# Pair-density wave signature observed by x-ray scattering in La-based
high-$T_{\rm c}$ cuprates

Jun-Sik Lee<EMAIL_ADDRESS>Stanford Synchrotron Radiation
Lightsource, SLAC National Accelerator Laboratory, Menlo Park, California
94025, USA  Steven A. Kivelson  Departments of Physics, Stanford University,
Stanford, CA 94305, USA  Tong Wang  Institute for Materials Research, Tohoku
University, Katahira 2-1-1, Sendai, 980-8577, Japan  Yoichi Ikeda  Institute
for Materials Research, Tohoku University, Katahira 2-1-1, Sendai, 980-8577,
Japan  Takanori Taniguchi  Institute for Materials Research, Tohoku
University, Katahira 2-1-1, Sendai, 980-8577, Japan  Masaki Fujita  Institute
for Materials Research, Tohoku University, Katahira 2-1-1, Sendai, 980-8577,
Japan  Chi-Chang Kao  SLAC National Accelerator Laboratory, Menlo Park,
California 94025, USA

Suggestive, but indirect evidence of the existence of pair-density wave (PDW)
order in several high-$T_{\rm c}$ cuprates has been reported 1 ; 2 ; 3 ; 4 ; 5
; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 . As this constitutes a new quantum phase
of matter, it is important to establish its existence at least somewhere in
the phase diagram 14 ; 15 ; 16 ; 17 . However, a direct correspondence between
experiment 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 20 ; 21 ;
22 ; A ; B ; C  and theory 23 ; 24 ; 25 ; 26 ; 27 ; 28 ; 29 ; 30 ; 31 ; 32 ;
33 ; D  has remained elusive. Here, we report the observation of a
theoretically predicted PDW bulk signature in two La-based cuprates, Sr-doped
La1.875Ba0.125CuO4 and Fe-doped La1.87Sr0.13CuO4, through a comprehensive
study that incorporates zero-magnetic field x-ray scattering, neutron
scattering, and transport measurements. Specifically, we observe the emergence
of so-called “1Q” order, which is to say subharmonic order associated with the
charge-density wave (CDW) stripes, in a range of temperatures in which
independent evidence suggests the co-existence of PDW long-range order and
fluctuating uniform superconducting order. The subharmonic order is most
pronounced around a half-integer $l$-vector, where the CDW diffraction peak is
also strongest. This is consistent with the theoretical proposal 23 ; 24 ; 25
; 26  that the cancellation of the Josephson coupling (“layer-decoupling”), is
a signature of PDW order and that it is commensurately locked to the density
wave stripes that are known to alternate orientation between adjacent layers.
Even if the PDW is not the “mother of all state” 14 ; 15 ; 16 ; 17 , it is at
least a close relative – possibly a second cousin.

Numerous studies have yielded highly suggestive but not yet universally
convincing evidence that a novel form of superconducting order – PDW order –
appears in some range of doping and temperature in certain high-$T_{\rm c}$
cuprates 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16
; 17 ; 20 ; 21 ; 22 ; 23 ; 24 ; 25 ; 26 ; 27 ; 28 ; 29 ; 30 ; 31 ; 32 ; 33 ; A
; B ; C ; D . The 2007 experimental observation in La1.875Ba0.125CuO4 (LBCO)
by Li et al. 3  of “layer-decoupling” – i.e., superconductivity that appears
to be confined to individual Cu-O layers accompanied by a spectacular absence
of any detectable interlayer Josephson coupling – initiated an exciting area
of study. Subsequent studies revealed similar transport phenomena in other La-
based cuprates, including La2-x-y$X$ySrxCuO4 with $X$ = Nd, Eu, and Ca A ; B ;
C , Fe and Zn-doped La2-xSrxCuO4 (LSCO) 11 ; 12 , and unadulterated LSCO with
$x$ = 0.1 in the presence of a magnetic field 6 , providing further support of
the existence of a PDW state.

The 2019 observation in scanning tunneling microscopy (STM) by Edkins et al 9
of a PDW signature in another classic cuprate, Bi2212, represented a
significant further breakthrough. They identified a subharmonic component of
the CDW order in a halo region about magnetic field induced vortices, with
ordering vector $\vec{K}/2$ that is half of the ordering vector ($\vec{K}$) of
the primary CDW order footnote1 . Thus, $\vec{K}/2$ is “subharmonic” order
which, as we will discuss below, is of the sort expected whenever there is
coexisting PDW order with $\vec{Q}=\vec{K}/2$ and uniform SC. This same theory
pertains, at least at mean-field level, to vortex cores under the assumption
that in a state with dominant uniform superconducting order, a subdominant PDW
order can emerge in the vortex core where the dominant order is suppressed 9 ;
33 ; 15 ; 17 . Moreover, similar patterns of CDW order are expected to arise
in regimes where the PDW and/or the uniform superconducting order are
fluctuating, assuming they have sufficiently long correlation lengths.

Nevertheless, several distinct aspects in Bi2212 compared with other cuprates
pose challenges to fully understanding the broader implications of the PDW
evidence. These aspects include the relatively short correlation length of CDW
order, at best marginally longer than the CDW period 34 ; 35 ; 36  (hereafter,
referred to as CDW short-range-order, CDW-SRO), the limited information
concerning SDW order at zero magnetic field 37 , and the lack of any
macroscopic evidence (e.g., layer-decoupling) of PDW order. More generally,
there have been no bulk (e.g., diffraction) measurements that show evidence of
a subharmonic signature in La-based , Bi2212, YBCO, or indeed any cuprate,
although various experiments 50 ; 51 ; 52 ; 53 ; 54  have been carried out in
regimes where it would be expected were PDW order ubiquitous.

