# Compact Michelson interferometers with subpicometer sensitivity

Jiri Smetana<EMAIL_ADDRESS>Institute for Gravitational Wave
Astronomy, School of Physics and Astronomy, University of Birmingham,
Birmingham B15 2TT, United Kingdom  Rebecca Walters  Institute for
Gravitational Wave Astronomy, School of Physics and Astronomy, University of
Birmingham, Birmingham B15 2TT, United Kingdom  Sophie Bauchinger  Institute
for Gravitational Wave Astronomy, School of Physics and Astronomy, University
of Birmingham, Birmingham B15 2TT, United Kingdom  Amit Singh Ubhi  Institute
for Gravitational Wave Astronomy, School of Physics and Astronomy, University
of Birmingham, Birmingham B15 2TT, United Kingdom  Sam Cooper  Institute for
Gravitational Wave Astronomy, School of Physics and Astronomy, University of
Birmingham, Birmingham B15 2TT, United Kingdom  David Hoyland  Institute for
Gravitational Wave Astronomy, School of Physics and Astronomy, University of
Birmingham, Birmingham B15 2TT, United Kingdom  Richard Abbott  LIGO
Laboratory, California Institute of Technology, Pasadena, CA 91125, USA
Christoph Baune  SmarAct Metrology GmbH & Co. KG, Rohdenweg 4, 26135
Oldenburg, Germany  Peter Fritchel  LIGO Laboratory, Massachusetts Institute
of Technology, Cambridge, Massachusetts 02139, USA  Oliver Gerberding
Institute of Experimental Physics, University of Hamburg, Luruper Chaussee
149, D-22761 Hamburg, Germany  Semjon Köhnke  SmarAct Metrology GmbH & Co. KG,
Rohdenweg 4, 26135 Oldenburg, Germany  Haixing Miao  Department of Physics,
Tsinghua University, Beijing, China 100084  Institute for Gravitational Wave
Astronomy, School of Physics and Astronomy, University of Birmingham,
Birmingham B15 2TT, United Kingdom  Sebastian Rode  SmarAct Metrology GmbH &
Co. KG, Rohdenweg 4, 26135 Oldenburg, Germany  Denis Martynov  Institute for
Gravitational Wave Astronomy, School of Physics and Astronomy, University of
Birmingham, Birmingham B15 2TT, United Kingdom

###### Abstract

The network of interferometric gravitational-wave observatories has
successfully detected tens of astrophysical signals since 2015. In this paper,
we experimentally investigate compact sensors that have the potential to
improve the sensitivity of gravitational-wave detectors to intermediate-mass
black holes. We use only commercial components, such as sensing heads and
lasers, to assemble the setup and demonstrate its subpicometer precision. The
setup consists of a pair of Michelson interferometers that use deep frequency
modulation techniques to obtain a linear, relative displacement readout over
multiple interference fringes. We implement a laser-frequency stabilisation
scheme to achieve a sensitivity of 0.3 $\text{pm}/\sqrt{\text{Hz}}$ above 0.1
Hz. The device has also the potential to improve other experiments, such as
torsion balances and commercial seismometers.

††preprint: APS/123-QED

##  I Introduction

Interferometric displacement readout has played a pivotal role in expanding
the boundaries of precision measurement. In the field of gravitational wave
(GW) astronomy, interferometric readout had shown its utility in detecting GW
signals Abbott et al. (2016, 2019, 2021). The cornerstone of its success, in
part, lies in two areas: (a) the exceptional sensitivity achievable over
distances significantly smaller than the wavelength of light used and (b)
versatility in design across vastly different length scales. To the first
point, contemporary GW detectors such as Advanced LIGO Aasi et al. (2015) and
Advanced Virgo Acernese et al. (2014) can measure some of the weakest signals
in the universe in the form of GW strain. In particular, Advanced LIGO is
capable of measuring displacements as fine as $2\times
10^{-20}\text{m}/\sqrt{\text{Hz}}$ in the most sensitive frequency region
around 100 Hz Buikema et al. (2020). To the latter point, interferometric
measurement sees widespread use in investigations of quantum phenomena but
also spans km-scales in terrestrial GW detection and million km scales in
LISA—the future space-based GW observatory Amaro-Seoane et al. (2017).

We present the investigation of a compact, interferometeric sensor for the
precise measurement of picometer-scale displacements of macroscopic objects.
Displacement sensing of gram-scale optics through to kilogram-scale test
masses is widespread in the GW community due to the strict requirements on
suppression of terrestrial displacement noise (such as seismic noise) in
ground-based GW detectors. Displacement sensing and control has been achieved
through numerous means, such as capacitive position sensors Aasi et al.
(2015), coil-readout geophones such as the Sercel L-4C and optical shadow
sensors such as the BOSEM Carbone et al. (2012). The displacement sensitivity
of these devices is limited across a broad band by the readout noise of these
sensors. When used in feedback control, the sensitivity of these instruments
imposes a fundamental floor on the level of suppression that is achievable.
For example, Advanced LIGO is limited in its sensitivity below 30 Hz by
controls noise Buikema et al. (2020), which can be improved with more
sensitive devices. The astrophysical case for expanding sensitivity towards
lower frequency in contemporary GW detectors is broad, ranging from improved
early-warning systems Magee et al. (2021) for multi-messenger detections
Branchesi (2016) to the routine observation of heavier astronomical objects
such as intermediate-mass black holes Amaro-Seoane et al. (2007).

The future role of interferometric displacement sensors in achieving the
sensitivity improvements necessary for observing these astrophysical phenomena
has already been explored in detail in Yu et al. (2018). The 5–30 Hz frequency
band in Advanced LIGO is dominated by a range of technical noises, such as
angular controls noise caused by low frequency ($<1$ Hz) ground vibrations
(see Buikema et al. (2020); Yu et al. (2018) and references therein). Inertial
sensors with an interferometeric readout, such as 6D seismometers Ubhi et al.
(2021) and next generation beam rotation sensors Venkateswara et al. (2014);
Ross et al. (2021), have the potential to significantly reduce the test mass
motion, simplify the lock acquisition process Staley et al. (2014), and reduce
the required bandwidth of angular control loops. Furthermore, interferometeric
sensors can efficiently damp the suspensions without limiting the design
sensitivity of GW detectors. The sensors achieve the required improvement of
sensing noise by a factor of 100 compared to the currently utilised shadow
sensors Carbone et al. (2012); Yu et al. (2018).

Outside the field of GW detection, compact interferometry has seen use across
a wide range of applications. It has been applied in the readout of torsion
balance experiments Ross et al. (2021), which can be used in investigations
into areas of fundamental physics, such as short-range force measurements or
bosonic dark matter searches Shaw et al. (2022). Compact interferometers have
also been used as reference sensors in the development of other displacement
measuring devices Thomas et al. (2019). The potential for compact
interferometers to serve as absolute distance measurement devices Yu et al.
(2019) has led to their use in alignment sensing at the Large Hadron Collider
Sosin et al. (2019).

