaystyle-\frac{a^{2}}{4}g^{\prime}(0^{+})$

if $g$ and $g^{\prime}$ vanish at $t\to\infty$. Negative $n$ gives a similar
result, with opposite sign, so that we have a total for the sum over $g$ in
Eq. (71) of

$\frac{a^{2}}{4}\left[g^{\prime}(0^{-})-g^{\prime}(0^{+})\right]\,.$ (74)

The result is a discretisation effect, proportional to the discontinuity in
the derivative of $g$ at $t=0$ but vanishing as $a\to 0$ as $a^{2}$.

We conclude that summing over lattice time slices to obtain
$\tilde{C}_{\mu\nu}$ (Eq. (9)) as an approximation to the time integral that
sets the photon on-shell introduces discretisation errors proportional to
$a^{2}$ at leading order. These come both from the trapezoidal integration
implied by the sum and from the oscillating terms in the correlation function
that arise from the use of staggered quarks. Such discretisation errors are
taken into account by our fit to the results for the form factor as a function
of lattice spacing and removed in our result for $F(0,0)$ in the continuum
limit.

A toy model illustrates this further. We take $f$ and $g$ to be single
exponentials, $f=\exp(-M_{\mathrm{n}}|t|)$ and $g=\exp(-M_{\mathrm{o}}|t|)$.
For $f$ we have

$\displaystyle\int_{-\infty}^{+\infty}dt\>e^{-M_{\mathrm{n}}|t|}$
$\displaystyle\approx a\sum_{j=-\infty}^{+\infty}e^{-aM_{\mathrm{n}}j}$
$\displaystyle=a\Big{(}-1+2\sum_{j=0}^{\infty}e^{-aM_{\mathrm{n}}j}\Big{)}$
(75)

on the lattice. The left-hand side of Eq. (A) evaluates to $2/M_{\mathrm{n}}$.
The sum on the right-hand side is a geometric series, so we have

$\displaystyle\mathrm{RHS}$
$\displaystyle=a\Big{(}-1+\frac{2}{1-e^{-aM_{\mathrm{n}}}}\Big{)}$ (76)
$\displaystyle=\frac{2}{M_{\mathrm{n}}}\Big{(}1+\frac{(aM_{\mathrm{n}})^{2}}{12}+\mathcal{O}((aM_{\mathrm{n}})^{4})\Big{)}\,.$

As expected the integral obtained in this manner is accurate up to
$(aM_{\mathrm{n}})^{2}$ errors.

We repeat this exercise for the oscillating exponential to obtain

$\displaystyle
a\Big{(}-1+2\sum_{j=0}^{\infty}(-1)^{j}e^{-aM_{\mathrm{o}}j}\Big{)}$
$\displaystyle=a\Big{(}-1+\frac{2}{1+e^{-aM_{\mathrm{o}}}}\Big{)}$ (77)
$\displaystyle=\frac{1}{M_{\mathrm{o}}}\Big{(}\frac{(aM_{\mathrm{o}})^{2}}{2}+\mathcal{O}((aM_{\mathrm{o}})^{4})\Big{)}.$

As expected the impact of the oscillatory contributions vanish in the
continuum limit as $a^{2}$ and the result matches that from Eq. (74).

We reach the same conclusions about discretisation error from the toy model as
from the more general $f$ and $g$ functions of Eq. (71).

##  Appendix B Fitting a subset of $\eta_{c}\to\gamma\gamma$ data

Figure 19: Fitted results for $F_{\text{latt}}(0,q_{2}^{2})$, comparing values
obtained from our full fit with those that use a subset of time separations,
$t$, between source and sink in $\tilde{C}_{\mu\nu}(t)$. Results are given for
the LOCAL set-up on set 1 (very-coarse, $a\approx$ 0.15 fm), set 3 (coarse,
$a\approx$ 0.12 fm, fit simultaneously with set 3A) and 5 (fine, $a\approx$
0.09 fm); all values are in lattice units. The values on the left give results
from the full fit to all $t$ values except those discarded at early times
(below $t_{\text{min}}$, see text in section II.2.5). The values on the right
give fit results for a selected 4, 5 or 6 $t$ values. We see that
uncertainties close to that of the full fit are possible even with 4 $t$
values. Even and odd $t$ values are needed and the spacing between $t$ values
and range covered is adjusted as a function of lattice spacing.

As remarked in Section II.2.5, it is possible to obtain accurate results for
the $\eta_{c}\to\gamma\gamma$ form factor with a lot less numerical work than
we have expended here. We showed, in Fig. 3, that our results converge very
rapidly to their final value as a function of the $t_{\text{width}}$ region
over with the 3-pt function is integrated to obtain $\tilde{C}_{\mu\nu}$. Here
we provide another test that significantly reduces the amount of computation
needed.

Figure 19 shows the results obtained if we restrict the fit of the two-point
function $\tilde{C}_{\mu\nu}(t)$ to a set of specific $t\equiv
t_{\eta_{c}}-t_{\gamma_{2}}$ values rather than fitting the full $t$ range.
The left-hand point shows the full fit that we use here and the right-hand
points show the results from selecting 4, 5 or 6 specific $t$ values. We need
a mix of even and odd $t$ values for an optimal fit because of the oscillating
terms from opposite parity states (see Eq. (24)). We must also adjust the
separation in lattice units between $t$ values as the lattice spacing changes.
On the very-coarse lattices a separation of 1 is appropriate, whereas for
coarse and fine lattices a separation of 3 or 5 gives a better range of $t$ to
reproduce the full fit. We conclude from Fig. 19 that we could obtain similar
uncertainties to our full fit by calculating correlation functions at only 4
or 5 time separations $t_{\eta_{c}}-t_{\gamma_{2}}$ if these were well-chosen.
This is possible because lattice data is correlated as a function of $t$.
Usually lattice two-point functions are calculated at all $t$ values because
there is no significant time saving in making a $t$ selection. Here, because
we actually calculate a three-point function, working with selected $t$ values
reduces the computational cost considerably and we will make use of this in
future. Here, however, we use the results from our full fit.