Here, we report the first bulk evidence of subharmonic order (i.e., a 1Q peak)
in two La-based near 1/8 doped cuprates, Sr-doped LBCO and Fe-doped LSCO. (For
details see the Methods section.) We employed resonant soft x-ray scattering
(RSXS) in conjunction with neutron scattering and transport measurements for a
comprehensive study of the PDW state. We found that the long-sought 1Q peak
appears as a new peak seen with RSXS for temperatures roughly midway between
$T_{\rm c}$ and $T_{\rm pdw}$, where $T_{\rm pdw}$ is, respectively, taken to
be the $T$ at which there is a sudden superconducting-like drop of the in-
plane resistivity, $\rho_{\rm ab}$, with no accompanying decrease in the
interplane resistivity, $\rho_{\rm c}$ (i.e., there is still no SC-LRO). The
new ordering vector $\vec{K}/2$ is half that of the dominant CDW order. These
results effectively address the challenges of PDW experiments (discussed
below) and provide a foundation for further investigation into PDW and its
implications in high-$T_{\rm c}$ cuprates.

Considerations of Intertwined orders: Several well-known 30  aspects of PDW
order and its relation to the other orders observed in the cuprates are worth
reviewing before turning to the principle experimental findings. The relevant
order parameters are the complex charge $2e$ fields, $\Delta_{0}$,
$\Delta_{\pm\vec{Q}}$, and $\Delta_{\pm\vec{Q}^{\prime}}$, that represent the
local amplitude of the uniform and PDW components of the superconducting (SC)
order, and the scalar and spin-vector fields, $\rho_{\vec{K}}$,
$\rho_{\vec{K}^{\prime}}$, ${\bf S}_{K_{\rm sdw}}$, and ${\bf S}_{K_{\rm
sdw}^{\prime}}$ that represent the local amplitude of the CDW and SDW orders.
Here the pairs of momenta [$\vec{Q}$ & $\vec{Q}^{\prime}$], [$\vec{K}$ &
$\vec{K}^{\prime}$], and [$\vec{K}_{\rm sdw}$ & $\vec{K}_{\rm sdw}^{\prime}$]
are related to each other by a $C_{4}$ rotation of the crystal axes.

Each of these is, in principle, an independent order parameter with their own
preferred ordering vector. In the present La-214 family of cuprates, the CDW
and SDW orders are always commensurately locked to each other, i.e.,
$\vec{K}\equiv 2\vec{K}_{\rm sdw}$ 11 ; 38 ; 43 ; 41 ; 57 , but this is not
necessarily the case universally in the other cuprates footnote2 . There is
compelling evidence that both SDW and CDW order strongly compete with uniform
SC order. However, PDW order, when mutually commensurate with the other
density wave orders (i.e., when $\vec{K}\equiv 2\vec{Q}$), appears able to
coexist with density wave order more easily – a point we revisit below.

As was known from previous studies, the phase diagrams of the La-cuprates
studied here, including crossovers associated with the onset of density wave
(DW) orders, have the same structure, shown schematically in Fig. 1, as that
found for LBCO by Li et al 3 . There is an onset of CDW-SRO below a high-
temperature crossover, a more sharply defined transition (or possibly two
distinct transitions) at a stripe ordering temperature to a state with clear
unidirectional CDW (below $T_{\rm cdw}$) and SDW “stripe” order (below $T_{\rm
sdw}\leq T_{\rm cdw}$) with long correlation lengths, a temperature at which
layer-decoupling (i.e. a precipitous drop of $\rho_{ab}$) develops at $T_{\rm
pdw}\leq T_{\rm sdw}$ (which we identify as the PDW transition temperature),
and finally a transition at $T_{\rm c}$ to a 3D superconducting state in which
$\rho_{c}$ also vanishes and the perfect diamagnetism of the Meissner phase is
apparent in measurements of the magnetization. The data showing this evolution
in our crystal of Sr doped LBCO is shown in Fig. 3; comparable data for the Fe
doped LSCO case were reported in the previous study 11 .

Theoretical considerations concerning a 1Q peak: Because the evidence of PDW
order has been somewhat hodge-podge, many discussions of cuprate physics do
not include any reference to such order at all. Among the theoretical
approaches that include it, there are two extremes in terms of perspective:
(i) The “subdominant order” view, in which the uniform SC, CDW and SDW orders
are the principle orders while PDW order appears - if at all - where the other
orders are not too strong. (ii) The “mother of all orders” view 14 ; 15 ; 16 ;
17 ; peter  which proposes that PDW ordering is the strongest, or one of the
strongest ordering tendencies. This latter perspective underlies the
suggestion that some form of fluctuating PDW order is much more robust than
uniform SC order, such that it gives rise to the pseudo-gap phenomena at
temperatures well above $T_{\rm c}$, and to the suggestion that all observed
CDW correlations arise as second harmonics of an underlying primary PDW order
– i.e., they arise when and only when PDW correlations are strong.

To better frame the issue, let us consider the case in which all the density
waves are assumed to be mutually commensurate, i.e., $\vec{K}\equiv
2\vec{Q}\equiv 2\vec{K}_{\rm sdw}$, and imagine that all the different order
parameters can be considered as independent dynamical variables footnote3 .
Then we expect the charge-density structure factor would have peaks at
$\vec{K}$ and $\vec{K}/2=\vec{Q}$ with intensities as follows:

$\displaystyle I(\vec{K})\sim\Big{|}\rho_{\vec{K}}+\alpha_{\rm
sc}\left[\Delta_{-\vec{Q}}^{\star}\Delta_{\vec{Q}}+c.c.\right]+\alpha_{\rm
sdw}\left[{\bf S}_{-\vec{K}_{\rm sdw}}^{\star}\cdot{\bf S}_{\vec{K}_{\rm
sdw}}+c.c.\right]\Big{|}^{2},$ $\displaystyle I(\vec{Q})\sim\left|\alpha_{\rm
sc}\right|^{2}\Big{|}\ \Delta_{0}^{\star}\Delta_{\vec{Q}}+c.c.\ \Big{|}^{2}\
,$ (1)

where $\alpha_{\rm sc}$ and $\alpha_{\rm sdw}$ are coupling constants that
depend on microscopic details. What is apparent here is that there could be
multiple contributions to the scattering intensity at $\vec{K}$: It could
primarily reflect CDW order if the fundamental CDW order is strong, or
conversely if there were no intrinsic CDW ordering tendency, it could be an
indirect signature of either PDW or SDW order. On the other hand, a $1\vec{Q}$
peak clearly signifies an interference effect between the PDW and the uniform
SC orders.