The history of compact interferometer development reaches back several decades
and includes a broad range of designs, topologies and varying levels of
sophistication based on the technology available at the time. A comprehensive
review of compact interferometry can be found in Watchi et al. (2018) and
expounds all of the different strategies undertaken in this sphere. Compact
interferometry is not a recent concept as exemplified by the device by de la
Rue et al. de la Rue et al. (1972) dating from 1972. However, the design of a
high performance compact interferometer is non-trivial and often falls into a
particular performance niche. The design can be tuned for high linearity, in
one case 5 pm, which was achieved by Weichert et al. Weichert et al. (2012).
Other designs are tuned for high sensitivity within the desired frequency
range, such as the LISA Pathfinder interferometer, achieving a sensitivity of
$1\text{pm}/\sqrt{\text{Hz}}$ at 10 mHz Armano et al. (2016).

Figure 1: Layout of the experiment and of the digital signal processing. A
digitally sourced modulation is applied to the laser, which is then connected
to both sensing heads. The raw DC diode signal is digitised and demodulated in
the digital domain at multiple harmonics. An algorithm is applied to extract
the displacement from the resulting signals.

The term ‘compact’ in reference to these devices can, however, be rather
misleading depending on the application. Many of the existing compact
interferometers still occupy an area of hundreds of square-centimeters, which
is unacceptably large for applications where space is at a premium, such as
the sensing of the Advanced LIGO quadruple suspension Robertson et al. (2002).
Some devices conform to the label of ‘compact’ better than others, for example
the design by Royer et al. Royer and Dieulesaint (1986) is confined to an area
of only $8\times 5\,\text{cm}$. Further development has led to interferometers
designed with the direct application to seismic displacement sensing in mind.
An example of this, the HoQI sensor Cooper et al. (2018), has shown
substantial improvements in the sensitivity of commercial geophones by
replacing the electronic coil readout and thus eliminating the dominant source
of noise Cooper et al. (2022).

The high sensitivity is in part due to the very sharp response of the
interferometer, being able to cover the full signal range in a mere quarter-
wavelength of displacement. A typical value of 266 nm for a simple Michelson
interferomter, is significantly smaller than that achievable by alternate
technology, such as the 0.7 mm linear range of the BOSEM Carbone et al.
(2012). The cost of this sharp response is a significant reduction in range,
limiting the use of many interferometric devices to specialised uses over sub-
wavelength range. However, we can implement multi-quadrature readout
techniques Isleif et al. (2019), coupled with a fringe-counting algorithm, to
expand the range over multiple interference fringes (read Watchi et al. (2018)
for an overview of the many multi-fringe readout techniques available). These
techniques maintain the precision of the instrument, whilst overcoming its
most significant limitation.

In this paper, we build a compact displacement sensor out of commercially
available products, such SmarAct C01 sensing heads, NKT Adjustik laser, and
Thorlabs fiber beam splitters. We implement a multi-quadrature readout scheme
with fringe-counting in the digital domain, which can be readily compared to
the analogue-domain equivalent demonstrated by numerous other multi-fringe
interferometers. The scheme makes use of a purely digital implementation of
the laser frequency modulation, subsequent signal demodulation, filtering and
arm phase extraction. The key strengths of this design are that the physical
system is highly compact, mechanically simple, robust and easy to deploy where
the necessary computational infrastructure is already present. We use only
commercial components, with the primary component, the sensing head, occupying
an area of merely $1.3\times 0.4\,\text{cm}$ (excluding any necessary mounting
brackets). We achieve a high level of sensitivity due to an appropriate choice
of laser and through additional frequency stabilisation. In Section II we
briefly outline the principles of deep frequency modulation and present the
experimental layout. Section III is devoted to the experimental results of our
fixed-mirror measurement with the accompanying noise budget and the subsequent
improvements we achieved using a frequency-stabilising feedback system.

##  II Optical layout

(a)

(b)

Figure 2: Panel (a) shows the DC sensitivity of the sensing heads without the
modulation-demodulation process (calibration factor is $2.85\times 10^{15}$
Hz/m in the setup). The DC sensitivity is compared with the laser frequency
noise that was independently measured by beating two identical lasers. The two
curves are consistent at low frequencies (below 1 Hz). The dashed lines
represent the inferred noise levels from the two largest noise sources near
the modulation frequency: laser amplitude noise and ADC noise. The former
noise is suppressed above 7 kHz by the anti-aliasing filter. The laser
frequency noise is below the level of $100\,{\rm Hz}/\sqrt{\rm Hz}$ and does
not limit the spectrum around the modulation frequency. Panel (b) shows the
total sensitivity of the sensing head with the heterodyne readout and
frequency noise stabilisation (FSS). The FSS-on curve represents the extend of
the sensitivity increase we achieve with the frequency stabilisation. The
curve is limited by the residual air currents below 100 mHz. The dashed lines
represent the noise limits of several prominent, incoherent noise sources and
show that our new noise floor is consistent with the laser amplitude noise
limit at 100 mHz - 10 Hz. The BOSEM sensitivity curve Carbone et al. (2012)
allows for a comparison between the sensitivity of the sensing head and
contemporary sensing devices.

The scheme for achieving our measurement is shown in Fig. 1. A single laser is
connected to two separate sensing heads mounted on separate base plates. Each
sensing head comprises a simple Michelson interferometer with an internal,
fixed-length reference arm and an open port, which forms the external
measurement arm. Each measurement beam is then reflected from a fixed mirror
mounted on the respective base plate and aligned to maximise the fringe
contrast. The interference pattern is detected by a photodiode with a
responsivity of 0.9 A/W. It is mounted on the output of a fiber circulator
which connects together the output of the laser, the sensing head and the
photodiode. The signal is passed through a 68 k$\mathrm{\Omega}$
transimpedance amplifier followed by a 7 kHz low-pass anti-aliasing filter. It
is then digitised at a sampling rate of 64 kHz by a 20 V peak-to-peak
analogue-to-digital converter (ADC). The signal is down-sampled to 32 kHz once
digitised for processing.

We modulate the laser frequency at $\omega_{m}=2\pi\times 700$ Hz and
amplitude of $A_{m}=0.8$ GHz with an internal PZT actuator. The power observed
at the photodetector is given by the equation Heinzel et al. (2010)

$P(t)=A[1+C\cos(\phi(t)+m\cos(\omega_{m}t))],$ (1)

where $A$ is the power amplitude, $C$ is the fringe contrast, $m=4\pi
A_{m}\Delta L/c$ is the unit-less modulation index, $\Delta L$ is the
longitudinal imbalance of the Michelson arms and $c$ is the speed of light.
The variable arm phase, $\phi$, is related to the test mass $x$ displacement
and laser wavelength, $\lambda$, via $\phi=4\pi x/\lambda$. The signal in Eq.
1 can be expressed in terms of its Fourier series expansion Heinzel et al.
(2010):

$P(t)=P_{0}+\sum_{n=1}^{\infty}2CAJ_{n}(m)\cos(\phi+n\pi/2)\cos(n\omega_{m}t),$
(2)

where $J_{n}$ is the $n^{\text{th}}$ order Bessel function of the first kind
and $P_{0}=A(1+CJ_{0}(m)\cos(\phi))$. The sum in Eq. 2 shows that the input
modulation at $\omega_{m}$ manifests in the output spectrum at $\omega_{m}$
and all higher order harmonics. Furthermore, displacements of the test mass
become up-converted to sidebands around each harmonic of $\omega_{m}$.