## References

  * Workman [2022] R. L. Workman (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Hatton  _et al._ [2020] D. Hatton, C. Davies, B. Galloway, J. Koponen, G. Lepage, and A. Lytle (HPQCD), Phys. Rev. D 102, 054511 (2020), arXiv:2005.01845 [hep-lat] .
  * Gaiser  _et al._ [1986] J. Gaiser _et al._ , Phys. Rev. D 34, 711 (1986).
  * Mitchell  _et al._ [2009] R. E. Mitchell _et al._ (CLEO), Phys. Rev. Lett. 102, 011801 (2009), [Erratum: Phys.Rev.Lett. 106, 159903 (2011)], arXiv:0805.0252 [hep-ex] .
  * Anashin  _et al._ [2014] V. V. Anashin _et al._ , Phys. Lett. B 738, 391 (2014), arXiv:1406.7644 [hep-ex] .
  * Haddadi [2017] Z. Haddadi,  _A study of the ground-state properties of charmonium via radiative transitions in $\psi^{\prime}\to\gamma\eta_{c}$ and $J/\psi\to\gamma\eta_{c}$_, Ph.D. thesis, U. Groningen (main) (2017).
  * Dudek  _et al._ [2006] J. J. Dudek, R. G. Edwards, and D. G. Richards, Phys. Rev. D 73, 074507 (2006), arXiv:hep-ph/0601137 .
  * Chen  _et al._ [2011] Y. Chen _et al._ , Phys. Rev. D 84, 034503 (2011), arXiv:1104.2655 [hep-lat] .
  * Becirevic and Sanfilippo [2013] D. Becirevic and F. Sanfilippo, JHEP 01, 028 (2013), arXiv:1206.1445 [hep-lat] .
  * Donald  _et al._ [2012] G. Donald, C. Davies, R. Dowdall, E. Follana, K. Hornbostel, J. Koponen, G. Lepage, and C. McNeile, Phys. Rev. D 86, 094501 (2012), arXiv:1208.2855 [hep-lat] .
  * Dudek and Edwards [2006] J. J. Dudek and R. G. Edwards, Phys. Rev. Lett. 97, 172001 (2006), arXiv:hep-ph/0607140 .
  * Chen  _et al._ [2016] T. Chen _et al._ (CLQCD), Eur. Phys. J. C 76, 358 (2016), arXiv:1602.00076 [hep-lat] .
  * Chen  _et al._ [2020] Y. Chen, M. Gong, N. Li, C. Liu, Y.-B. Liu, Z. Liu, J.-P. Ma, Y. Meng, C. Xiong, and K.-L. Zhang (CLQCD), Chin. Phys. C 44, 083108 (2020), arXiv:2003.09817 [hep-lat] .
  * Liu  _et al._ [2020] C. Liu, Y. Meng, and K.-L. Zhang, Phys. Rev. D 102, 034502 (2020), arXiv:2004.03907 [hep-lat] .
  * Meng  _et al._ [2021] Y. Meng, X. Feng, C. Liu, T. Wang, and Z. Zou, (2021), arXiv:2109.09381 [hep-lat] .
  * Czarnecki and Melnikov [2001] A. Czarnecki and K. Melnikov, Phys. Lett. B 519, 212 (2001), arXiv:hep-ph/0109054 .
  * Cronin-Hennessy  _et al._ [2012] D. Cronin-Hennessy _et al._ (CLEO), Phys. Rev. D 86, 072005 (2012), arXiv:1104.3265 [hep-ex] .
  * Shifman [1980] M. A. Shifman, Z. Phys. C 4, 345 (1980), [Erratum: Z.Phys.C 6, 282 (1980)].
  * Follana  _et al._ [2007] E. Follana, Q. Mason, C. Davies, K. Hornbostel, G. Lepage, J. Shigemitsu, H. Trottier, and K. Wong (HPQCD, UKQCD), Phys. Rev. D 75, 054502 (2007), arXiv:hep-lat/0610092 .
  * Ji and Jung [2001a] X.-d. Ji and C.-w. Jung, Phys. Rev. Lett. 86, 208 (2001a), arXiv:hep-lat/0101014 .
  * Ji and Jung [2001b] X.-d. Ji and C.-w. Jung, Phys. Rev. D 64, 034506 (2001b), arXiv:hep-lat/0103007 .
  * Borsanyi  _et al._ [2012] S. Borsanyi _et al._ , JHEP 09, 010 (2012), arXiv:1203.4469 [hep-lat] .
  * Dowdall  _et al._ [2013] R. Dowdall, C. Davies, G. Lepage, and C. McNeile, Phys. Rev. D 88, 074504 (2013), arXiv:1303.1670 [hep-lat] .
  * Monahan  _et al._ [2013] C. Monahan, J. Shigemitsu, and R. Horgan, Phys. Rev. D 87, 034017 (2013), arXiv:1211.6966 [hep-lat] .
  * Bazavov  _et al._ [2010] A. Bazavov _et al._ (MILC), Phys. Rev. D82, 074501 (2010), arXiv:1004.0342 [hep-lat] .
  * Bazavov  _et al._ [2013] A. Bazavov _et al._ (MILC), Phys. Rev. D87, 054505 (2013), arXiv:1212.4768 [hep-lat] .
  * Aubin  _et al._ [2004] C. Aubin, C. Bernard, C. E. DeTar, J. Osborn, S. Gottlieb, E. Gregory, D. Toussaint, U. Heller, J. Hetrick, and R. Sugar (MILC), Phys. Rev. D 70, 114501 (2004), arXiv:hep-lat/0407028 .
  * Sachrajda and Villadoro [2005] C. Sachrajda and G. Villadoro, Phys. Lett. B 609, 73 (2005), arXiv:hep-lat/0411033 .
  * Guadagnoli  _et al._ [2006] D. Guadagnoli, F. Mescia, and S. Simula, Phys. Rev. D 73, 114504 (2006), arXiv:hep-lat/0512020 .
  * Hatton  _et al._ [2019] D. Hatton, C. T. H. Davies, G. P. Lepage, and A. T. Lytle (HPQCD), Phys. Rev. D 100, 114513 (2019), arXiv:1909.00756 [hep-lat] .
  * Lepage [tter] G. P. Lepage, Corrfitter Version 8.2 (github.com/gplepage/corrfitter), 10.5281/zenodo.5733391.
  * Dowdall  _et al._ [2019] R. J. Dowdall, C. T. H. Davies, R. R. Horgan, G. P. Lepage, C. J. Monahan, J. Shigemitsu, and M. Wingate, Phys. Rev. D 100, 094508 (2019), arXiv:1907.01025 [hep-lat] .
  * Lepage [qfit] G. P. Lepage, lsqfit Version 12.0.3 (github.com/gplepage/lsqfit), 10.5281/zenodo.5777652.
  * Lepage  _et al._ [2002] G. P. Lepage, B. Clark, C. T. H. Davies, K. Hornbostel, P. B. Mackenzie, C. Morningstar, and H. Trottier, Nucl. Phys. B Proc. Suppl. 106, 12 (2002), arXiv:hep-lat/0110175 .
  * Bazavov  _et al._ [2018] A. Bazavov _et al._ , Phys. Rev. D 98, 074512 (2018), arXiv:1712.09262 [hep-lat] .
  * Chakraborty  _et al._ [2015] B. Chakraborty, C. Davies, B. Galloway, P. Knecht, J. Koponen, G. C. Donald, R. J. Dowdall, G. P. Lepage, and C. McNeile, Phys. Rev. D 91, 054508 (2015), arXiv:1408.4169 [hep-lat] .
  * McLean  _et al._ [2019] E. McLean, C. Davies, A. Lytle, and J. Koponen, Phys. Rev. D 99, 114512 (2019), arXiv:1904.02046 [hep-lat] .
  * Ablikim  _et al._ [2013] M. Ablikim _et al._ (BESIII), Phys. Rev. D 87, 032003 (2013), arXiv:1208.1461 [hep-ex] .
  * Wang and Yuan [2022] H. Wang and C.-Z. Yuan, Chin. Phys. C 46, 071001 (2022), arXiv:2112.08584 [hep-ph] .
  * Asner  _et al._ [2004] D. M. Asner _et al._ (CLEO), Phys. Rev. Lett. 92, 142001 (2004), arXiv:hep-ex/0312058 .
  * Lees  _et al._ [2010] J. P. Lees _et al._ (BaBar), Phys. Rev. D 81, 052010 (2010), arXiv:1002.3000 [hep-ex] .
  * Ablikim  _et al._ [2019] M. Ablikim _et al._ (BESIII), Phys. Rev. D 100, 012003 (2019), arXiv:1903.05375 [hep-ex] .
  * Adams  _et al._ [2008] G. S. Adams _et al._ (CLEO), Phys. Rev. Lett. 101, 101801 (2008), arXiv:0806.0671 [hep-ex] .
  * PDG [2022] PDG (Particle Data Group), private communication (2022).
  * Feng  _et al._ [2017] F. Feng, Y. Jia, and W.-L. Sang, Phys. Rev. Lett. 119, 252001 (2017), arXiv:1707.05758 [hep-ph] .
  * Bodwin and Chen [2001] G. T. Bodwin and Y.-Q. Chen, Phys. Rev. D 64, 114008 (2001), arXiv:hep-ph/0106095 .
  * Brambilla  _et al._ [2018] N. Brambilla, H. S. Chung, and J. Komijani, Phys. Rev. D 98, 114020 (2018), arXiv:1810.02586 [hep-ph] .
  * Penin  _et al._ [2004] A. A. Penin, A. Pineda, V. A. Smirnov, and M. Steinhauser, Nucl. Phys. B 699, 183 (2004), [Erratum: Nucl.Phys.B 829, 398–399 (2010)], arXiv:hep-ph/0406175 .
  * Kiyo  _et al._ [2010] Y. Kiyo, A. Pineda, and A. Signer, Nucl. Phys. B 841, 231 (2010), arXiv:1006.2685 [hep-ph] .
  * Landsberg [1985] L. G. Landsberg, Phys. Rept. 128, 301 (1985).
  * Eichten  _et al._ [2008] E. Eichten, S. Godfrey, H. Mahlke, and J. L. Rosner, Rev. Mod. Phys. 80, 1161 (2008), arXiv:hep-ph/0701208 .
  * Khodjamirian [1984] A. Y. Khodjamirian, Sov. J. Nucl. Phys. 39, 614 (1984).
  * Pineda and Segovia [2013] A. Pineda and J. Segovia, Phys. Rev. D 87, 074024 (2013), arXiv:1302.3528 [hep-ph] .
  * Ebert  _et al._ [2003] D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Rev. D 67, 014027 (2003), arXiv:hep-ph/0210381 .
  * Brambilla  _et al._ [2006] N. Brambilla, Y. Jia, and A. Vairo, Phys. Rev. D 73, 054005 (2006), arXiv:hep-ph/0512369 .
  * Guo  _et al._ [2020] S.-P. Guo, Y.-J. Sun, W. Hong, Q. Huang, and G.-H. Zhao, Nucl. Phys. B 955, 115053 (2020), arXiv:1912.00836 [hep-ph] .
  * Deng  _et al._ [2017] W.-J. Deng, H. Liu, L.-C. Gui, and X.-H. Zhong, Phys. Rev. D 95, 034026 (2017), arXiv:1608.00287 [hep-ph] .
  * Gu  _et al._ [2019] L.-M. Gu, H.-B. Li, X.-X. Ma, and M.-Z. Yang, Phys. Rev. D 100, 016018 (2019), arXiv:1904.06085 [hep-ph] .
  * Hatton  _et al._ [2021] D. Hatton, C. T. H. Davies, J. Koponen, G. P. Lepage, and A. T. Lytle, Phys. Rev. D 103, 054512 (2021), arXiv:2101.08103 [hep-lat] .
  * Heister  _et al._ [2002] A. Heister _et al._ (ALEPH), Phys. Lett. B 530, 56 (2002), arXiv:hep-ex/0202011 .
  * [61] MILC Code Repository, https://github.com/milc-qcd.


# Dijet Acoplanarity in CUJET3 as a Probe of the Nonperturbative Color
Structure of QCD Perfect Fluids

M. Gyulassy  P.M. Jacobs  J. Liao  S. Shi  X.N. Wang  F. Yuan  Nuclear Science
Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Institute of Particle Physics, Central China Normal University, Wuhan, China
Physics Department and Center for Exploration of Energy and Matter, Indiana
University,
2401 N Milo B. Sampson Lane, Bloomington, IN 47408, USA  Dept. of Physics,
McGill University, 3600 rue University, Montreal, QC H3A 2T8, Canada

###### Abstract

Using the CUJET3=DGLV+VISHNU jet-medium interaction framework, we show that
dijet azimuthal acoplanarity in high energy $A+A$ collisions is sensitive to
possible non-perturbative enhancement of the jet transport coefficient,
$\hat{q}(T,E)$, in the QCD crossover temperature $T\sim 150-300$ MeV range.
With jet-medium couplings constrained by global RHIC& LHC $\chi^{2}$ fits to
nuclear modification data on $R_{AA}(p_{T}>20)\;{\rm GeV}$, we compare
predictions of the medium induced dijet transverse momentum squared,
$Q_{s}^{2}\sim\langle\hat{q}L\rangle\sim\Delta\phi^{2}E^{2}$, in two models of
the temperature, $T$, and jet energy $E$ dependence of the jet medium
transport coefficient, $\hat{q}(T,E)$. In one model, wQGP, the chromo degrees
of freedom (dof) are approximated by a perturbative dielectric gas of quark
and gluons dof. In the second model, sQGMP, we consider a nonperturbative
partially confined semi-Quark-Gluon-Monopole-Plasma with emergent color
magnetic dof constrained by lattice QCD data. Unlike the slow variation of the
scaled jet transport coefficient, $\hat{q}_{wQGP}/T^{3}$, the sQGMP model
$\hat{q}_{sQGMP}/T^{3}$ features a sharp maximum in the QCD confinement
crossover $T$ range. We show that the dijet path averaged medium induced
azimuthal acoplanarity, $\Delta\phi^{2}$, in sQGMP is robustly $\sim 2$ times
larger than in perturbative wQGP. even though the radiative energy loss in
both models is very similar as needed to fit the same $R_{AA}$ data. Future
A+A dijet acoplanarity measurements constrained together with single jet
$R_{AA}$ and $v_{n}$ measurements therefore appears to be a promising strategy
to search for possible signatures of critical opalescence like phenomena in
the QCD confinement temperature range.

###### keywords:

Quark Gluon Plasmas, Heavy Ion Collision, Jet Quenching, Dijet Acoplanarity

##  1 Introduction and Conclusions

Dijet relative azimuthal angle acoplanarity,
$\Delta\phi^{2}=(\pi-\phi_{1}+\phi_{2})^{2}=(Q_{vac}^{2}+Q_{s}^{2})/E^{2}$, as
a probe of Quark Gluon Plasmas (QGP) produced in high energy nuclear
collisions, has a long history, see e.g. [1, 2, 4, 3, 5, 6, 7]. In both $p+p$
and $A+A$ collisions this observable is always dominated by
$Q_{vac}^{2}/E^{2}\sim\alpha_{s}^{2}\sim 0.1$ due to (Sudakov) multiple gluon
emission into the vacuum associated with all hard QCD processes[8].