Motivated by the theoretical framework outlined above, we devised an
experimental strategy to identify a putative 1Q peak within our crystals (see
details in the Method section). Specifically, we have chosen materials that
exhibit layer-decoupling, indicative of the presence of significant PDW order,
and have focused on two such materials that have relatively higher values of
$T_{\rm c}$’s, suggestive of a somewhat stronger tendency to develop uniform
SC order. We have used the fact that there is little evidence of any density
wave or SC order with significant correlation lengths at temperatures above
$T_{\rm cdw}$ to identify high temperature scattering data as the
uninteresting background signal (Extended data Fig. 1), which we can then
subtract from the lower temperature data to obtain information about the
interesting CDW correlations. Finally, we have chosen our scattering geometry
to optimize the intensity of the 1Q peak. Together, these features have
allowed us to extract clear evidence of a 1Q peak – a subharmonic of the
dominant CDW order – that appears in the midst of the temperature range
$T_{\rm c}<T<T_{\rm pdw}$, where it is theoretically expected.

Experimental search for a 1Q signature in Fe-doped LSCO: The first set of new
results reported here (see Fig. 2) involve x-ray scattering from a previously
studied crystal of Fe-doped LSCO 11  which exhibits bulk SC below $T_{\rm c}$
$\approx$ 6 K and layered-decoupling (2D-SC) below $T_{\rm pdw}$ $=$ 32 K (see
detailed in the Method section). At $T$ = 60 K (see Fig. 2a), the RSXS map
exhibits a weak signal indicative of CDW-SRO with a correlation length (21
${\rm\AA}$) at $h$ $\approx$ 0.24 r.l.u. (i.e., $\vec{K}$). As the temperature
decreases to $T$ = 43 K (below $T_{\rm sdw}$ $\approx$ 50 K), the CDW signal
becomes considerably stronger. Further cooling to $T$ = 25 K (situated below
$T_{\rm pdw}$), results in a distinctly sharper CDW stripe ordering peak,
characterized by a correlation length ($\xi_{\vec{K}}$ $\approx$ 61
${\rm\AA}$) that surpasses 8 unit cells (8$a$ $\approx$ 30 ${\rm\AA}$) along
the $a$-axis, in line with prior studies 11 ; 41 . In addition, at $T$ = 25 K,
an excess of intensity distribution around $\vec{K}/2$ is observed (indicated
by the arrow in the figure). Figure 2b shows the intensity of $h$-scans
integrated over the $k$-direction at each temperature. As the CDW signal grows
with decreasing temperature, the intensity around the CDW peak’s tail at
$h\sim 0.2$ r.l.u., marked as an arrow in the figure, increases. In addition,
below $T_{\rm pdw}$, a discernible background variation (indicated by the
green-colored dashed line in the figure), also emerges, which is distinct from
the contributions of the CDW tail and reflection intensities. Figure 2c
further illustrates this variation with an extended $h$ value scale. The
excess intensity distribution emerges around $\vec{K}/2$, where the 1Q peak is
expected.

Direct interpretation of the measured scattering intensity is complicated by
the presence of a large background signal arising from fluorescence and
specular reflection from the sample surface (Extended Data Fig. 1). However,
these are expected to be largely independent of temperature, so can be removed
by subtracting the RSXS maps measured at two different temperatures, as shown
in Figs. 3a and 3b. These subtractions provide direct information about the
temperature dependent changes in the scattering intensities – A more detailed
justification of this subtraction is described in the Method section. The
results reveal a 1Q signature in this crystal that is weaker than, but
coexists with the primary CDW stripe signal. Additional information concerning
the $T$ dependence of this structure can be gleaned by comparing the
difference signal between a pair of temperatures just above and just below
$T_{\rm pdw}$ (i.e., 25 K and 43 K) and two temperatures both above $T_{\rm
pdw}$ (i.e., 43 K and 60 K), as depicted in Fig. 3c. We observe that a
distinct ordering feature at $\textit{h}\approx 0.131\pm$ 0.007 r.l.u.
($=\vec{K}/2=\vec{Q}$) emerges below $T_{\rm pdw}$ (with a peak width
corresponding to a correlation length, $\xi_{\vec{Q}}$ $\approx$ 43.47
${\rm\AA}$), while the feature at $\textit{h}\approx 0.235\pm$ 0.005 r.l.u.
($=\vec{K}=2\vec{Q}$) is clear both above and below $T_{\rm pdw}$. Moreover,
by examining the x-ray photon energy dependence (Fig. 3d and Extended Data
Fig. 2), we found that the ordering features are associated with the Cu ions.

1Q signature in other La-based cuprates: To rule out the possibility that the
(relatively weak) 1Q peak we have seen in Fe-doped LSCO is a peculiarity of
one particular material, we conducted additional scattering experiments on
another high-$T_{\rm c}$ cuprate, Sr-doped LBCO. This choice was intentional
as LBCO is the archetypal system in PDW research. Furthermore, based on the
previous work 38 ; 43 , the light Sr-doping leads to an enhancement of $T_{\rm
c}$ and the suppression of the CDW compared to pristine LBCO; both features
are expected to help in resolving signatures of subharmonic order.