The measured signal passes through a transimpedance amplifier and the raw
signal is then digitised. The procedure gives us access to the raw power
signal (up to the input anti-aliasing filter’s cutoff frequency at 7 kHz) as
given in Eq 1. The extraction of the signal from the modulation harmonics
relies on accurate demodulation, which in turn relies on a local oscillator
that corresponds to the modulation source with high fidelity. The key
advantage of the digital scheme is that the local oscillator is simultaneously
used for the signal demodulation and laser modulation, which leads to a form
of digital homodyne readout. This compensates for timing jitter that could
otherwise introduce a discrepancy between the modulation source and the local
oscillator.

This signal is then demodulated at multiple harmonics of the input modulation
frequency by multiplying the raw input signal by $\sin(k\omega_{m}t+\psi_{k})$
and $\cos(k\omega_{m}t+\psi_{k})$ for $k\in\\{1,2,3\\}$. The demodulation
phase, $\psi_{k}$, represents the phase difference between the modulation
signal at its generation and at the point of digitisation together with any
time-varying fluctuations such as jitter noise. We tune $\psi_{k}$ to maximise
the signal in the cosine quadrature and reduce the timing fluctuations to a
second order effect. The output is given by the equation

$S_{k}=CAJ_{k}(m)\cos(\phi+k\pi/2)$ (3)

We take the ratio of two harmonics to construct our elliptical Lissajous
figure. We derive the arm phase for the first sensing head from the ratio of
the first two signals, which is given by

$\frac{S_{1}}{S_{2}}=\frac{J_{1}(m)}{J_{2}(m)}\tan(\phi).$ (4)

To simplify the process further, we tune the modulation depth, $A_{m}$, to
achieve the ratio of the Bessel functions in Eq. 4 equal to unity (circular
Lissajous figure). This leads to improvements in linearity of the measurement
as well as preventing the introduction of non-stationary noise. Since we can
tune $A_{m}$ only for one sensing head, we tuned the length between the second
sensing head and the measured mirror to achieve the ratio of the Bessel
functions for the second and third harmonics of the sensing head close to
unity. The arm phase is extracted in real time through Eq. 4. Through digital
demodulation, we obtain signals $S_{1}$ and $S_{2}$ in real time. We then
apply the four-quadrant arctangent function to this pair of values per sample
on the assumption that the Lissajous figure is circular. Fluctuations and
drifts in the parameters of the Lissajous figure introduce non-linear effects
in the real-time phase readout. We analysed these effects by applying an
ellipse fitting algorithm in post-processing to compare to the performance of
the real-time phase extraction scheme. This comparison is addressed in Section
III.4. A more complex real-time phase extraction scheme that does not rely on
fixed values of the ellipse parameters could later be adopted if more sensing
heads need to be probed with various longitudinal imbalances or drifts become
an issue Heinzel et al. (2010).

##  III  Experimental results

The sensitivity analysis of our interferometric sensing heads was performed on
a fixed test mass in order to effectively suppress the presence of ground
motion and ‘zero out’ the displacement signal. The observed sensitivity to
displacement is shown in Fig. 2. The achieved sensitivity is $3\times
10^{-13}\text{m}/\sqrt{\text{Hz}}$ at 1 Hz and is a factor of 300 better
compared to the BOSEM, one of the currently used sensors in Advanced LIGO
Carbone et al. (2012). In this section, we discuss the noise analysis of the
measured sensitivity.

###  III.1 Environmental disturbances

We mitigated the perturbation of the laser beams via air currents, which
otherwise form the dominant noise source at frequencies below 1 Hz. We
suppressed these effects by placing the components into a sealed, foam-padded
box. The box has reduced the air currents by three orders of magnitude around
1 Hz. However, the noise is still seen below 100 mHz in Fig. 2. At these
frequencies, our insulation becomes less efficient due to the air pressure
fluctuations in the laboratory. We expect that in-vacuum operation of the
sensing heads will further suppress the noise below 100 mHz.

Seismic noise dominates the measured spectrum at 10–100 Hz because the common-
mode-rejection of our setup vanishes above 10 Hz. The sensing heads do not
share the same base plate and are physically separated by 20 cm. Above 100 Hz,
our passive isolation system suppresses the coupling of the ground vibrations.
We achieve the isolation by positioning the sensing heads on metal plates
supported by rubber pads. The sealed box is also supported by another set of
rubber pads.

###  III.2 Laser frequency noise

Our displacement sensitivity is strongly limited by the laser frequency noise
below 10 Hz as shown in Fig. 2. The figure also shows the readout signal
calibrated to the units of Hz/$\sqrt{\rm Hz}$ when no modulation is provided
to the laser and the Michelson interferometer is in the gray state: the
readout power is at half maximum. In this state, the detector is sensitive to
the displacement noise of the test mass, laser frequency and intensity noise
as well as the readout electronics.

The frequency noise rises sharply towards lower frequencies and causes
significant drifts in the displacement measurement over long periods of time.
It is the most significant noise source in our region of interest and poses a
significant limit to improvements beyond existing displacement sensor
performance. We thus devoted our attention to suppressing this noise to
improve the performance of the sensor in this key region.

Typically, the laser sensing noise in Michelson interferometers is suppressed
by reducing the longitudinal imbalance between the two arms ($\Delta L=0$ m).
However, our scheme intentionally uses the imbalance to read out the signal at
multiple harmonics of the modulation frequency, $\omega_{m}$. Therefore,
instead of reducing the imbalance, we follow the approach from Gerberding et
al. (2017); Gerberding and Isleif (2021) and feed the reference head readout
channel back to the laser modulation servo as shown in Fig. 1. This has the
effect of rejecting signal, common to both sensing heads, from the second
sensing head (the ‘measurement’ head). The displacement spectrum obtained in
closed-loop operation is shown in Fig. 2. Any true motion of the test mass
that is detected by the measurement head will be independent of the reference
head readout and thus will not be suppressed by the frequency stabilisation
servo. Therefore, the noise suppression achieved in Fig. 2 represents a true
improvement in the signal-to-noise ratio.

###  III.3 Down-converted noises

Aside from noises that naturally sit at the frequencies measured by the
sensing head, there are noise sources which are shifted in frequency according
to the choice of modulation frequency. The former noise types, such as laser
frequency noise, manifest in the same way as the displacement signal, $\phi$,
in Eq. 1. Therefore, they undergo the same shift from sidebands around DC up
to sidebands around $\omega_{m}$ and then shift back to DC upon demodulation.
The latter types of noise, such as analogue-to-digital converter noise, laser
intensity noise, transimpedance amplifier noise, and shot noise, couple
directly into the series terms in Eq. 2. These noises, therefore, only undergo
the down-conversion from the demodulation, which leads to displacement noise
coupling of $n_{x}(\Omega)\propto n_{i}(\omega_{m}\pm\Omega)$ for a source $i$
of this type.