The BDMS[2] medium dependent “saturation” scale,
$Q_{s}^{2}=\langle\hat{q}L\rangle=\langle\int dt\;\hat{q}(T(t))\rangle$, is a
jet path averaged measure of jet straggling transverse to its initial
direction $\hat{\mathbf{n}}(\phi_{0})$. It is the average transverse momentum
squared accumulated over a path length $L$. For a jet with color,flavor $a$,
the jet transport coefficient $\hat{q}_{a}$ in a QCD fluid of temperature
$T=T({\mathbf{x}},t)$, $\hat{q}_{a}(T)=\langle q^{2}/\lambda\rangle_{a}=\int
dq^{2}\;q^{2}\Gamma_{a}(q^{2},T)$. It depends on the composition densities ,
$\\{\rho_{b}(T);b=q,g,m,\cdots\\}$, of effective color electric and magnetic
degrees of freedom (dof) as well as on the microscopic differential scattering
rates, $\Gamma_{a}(q^{2},T)\equiv\sum_{b}\rho_{b}(T)d\sigma_{ab}(T)/dq^{2}$.

In fact, $Q_{s}^{2}(\hat{\mathbf{n}},E_{fin})=\langle\int
dt\;\hat{q}(T({\mathbf{z}}(t),t),E_{ini})\rangle_{\\{E_{ini},{\mathbf{z}}(0)\\}}$,
is only one of a large set of jet path line integral functionals depending on
jet paths, ${\mathbf{z}}(t)={\mathbf{x}}_{0}+t\;\hat{\mathbf{n}}(\phi_{0})$
that control the medium modification of jets in $A+A$. In particular, the
correct geometric averaging of $\langle Q_{s}^{2}(E_{fin})\rangle$ for given
observed jet energy and direction requires simultaneous calculation of
radiative and elastic energy loss functionals as well: $\Delta E_{rad}\sim\int
dt\;t\;\hat{q}(t)\;{\cal F}_{rad}(t,E)$ and $\Delta E_{el}\sim\int
dt\;\hat{q}(t)/T(t)$. This is because jet quenching strongly biases the
spatial distribution of jet initial production points, ${\mathbf{x}}_{0}$, to
a sub region of the medium transverse geometry from which jets in a given
direction $\hat{n}(\phi_{0})$ emerge with given final energy $E_{fin}$. Note
that in the asymptotic $E\rightarrow\infty$ BDMS limit $\Delta
E_{rad}^{BDMS}\sim\int dt\;t\hat{q}\approx\langle\hat{q}L^{2}\rangle/2$
because ${\cal F}_{BDMS}(t,\infty)\equiv 1$. However, for non-asymptotic
$E<100$ GeV jet energies of interest here, the DGLV[15] formalism predicts
that $\Delta E_{rad}^{DGLV}=\int dt\int
d^{2}{\mathbf{q}}\;\Gamma_{a}({\mathbf{q}},T(t))\;\left\\{\int
dxd^{2}{\mathbf{k}}\;A({\mathbf{q}},{\mathbf{k}},M^{2}(x,T))\left((1-\cos[t(({\mathbf{k}}-{\mathbf{q}})^{2}+M^{2}(x,T))/(2xE)]\right)\right\\}$.
We found numerically that only in the high energy ($E>100$ GeV) limit can we
approximate ${\cal F}_{rad}(t,E>100)\approx 1$. In the CUJET framework the
DGLV energy loss integrals $\int dtd^{2}{\mathbf{q}}dx{\mathbf{k}}\cdots$
needed to compute $\Delta E_{rad}^{DGLV}\neq\int dt\;t\hat{q}_{DGLV}$ and
hence $\Delta E_{rad}^{DGLV}$ does not simply scale for moderate energy jets
with $\hat{q}$ as does $Q_{s}^{2}[elas]$ by definition.

As emphasized in [1] long ago, dijet acoplanarity , as a stand alone
observable cannot uniquely discriminate between different models of the color
dof $\rho_{b}(T)$ and the microscopic $d\sigma_{ab}$ cross sections. This
ambiguity is further amplified by the strong dependence of all jet path
functionals on the non static, inhomogeneous, anisotropic temperature field ,
$T({\mathbf{z}},t)$ produced in finite $A+A$ collisions. The unavoidable
geometric bias caused by jet quenching implies that the triple set of hard jet
observables $\\{R_{AA},\;v_{n},\;\Delta\phi\\}$ must be strongly correlated.
Hence, measuring the correlation between these three observables should
enhance the discriminating power of hard jet and dijet observables to probe
the color structure of QCD fluids, as we emphasized in [6, 9].

Current interest in $A+A$ dijet acoplanarity observables is motivated by the
first preliminary data from RHIC[25, 26] and LHC[27] that suggest [10, 11]
that future higher statics measurements of the acoplanarity distribution in
the “sweet spot” $20<E_{fin}<80$ GeV jet energy range will be able to resolve
medium induced corrections, $\Delta\phi_{med}^{2}=Q_{s}^{2}/E^{2}$ from the
dominant Sudakov source of dijet acoplanarity[8] that can be directly measured
in $p+p$.

Another important motivation for our focus on dijet acoplanarity here is that
there exist currently several independent frameworks[12, 13, 14] with rather
different combinations of quenching dynamics and viscous hydrodynamics
modeling that have tested equally well at the $\chi^{2}/dof<2$ level against
currently available soft and hard $R_{AA}$ and $v_{n}$ data in $A+A$ at RHIC
and LHC. This work is thus also motivated by the question: “Can dijet
acoplanarity help experimentally to break the current degeneracy between
soft+hard modeling of $A+A$?”

The CUJET3 framework used here utilized the temperature and flow velocity
fields predicted by VISHNU2+1 [13] code with Glauber Initial Conditions. The
DGLV jet quenching theory[15] is then applied to evaluate both $\Delta E$ and
$Q_{s}^{2}$ jet path functionals in the VISHNU2+1 viscous hydrodynamic fluid
fields. See Refs.[16, 17, 18, 19, 20, 21] for details. Our global
$\chi^{2}/dof<2$ fit[16] to available soft+hard data constrained the two free
parameters of CUJET3: the maximum of the running coupling $\alpha_{c}=0.9\pm
0.1$ and the ratio of magnetic to electric screening scales,
$c_{m}=\mu_{M}(T)/(g(T)\mu_{D}(T))\approx 0.25\pm 0.03$. We use the same
values of the two parameters to compute $Q_{s}^{2}(E)$ here.

The CUJET3 jet path functionals are evaluated in two models, wQGP and sQGMP,
of the color composition of the QCD fluid . The wQGP composition model assumes
the color structure of the QCD fluid can be approximated by perturbative two
component color di-electric model with one loop dynamically screened quark and
gluon dof. However, for consistency with lattice QCD equation of state, the
Stefan-Boltzmann partial pressures, $P_{b}^{SB}(T)=T\rho^{SB}_{b}(T)$, are
scaled down by the ratio of the nonperturbative lattice QCD pressure,
$P_{lat}(T)$, to the ideal gas pressure $T(\rho^{SB}_{q}+\rho^{SB}_{g})$.

The sQGMP composition model includes emergent color magnetic monopole (cmm)
degrees of freedom, as proposed in [22, 23, 24] to solve the $R_{AA}\times
v_{2}$ puzzle. In CUJET3, the sQGMP model further generalizes wQGP by not only
adding the monopole dof but also by further reducing the q and g dof partial
pressures by powers of the nonperturbative lattice Polyakov loop, $L(T)$,
and/or the light quark susceptibility, $\chi^{u}_{s}(T)$, data as proposed by
[28]. See [16, 17, 18, 19, 20, 21] for further details.

In Fig.1 (Left panel) we plot the the global $R_{AA}$ $\chi^{2}$ data
constrained quark jet transport fields, $\hat{q}_{sQGMP}(T,E)$ and
$\hat{q}_{wQGP}(T,E)$. Note that $\hat{q}_{sQGMP}(T,E)$ is strongly enhanced
relative to $\hat{q}_{wQGP}(T,E)$ in the QCD crossover temperature range
$160<T<320$ MeV. This is due to enhanced jet-monopole interactions with
$d\sigma_{qm}\propto\alpha_{E}\alpha_{M}=1\gg
d\sigma_{qg}\propto\alpha_{E}^{2}$. The question addressed here is whether
$R_{AA}$ constrained acoplanarity could serve to search for such “critical
opalescence” like phenomena near the confinement temperature range. Our answer
is positive, as we show below.

In the Middle panel of Fig.1, the spacetime isochrone evolution of the VISHNU
temperature field in central 0-10% $Pb+Pb$ 5 ATeV is shown. In the Right panel
of Fig.1 the isochronous evolution of $\hat{q}_{wQGP}$ and $\hat{q}_{sQGMP}$
are compared as a function of the reaction plane $x$ coordinate with at $y=0$.
The emergent monopole degrees in the crossover temperature range near the
freeze-out surface $T=160$ MeV and at late times are seen to enhance
$\hat{q}_{sQGMP}$ by a factor $\sim 4$. The enhancement of $\hat{q}$ near the
crossover surface regions plays the decisive role , as proposed in [22, 23,
24], in enhancing the CUJET3 predicted elliptic azimuthal asymmetry, $v_{2}$,
in agreement with data. .

Our main new result shown in Fig.2 is that with charged hadron $R_{AA}$
constrained $\hat{q}(x,t)$ transport fields, the medium induced single jet
acopanarity broadenning $\Delta\phi^{2}$ is robustly a factor of $\sim 2$
larger in a QGP fluid with magnetic monopole degrees of freedom than in a
purely di-electric (pQCD/HTL type) “wQGP” fluid. .

Fig. 1: (color online) (Left) The CUJET3.1 $R_{AA}$ constrained [16, 17, 18,
19] jet transport field, $\hat{q}_{F}(T,E)$ for quark jets with
$E_{ini}=5,20,100$ GeV are compare to wQGP and sQGMP models of the chromo
electric and magnetic dof in the QCD fluid. Dashed curves for wQGP assume only
color di-electric dof while solid curves for sQGMP assume that the color
electric quark and gluon dof are suppressed by lattice Polyakov loop and quark
susceptibility factors, $\chi_{T}^{u}$, due to only partial confinement in
$160<T<320$ MeV QCD transition range. In sQGMP the remaining dof are assumed
to be color magnetic monopole quasi-parton dof. (Center) The isochronous
evolution of temperature field, $T(x,0,t)$,from VISHNU2+1 viscous
hydrodynamics[13] for 0-10% $Pb+Pb$ 5.02ATeV is shown. (Right) The isochronous
evolution of the jet transport coefficients, $\hat{q}_{wQGP}$ (Blue) and
$\hat{q}_{sQGMP}$ (Red), for $E=20\;{\rm GeV}$ are compared at given $x,y=0$
at times $t=0.6,\cdots,10$ fm/c . Note that $\hat{q}_{sQGMP}$ is strongly
enhanced compared to $\hat{q}_{wQGP}$ in the surface regions and in interior
at late times.