In our initial assessment of this material’s suitability for our study, we
examined DWs and superconducting properties to verify that the basic structure
of the phase diagram is as in Fig. 1. Figures 4a and 4b show the CDW measured
by RSXS and SDW measured by neutron scattering, respectively (see the Method).
Similar to undoped LBCO, both DWs exhibit clear features as temperature
decreases. The scattering data were analyzed and summarized in Figs. 4c and
4d. Consistent with the intertwined DW phenomena discussed earlier, the doped
LBCO system undergoes a transition from CDW-SRO to a CDW stripe, accompanied
by the emergence of SDW at $T_{\rm sdw}$ $\approx$ 35.4 K. At this
temperature, the correlation length of the CDW stripe order ($\xi_{\rm cdw}$
$\approx$ 90 $\rm\AA$) already exceeds 8$a$ ($\approx$ 30 ${\rm\AA}$). The in-
plane resistivity as shown in Fig. 4e confirms the emergence of significant 2D
SC correlations, also below $T$ $\approx$ 35.4 K (i.e., $T_{\rm pdw}\approx
T_{\rm sdw}$), which is distinctly different from the bulk superconducting
transition temperature ($T_{\rm c}$ $\approx$ 10 K) determined by magnetic
susceptibility. In addition, the relatively suppressed strength of the overall
CDW order in the Sr-doped LBCO relative to the pristine material, as well as
the correspondingly weak background, which are consistent with findings of
previous studies 38 ; 43 , are helpful for detecting subharmonic order below
$T_{\rm pdw}$.

Figure 5a presents the RSXS maps obtained from Sr-doped LBCO; with additional
experimental details reported in the Method section. At $T$ $=$ 42.5 K (upper
panel in Fig. 5a), where the stripe phase has not yet developed, the
scattering map shows the CDW-SRO signal. Consistent with the discussion in
Fig. 4, the RSXS map measured at 26.5 K (i.e., below $T_{\rm pdw}$) exhibits
pronounced CDW intensity at h $\approx$ 0.235 $\pm$ 0.005 r.l.u., indicating
the development of the stripe phase. Furthermore, there appears to be enhanced
intensity at h $\approx$ 0.12 $\pm$ 0.007 r.l.u.

Adopting the same method of analysis we used for Fe-doped LSCO, we exhibit the
difference between the RSXS maps measured at $T=26.5$ K and 42.5 K in Fig. 5b.
The subtracted (i.e., difference) map reveals intensity in the expected
regions. Not only is the CDW stripe clearly evident, but a notable emergence
of 1Q order is also apparent below $T_{\rm pdw}$. These results bear a
striking resemblance to the observations made in the context of the Fe-doped
LSCO cuprate, confirming the robustness of the findings.

Figures 5c and 5d show the projected intensities derived from the difference
map along various cuts through reciprocal space. Recalling that $\vec{K}$
$\approx$ $(h=1/4,~{}k=0)$ and $\vec{Q}=$ $\vec{K}/2$ $\approx$
$(h=1/8,~{}k=0)$, in Fig. 5c we display the scattering intensity along the $k$
direction integrated over 0.10 $<$ $h$ $<$ 0.15 r.l.u. and 0.21 $<$ $h$ $<$
0.26 r.l.u. (shown as the green circles and the solid blue line,
respectively). In common with what is seen at low doping in LSCO 41 ; 44 ; 45
, the CDW peak in Sr-doped LBCO is relatively broad in this direction with a
slight two-peak structure along the $k$-direction at $h\approx 1/4$. This
feature reflects the coexistence of the LTO and LTT structural phases 38 ; 43
. As anticipated in Eq.(1)., a comparable splitting effect is discernible in
the subharmonic order along the corresponding cut through $k$-space. In Fig.
5d, we show data taken at different detector-angles (2$\theta$). A subtlety of
the way the experiments are carried out is that all cuts through $q-$space
traverse a trajectory that has a fixed 2$\theta$ relative to the $ab$-plane as
shown in the extended data Fig. 3. This means there is an implicit $l$
dependence of the data. All the density wave orders seen in these materials
are quasi-2D, which is to say that the $l$ dependence is relatively smooth,
but none-the-less, the primary CDW order has been long known to have maximal
intensity at half-integer $l$ (Extended data Fig. 3a). For
2$\theta=123^{\circ}$, the $q$-space trajectory passes through the principle
stripe-ordering vector $\vec{K}$ at $l\approx$ 1.5 r.l.u., while at the
position of the 1Q peak, $l\approx$ 1.3 r.l.u. Conversely, the trajectory with
2$\theta=106^{\circ}$ passes through the stripe ordering vector at $l\approx$
1.7 r.l.u., and through the 1Q peak at $l\approx$ 1.5 r.l.u. Both cuts show
the principle stripe-ordering peak clearly, albeit the first cut shows it more
clearly. But because it is weak, it is only in the second cut where conditions
are optimized that the 1Q peak is evident. In addition to providing
information pertinent to avoiding a possible failure route in future studies
of this order, this observations demonstrates that the 1Q peak shares the same
interlayer structure as the principle stripe-order peak.

Discussion: In this work, we have observed subharmonic (1Q) CDW order with a
structure that links it with the fundamental quasi-2D CDW order. The
fundamental (2Q) is always much stronger, and is seen over a wide range of
$T$, while the subharmonic is only observable below $T_{\rm pdw}$, where
observed layer-decoupling phenomena already provided strong indirect evidence
of the existence of PDW order. Together, these results provide compelling
evidence of the theoretically conjectured state consisting of mutually
commensurate CDW, SDW, and PDW orders.