This important distinction is a key practical consideration in our choice of
modulation frequency. The region around the modulation frequency and its
harmonics must be intrinsically lower in noise compared to the signal in the
region of interest. Otherwise, the down-converted noise will dominate over the
signal in the readout channel. For this very reason, we avoided placing the
modulation frequency in the intermediate frequencies dominated by the forest
of mechanical resonances. We choose the modulation frequency of 700 Hz because
the laser frequency noise and seismic disturbances are below the broadband
laser amplitude noise at this frequency as shown in Fig. 2. In the
unconstrained mirror application, such as in the readout of a multiple-degree-
of-freedom seismometer Ubhi et al. (2021), this factor will be of particular
importance as the seismic measurement will become dominant over other noise
sources and will itself be down-converted and potentially obscure the
measurement at other frequencies. We are interested in enhancing the precision
towards the low frequency limit and, fortunately, the ground motion inherently
decreases towards the high frequencies. Thus, if we place the modulation at an
appropriately high frequency, the down-converted ground motion should not pose
an issue.

(a)

(b)

Figure 3: Panel (a) shows the residual displacement from non-linear coupling
observed when performing a sweep over the laser wavelength using temperature
tuning. The circular fit represents a phase extraction algorithm that assumes
balanced harmonics. The elliptical fit shows the improved linearity that we
obtained through a more advanced algorithm that applies ellipse fitting to the
Lissajous figure. Panel (b) shows the phase spectrum of a 1 Hz excitation of
the laser wavelength using the PZT. The measured data represents the upper
limit for the distribution of signal to higher harmonics due to non-linear
coupling. The simulated curve represents our estimate of the spectrum that
would correspond to the elliptical fit in panel (a).

###  III.4 Linearity

Interferometric devices have an exceedingly narrow span of linearity in their
raw readout (less than a quarter-wavelength). The long-range linearity is
recovered via the construction of a Lissajous figure using the aforementioned
dual-quadrature readout scheme. This technique relies on the correct
determination of the centre position and eccentricity of the Lissajous figure
in the phase extraction using the four-quadrant arctangent. If these ellipse
parameters are determined poorly, the phase signal couples non-linearly into
the readout channel, making non-linearities a particular area of concern in
the literature Watchi et al. (2018).

A key source of non-linear coupling we can demonstrate is an imbalance in the
amplitude of the quadrature signals, which produces a non-circular Lissajous
figure. This can occur through a fluctuation in the modulation depth, $A_{m}$,
which disrupts the equal ratio of the Bessel functions in the first two
harmonics ($J_{1}(m)\neq J_{2}(m)$ in Eq. 4). We investigated the long-range
linearity of the system by sweeping through the laser wavelength using the
laser’s temperature control. We swept over a range of 1 nm corresponding to 62
full rotations of the Lissajous figure. We applied two phase extraction
algorithms in post-processing, shown in Fig. 3, using a circular and
elliptical fit. The former algorithm mimics our real-time phase extraction
technique that assumes balanced harmonics, giving a residual RMS displacement
of 0.68 nm over an effective displacement of 48 m. The latter algorithm
applies an elliptical fit to the many-period Lissajous figure constructed from
the full sweep in order to correct for non-zero eccentricity and offsets in
the centre of the Lissajous figure. Algorithms developed in Watkins and
Collett (2014); Collett and Watkins (2015) can be readily adopted to perform
the ellipse fitting. The residual RMS in this case was reduced to 0.25 nm.

We also approached the investigation of non-linear coupling through another
method, consisting of a controlled 1 Hz excitation of the laser frequency
using the laser PZT. The amplitude of the oscillation was set at approximately
half the Lissajous period in order to maximise the non-linear effect (we were
limited in our choice of amplitude by the range of the PZT). The amplitude
spectrum of the phase derived from the real-time readout, shown in Fig. 3,
contains a forest of peaks at multiples of 1 Hz, which represent the non-
linear coupling of the pure 1 Hz input signal. The measured curve should be
treated as an upper limit to the non-linearity of the system, as the non-
linear response of the PZT is dominant over the non-linearity of the sensing
head readout. We define a measure of the single-period non-linearity as the
ratio of the RMS contribution from the non-linear frequency components to the
total RMS of the signal. Under this definition, the measured spectrum shows a
non-linear measure of 1.2%. The value is similar to the non-linearity observed
in Isleif et al. (2016) for TLB 6700 Velocity laser from Newport. The
simulated curve in Fig. 3 contains the estimated spectrum for an excitation
with a non-linear RMS contribution that has been matched to the RMS of the
elliptical fit in Fig. 3. This model exhibits a non-linear measure of 0.12%.

In applications that experience larger displacements, such as the position
sensing of a suspension, the range of the signals will span multiple
interference fringes, making the coupling of non-linearities more prominent
than in the short displacement range of our fixed-mirror measurement.
Naturally, this makes the topic of linearity particularly relevant.

##  IV Conclusion

Compact, interferometric position sensors are well positioned as candidates
for the next generation of displacement readout in precision applications. Two
such examples, the 6D seismometer Ubhi et al. (2021) and the Advanced LIGO
quadruple suspensions Robertson et al. (2002), can benefit from compact
interferometers.

We presented a commercially available, interferometric displacement sensor
which makes use of the sharp sub-fringe response of a Michelson
interferometer. Using deep frequency modulation techniques, we were able to
linearise the response over the range of a full fringe and a simple fringe-
counting algorithm was implemented to extend the range over multiple fringes.
Our fixed mirror measurement showed an improved level of sensitivity over
existing shadow sensors. This, together with the sensor’s compact design,
means it is a suitable candidate for future upgrades and can be readily
implemented with little impact on existing infrastructure. The complexity has
been effectively shifted to the digital domain. This does make the sensors
more difficult to implement in situations where existing computational systems
are lacking, but in the case of current GW detectors, this will not be an
issue.

We were able to overcome the most significant initial sensitivity limitation,
the coupling of laser frequency noise, through the use of one sensing head as
a frequency reference. Particularly in multiple degree-of-freedom readout
situations, where all sensing heads can receive input from the same laser
source, this is a very low-impact solution to improving upon current
displacement sensing methods. In a future upgrade, we will read out the signal
from the anti-symmetric port of the Michelson interferometer. The approach
will help suppress extra noise from the fiber between the sensing head and the
photodetector Fleddermann et al. (2018).

Finally, we have discussed the occurrence of non-linearities in the presence
of phase extraction algorithms that apply circular and fitted, static ellipse
parameters. The application of our displacement sensor in, for example,
suspended test masses may require the routine correction of the parameters of
the Lissajous fit. This may arise due to long-term effects such as drifts in
the modulation depth and timing jitter, but also due to a transformation of
the Lissajous figure over high-RMS displacements, which will vary
significantly depending on the specifics of the application.