Fig. 2: (Left) Comparison of $R_{AA}^{ch}$ constrained CUJET3.1 predictions
for single parton jet level $Q_{s}^{2}$ in 20-30% centrality Pb+Pb 5.02 ATeV.
The final quenched energy , $E_{fin}$, dependence of the average BDMS
transverse momentum squared scale is compared for the three models of the
color structure of QCD fluids as in Fig.1a but using the evolving VISHNU fluid
$T(x,t)$ filed 20-30% centrality class. Green curves show predictions in wQGP
fluids, blue curves show sQGMP with semi quark and and semi gluon degrees of
freedom suppressed by powers, $L(T)^{1}$ and $L(T)^{2}$ resp., of lattice
Polyakov loop data. The red curves show results assuming an sQGMP fluid with
semi quarks suppressed by lattice light quark susceptibility data on
$\chi^{u}_{2}(T)$, while semi-gluons are suppressed by $L(T)^{2}$ (see [16]
for details). (Right) CUJET3.1 predictions for medium induced azimuthal angle
broadening width squared $\Delta\phi^{2}=Q_{s}^{2}/\langle
E_{\mathrm{ini}}\rangle^{2}$ at the single parton level averaged over both q
and g jets. The abscissa is scaled up by a factor 100 for clarity.

Our previous study[6] of dijet acoplanarity concentrated on the tails of the
$dN/\Delta\phi$ distributions in the $2.4<\Delta\phi<3$ range and showed that
future experiments must reach sub-percent levels of precision to discriminate
between medium dependent BDMS Gaussian and DGLV, power law like Rutherford
tails convoluted on top of the dominant Sudakov vacuum radiation tails. The
present study[9], summarized in Fig.2, utilized the CUJET3=DGLV+VISHNU
framework [16, 17, 18, 19] to compute, at leading partonic level, the elastic
$Q_{s}^{2}[E_{fin}]=\langle\int dt\hat{q}(T(t),E_{fin}+\Delta
E({\mathbf{x}}_{0},\phi_{0})\rangle$, taking into account the unavoidable
geometric ”sunny side up” bias due to jet energy loss via $\Delta
E({\mathbf{x}}_{0},\phi_{0})=\Delta E_{rad}+\Delta E_{elas}$ , that we
constrained by global fits to data on nuclear modification os high $p_{T}$
hadron fragments, $R_{AA}^{ch}(p_{T})$. We compared different models of the
temperature dependence of the color dof composition of the QCD fluid
constrained not only by $R_{AA}^{ch}$ data but also by numerical lattice QCD
equation of state data.

Our main new result, shown in Fig.2a, is that elastic scattering $Q_{s}^{2}$
is predicted to be robustly $\sim 2$ times larger in sQGMP than in wQGP models
of the color structure[7]. Future work must next resolve the current debate on
the sign and magnitude of radiative corrections, $\Delta Q_{s}^{2}[rad]$ to
elastic $Q_{s}^{2}[elas]$ [29, 30, 31]. Our preliminary estimates, to be
reported elsewhere [31], agree with Ref. [30] that $\Delta Q_{s}^{2}[rad]$
reduces moderately the magnitude of elastic scattering induced acoplanarity.

Acknowledgments. SS is supported by the Natural Sci. and Engin.Research
Council of Canada, JL by NSF PHY-1913729, XNW and FY by DOE DE-AC02-05CH11231,
PJ by DOE Contract DE-AC02-05CH1123, and MG and XNW by NSFC grants 11935007,
11221504 and 11890714.

## References

  * [1] D. A. Appel, Phys. Rev. D 33, 717 (1986); J. P. Blaizot and L. D. McLerran, Phys. Rev. D 34, 2739 (1986).
  * [2] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiff, Nucl. Phys. B 484, 265 (1997).
  * [3] A. Majumder et al., Phys. Rev. C 80, 031902 (2009); 125020 (2009).
  * [4] F. D’Eramo et al.JHEP 1901, 172 (2019); JHEP 1305, 031 (2013).
  * [5] T. Luo, S. Cao, Y. He and X. N. Wang, Phys. Lett. B 782, 707 (2018).
  * [6] M. Gyulassy, P. Levai, J.Liao, S. Shi, F. Yuan and X. N. Wang, Nucl. Phys. A 982, 627 (2019)
  * [7] S. Shi, J.Liao, M.Gyulassy, P.M. Jacobs, X.-N. Wang, in prep.
  * [8] S. D. Ellis and W. J. Stirling, Phys. Rev. D 23, 214 (1981).
  * [9] M. Gyulassy, Quark Matter 2019, https://indico.cern.ch/event/792436/contributions/3535718/attachments/
1939248/3215717/MGyulassy_110619_QM19_vff.pdf

  * [10] A. H. Mueller, B. Wu, B. W. Xiao and F. Yuan, Phys. Lett. B 763, 208 (2016); Phys. Rev. D 95, 034007 (2017).
  * [11] L. Chen, G. Y. Qin, S. Y. Wei, B. W. Xiao and H. Z. Zhang, Phys. Lett. B 773, 672 (2017) [arXiv:1607.01932 [hep-ph]].
  * [12] J. Noronha-Hostler, B. Betz, J. Noronha and M. Gyulassy, Phys. Rev. Lett. 116, no. 25, 252301 (2016)
  * [13] C. Shen, Z. Qiu, H. Song, J. Bernhard, S. Bass and U. Heinz, Comput. Phys. Commun. 199, 61 (2016)
  * [14] H. Niemi, K. J. Eskola and R. Paatelainen, Phys. Rev. C 93, no. 2, 024907 (2016)
  * [15] M. Gyulassy, P. Levai and I. Vitev, Phys. Rev. D66, 014005(2002); M. Djordjevic et al., Nucl. Phys. A733, 265(2004)
  * [16] S. Shi, J. Liao and M. Gyulassy, Chin. Phys. C 43, no. 4, 044101 (2019) [arXiv:1808.05461 [hep-ph]].
  * [17] S. Shi, J. Liao and M. Gyulassy, Chin. Phys. C42 (2018) 104104, arXiv:1804.01915 [hep-ph];
  * [18] S. Shi, J. Xu, J. Liao and M. Gyulassy, Nucl. Phys. A 967, 648 (2017) [arXiv:1704.04577 [hep-ph]].
  * [19] J. Xu, J. Liao and M. Gyulassy, JHEP 1602, 169 (2016) [arXiv:1508.00552 [hep-ph]].
  * [20] J. Xu, J. Liao and M. Gyulassy, Chin. Phys. Lett. 32, no. 9, 092501 (2015) [arXiv:1411.3673 [hep-ph]];
  * [21] J. Xu, A. Buzzatti and M. Gyulassy, JHEP 1408, 063 (2014) [arXiv:1402.2956 [hep-ph]].
  * [22] J. Liao and E. Shuryak, Phys. Rev. C 75, 054907 (2007) [hep-ph/0611131].
  * [23] J. Liao and E. Shuryak, Phys. Rev. Lett. 101, 162302 (2008) [arXiv:0804.0255 [hep-ph]].
  * [24] J. Liao and E. Shuryak, Phys. Rev. Lett. 102, 202302 (2009) [arXiv:0810.4116 [nucl-th]].
  * [25] PHENIX, A. Adare et al. , PRC77, 011901 (2008);STAR L. Adamczyk et al., Phys. Lett. B 760, 689 (2016).
  * [26] L. Adamczyk et al. [STAR Collaboration], Phys. Rev. C 96, no. 2, 024905 (2017)
  * [27] J. Adam et al. [ALICE Collaboration], JHEP 1509, 170 (2015)
  * [28] Y. Hidaka and R. D. Pisarski, Phys. Rev. D 78, 071501 (2008)
  * [29] J. P. Blaizot and F. Dominguez, Phys. Rev. D 99, no. 5, 054005 (2019)
  * [30] B. G. Zakharov, Zh. Eksp. Teor. Fiz. 156, 615 (2019) [J. Exp. Theor. Phys. 129, no. 4, 521 (2019)]
  * [31] S.Shi, J.Liao, and M.Gyulassy, in preparation.


# Probing the Quantum Noise of the Spinon Fermi Surface with NV Centers

Jun Yong Khoo  Institute of High Performance Computing, Agency for Science,
Technology, and Research, Singapore 138632  Max-Planck-Institut für Physik
komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany  Falko Pientka
Institut für Theoretische Physik, Goethe-Universität, 60438 Frankfurt a.M.,
Germany  Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße
38, 01187 Dresden, Germany  Patrick A. Lee  Department of Physics,
Massachusetts Institute of Technology, Cambridge Massachusetts 02139, USA
Inti Sodemann Villadiego  Institut für Theoretische Physik, Universität
Leipzig, Brüderstraße 16, 04103, Leipzig, Germany  Max-Planck-Institut für
Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany

###### Abstract

We study the transverse electrical conductivity and the corresponding magnetic
noise of a two-dimensional U(1) spin liquid state with a spinon Fermi surface.
We show that in the quasi-static regime these responses have the same wave-
vector dependence as that of a metal but are reduced by a dimensionless pre-
factor controlled by the ratio of orbital diamagnetic susceptibilities of the
spinons and chargons, correcting previous work. We estimate that this quasi-
static regime is comfortably accessed by the typical NV center splittings of a
few GHz and estimate that the expected $T_{1}$ times for an NV center placed
above candidate materials, such as the organic dmit and ET salts, monolayer
1T-TaS2/Se2, would range from several tens to a few hundred milliseconds.

###### pacs:

##  I Introduction

A recent work Khoo  _et al._ (2021) has shown that even when a metallic Fermi
surface has a complex shape, the dissipative real part of the transverse
conductivity is a universal quantity controlled entirely by its geometric
shape, which in turn gives rise to a universal $T_{1}$ time for an NV center
controlled only by the length of the perimeter of the Fermi surface (in the
regime where the NV center distance $z$ satisfies $p_{\rm F}^{-1}\ll z\ll
l_{\rm mfp}$). However, while the conclusions of Ref. Khoo  _et al._ (2021) on
metallic Fermi surface states remain valid, the same work concluded
incorrectly that the spinon Fermi surface state would have the same value for
the low-frequency limit of the transverse conductivity and the associated
$T_{1}$ time as that of a metallic Fermi liquid.