An intuitive real-space image for such a state can be visualized starting with
the familiar “stripe” image (suggestive Extended data Fig. 3b) 24  of regions
of N$\acute{\rm e}$el antiferromagnetic order separated by a periodic array of
anti-phase boundaries at which the density of itinerant (doped) holes is
maximal. The PDW order would then be expected to be shifted by half a period
relative to the SDW order, such that the pair-field has maximum magnitude
where the itinerant hole density is largest, and vanishes where the pair-field
changes sign in the midst of the spin-ordered regions. This complex state,
while theoretically conjectured, remains challenging to directly visualize.
Nevertheless, a state with PDW order but no long-range uniform SC order is
distinct by symmetry from any usual SC state, including states with coexisting
CDW and SDW order. It is a distinct phase of matter. Thus, establishing its
existence is significant, independent of any broader importance of this result
for the physics of the high-$T_{\rm c}$ cuprates.

Turning to the cuprates, the present results extend the evidence that a
central feature of the physics 30  is the existence of a large number of
distinct forms of order all of which onset in the same range of temperatures
(roughly from 30K - 150K), and which are strongly “intertwined” in the sense
that they are strongly coupled to each other but neither inevitably coexist
nor universally compete with one another. Manifestly, PDW order can now be
added to the list of established intertwined orders – d-wave SC, CDW, SDW, and
possibly nematic.

Somewhat more speculatively, the present results support the conclusion that
PDW order is typically subdominant relative to competing uniform SC order, and
occurs only where the dominant order is in some way significantly weakened.
For instance, investigations into a 1Q peak in YBCO have been carried out over
a range of magnetic fields and strain where global SC order coexists with
pronounced signals of charge-density wave (CDW) order (at $\vec{K}=2\vec{Q}$).
If CDW order were primarily a harmonic of PDW order, a prominent 1Q peak would
have been anticipated alongside the CDW signal 50 ; 51 . However, in contrast
to the outcomes observed in La-based cuprates here, the emergence of a 1Q peak
has not been reported. Based on the considerations discussed, our assessment
suggests that the perspective of a “mother of all orders,” while intriguing,
lacks consistency with a variety of observations in cuprates. Indeed, the
subdominant PDW order perspective offers a plausible explanation for the
elusiveness of the 1Q peak; to see it one has to identify circumstances in
which the uniform SC component, which typically prevents PDW order, is
relatively weak but not entirely eliminated. Importantly, although CDW and SDW
order likely also compete with PDW order Tranquada , the real-space picture
presented above makes it plausible that this competition is much less strong
for PDW than for uniform SC order. In the sense that “the enemy of my enemy is
my friend,” one can anticipate that relatively strong CDW and/or SDW order may
effectively enhance PDW tendencies by suppressing the more strongly
competitive uniform SC order.

Finally, it is worth highlighting the future experimental opportunities that
arise from our current findings. Since our present experimental results lack
phase-sensitive information, they do not directly corroborate certain aspects
of the “stripe” image or conclusively address questions like whether the
intertwined order is commensurate (i.e., precisely period 8), and if so,
whether it is “site-centered” or “bond-centered.” Therefore, further
investigations that explore the polarization and azimuthal dependence of this
1Q peak are warranted. In addition, conducting experiments in the presence of
an external magnetic field holds the promise of yielding valuable insights
into the characteristic behavior of the observed peak and its correlation with
the 2D and 3D superconducting states. Furthermore, expanding the scope of
x-ray measurements to encompass other cuprate materials, based on the criteria
established in this study, would offer an enriching avenue for research. These
efforts will undoubtedly contribute to a deeper understanding of the complex
interplay between various orders in the high-$T_{\rm c}$ cuprates.

##  I Methods

Synthesis of single crystals: High-quality single crystals of both Sr-doped
LBCO (La2-1/8Ba1/8-0.06Sr0.06CuO4) and Fe-doped LSCO
(La1.87Sr0.13Cu0.99Fe0.01O4) were grown by the traveling solvent floating zone
method. The grown crystal was annealed in one bar of $O_{2}$ gas to minimize
oxygen deficiencies. The typical growth rate was 1.0 mm/hr and a 50–60 mm long
crystal rod was successfully obtained for each concentration. Before the
scattering measurements, using a Quantum Design PPMS we characterized their
superconducting $T_{\rm c}$ as the mid-point of the transition. To check the
crystal properties, the in-plane electrical resistivity and magnetic
susceptibility measurements of the crystal was measured by the conventional
four-point probe method at the Institute for Materials Research, Tohoku
University. For elastic neutron scattering measurements, a crystal of Sr-doped
LBCO (Fe-doped LSCO) with the length of 25mm (35mm) was used. For the RSXS
measurements, we prepared all the samples with a typical dimension of 1.5 mm
$\times$ 1.5 mm $\times$ 2.5 mm ($a$ $\times$ $b$ $\times$ $c$ axis). To
achieve a fresh $c$-axis normal surface, each sample was ex-situ cleaved,
before transported into the ultra-high vacuum chamber (base pressure = 8
$\times$ 10-10 Torr).

RSXS measurement: The zero-magnetic field x-ray experiments were conducted at
beamline 13-3 of the Stanford Synchrotron Radiation Lightsource (SSRL). The
sample was positioned on an in-vacuum 4-circle diffractometer and maintained
at a controlled temperature using an open-circle helium cryostat (base
$T\approx$ 25 K). To avoid distortions of the cryostat and mechanical
components of the diffractometer typically occurred during initial cool down
or large temperature changes, we ensured that the setup had sufficient time to
stabilize at each measurement temperature and the temperature did not exceed
100K during the whole data collection. Incident photon polarization was
consistently set as vertical linear ($\sigma$) polarization. The precise ($h$,
0, $l$) scattering plane alignment was achieved using the measured (0, 0, 2),
(-1, 0, 1), and (1, 0, 1) structural Bragg reflections at a photon energy of
approximately 1770 eV.