A more detailed study on the impact of drifts in the parameters of the
elliptical fit will be left for future work. Our simplified real-time phase
extraction technique may not be suitable for high-RMS applications such as the
displacement sensing of a suspended mass with no seismic pre-isolation. In
this case, however, it is possible to employ a more advanced, dynamic
algorithm that can apply parameter corrections in real-time and maintain
linearity more robustly Heinzel et al. (2010). We are in the process of
testing the effectiveness of such an algorithm applied to the sensing heads
presented.

The impact of non-linearities in the phase readout will be suppressed in
applications such as the Advanced LIGO quadruple suspensions. This is due to
their significantly lower RMS displacement, which is achieved by the previous
stages of seismic isolation Matichard et al. (2015). In this case, the RMS
displacement should be lower than that simulated by our frequency sweep in
Section III.4, which makes our set-up suitable for such an application
already, without significant expansion of the phase extraction scheme.

## Acknowledgements

We thank members of the LIGO SWG group for useful discussions and Michael Ross
for his valuable feedback. The authors acknowledge the support of the
Institute for Gravitational Wave Astronomy at the University of Birmingham,
STFC 2018 Equipment Call ST/S002154/1, STFC ’Astrophysics at the University of
Birmingham’ grant ST/S000305/1. A.S.U. and J.S. are supported by STFC
studentships 2117289 and 2116965. O.G. is supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy—EXC 2121 “Quantum Universe”—390833306, and by the German
Federal Ministry of Education and Research (BMBF, Project 05A20GU5).

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# Quantum-critical resistivity of strange metals in a magnetic field

Chandra M. Varma  111Recalled ProfessorPhysics Department, University of
California, Berkeley, Ca. 94704
222Emeritus Distinguished ProfessorPhysics Department, University of
California, Riverside, Ca. 92521

###### Abstract

Resistivity in the quantum-critical fluctuation region of several metallic
compounds such as the cuprates, the heavy-fermions, Fe-chalogenides and
pnictides, Moiré bi-layer graphene (MBLG) and WSe2, is linear in temperature
$T$ as well as in the magnetic field $H_{z}$ perpendicular to the planes.
Scattering of fermions by the fluctuations of a time-reversal odd polar vector
field ${\bf\Omega}$ has been shown to give a linear in T resistivity and other
marginal Fermi liquid properties. An extension of this theory to an applied
magnetic field is presented. Magnetic field is shown to generate a density of
vortices in the field ${\bf\Omega}$ proportional to $H_{z}$. The elastic
scattering of fermions from the vortices gives a resistivity linear in $H_{z}$
with the coefficient varying as the marginal fermi-liquid susceptibility
$\ln(\frac{\omega_{c}}{T})$. Quantitative comparison with experiments is
presented for cuprates and MBLG .

High temperature cuprates Ginsburg (1994) Varma (2020) have a linear in $T$
resistivity for doping in the region above Tc which is bounded by a phase with
a ”pseudo-gap” on one side and cross-over to a Fermi-liquid on the other. This
and related anomalies Varma (1994) in this region suggested a quantum-critical
origin for the anomalies Varma (1997) Simon and Varma (2002) and the
prediction that the pseudogap phase breaks time-reversal and inversion
symmetries. Linear in T resistivity and other anomalies, similar to those in
the cuprates, are also found in several Fe-based compounds in the fluctuation
region of their antiferromagnetic (AFM) quantum-critical point Ishida  _et
al._ (2009); Shibauchi  _et al._ (2014); Kim  _et al._ (2015); Lv  _et al._
(2009), in heavy fermion compounds Löhneysen  _et al._ (2007); Gegenwart  _et
al._ (2007), as well as more recently in twisted bi-layer graphene Cao  _et
al._ (2019); Lu  _et al._ (2019) and in the twisted bi-layer compound WSe2
Ghiotto  _et al._ (2021) with hitherto un-discovered order parameters. An
important recent discovery Hayes  _et al._ (2014) Giraldo-Gallo  _et al._
(2018) Jaoui  _et al._ (2021) Ghiotto  _et al._ (2021) is that in all of them
the resistivity is linear also in an applied magnetic field $|H|$. The
magnitude of the magneto-resistivity is similar to the zero-field resistivity
at temperature $T$ for $\mu_{B}H$ of $O(k_{B}T)$. Where investigated Hayes
_et al._ (2018), 333G. Boebinger and A. Shekhter - private communication 2020
the linear in $|H|$ resistivity is found only for the component $H_{z}$
applied perpendicular to the planes.

Three important general points should be noted: First, a transport scattering
rate linear in $|H|$ and (nearly) independent of temperature can only be due
to elastic scattering of fermions from time-reversal odd axial objects induced
by the magnetic field. Second, the fact that only the component of the field
orthogonal to the high conducting plane in all these metals is effective
excludes magnetic moment due to spins in favor of magnetic moments due to
orbital loop currents. (The inevitable coupling to spins produces a magneto-
resistance $\propto H^{2}$, which from measurements away from criticality is
estimated at 50 Tesla to be few percent of the experimental value Giraldo-
Gallo  _et al._ (2018).) Third, the magnitude mentioned above implies that the
theory of the linear in $|H|$ resistivity must be closely related to the
theory which gives linear in $T$ resistivity.

A theory that gives linear in $T$ resistivity and other anomalies in cuprates
rests on the theory of quantum critical fluctuations Aji and Varma (2007,
2010) which are prelude to a state of loop current order. The new experiments
invite extension of this theory to the effects of a magnetic field. The
occurrence of the linear in $T$ and in $H$ resistivity as well as the
associated $T\leavevmode\nobreak\ \ln(\omega_{c}/\pi T)$ entropy in the
quantum-critical regions in at least all the other compounds where results are
available Löhneysen  _et al._ (2007); Shibauchi  _et al._ (2014) is to be
expected if their quantum-criticality is described by a model which maps to
the quantum-xy model coupled to fermions (QXY-F). The mapping has been shown
Varma (2012) for the planar ferro or antiferro-magnetic model or an
incommensurate Ising model. Here, I will first present a theory for the
magnetic field dependence of the resistivity in the cuprate compounds for
which more quantitative information is available than the other compounds and
briefly comment on the other cases.

Loop-current order in cuprates can be represented as a time-reversal odd polar
vector ${\bf\Omega}$ on a lattice, sketched in Fig. (1 \- A). Using
conservation laws alone Else and Senthil Else and Senthil (2021) have recently
shown that to get resistivity proportional to $T$ for $T\to 0$ in the pure
limit, the critical fluctuations must be of an order parameter of such a
symmetry. Such an order parameter has indeed been found to be consistent with
experiments using a variety of different techniques Kaminski  _et al._ (2002);
Bourges and Sidis (2011); Zhao  _et al._ (2016); Lubashevsky  _et al._ (2014);
Zhang  _et al._ (2018).