The reason for this discrepancy is that Ref. Khoo  _et al._ (2021)
inadvertently missed the diamagnetic contributions of the spinons. As we will
discuss here, the spinon transverse conductivity assumes a standard form of a
Fermi liquid given by (in the limit $\omega\ll v_{\rm F}q\ll\epsilon_{\rm
F}$)Pines and Nozières (2018); Giuliani and Vignale (2005)

$\displaystyle\sigma^{\rm
spinon}_{\perp}(q,\omega)\approx\frac{e^{2}}{h}\frac{p_{\rm
F}}{q}+\chi_{s}\frac{q^{2}}{i\omega}.$ (1)

Moreover, we will show that the expression for the physical transverse
electrical conductivity of the spinon Fermi surface state (in the low
frequency limit) is given by:

$\displaystyle\sigma_{\perp}(q,\omega)\approx\left(\frac{\chi_{c}}{\chi_{s}+\chi_{a}+\chi_{c}}\right)^{2}\frac{e^{2}}{h}\frac{p_{\rm
F}}{q}+\chi\frac{q^{2}}{i\omega}.$ (2)

The above expressions apply in the zero temperature and clean limit of a spin-
degenerate, circular Fermi surface with radius $p_{\rm F}$. $\chi_{s,c}$ are
spinon and chargon orbital diagmagnetic susceptibilities, $\chi_{a}$ is a
diamagnetic susceptibility associated with the Maxwell term of the emergent
U(1) gauge field and $\chi^{-1}=(\chi_{s}+\chi_{a})^{-1}+\chi^{-1}_{c}$.
Therefore, we see that while the transverse conductivity has the same form as
that of an ordinary metal in this quasi-static regime, its real (dissipative)
part is corrected by the dimensionless factor
$\chi_{c}^{2}/(\chi_{s}+\chi_{a}+\chi_{c})^{2}$. As a consequence, the
remarkable result of Ref. Khoo  _et al._ (2021) that the relaxation rate
$1/T_{1}$ of an NV center coupled to a two-dimensional spinon Fermi surface
state has the same dependence on distance, frequency, and temperature as in
the metallic case remains valid, albeit with an extra pre-factor,
$\chi_{c}^{2}/(\chi_{s}+\chi_{a}+\chi_{c})^{2}$ (see Table 1 for specific
estimates).

##  II Spinon Fermi surface low-energy theory

The following low energy effective Lagrangian captures the orbital coupling of
the spinon Fermi surface state to the physical electromagnetic fields (denoted
by
$\boldsymbol{E}=-\boldsymbol{\partial}_{\boldsymbol{r}}\varphi-\partial_{t}\boldsymbol{A}$
and
$\boldsymbol{B}=\boldsymbol{\partial}_{\boldsymbol{r}}\times\boldsymbol{A}$):

$\mathcal{L}=\mathcal{L}_{\rm
spinon}(\boldsymbol{p}-\boldsymbol{a})+\mathcal{L}_{\rm
chargon}(\boldsymbol{p}-\boldsymbol{A}+\boldsymbol{a})\\\
+\frac{\epsilon_{a}}{2}\boldsymbol{\mathsf{e}}^{2}-\frac{\chi_{a}}{2}\boldsymbol{\mathsf{b}}^{2}+\cdots.$
(3)

Let us briefly explain the physical meaning of the various terms. Here we are
imagining that deconfinement of the U(1) gauge field has taken place and thus
the emergent electric and magnetic fields, denoted by
$\boldsymbol{\mathsf{e}}=-\boldsymbol{\partial}_{\boldsymbol{r}}\phi-\partial_{t}\boldsymbol{\mathsf{a}}$
and
$\boldsymbol{\mathsf{b}}=\boldsymbol{\partial}_{\boldsymbol{r}}\times\boldsymbol{\mathsf{a}}$,
can be taken to be non-compact. The emergent gauge field now acquires dynamics
which is described by a Maxwell-like term and parametrized by emergent
dielectric and diamagnetic constants $\epsilon_{a}$ and $\chi_{a}$.
$\mathcal{L}_{\rm spinon}(\boldsymbol{p}-\boldsymbol{a})$ is a short-hand for
the Lagrangian of a fermion (the spinon) minimally coupled to the emergent
U(1) gauge field $\boldsymbol{a}$, but neutral under the physical gauge field
$\boldsymbol{A}$. On the other hand $\mathcal{L}_{\rm
chargon}(\boldsymbol{p}-\boldsymbol{A}+\boldsymbol{a})$ describes the
Lagrangian of a boson (the chargon) that minimally couples to the physical
field and carries opposite gauge charge to the spinon with respect to the
emergent field. The above Lagrangian can be motivated from a slave-boson (or
the closely related slave-rotor Florens and Georges (2004)) parton
decomposition of the electron operator which is viewed as the composite of the
spinon and chargon (see e.g. Lee  _et al._ (2006); Senthil (2008)).
Nevertheless, we would like to emphasize that in the above Lagrangian the
spinon and chargon should not be viewed strictly speaking as the same objects
as the unphysical UV partons, but rather as low energy deconfined physical
quasiparticles. In fact, due to the presence of the Maxwell term, if we were
to change the Lagrangian in Eq.(3) by coupling the spinon to the physical
gauge field instead of the chargon, one would obtain different predictions for
phyisical gauge-invariant observables. Therefore, in this sense, we no longer
have the freedom to assign the coupling of the electromagnetic field to either
the chargon or the spinon and get the same answer.

Because the chargons are gapped they can be integrated out for the purpose of
the low energy description, and thus their lagrangian can be replaced by a
Maxwell term of the form:

$\displaystyle\mathcal{L}_{\rm
chargon}(\boldsymbol{p}-\boldsymbol{A}+\boldsymbol{a})\approx\frac{\epsilon_{c}}{2}(\boldsymbol{\mathsf{e}}-\boldsymbol{E})^{2}-\frac{\chi_{c}}{2}(\boldsymbol{\mathsf{b}}-\boldsymbol{B})^{2}.$
(4)

The effective dielectric and diamagnetic constants of the chargons,
$\epsilon_{c}$ and $\chi_{c}$, will depend on their detailed microscopic
dispersion. For example, within a relativistic boson model of the chargon
dispersion, one obtains $\chi_{c}=1/(24\pi m_{c})$, where
$m_{c}=\Delta_{c}/v_{c}^{2}$, is the effective mass of the chargons with a
speed $v_{c}$ and a gap $\Delta_{c}$, as shown in Ref. Dai  _et al._ (2020).

On the other hand for the spinons, one can keep track of their occupation of
momentum states only within a narrow sliver around the Fermi surface while
integrating out higher energy modes. This amounts to replacing
$\mathcal{L}_{\rm spinon}$ by the following Lagrangian:

$\displaystyle\mathcal{L}_{\rm
spinon}(\boldsymbol{p}-\boldsymbol{a})\approx\mathcal{L}^{FS}_{\rm
spinon}(\boldsymbol{p}-\boldsymbol{a})+\frac{\epsilon_{s}}{2}\boldsymbol{\mathsf{e}}^{2}-\frac{\chi_{s}}{2}\boldsymbol{\mathsf{b}}^{2}.$
(5)

The Maxwell terms for the emergent electromagnetic fields arise from
integrating out the modes away from the Fermi surface, and in particular the
magnetic term accounts for the spinon diamagnetism ($\chi_{s}$). These terms
were missed in Ref. Khoo  _et al._ (2021). For a non-relativistic parabolic
dispersion for the spinons one obtains $\chi_{s}=g_{s}/(24\pi m_{s})$ (namely
the standard Landau diagmagnetic constant with $g_{s}=2$ accounting for the
spin degeneracy).

Next we follow the classic work of Ioffe and Larkin Ioffe and Larkin (1989) to
derive a relation between the physical conductivity and the spinon and chargon
conductivities. Due to the appearance of the Maxwell term in Eq. (3), which
was absent in the original Ioffe-Larkin paper, the derivation is a little
different and the result slightly modified. First, the Euler-Lagrange
equations of motion that follow from Eq. (3), by taking its variational
derivatives with respect to $\boldsymbol{a}$,
$\delta\mathcal{L}/\delta\boldsymbol{a}=0$, is given by:

$\chi_{a}\boldsymbol{\partial}_{\boldsymbol{r}}\times\boldsymbol{\mathsf{b}}-\epsilon_{a}\partial_{t}\boldsymbol{\mathsf{e}}=\boldsymbol{j}_{\rm
s}-\boldsymbol{j}_{\rm c}.$ (6)

where ${\boldsymbol{j}}_{s}=\delta\mathcal{L}_{\rm
spinon}/\delta\boldsymbol{a}$ and
${\boldsymbol{j}}_{c}=-\delta\mathcal{L}_{\rm chargon}/\delta\boldsymbol{a}$
are the spinon and chargon current densities respectively. Second, by taking
functional derivatives with respect to the physical gauge field, we find that:

${\boldsymbol{j}}_{e}=\frac{\delta\mathcal{L}}{\delta\boldsymbol{A}}={\boldsymbol{j}}_{c},$
(7)

so that the physical electric current density ${\boldsymbol{j}}_{e}$ is the
same as the chargon current ${\boldsymbol{j}}_{c}$ but is modified from the
spinon current. To obtain explicit dependence in the electric fields, we use
Faraday’s law
$\boldsymbol{\partial}_{\boldsymbol{r}}\times\boldsymbol{\mathsf{e}}=-\partial_{t}\boldsymbol{\mathsf{b}}$
to eliminate $\boldsymbol{b}$ in Eq. (6). It is convenient to decompose the
currents into perpendicular and transverse components and define the
corresponding conductivity as:

$\displaystyle\begin{split}&{\boldsymbol{j}}_{e\perp}=\sigma_{\perp}\boldsymbol{E}_{\perp},\\\
&{\boldsymbol{j}}_{s\perp}=\sigma_{s\perp}\boldsymbol{\mathsf{e}}_{\perp},\\\
&{\boldsymbol{j}}_{c\perp}=\sigma_{c\perp}(\boldsymbol{E_{\perp}-\mathsf{e}}_{\perp}),\\\
\end{split}$ (8)

and similarly for the longitudinal components. Going to Fourier space, we find
the relations between $\boldsymbol{e}$ and $\boldsymbol{E}$:

$\displaystyle\begin{split}&(\sigma_{c\|}+\sigma_{s\|}+i\epsilon_{a}\omega)\boldsymbol{\mathsf{e}}_{\|}=\sigma_{c\|}\boldsymbol{E}_{\|},\\\
&\left(\sigma_{c\perp}+\sigma_{s\perp}+i\epsilon_{a}\omega+\chi_{a}\frac{q^{2}}{i\omega}\right)\boldsymbol{\mathsf{e}}_{\perp}=\sigma_{c\perp}\boldsymbol{E}_{\perp},\\\
\end{split}$ (9)

here the conductivities are understood to be frequency and wave-vector
dependent. It is convenient to introduce:

$\displaystyle\begin{split}&\sigma^{\prime}_{s\|}\equiv\sigma_{s\|}+i\epsilon_{a}\omega,\\\
&\sigma^{\prime}_{s\perp}\equiv\sigma_{s\perp}+i\epsilon_{a}\omega+\chi_{a}\frac{q^{2}}{i\omega}.\end{split}$
(10)

Then, by using Eqs. (7) and (LABEL:perpconducs) to compute $\sigma$ and using
Eq. (LABEL:evsE) to eliminate $\boldsymbol{\mathsf{e}}$, we arrive at a result
with the same form as the classic Ioffe-Larkin formula,
$\sigma^{-1}=\sigma_{c}^{-1}+\sigma_{s}^{\prime-1}$. The longitudinal and
transverse components of the physical conductivity $\sigma$ are displayed
below:

$\displaystyle\sigma_{\|}^{-1}(q,\omega)=\sigma_{c\|}^{-1}(\omega)+(\sigma_{s\|}(q,\omega)+i\omega\epsilon_{a})^{-1},$
(11)
$\displaystyle\sigma_{\perp}^{-1}(q,\omega)=\sigma_{c\perp}^{-1}(q,\omega)+\left(\sigma_{s\perp}(q,\omega)+i\omega\epsilon_{a}+\chi_{a}\frac{q^{2}}{i\omega}\right)^{-1}$
(12)

Compared with the stand Ioffe-Larkin formula, the spinon conductivity has
extra terms which are proportional to $\epsilon_{a}$ and $\chi_{a}$.