Detecting the RSXS signals used a 256 $\times$ 1024 pixel area detector. To
ensure the maximized scattering intensity, we set the detector at a fixed
position of 2$\theta=106^{\circ}$ in which the scattering from $\vec{Q}$ is at
$l$ $\approx$ 1.5. We rotated the sample in small steps ($\theta$-scan) to
gather information about the $h$-dependence. Throughout the $\theta$-scan,
each CCD image included scattering information from the well-aligned ($h$, 0,
$l$) scattering plane near the center of the detector and from off-scattering
planes ($h$, $\pm k$, $l$) at the top and bottom parts of the CCD 41 . For
data analysis, each CCD pixel was translated into an $hkl$ reciprocal space
index. The resulting three-dimensional scattering intensity data were then
projected onto different planes/directions for further analysis.

It is worth noting that the single scattering data collection at each
temperature point involved a total of 8 hours of measurements. Due to the
critical role of background information in shaping our data analysis strategy,
meticulous signal-to-noise ratio reduction and x-ray footprint control were
paramount considerations (see the section below - Strategy for detecting a 1Q
peak).

Furthermore, it is important that the incident polarization for the experiment
was chosen to be vertical because using linear polarization parallel to the
scattering plane poses additional challenges. Specifically, when the 1Q’s
$l$-vector is around half-integer $l$, the 2$\theta$ angle approaches
Brewster’s angle. In other words, the charge scattering signal (i.e.,
$\pi^{{}^{\prime}}$$\cdot$$\pi$) is reduced by approximately 1 $-$
cos(2$\theta$) without considering details of the scattering tensors’ matrix
elements based on the Cu orbital state in the CuO2 plane. In this scenario,
the expected 1Q signal in $\pi$-polarization could be even less, approximately
70% lower than that in $\sigma$-polarization. Although other backgrounds, such
as specular and CDW tails, would also be reduced, but it becomes more
challenging when attempting to detect an intrinsically smaller 1Q signal in
this type of experiment.

Neutron measurement: In order to investigate the SDW order, we performed the
elastic neutron scattering measurements on the cold triple-axis spectrometer
HER and the thermal neutron triple-axis spectrometer TOPAN located at the
research reactor JRR-3. The most of measurements were performed at HER with
the incident and scattered neutron energies of 5 meV. Additional elastic
scattering measurements were carried out using the thermal neutron triple-axis
spectrometer TOPAN at JRR-3. The horizontal collimations sequences at HER and
TOPAN were 32’-100’-Be-S-80’-80 and 30’-Sap-30’-PG-S-PG-30’-100’,
respectively. Here Be, Sap and PG represent the beryllium, sapphire and
pyrolytic graphite filters, respectively, to reduce contamination from the
higher-order wavelength neutron in the beam. S denotes the sample. The single
crystal was sealed in an aluminum can with Helium gas for thermal exchange.
The aluminum can was attached to the closed-cycle cryostat to control the
temperature.

Strategy for detecting a 1Q peak: To optimize conditions for observing a 1Q
peak, we begin by focusing on members of the 214 family in which clear
signatures of layer-decoupling provide strongly suggestive evidence of
significant PDW order. In the poster-child material for layer-decoupling, 1/8
doped LBCO, the CDW and SDW order are particularly strong, while $T_{\rm c}$
is 4 K or less, suggestive that the uniform SC order is all but absent in the
range of temperatures that are accessible to us in RSXS. We therefore have
looked at two closely related materials – Fe-doped LSCO and partially Sr
substituted LBCO – both of which show a clear range of $T$ in which layer-
decoupling is apparent, but which have relatively higher values of $T_{\rm c}$
and somewhat weaker scattering at $\vec{K}$.

In resonant x-ray scattering, there are three purely technical challenges to
observing a weak 1Q peak. The first is that the strength of the scattering off
the principle CDW order implies technical challenges in observing a putative
1Q peak on the background coming from the tail of the $\vec{K}$ peak
(hereafter, CDW-tail). Given that this tail signal typically persists even at
significantly off-$q$ positions (e.g., $\vec{Q}$), it would also be
advantageous if the material exhibits a somewhat weaker scattering signal at
$\vec{K}$, as discussed above. This further supports the rationale for our
choice of these cuprates from a scattering perspective.

The second is that there is always a strong elastic background that has
nothing to do with the desired diffraction signal that comes from a
combination of specular scattering from the sample surface (i.e., reflection
and its tail), and florescence (including the geometrical effect of a detector
position). Fortunately, these two background sources are expected to be
approximately temperature independent. Thus, we can remove it by subtracting
the signal taken at relatively high temperatures, $T$ $=$ 60 K (Fe-LSCO case),
from the lower temperature signals as shown in Figs. 2 and 5. Since this is
well above $T_{\rm strp}$ $\approx$ 40 K, we do not expect any significant CDW
component to the high $T$ signal, so this subtraction should be reasonably
reliable.

The last arises from the expected dependence of the scattering intensity on
the $c$-axis component of the scattering vector, $l$. It is known that the CDW
scattering produces quasi-2D rods of scattering, so that one generally treats
$\vec{K}$ as a 2D vector. In this context, the CDW intensity is greatest at
$l$ $=$ 1/2 in reciprocal lattice units (r.l.u.), where is a doubled $c$-axis
unit cell corresponding to a 4-Cu-O plane period. Assuming that the same is
true of the PDW order, then it would be expected that the 1Q scattering
intensity would also be strongest at half-integer $l$. However, as shown in
the extended data Fig. 1b, RSXS data is measured by rotating the sample at the
fixed the detector in the scattering plane, meaning that as the scattering
vector, $q$ $=$ ($h$, $k$, $l$), traverses a given trajectory in the $ab$
plane (for instance, variable $h$ for fixed $k$), the value of $l$ is an
implicit function of the position along this trajectory. Thus, it is customary
to choose a trajectory such that $l$ is half integer when the in-plane
component of $q$ equals $\vec{K}$. However, in this case, the value of $l$ is
typically far from a half-integer value when the in-plane component of $q$
equals $\vec{Q}$. If, on the other hand, the trajectory is chosen so that $l$
is half-integer for the in-plane component of $q$ equal to $\vec{Q}$, one
would expect enhanced 1Q scattering intensity, and reduced intensity at
$\vec{K}$ (see the extended data Fig. 3a). That this expectation is realized
can be seen by comparing the results along two such trajectories in the
following experimental results shown in Figs. 2 and 5.