Consider the orbital magnetic susceptibility of the model in its quantum-
critical regime. This is obtained from the critical fluctuations already
derived in Refs. Aji and Varma (2007); Zhu  _et al._ (2015, 2016); Hou and
Varma (2016). The model at $H=0$ is specified by the interaction energy of the
angles $\theta_{i,\tau}$ of ${\bf\Omega}$ at neighboring sites, by the kinetic
energy due to their angular momentum $(\partial\theta_{i\tau}/\partial\tau)$,
and the coupling of spatial and temporal fluctuations in $\theta_{i,\tau}$ to
the fermions. The action is Aji and Varma (2007)

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int_{0}^{\beta}d\tau\leavevmode\nobreak\
J\sum_{i<j}\cos\Big{(}{\bf\theta}_{i,\tau}-{\bf\theta}_{j,\tau}\Big{)}+C^{-1}\sum_{i}\Big{(}\frac{d{\bf\theta}_{i}}{d\tau}\Big{)}^{2}+S_{CF}.$
(1) $\displaystyle S_{CF}$ $\displaystyle=$
$\displaystyle\int_{0}^{\beta}d\tau\sum_{i}i\alpha\leavevmode\nobreak\
\nabla\theta_{i,\tau}\cdot\psi^{+}{\bf j}_{i,\tau}\psi$ (2)

$S_{CF}$ is the coupling of the fluctuations to the fermions Varma (2012)
through the coupling of collective mode current $\propto i\nabla\theta$ to the
fermions current operator ${\bf j}$. $\psi,\psi^{+}$ are the fermion field
operators. We have neglected a term for the four-fold anisotropy of $\theta$
on the lattice because it has been shown to be irrelevant in the quantum-
critical region Aji and Varma (2007).

The QXY-F model, just as the classical XY model, does not belong to the
universality class of the Ginzburg-Landau-Wilson theories and their quantum
extensions. The quantum-critical fluctuations are driven by proliferation of
topological defects, 2D spatial vortices, and warps which are spatially local
events interacting logarithmically in imaginary time Aji and Varma (2007). The
critical correlations of (1) after integrating over the fermions, $C({\bf
r},\tau)\equiv<e^{-i\theta({\bf r},\tau)}e^{i\theta(0,0)}>$, have been
obtained by quantum-monte-carlo calculations Zhu  _et al._ (2015, 2016) as
well as derived by renormalization group Hou and Varma (2016). It is shown in
an Appendix in Ref. Aji  _et al._ (2010) that the orbital magnetic
susceptibility $\chi_{LL}({\bf r},\tau)$ defined by (3), are proportional to
those of $C({\bf r},\tau)$. Near criticality the (dimensionless) dynamic
orbital magnetic susceptibility is

$\displaystyle\chi_{LL}({\bf r},\tau)\equiv\mu_{B}^{2}<{\bf L}^{+}_{z}({\bf
r},\tau){\bf L}_{z}({\bf
0},0)>=\mu_{B}^{2}<L_{z}^{2}>\frac{\tau_{c}^{2}}{\tau}e^{-(\tau/\xi_{\tau})^{1/2}}\ln\frac{r}{a}e^{-r/\xi_{r}}.$
(3)

$\mu_{B}^{2}<L_{z}^{2}>$ is the expectation value of the square of the
magnitude of the orbital magnetic moment per unit-cell volume. We take this to
be given by the amplitude of the measured Bourges and Sidis (2011) ordered
staggered moment per unit-cell $(\ell_{z}\mu_{B})^{2}$. Since in the 2 d xy
model, the dynamics are purely due to the phase fluctuations, the amplitude
factor $<L_{z}^{2}>$ is nearly temperature independent in the region of
interest. $\tau_{c}$ is the short time cut-off obtainable from experiments and
has a value of $O(10^{3}K)^{-1}$ Aji  _et al._ (2010); Varma (2020), $a$ is
the lattice constant. An important result that the spatial correlation length
normalized to the lattice constant $\xi_{r}/a\approx
log(\xi_{\tau}/\tau_{c})$, the normalized temporal correlation. So the
dynamical critical exponent $z=\infty$ and the problem is effectively local in
space. The spectral function (3) is of the form proposed phenomenologically
Varma  _et al._ (1989) to give the fluctuations of marginal Fermi-liquid,
rather than the $1/\tau^{2}$ of the Landau Fermi-liquid. In terms of the
frequency $\omega$ and temperature $T$,

$\displaystyle\chi_{LL}(\omega,T)=\frac{\mu_{B}^{2}<L_{z}^{2}>}{\omega_{c}}\Big{(}\ln\big{|}\frac{\omega_{c}}{max(\omega,\pi
T)}\big{|}-i\tanh\frac{\omega}{2T}\Big{)},$ (4)

at criticality. $\omega_{c}=1/\tau_{c}$ is the ultra-violet cut-off. This
functional form is also the principal result of theories on interesting models
of mathematical interest such as the SYK model Chowdhury  _et al._ (2021),
holographic models Faulkner  _et al._ (2011) and of other models Dumitrescu
_et al._ (2021), Varma (1994); Ruckenstein and Varma (1991); Mousatov  _et
al._ (2020).

Figure 1: Representation of the current distribution in the cu-o unit cell
for, A: the vector field ${\bf\Omega}$ which has one of four possible angles
$\theta_{i}$ in the unit-cell $i$, B: the angular momentum ${\bf\ell}_{z}$
which is a generator of rotations of ${\bf\Omega}$ and has a magnetization at
its core. C: Represents a fluctuation of ${\bf\Omega}$ over regions of many
cells. A current represented by the green arrows runs at the boundary between
any two orientations of ${\bf\Omega}$. At all corners of the variations in
${\bf\Omega}$ a vortex or ${\bf\ell}_{z}$, represented by the black dot, is
required to exist. At $H=0$, the vortices are of equally up and down
orientations. But an applied finite $H$ leads to a net orbital angular
momentum due to unequal density of vortices of different orientation.

The Hamiltonian for the coupling of a magnetic field couples to the angular
momentum is

$\displaystyle H_{B}=-\mu_{B}\sum_{i}\leavevmode\nobreak\ {\bf H\cdot
L}_{iz}(\tau),$ (5)

where ${\bf L}_{iz}\equiv i(\partial/\partial\theta_{i})$. In the quantum-
critical regime $H_{z}$ induces a static macroscopic $<L_{z}>$ given by

$\displaystyle\mu_{B}<L_{z}>=\chi^{\prime}_{LL}H_{z},\leavevmode\nobreak\
\leavevmode\nobreak\
\chi^{\prime}_{LL}(T)=\frac{\mu_{B}^{2}\ell_{z}^{2}}{\omega_{c}}\log\big{(}\frac{\omega_{c}}{\pi
T}\big{)}.$ (6)