Now we apply Eq. (12) to the problem at hand. Since we are interested in the
limit $\omega\ll v_{\rm F}q\ll\epsilon_{\rm F}$) we can drop the term
$i\omega\epsilon_{a}$ in Eq. (12). We first treat the spinon as a free Fermi
liquid and use the form given by Eq. (1) for $\sigma_{s\perp}$. Since the
chargon is gapped, we set $\sigma_{c\perp}=\chi_{c}\frac{q^{2}}{i\omega}$.
Plugging into Eq. (12) we obtain our main result given by Eq. (2). The first
term is the real (dissipative) part that enters into the calculation of the
relaxation rate of the NV center. As noted in the introduction, it is reduced
from that of a metal by the factor
$\chi_{c}^{2}/(\chi_{s}+\chi_{a}+\chi_{c})^{2}$. We can interpret this result
as originating from the fact that the physical fluctuating current is reduced
from that of a free Fermi sea by the factor
$F=\chi_{c}/(\chi_{s}+\chi_{a}+\chi_{c})$ because its flow is restricted by
coupling to the emergent gauge field. The conductivity is suppressed by
$F^{2}$ because it is proportional to a product of two current operators. It
is also worth noting that when compared with the standard Ioffe-Larkin
formula, $\chi_{s}$ is replaced by $\chi_{s}+\chi_{a}$. Thus the appearance of
the Maxwell term in the gauge field Lagrangian can be absorbed as a re-
definition of the spinon diamagnetic susceptibility if one chooses to use the
original Ioffe-Larkin formula.

In Ref. Khoo  _et al._ (2021) the spinon conductivity was treated more
accurately including Fermi liquid corrections due to the scattering of quasi-
particles near the Fermi surface as well as accounting for Landau parameters.
For completeness we give the relevant formulas below which can be plugged into
Eq. (12) to give the physical conductivity. Nevertheless, as we will see, in
the low-frequency limit $\omega\ll v_{\rm F}q\ll\epsilon_{\rm F}$ and clean
limit $q\gg l_{\rm mfp}^{-1}$ (see upcoming discussion in Section III), the
same reduction factor $F^{2}$ appears in the dissipative part of the
conductivity which enters into the relaxation rate of the NV center, and
therefore one recovers the same limit as in Eq. (2) even within this more
accurate description of the spinon conductivity.

The contribution to the spinon density ($\rho_{\rm FS}$) and current
($\boldsymbol{j}_{\rm FS}$) arising from fluctuations in the vicinity of the
Fermi surface can be expressed in terms of the change of the spinon occupation
near the Fermi surface ($\delta n_{\boldsymbol{p}}$) as follows:

$\displaystyle\rho_{\rm FS}$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{A}}\sum_{\boldsymbol{p}}\delta
n_{\boldsymbol{p}},\ \boldsymbol{j}_{\rm
FS}=\frac{1}{\mathcal{A}}\sum_{\boldsymbol{p}}\boldsymbol{v}_{\boldsymbol{p}}\delta\bar{n}_{\boldsymbol{p}},$
(13) $\displaystyle\delta\bar{n}_{\boldsymbol{p}}$ $\displaystyle=$
$\displaystyle\delta
n_{\boldsymbol{p}}+\sum_{\boldsymbol{p^{\prime}}}f_{\boldsymbol{p}\boldsymbol{p^{\prime}}}\delta(\epsilon_{\boldsymbol{p}}-\epsilon_{\rm
F})\delta n_{\boldsymbol{p^{\prime}}},$ (14)

where
$\boldsymbol{v}_{\boldsymbol{p}}=\partial_{\boldsymbol{p}}\epsilon({\boldsymbol{p}})$
is the spinon quasiparticle velocity and
$f_{{\boldsymbol{p}},{\boldsymbol{p^{\prime}}}}$ accounts for Landau
parameters. The spinon distribution function obeys the linearized kinetic
equation

$\displaystyle\partial_{t}\delta n_{\boldsymbol{p}}$
$\displaystyle+\boldsymbol{v}_{\boldsymbol{p}}\cdot\boldsymbol{\partial}_{\boldsymbol{r}}\delta\bar{n}_{\boldsymbol{p}}+\boldsymbol{\mathsf{e}}\cdot\boldsymbol{v}_{\boldsymbol{p}}\delta(\epsilon_{\boldsymbol{p}}-\epsilon_{\rm
F})=I[\delta n_{\boldsymbol{p}}],$ (15)

where $I[\delta n_{\boldsymbol{p}}]$ accounts for momentum relaxing and
momentum preserving collisions with respective scattering rates denoted by
$\Gamma_{1}$ and $\Gamma_{2}$ based on the model for collisions described in
Refs. Khoo  _et al._ (2021); Levitov and Falkovich (2016); Guo  _et al._
(2017); Alekseev and Semina (2018). By solving the kinetic equation for a
circular Fermi surface, one obtains the following expressions for the
different conductivities:

$\displaystyle\sigma_{s\|}=i\frac{n}{m}\left[\frac{2}{\frac{2in}{m}\rho_{*}(q,\omega)+F_{1}\omega_{-}-\omega_{+}-2i\Gamma_{2}}\right]+i\omega\epsilon_{s},$
(16)
$\displaystyle\sigma_{s\perp}=i\frac{n}{m}\left[\frac{2}{F_{1}\omega_{-}-\omega_{+}-2i\Gamma_{2}}\right]+i\omega\epsilon_{s}+\chi_{s}\frac{q^{2}}{i\omega},$
(17)
$\displaystyle\rho_{*}(q,\omega)=-i\frac{1}{n^{2}\kappa}\frac{q^{2}}{\omega},\quad\kappa=\frac{1}{nE_{\rm
F}}\frac{1}{1+F_{0}},$ (18)
$\displaystyle\omega_{\pm}=\omega-i(\Gamma_{1}+\Gamma_{2})\pm\sqrt{\left[\omega-i(\Gamma_{1}+\Gamma_{2})\right]^{2}-(v_{\rm
F}q)^{2}},$ (19) $\displaystyle\sigma_{c\|}=i\omega\epsilon_{c},\
\sigma_{c\perp}=i\omega\epsilon_{c}+\chi_{c}\frac{q^{2}}{i\omega},$ (20)

where $n=p_{\rm F}^{2}/4\pi$ is the spinon density, $E_{\rm
F}=p_{F}^{2}/2m^{*}$ is the Fermi energy, and we have included only two non-
zero Landau parameters $F_{0}$ and $F_{1}$. The Landau parameter $F_{1}$
determines the renormalization of the spinon quasiparticle mass $m^{*}$
relative to its transport mass $m$: $m=m^{*}/(1+F_{1})$. The above expressions
from Eqs. (20) can then be plugged onto Eqs. (12).

Although we are focusing on the properties of the transverse conductivity in
this study, we mention in passing that the optical conductivity peaks at some
typical scale that can be viewed as the optical pseudo-gap associated with the
Mott scale (see Fig. 1 and further information in Appendix B). The expression
for this scale is:

$\bar{\omega}=\sqrt{\frac{n}{m(\epsilon_{c}+\epsilon_{s}+\epsilon_{a})}}.$
(21)

##  III Conductivities and magnetic noise of the Spinon Fermi surface state

As discussed in Ref. Khoo  _et al._ (2021), the clean limit of the quasi-
static conductivity is attained for wave-vectors above an inverse spinon mean-
free path, $q\gg l_{\rm mfp}^{-1}$, given by

$\displaystyle l_{\rm mfp}^{-1}$ $\displaystyle={\rm max}(q_{C},q_{D}),$ (22)
$\displaystyle v_{\rm F}q_{D}$ $\displaystyle=\frac{2\Gamma_{1}}{(1+F_{1})},\
v_{\rm F}q_{C}=(\Gamma_{1}+\Gamma_{2}).$ (23)

In this wave-vector regime, Eq. (2) can be obtained by taking the limit
$\omega\rightarrow 0$, from the expressions in Eqs. (11)-(20). At fixed wave-
vector but finite frequency, the real part of the transverse conductivity
vanishes above a typical frequency window $\Delta\omega_{s}$, as illustrated
in Fig. 1, which can be estimated to be:

$\displaystyle\frac{\Delta\omega_{s}}{2\pi}$ $\displaystyle\simeq$
$\displaystyle\frac{(\chi_{c}+\chi_{s}+\chi_{a})q^{3}}{p_{\rm F}}.$ (24)

Hence, measuring the quasi-static conductivity requires experiments to operate
at frequencies $\omega\ll\Delta\omega_{s}$. NV centers have a level splitting
of about $3$GHz Casola  _et al._ (2018), well below typical values for
$\Delta\omega_{s}$ listed in Table 1, making them ideally suited to probe the
quasi-static regime of the conductivity. As we will see later on, the main
challenge is their rather weak coupling to the spin liquid which leads to
relatively long $T_{1}$ times.