Finally, to properly implement the aforementioned strategy, it is imperative
to maintain high-quality data throughout prolonged acquisition times. The
following steps were taken to ensure that the scattering signal was
consistently originated from the same spot on the sample. First, two optical
cameras continuously tracked both the x-ray beam and sample positions with
micron-level resolution. Second, before and after each angle scan, we veriﬁed
the positions of structural Bragg reﬂections (0, 0, 2), (-1, 0, 1), and (1, 0,
1). This ensured that the X-rays consistently probed the same domain structure
and sample volume. Third, during the acquisition of photon signals (i.e.,
angular scan), we simultaneously recorded the secondary electron signal, known
as total electron yield (TEY). The intensity of TEY is highly sensitive to
incident x-ray beam angle with respect to the sample and beam footprint on the
sample. We utilized the angular dependence of the TEY signal as a real-time
indicator to ensure reproducibility.

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Data availability
All data that support the plots and other findings of this study are available
from the corresponding authors upon reasonable request.

Acknowledgements
We thank John M. Tranquada, Peter Abbamonte, and David Hawthorn for insightful
discussions and T. Masuda, S. Asai, and M. Ohkawara for supporting the neutron
measurements at HER and TOPAN. All soft X-ray experiments were carried out at
the SSRL (beamline 13-3), SLAC National Accelerator Laboratory, supported by
the U.S. Department of Energy, Office of Science, Office of Basic Energy
Sciences under Contract No. DE-AC02-76SF00515. The neutron scattering
experiments were performed under the Joint-Use Research Program for Neutron
Scattering, Institute for Solid State Physics, University of Tokyo, at the
Japan Research Reactor JRR-3 (Program Nos. 21527 and 22402). We acknowledge
the support from Center of Neutron Science for Advanced Materials, Institute
for Materials Research, Tohoku University. S.A.K. acknowledges the support by
Department of Energy, Office of Basic Energy Sciences, under contract no. DE-
AC02-76SF00515 at Stanford. M.F. is supported by Grant-in-Aid for Scientific
Research(S) (GrantNo.21H04987).

Author contributions
J.-S.L. and C.-C.K carried out the RSXS experiment and analyzed the data. T.W.
Y.I., T.T., and M.F. synthesized the materials and carried out the neutron
scattering. S.A.K., C.-C.K., and J.-S.L. wrote the manuscript with input from
all authors. J.-S.L. coordinated the project.

Competing interests
The authors declare no competing interests.