From the experimental observations Bourges and Sidis (2011) that the ordered
staggered moment per-cell is about $0.1\mu_{B}$ , and $\omega_{c}\approx
2000K$ Aji  _et al._ (2010); Varma (2020), $\chi^{\prime}_{LL}$ is estimated
to be about $10^{-5}\mu_{B}^{2}/(Kelvin-cell)$. So a magnetic field of 50
Tesla can be estimated to produce a static magnetization $\approx 5\times
10^{-4}\mu_{B}$’s, not including the numerical factor due to the logarithmic
temperature dependence in $\chi_{LL}$. An important question in the present
context is how such a moment would be distributed. To think of this, it is
useful to know the physical description of $\ell_{z}$, the quasi-quantized
unit of orbital angular momentum in the present problem. A loop-current
carrying the lattice representation of angular momentum is shown in Fig. (1 \-
B) Sun and Fradkin (2008). It has been shown Aji  _et al._ (2010); He and
Varma (2012); He  _et al._ (2012) to be the generator of rotations of the
magneto-electric vector ${\bf\Omega}$ in the plane, from one of its four
orientations to the clockwise or anti-clockwise orientation:

$\displaystyle
e^{i(\pi/2){\bf\ell}_{z}}|\hat{{\bf\Omega}}>=|\hat{\bf{\Omega}}+\pi/2>.$ (7)

The pictorial representation in Fig. (1 \- c) of $L_{z}$ corresponds to a
vortex in the vector field ${\bf\Omega}$ with quantized angle but a magnetic
moment given by the area and the current carried by the core cell around which
the four-orientations of ${\bf\Omega}$ meet. Over long wave-lengths, one may
ignore the granularity of the lattice so that $L_{z}$ is similar to the vortex
in more familiar $U(1)$ fields such as superconductors in a magnetic field or
superfluids in rotation. Instead of quantization of the magnetic moment in
terms of fundamental constants, it is non-universal and given by the magnitude
of the vectors $\Omega$ which have very weak temperature dependence. The
magnitude of an individual moment $\ell_{z}$ is similar to the measured value
of ${\bf\Omega}$ in the ordered state of about $0.1\mu_{B}$ Bourges and Sidis
(2011). Therefore the density of such moments $n_{L}$ is about $5\times
10^{-3}$/unit-cell for a field of 50 Tesla so that their separation is about
50 unit-cells. In an ordered state of ${\bf\Omega}$, such moments would
crystallize at low enough temperature due to their long-range mutual
interactions. But we are considering the region in which they live in a bath
of ${\bf\Omega}$’s quantum-fluctuating in time and space. Therefore such
moments would remain disordered at the temperatures of interest and diffuse at
a very slow rate because of their enormous effective mass. If the motion of
$<L_{z}>$ is very slow compared to the motion of fermions with which they
scatter, the scattering should be considered elastic.

Figure 2: (a) Elastic scattering of fermions by vortices of angular momentum
$<L_{z}>$, (b) Inelastic scattering of fermions by fluctuations
$\chi"(\omega,T,{\bf q})$.

The s-wave scattering rate $1/\tau_{L}$ of such local magnetic field generated
defects can be easily estimated, See Fig. (2-a).

$\displaystyle 1/\tau_{L}$ $\displaystyle=$ $\displaystyle
2\pi\leavevmode\nobreak\ n_{L}(g_{0}\mu_{B}\ell_{z})^{2}\leavevmode\nobreak\
N(0),$ (8) $\displaystyle n_{L}$ $\displaystyle=$
$\displaystyle\frac{\chi^{\prime}_{LL}H_{z}}{\mu_{B}\ell_{z}}\leavevmode\nobreak\
\approx\leavevmode\nobreak\
\ell_{z}\frac{\mu_{B}H_{z}}{\omega_{c}}\ln(\omega_{c}/\pi T).$ (9)

$N(0)$ is the density of states of fermions at the chemical potential and
$g_{0}$ is the coupling energy Aji  _et al._ (2010) of the fermions to a
vortex with orbital moment $\mu_{B}\ell_{z}$. This is to be compared with the
inelastic scattering of fermions by the fluctuations $\chi"(\omega,T)$, See
Fig. (2-b). This is calculated from the analytic continuation of the imaginary
part of the self-energy at zero-frequency which has been often derived, Varma
_et al._ (1989); Pelzer (1991); Varma  _et al._ (2002)

$\displaystyle 1/\tau(T)$ $\displaystyle=$ $\displaystyle 2UIm\Sigma"(0,T),$
(10) $\displaystyle\Sigma(i\omega_{n})$ $\displaystyle=$ $\displaystyle
g_{0}^{2}\sum_{\omega_{m},\bf k}G({\bf
k},i\omega_{m})\chi(i\omega_{n}-i\omega_{m}),$ (11)

$U$ is a dimensionless Umklapp factor, which is necessary for finite
resistivity. Recently, in an asymptotically exact theory for resistivity due
to fluctuations of the QXY-CF model, it has been shown that $U$ is temperature
independent 444H. Maebashi, C.M. Varma - unpublished. A way to estimate $U$ is
to compare the transport scattering rate with the imaginary part of the self-
energy in the direction on the Fermi-surface of maximum velocity. This gives
$U$ of $O(1)$ Varma (2020) for the cuprates where both have been measured. Eq.
(10) gives

$\displaystyle 1/\tau(T)\approx$ $\displaystyle\leavevmode\nobreak\ \pi
U\leavevmode\nobreak\ (g_{0}\mu_{B}\ell_{z})^{2}\leavevmode\nobreak\
N(0)\frac{k_{B}T}{\omega_{c}}.$ (12)

$1/\tau_{L}$ and $1/\tau(T)$ are of similar magnitude at $\mu_{B}H/k_{B}T$ of
$O(1)$ for $\ln(\omega_{c}/\pi T)\approx 1$. They are similar because the
inelastic scattering rate comes from the imaginary part of the same
fluctuations whose real part gives $n_{L}$ to give the elastic scattering rate
and the coupling energy to fermions is identical. Specifically the ratio of
the scattering rates is

$\displaystyle(1/\tau_{L})\div(1/\tau(T))\approx\frac{2\ell_{z}}{U}\frac{\mu_{B}H}{k_{B}T}\ln(\omega_{c}/\pi
T).$ (13)

Figure 3: The temperature dependence of the linear in H resistivity in
$La_{2-x}Sr_{x}CuO_{4}$, for $x=0.19$. The data shown as red dots is taken
from Fig. 1 b of Ref. (Giraldo-Gallo  _et al._ (2018)). $\beta(T)$ is obtained
from the fit to the resistivity (after subtracting a small residual value)
$\rho(T,H)-\rho_{0}=\alpha\leavevmode\nobreak\ T+\beta(T)\leavevmode\nobreak\
H$ at $H$ = 70 Tesla. $A(T)=0.14\ln\big{|}\frac{\omega_{c}}{\pi T}\big{|}$,
with $\omega_{c}\approx 1600K$.

The result (13) is subject to a cut-off at low temperatures if one is not at
critical parameters, (the critical point is also expected to shift in a
magnetic field if the usual magnetic susceptibility of the system is different
on the two sides of critical point) and a high temperature cut-off on the
scale of the upper cut-off $\omega_{c}$.