| $k_{\rm F}^{-1}$ | $\epsilon_{\rm F}$ | $\omega_{p}$ | $\Delta\omega_{s}/2\pi$ | $T_{1}$
---|---|---|---|---|---
| (A) | (meV) | (meV) | ${\rm(GHz)}$ | $({\rm ms})$
dmit | 2.4 | 59 | 80 Rao and Sodemann (2019) | 215 | 58
$\kappa$-ET | 3.2 | 98 | 87 Rao and Sodemann (2019) | $2\times 10^{3}$ | 78
1T-TaS2/Se2 | 4.4 | - | 200 Rao and Sodemann (2019) | - | 109
WTe2 | 26 | 29 | 60 Wang  _et al._ (2021) | $6\times 10^{4}$ | 630
Table 1: Order-of-magnitude estimates for material candidates (dmit stands for
EtMe3Sb[Pd(dmit)2]2 and $\kappa$-ET for $\kappa$-(ET)2Cu2(CN)3). We took
$c=0.5v_{\rm F}$, $F_{1}=0$ and $\chi_{c}=\chi_{s}+\chi_{a}$, and for
simplicity $\epsilon_{s}=\epsilon_{a}=0$, so that the Mott scale,
$\bar{\omega}$, is determined by the chargon dielectric constant
$\epsilon_{c}$. We have taken the temperature to be $T=10K$, and an NV center-
to-sample distance of $z=1nm$ and the NV center splitting to be $\omega=3$GHz
(we take $q=1/z$ to estimate $\Delta\omega_{s}$). The $\epsilon_{\rm F}$ and
$\Delta\omega_{s}$ of 1T-TaS2 are left blank as they are still uncertain (see
discussion in main text). These values correct those of Table. 2 of Ref. Khoo
_et al._ , 2021. Notice that the above are estimates for a single layer of
each material. For multi-layered or bulk samples we can add up the
contributions of each layer and there will be reduction of the $T_{1}$ time by
a factor of the order $T_{1}\rightarrow T_{1}\frac{d}{z\log(l_{\rm
mfp}/z)}\sim T_{1}\frac{d}{z}$, where $d$ is the interlayer distance (assuming
$l_{\rm mfp}\gg z\gg d$). Figure 1:  Comparison of the conductivity of the
spinon Fermi surface state (labeled SFSS in blue) and a metal (black). This
corrects Fig. 3 of Ref. Khoo  _et al._ , 2021 by including the spinon Landau
diamagnetism. (a) Re $\sigma_{\perp}(q,\omega)$ for $q=0.3p_{\rm F,0}$ [along
solid vertical line cut in (b)]. (b) Dispersion of collective modes and
particle-hole excitations in the spinon Fermi surface state ($F_{l}=0$). (c)
Re $\sigma_{\|}(q\rightarrow 0,\omega)$ with weak collisions:
$\Gamma_{1}=\Gamma_{2}=0.1v_{\rm F}p_{\rm F,0}$. (b,c) The optical pseudo-gap
in this case is $\bar{\omega}$ [see Appendix. B of the Supplementary Material
sup  for details]. (d) Re $\sigma_{\perp}(q,\omega)$ at different (small)
frequencies. We take here $\chi_{s}+\chi_{a}=\chi_{c}$ and
$\epsilon_{s}=\epsilon_{a}=0$.

In the more general case of a noncircular spinon Fermi surface, the real part
of the quasi-static transverse conductivity (at low temperatures and for
$p_{\rm F}^{-1}\gg q\gg 1/l_{\rm mfp}$) is given by:

${\rm
Re}~{}\sigma_{\perp}=\left(\frac{\chi_{c}}{\chi_{s}+\chi_{a}+\chi_{c}}\right)^{2}(2S+1)\frac{e^{2}}{2hq}\sum_{i}\mathcal{R}_{\rm
F}|_{\boldsymbol{p}_{i}^{*}(\hat{q})},$ (25)

where $(2S+1)$ is the spin degeneracy factor,
$\left\\{\boldsymbol{p}^{*}_{i}\right\\}$ is the set of points on the Fermi
surface at which the Fermi velocity is orthogonal to $\hat{q}$, and
$\mathcal{R}_{\rm F}|_{\boldsymbol{p}_{i}^{*}(\hat{q})}$ the absolute value of
the local radius of curvature of the Fermi surface at
$\boldsymbol{p}_{i}^{*}$. Therefore, the conductivity only depends on the
geometry of the Fermi surface and is reduced by the same factor,
$\chi_{c}^{2}/(\chi_{s}+\chi_{a}+\chi_{c})^{2}$, relative to the case of
metals described in Ref Khoo  _et al._ (2021).

The above quasi-static transverse conductivity can be probed by measuring the
$T_{1}$-time of a single spin (NV center) placed above the sample at distance
$z$ Casola  _et al._ (2018); Agarwal  _et al._ (2017); Chatterjee  _et al._
(2019). This $T_{1}$-time is inversely proportional to the imaginary part of
the magnetic field autocorrelation function (magnetic noise) at the NV center
location, ${\rm Im}\chi_{B_{\mu}B_{\nu}}(z,\omega)$, and at the frequency
$\omega$ given by the energy splitting of the NV center Casola  _et al._
(2018); Langsjoen  _et al._ (2012). For the circular Fermi surface, we have:

$\displaystyle{\rm Im}\chi_{B_{z}B_{z}}(z,\omega)$
$\displaystyle=\frac{\mu_{0}^{2}\omega}{8\pi}\int qdqe^{-2qz}{\rm
Re}~{}\sigma_{\perp}(q,\omega).$ (26)

Figure 2:  Magnetic noise of the spinon Fermi surface state (in blue and
labeled SFSS) analogous to those shown in Fig. 5 of Ref. Khoo  _et al._ ,
2021, but including the Landau diamagnetism, for $\epsilon_{s}=\epsilon_{a}=0$
and $\chi_{s}=\chi_{c}$. The analogous behavior of the metals is shown in the
black and red curves.

More general expressions are discussed in Ref. Khoo  _et al._ (2021). With our
complete expressions for the conductivity from Eqs. (11)–(20), we have plotted
the above autocorrelation function at low frequencies in Fig. 2 as a function
of the NV center distance $z$. We see that when the distance of the NV center
is smaller than the spinon mean free path ($p_{\rm F}^{-1}\ll z\ll l_{\rm
mfp}$) the magnetic noise approaches the following universal value at low
frequencies:

$\displaystyle{\rm Im}\chi_{B_{z}B_{z}}$
$\displaystyle\simeq\left(\frac{\chi_{c}}{\chi_{s}+\chi_{a}+\chi_{c}}\right)^{2}\frac{e^{2}\mu_{0}^{2}}{16\pi
h}\frac{\omega}{z}\frac{(2S+1)}{2\pi}\mathcal{P}_{\rm FS},$ (27)

with subleading corrections to the above starting at
$\mathcal{O}(\omega^{3})$. The above formula applies to Fermi surfaces of
arbitrary shapes and here $\mathcal{P}_{\rm FS}$ is the length of the
perimeter of the spinon Fermi surface, which in the case of the circle is
$\mathcal{P}_{\rm FS}=2\pi p_{\rm F}$. The above expression contains the
dimensionless prefactor, $\chi_{c}^{2}/(\chi_{s}+\chi_{a}+\chi_{c})^{2}$,
which was missing in Ref. Khoo  _et al._ (2021). From the above we have
estimated the $T_{1}$ times in Table 1 using the following formula Casola  _et
al._ (2018); Langsjoen  _et al._ (2012):

$\displaystyle\frac{1}{T_{1}}=\frac{\mu_{B}^{2}}{2\hbar}\coth\Bigl{(}\frac{\beta\hbar\omega}{2}\Bigr{)}{\rm
Im}\chi_{B_{z}B_{z}}(z,\omega),$ (28)

where here we have simply set the magnetic moment of the NV to be the Bohr
magneton for purposes of order-of-magnitude estimates. For Table 1 we have
taken $z=1$nm, $\omega=3$GHz and $T=10$K as parameters for the NV center
Casola  _et al._ (2018). The search for potential spin liquids states in
1T-TaS2 and 1T-TaSe2 Law and Lee (2017); He  _et al._ (2018); Ruan  _et al._
(2021); Chen  _et al._ (2021) might be simpler in monolayers compared to bulk
samples, as there are two possible types of surfaces for the latter: unpaired
single layer or paired double layer Butler  _et al._ (2020). Unfortunately so
far the specific heat has only been measured in bulk samples, which
complicates the extraction of the spinon Fermi energy from bulk measurements,
such as the specific heat Ribak  _et al._ (2017). This is why we have not
indicated a value for $\epsilon_{\rm F}$ and $\Delta\omega_{s}$ in Table 1.
Nevertheless, the estimate of the $T_{1}$ time can be performed without
detailed knowledge of the spinon Fermi energy scale, thanks to its simple
dependence on the length of the perimeter of the Fermi surface (which can be
estimated from the spinon density). This is why we have been able to provide
an estimate for the expected $T_{1}$ time of 1T-TaS2/Se2 in Table 1.

We also would like to mention in passing that the discussion of this work does
not necessarily apply to the “pseudo-scalar” spinon Fermi surface introduced
in Ref. Sodemann Villadiego, 2021 to explain the oscillations of thermal
conductivity in $\alpha$-RuCl3 Czajka  _et al._ (2021); Bruin  _et al._
(2021).

##  IV Summary and outlook

We have shown that in the regime of wave-vectors $l^{-1}_{\rm mfp}\ll q\ll
p_{\rm F}$, the dissipative part of the transverse electrical conductivity of
a spinon Fermi surface state has the same form as that of a metal, but it is
multiplied by the overall prefactor
$\chi_{c}^{2}/(\chi_{s}+\chi_{a}+\chi_{c})^{2}$, where $\chi_{s/c}$ are the
spinon/chargon orbital diamagnetic susceptibilities and $\chi_{a}$ is the
diamagnetic susceptibility of the emergent U(1) gauge field, correcting the
results of Ref. Khoo  _et al._ (2021). Interestingly this prefactor also
controls the effective magnetic field that the spinons experience in response
to a physical magnetic field and consequently the period of their quantum
oscillations Motrunich (2006); Chowdhury  _et al._ (2018); Sodemann  _et al._
(2018).

The $1/T_{1}$ decay rate of an NV center placed near the spin liquid is
therefore determined by the product of this prefactor and the length of the
perimeter of the spinon Fermi surface, according to Eqs. (27)–(28). We have
estimated $T_{1}$ times for several spinon Fermi surface candidate materials
and found that it ranges from a few tens to a few hundred milliseconds, as
summarized in Table 1. The remaining challenge therefore, is to attain the
experimental conditions required to measure these relatively long $T_{1}$
times. Nonetheless, NV centers are ideally suited for measuring the quasi-
static regime of the transverse conductivity as they probe frequencies of the
order of GHz, which are very small compared to the typical energy scales of
these systems.

Acknowledgements. P.L. acknowledges the support by DOE office of Basic
Sciences Grant No. DE-FG0203ER46076.

## References

  * Khoo  _et al._ (2021) J. Khoo, F. Pientka, and I. Villadiego, New Journal of Physics 23 (2021), 10.1088/1367-2630/ac2dab.
  * Pines and Nozières (2018) D. Pines and P. Nozières,  _Theory of Quantum Liquids: Normal Fermi Liquids_ (CRC Press, 2018).
  * Giuliani and Vignale (2005) G. Giuliani and G. Vignale,  _Quantum Theory of the Electron Liquid_ (Cambridge University Press, 2005).
  * Florens and Georges (2004) S. Florens and A. Georges, Physical Review B 70, 035114 (2004).
  * Lee  _et al._ (2006) P. A. Lee, N. Nagaosa, and X.-G. Wen, Reviews of modern physics 78, 17 (2006).
  * Senthil (2008) T. Senthil, Physical Review B 78, 045109 (2008).
  * Dai  _et al._ (2020) Z. Dai, T. Senthil, and P. A. Lee, Physical Review B 101, 064502 (2020).
  * Ioffe and Larkin (1989) L. B. Ioffe and A. I. Larkin, Phys. Rev. B 39, 8988 (1989).
  * Levitov and Falkovich (2016) L. Levitov and G. Falkovich, Nature Physics 12, 672 (2016).
  * Guo  _et al._ (2017) H. Guo, E. Ilseven, G. Falkovich, and L. S. Levitov, Proceedings of the National Academy of Sciences 114, 3068 (2017).
  * Alekseev and Semina (2018) P. S. Alekseev and M. A. Semina, Phys. Rev. B 98, 165412 (2018).
  * Casola  _et al._ (2018) F. Casola, T. van der Sar, and A. Yacoby, Nature Reviews Materials 3, 17088 (2018).
  * Rao and Sodemann (2019) P. Rao and I. Sodemann, Phys. Rev. B 100, 155150 (2019).
  * Wang  _et al._ (2021) P. Wang, G. Yu, Y. Jia, M. Onyszczak, F. A. Cevallos, S. Lei, S. Klemenz, K. Watanabe, T. Taniguchi, R. J. Cava, L. M. Schoop, and S. Wu, Nature 589, 225 (2021).
  * (15) _See Supplementary Material_.
  * Agarwal  _et al._ (2017) K. Agarwal, R. Schmidt, B. Halperin, V. Oganesyan, G. Zaránd, M. D. Lukin, and E. Demler, Phys. Rev. B 95, 155107 (2017).
  * Chatterjee  _et al._ (2019) S. Chatterjee, J. F. Rodriguez-Nieva, and E. Demler, Phys. Rev. B 99, 104425 (2019).
  * Langsjoen  _et al._ (2012) L. S. Langsjoen, A. Poudel, M. G. Vavilov, and R. Joynt, Phys. Rev. A 86, 010301 (2012).
  * Law and Lee (2017) K. T. Law and P. A. Lee, Proceedings of the National Academy of Sciences 114, 6996 (2017).
  * He  _et al._ (2018) W.-Y. He, X. Y. Xu, G. Chen, K. T. Law, and P. A. Lee, Physical review letters 121, 046401 (2018).
  * Ruan  _et al._ (2021) W. Ruan, Y. Chen, S. Tang, J. Hwang, H.-Z. Tsai, R. L. Lee, M. Wu, H. Ryu, S. Kahn, F. Liou, _et al._ , Nature Physics 17, 1154 (2021).
  * Chen  _et al._ (2021) C. Chen, I. Sodemann, and P. A. Lee, Physical Review B 103, 085128 (2021).
  * Butler  _et al._ (2020) C. Butler, M. Yoshida, T. Hanaguri, and Y. Iwasa, Nature communications 11, 1 (2020).
  * Ribak  _et al._ (2017) A. Ribak, I. Silber, C. Baines, K. Chashka, Z. Salman, Y. Dagan, and A. Kanigel, Phys. Rev. B 96, 195131 (2017).
  * Sodemann Villadiego (2021) I. Sodemann Villadiego, Phys. Rev. B 104, 195149 (2021).
  * Czajka  _et al._ (2021) P. Czajka, T. Gao, M. Hirschberger, P. Lampen-Kelley, A. Banerjee, J. Yan, D. G. Mandrus, S. E. Nagler, and N. Ong, Nature Physics , 1 (2021).
  * Bruin  _et al._ (2021) J. Bruin, R. Claus, Y. Matsumoto, N. Kurita, H. Tanaka, and H. Takagi, arXiv preprint arXiv:2104.12184 (2021).
  * Motrunich (2006) O. I. Motrunich, Physical Review B 73, 155115 (2006).
  * Chowdhury  _et al._ (2018) D. Chowdhury, I. Sodemann, and T. Senthil, Nature communications 9, 1 (2018).
  * Sodemann  _et al._ (2018) I. Sodemann, D. Chowdhury, and T. Senthil, Physical Review B 97, 045152 (2018).

##  Appendix A Derivation of spinon Fermi surface state quasi-static
transverse conductivity

The transverse conductivity in the quantum (collisionless) regime $q\gg q_{*}$
and at low frequencies $\omega\ll v_{F}q$ can be approximated by

$\displaystyle\sigma_{\perp}(q,\omega)$
$\displaystyle\approx\biggl{[}\Bigl{(}\frac{p_{\rm F}}{2\pi
q}+(\chi_{s}+\chi_{a})\frac{q^{2}}{i\omega}\Bigr{)}^{-1}+\Bigl{(}\chi_{c}\frac{q^{2}}{i\omega}\Bigr{)}^{-1}\biggr{]}^{-1}$
(29) $\displaystyle\approx\frac{\Bigl{(}\frac{p_{\rm F}}{2\pi
q}+(\chi_{s}+\chi_{a})\frac{q^{2}}{i\omega}\Bigr{)}\Bigl{(}\chi_{c}\frac{q^{2}}{i\omega}\Bigr{)}}{\frac{p_{\rm
F}}{2\pi q}+(\chi_{c}+\chi_{s}+\chi_{a})\frac{q^{2}}{i\omega}}.$ (30)

where we have set $g_{S}$, $\hbar$ and $e$ to one. The real part is a
Lorentzian centered around zero frequency

$\displaystyle{\rm Re}\,\sigma_{\perp}(q,\omega)\approx\frac{p_{\rm F}}{2\pi
q}\frac{\chi_{c}^{2}}{\Bigl{(}\frac{p_{\rm F}\omega}{2\pi
q^{3}}\Bigr{)}^{2}+(\chi_{c}+\chi_{s}+\chi_{a})^{2}}.$ (31)

At zero frequency this yields the result

$\displaystyle{\rm
Re}\,\sigma_{\perp}(q,0)=\frac{\chi_{c}^{2}}{(\chi_{c}+\chi_{s}+\chi_{a})^{2}}\frac{p_{\rm
F}}{2\pi q}.$ (32)

The half width of the peak $\Delta\omega_{s}$, for which
$\sigma_{\perp}(q,\Delta\omega_{s})=\sigma_{\perp}(q,0)/2$, is given by

$\displaystyle\Delta\omega_{s}=\frac{2\pi(\chi_{c}+\chi_{s}+\chi_{a})q^{3}}{p_{\rm
F}}$ (33)

##  Appendix B Derivation of the optical pseudo-gap

The conductivity in response to spatially uniform electric fields, obtained in
the limit $\bf{q}\rightarrow 0$ of Eq. (11) or Eq. (12) of the main text,
which governs optical and transport properties, is given by

$\displaystyle\sigma(\omega)=\sigma_{\|}(0,\omega)=\sigma_{\perp}(0,\omega)=i\frac{n}{m}\frac{\omega(\omega^{2}-\bar{\omega}_{s}^{2}-i\omega\Gamma_{1})}{(\omega^{2}-\bar{\omega}^{2}-i\omega\Gamma_{1})},$
(34)
$\displaystyle\bar{\omega}=\sqrt{\frac{n}{m(\epsilon_{c}+\epsilon_{s}+\epsilon_{a})}},\
\bar{\omega}_{s}=\sqrt{\frac{n}{m(\epsilon_{s}+\epsilon_{a})}}.$ (35)

As shown in Fig. 1(b), this conductivity features a peak at
$\omega=\bar{\omega}$. This peak can be viewed as the optical pseudo-gap or
Mott optical lobe of these correlated states.

##  Appendix C Diamagnetic correction to Landau Fermi liquid theory

Landau Fermi liquid theory is unable to properly capture the diamagnetic
response of a Fermi liquid (see discussion around Eq. [4.194] and Eq [5.86] of
Ref. Pines and Nozières, 2018 as well as the absence of the diamagnetic
response term in Eq [8.75] of Ref. Giuliani and Vignale, 2005). As discussed
in Ref. Pines and Nozières, 2018, one supplements the diamagnetic correction,
obtained from a microscopic model (e.g. RPA), to the transverse current-
current response tensor $\chi_{\perp}^{\rm LFL}(q,\omega)$ obtained from
Landau Fermi liquid theory.

More precisely, the diamagnetic term for a non-interacting electrons with
$g_{S}$ spin components in an isotropic 2D system is given by Pines and
Nozières (2018); Giuliani and Vignale (2005)

$\chi^{\rm DM}_{\perp}(q,\omega\rightarrow
0)=g_{S}\frac{2}{3}\frac{n}{m}\left(\frac{q}{2p_{\rm
F}}\right)^{2}=g_{S}\frac{1}{6}\frac{n}{m}\frac{q^{2}}{p_{\rm F}^{2}},$ (36)

so that the full response function can be approximated as

$\chi_{\perp}^{\rm full}(q,\omega)\approx\chi_{\perp}^{\rm
LFL}(q,\omega)+\chi^{\rm DM}_{\perp}(q,\omega\rightarrow 0).$ (37)

Equivalently, through the relation between conductivity and response function,
$\sigma(q,\omega)=e^{2}\chi(q,\omega)/i\omega$, one obtains the full
transverse conductivity,

$\displaystyle\sigma_{\perp}(q,\omega)$ $\displaystyle\approx$
$\displaystyle\sigma_{\perp}^{\rm
LFL}(q,\omega)+g_{S}\frac{e^{2}}{i\omega}\frac{n}{6m}\frac{q^{2}}{p_{\rm
F}^{2}},$ (38) $\displaystyle\sigma_{\perp}^{\rm LFL}(q,\omega)$
$\displaystyle=$ $\displaystyle
i\frac{n}{m}\frac{2}{F_{1}\omega_{-}-\omega_{+}-2i\Gamma_{2}}.$ (39)

However, since the diamagnetic term is purely imaginary, it does not
contribute to the dissipative transverse conductivity in metals. Consequently,
the plots in Fig. 2(a) and Fig. 2(d) of Ref. Khoo  _et al._ , 2021, and most
importantly its main result on the universal transverse conductivity of
metals, remain unchanged.