We can compare the result in Eq. (13) quantitatively with experiments. The
data for the resistivity in the most extensively investigated case, for a
cuprate near criticality, is represented in Ref. Giraldo-Gallo  _et al._
(2018) by $\rho(T,H)=\alpha k_{B}T+\beta(T)\mu_{B}H$. We can write using Eqs.
(12) and (13) that $\beta(T)=\alpha\frac{2\ell_{z}}{U}\ln(\omega_{c}/\pi T)$.
$\beta(T)$ from low $T$ to the highest available temperature, $180K$, and a
logarithmic fit to it by $0.14\ln(1500/\pi T)$ are given in Fig. (3). The
coefficient $0.14$ should be compared with $0.19$ that is estimated from
parameters above and the value of $\alpha\approx 1.1$ deduced in the
experiment Giraldo-Gallo  _et al._ (2018). A logarithmic fit appears
reasonable for $T\gtrsim 30K$ below which the data saturates. The parameter
$\omega_{c}$ is about $1600K$, which may be compared with the $O(3000)K$
deduced Varma (2020) from the fit to the logarithmic $C_{v}/T$ measured Michon
_et al._ (2019) between $0.3K$ and $10K$. The peril of deducing a number from
a logarithm in a range far above the data should be kept in mind. The data in
Fig. 1a in Giraldo-Gallo  _et al._ (2018) shows systematic rounding towards
zero below about 30 K even in a field of $70$ Tesla. One may be tempted to
ascribe it to not being very close to criticality, but a closer look at all
the data at various fields suggests a more mundane reason. The data shows a
large region of rounding from the zero-field transition temperature ($\approx
41K$) towards zero resistivity at low temperatures even in large fields. This
is generally the rule in 2d strongly type II superconductors or
superconducting films due to an enhanced region of phase fluctuations in a
field.

An independent way to test the prediction made here is to see if a direct
measurement of magnetization in the range in which the resistivity satisfies
Eqn. (13) shows the same logarithmic enhancement.

I now briefly discuss the other compounds, to begin with those for which the
quantum-criticality is that of antiferromagnetism. Significantly, the
important critical fluctuations for planar and incommensurate Ising
ferromagnets or anti-ferromagnets (or charge density waves) are of the phase
variable given by the xy model McMillan (1975); Varma (2012). It is very
interesting to note that the measured spectral functions for the quantum-
critical fluctuations for planar antiferromagnetism in
$BaFe_{1.85}Co_{0.15}As_{2}$ Inosov  _et al._ (2010) or incommensurate anti-
ferromagnetism in the heavy fermion $CeCu_{6}$ Schröder and et al. (1998,
2000) are consistent with the product form in momentum and energy Varma  _et
al._ (2015) as in Eq. (3) for the QXY-SF model.

The data Hayes  _et al._ (2014) in $BaFe_{2}(As_{1-x}P_{x})_{2}$ with
$T_{c}\approx 30K$, is available only to 60 K with fields up to 59 Tesla has a
severe rounding of resistivity towards zero at low temperatures for fields
less than 50 Tesla so that linearity of $H$ above this field is observed only
in a narrow range of temperatures. We therefore cannot usefully compare the
data in $BaFe_{2}(As_{1-x}P_{x})_{2}$. The fit of the resistivity data made as
$\propto\sqrt{(\mu_{B}H)^{2}+(k_{B}T)^{2}}$ made earlier Hayes  _et al._
(2014) is not good under closer examination of the detailed data kindly
received from the authors (I. Hayes - private communication - Dec. 2021). That
fit also does not work for the cuprate or for the twisted bi-layer graphene
Jaoui  _et al._ (2021), as stated by the authors. However, an $H^{2}$
dependence of magneto-resistance at low fields is conventional and well
understood and there is no reason why it should be completely absent in the
metals under discussion.

Figure 4: The temperature dependence of the linear in H resistivity in twisted
bi-layer graphene at the dopings where the resistivity at $H=0$ is most nearly
linear in $T$ down to the lowest temperature measured. The data is from Ref.
(Jaoui  _et al._ (2021)) and replotted by the authors of that paper and
provided to me. $A_{B,1}\equiv\beta(T)$ is obtained just as for the cuprate
compound in Ref. (3).

The relevant order parameter for Twisted bi-layer (TB)-graphene and
TB-$WSe_{2}$ is not known yet from experiments, although there are theoretical
calculations suggestive of loop-current ordered states Bultinck  _et al._
(2020); Hofmann  _et al._ (2021) in TB-graphene. TB-$WSe_{2}$ is similar
except for the large spin-orbit coupling. Their structure has a triangular
motif and it is expected that the nearest neighbor repulsion is comparable to
the kinetic energy. In this situation, loop-current order is a likely
instability Weber  _et al._ (2014); Nandkishore and Levitov (2011); Zhu  _et
al._ (2013). It should be ascertained if only the component of the magnetic
field perpendicular to the plane is responsible for the resistance linear in
the field. If this holds, more experiments to test the time-reversal,
inversion and possible chirality given by loop-currents are suggested to
decipher their long-range order. The data on $WSe_{2}$ is not yet detailed
enough to compare with theory for $\beta(T)$, but it is for TB-graphene Jaoui
_et al._ (2021). This is plotted in (4). The logarithmic fit is found with a
coefficient varying from about 0.5 to about 1 and an upper cut-off
$\omega_{c}$ varying from about 50 K to about 100 K. There are no independent
numbers from other experiments to compare. But the scale of the fluctuation
energies an order of magnitude smaller than the cuprates appears reasonable
and the magnitude suggests that the magnitude of the orbital moment in the
ordered state is similar to that in the cuprates. The saturation at the lowest
point at $40mK$ is almost certainly due to rounding of resistivity due to
impending superconductivity.

The experiments in a magnetic field test a crucial microscopic aspect
underlying the applicability of the theory of quantum-fluctuations of the xy-
model. They identify that the xy model pertains to a vector which is odd in
time-reversal and inversion. Only then can the magnetic field couple to the
magnetic moment which is a generator of rotations of the vector characterizing
in-plane order. It is already understood that d-wave superconductivity is not
possible if the self-energy of the fermions is angle-independent as it is in
cuprates without the fermions coupling to the fluctuations of angular momentum
Aji  _et al._ (2010); Varma (2020). This is consistent with the information
about the physics of d-wave pairing in cuprates obtained from angle-resolved
photoemission Bok  _et al._ (2016). To conclude, one might also add that the
mechanism of superconductivity in all these systems is inevitably related to
the fluctuations which give resistivity linear in $T$ and in $H$.

Acknowledgements: I have benefitted enormously from discussions with Arkady
Shekhter both of the experimental data and the theory. Discussions of the
theory with Lijun Zhu and Vivek Aji have been invaluable. I have also
benefitted by discussions of the state of experiments with Greg Boebinger on
cuprates, James Analytis and Ian Hayes on the Fe-compound, with Augusto
Ghiotto and Abhay Pasupathy on WSe2, with A. Jaoui on MBLG, and with Erez Berg
on the theory for TB-graphene. This work was partially performed at the Aspen
Center for Physics, which is supported by National Science Foundation grant
PHY-1607611.

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