Figure 1: Evolution of intertwined orders. A peak in the charge-density
structure factor – generally identified with fluctuating CDW order (CDF) – is
detectable up to relatively high temperatures Giacomo . With decreasing
temperature, CDW-SRO grows significantly stronger below a crossover
temperature, $T_{\rm cdw}$, which is greater than or equal to the temperature,
$T_{\rm sdw}$, below which unidirectional SDW (stripe) order onsets. We
associate the onset of PDW order with the temperature $T_{\rm pdw}\leq T_{\rm
sdw}$ at which the in-plane resistivity, but not the interplane resistivity,
drops precipitously, indicative of 2D SC. The 3D SC (Meissner) state, which
exhibits perfect diamagnetism and in which all components of the resistivity
vanish, appears below a much suppressed $T_{\rm c}$. The 1Q peak is detected
at temperatures roughly midway between $T_{\rm pdw}$ and $T_{c}$, where it is
plausible that some degree of PDW and uniform (long-range-order, LRO) SC
(possibly fluctuating) orders coexist. The cartoon below represents a real-
space visualization of the nature of the intertwined orders in various ranges
of $T$. It is plausible that the states with coexisting orders are
inhomogeneous, but as long as the resulting domains in which one or another
order dominates are sufficiently small and the orders overlap in significant
interfacial regimes, this does not qualitatively change the analysis. Figure
2: Cu L-edge resonant x-ray scattering on Fe-doped LSCO. a, RSXS maps at
temperatures of $T$ = 60 K (top-panel), 43 K (middle), and 25 K, below $T_{\rm
pdw}$ = 32 K, (bottom), respectively. All data were collected with the
detector at an angle of 106∘. The dashed lines indicate the CDW order at
$\vec{K}$. The gray-colored arrow in data at $T=25$ K denotes an excess of
intensity distribution around $\vec{K}/2$. b, h-scan projected along the
k-direction in RSXS maps measured at each temperature shown in (a). With
decreasing the sample temperature, CDW order are enhanced. The blue-colored
arrow indicates the increase in CDW background with increasing CDW order,
while the reflection tail, comes from a $h\sim$ 0, remains temperature
independent. The green-colored dashed line marks the putative 1Q peak. c, The
extended h-scan for potential subharmonic order area (i.e., around
$\vec{K}/2$). The green-colored arrow indicates the intensity increment at
$T=25$ K, compared to $T$ = 60 K and 43 K. The green circles with the cap-bar
denote the statistic error between $T$ = 60 K and 43 K. based on the 1
standard deviation (1 SD) Figure 3: A signature of the putative 1Q peak in Fe-
doped LSCO. a,b, Differences between RSXS maps measured at 25 K, 43 K, and 60
K. c, Comparison of h-scans projected along the k-direction, which were
extracted from difference maps ($I_{25\rm K}-I_{43\rm K}$, below $T_{\rm
pdw}$) and ($I_{43\rm K}-I_{60\rm K}$, above $T_{\rm pdw}$). The solid line
represents the fitting result, while dashed lines and shading serve as visual
guides. All error bars denote the statistical error, including the
experimental resolutions. d, Comparison of differences extracted below and
above Cu $L$-edge resonant photon energy (933.1 eV). The red dashed line and
shade are guides to the eye. Figure 4: DWs and superconducting properties on
Sr-doped LBCO. a,b Intensity profiles of CDW measured at q = ($h$, 0, $\sim
1.5$) through RSXS and SDW measured at q = ($h$, 0.622, 0) through neutron
scattering as a function of temperature, respectively. The solid lines are
Lorentzian fits. The error bars represent 1 SD of the fit parameters. c,d,
Integrated intensity plots by using the fitted CDW and SDW results,
respectively. All dashed and solid lines are guides to the eye. e,
Superconductivity measurements as a function of temperature. Magnetic
susceptibility for bulk superconductivity (left y-axis). The zero-field
resistivity ($\rho_{\rm ab}$) along the in-plane (right y-axis). Figure 5: Cu
L-edge resonant x-ray scattering on Sr-doped LBCO. a, RSXS maps at T = 42.5 K
(upper panel) and 26.5 K (lower panel). b, Difference map between the two
temperatures (26.5 K and 42.5 K, below $T_{\rm pdw}$). The brackets indicate
the positions of the CDW and subharmonic. c, $k$-scans intensity integrated at
the fixed regions, 0.10 $<$ $h$ $<$ 0.15 r.l.u. (indicating the subharmonic
area) and 0.21 $<$ $h$ $<$ 0.26 r.l.u. (CDW area). The line and circled date
correspond to the subharmonic and CDW regions, respectively. The inset shows
the extended CDW region. d, Intensity of $h$-scans (green-colored circles)
integrated over the $k$-direction in the difference RSXS map. The data marked
by the blue-colored circles shows a different two-theta (detector angle,
2$\theta$ = 123∘), where $l$ vector at $\vec{Q}$ is $\sim$ 1.7 r.l.u. Whereas,
the green-colored circles were measured at 2$\theta$ = 106∘, $l$ $\sim$ 1.5
r.l.u. The solid line represents the fitting result, while dashed lines and
shading serve as visual guides. Figure – Extended data 1: Background issue in
x-ray scattering measurement. a, Massive backgrounds at the expected
subharmonic positions ($\vec{Q}$ and $\vec{-Q}$). There are three dominant
components – 1) a scattering intensity from specular (elastic) reflection of
the sample surface (i.e., from $h$ = 0), 2) fluorescence (inelastic)
contribution, and 3) another elastic signal from the CDW stripe (i.e., from
$\vec{K}$ and $\vec{-K}$). The 1) and 2) components are expected to be largely
independent of temperature. The 3) component has a temperature dependence. b,
Schematic carton for our RSXS experimental configuration. The scattering plane
is $ac$-plane. While rotating the sample angle ($\theta$), the so-called
$\omega$-scan, the RSXS intensity is accumulated at a fixed detector angle
(2$\theta$). Due to the resonant nature, the scattered x-rays include not only
the elastic component (e.g., DWs and specular) but also the inelastic
component (e.g., fluorescence). Figure – Extended data 2: Resonant profile on
Fe-doped LSCO. a, X-ray absorption spectroscopy (XAS) from the sample. It
becomes a source of fluorescence background discussed in Extended Data Fig. 1.
The dashed line denotes the absorption maximum. b, Incident photon energy vs.
$h$. At slightly lower energy than the absorption maximum, both the CDW and
subharmonic orders are resonated. c, Projected intensity from the map
regarding both subharmonic (green colored) and CDW (green colored) areas where
are from two $h$ regions marked in the Extended Data Fig. 2b. Im[$F_{\rm r}$]
is XAS, and Re[$F_{\rm r}$] is a result driven by Kramers-Kronig relation
(KKR) of Im[$F_{\rm r}$] term, which is related with the electron density
(i.e., charge contribution). The resonant x-ray scattering signal is related
with a combination of both terms, ${\rm Re}[F_{\rm r}]$ and ${\rm Im}[F_{\rm
r}]$. The observed DWs signals tend to move to the lower energy side compared
to the electron absorption energy. Figure – Extended data 3: Understanding
$q$-space and real-space. a, The $h/l$-vectors trajectory through an
$\omega$-scan at the fixed detector angle (2$\theta$). The blue trajectory
(2$\theta=$ 123∘) represents the CDW (i.e., $\vec{K}$) optimized at $l\sim$
1.5 r.l.u., while the green trajectory (2$\theta=$ 106∘) represents the 1Q
peak (i.e., $\vec{Q}$) also optimized at $l\sim$ 1.5 r.l.u. b, Schematic
cartoon illustrating the suggestive stripes image in real-space. Within each
CuO2 plane, a CDW forms with 4$a$ periodicity. The SDW (i.e., N$\acute{\rm
e}$el antiferromagnetic order) is separated by a periodic array of anti-phase
boundaries where the density of doped holes is maximal. With intertwined DWs,
the PDW order is expected to be shifted by half a period relative to the SDW
order, resulting in an 8$a$ periodicity. Along the $c$-axis, the inter CuO2
planes form the 90∘ rotation, representing “layer-decoupling.” With a double
unit-cell along the $c$-direction (i.e., 2$c$), the same wave pattern emerges.


# The quantum Hall effect in the absence of disorder

Kyung-Su Kim<EMAIL_ADDRESS>Steven A. Kivelson

###### Abstract

It is widely held that disorder is essential to the existence of a finite
interval of magnetic field in which the Hall conductance is quantized, i.e.
for the existence of ‘plateaus’ in the quantum Hall effect. Here, we show that
the existence of a quasi-particle Wigner crystal results in the persistence of
plateaus of finite extent even in the limit of vanishing disorder. Several
experimentally detectable features that characterize the behavior in the zero
disorder limit are also explored.

Introduction.— In the absence of disorder, the electrostatic force exerted by
an electric field $\mathbf{E}$ on an electron fluid balances the Lorentz force
from a magnetic field $\mathbf{B}$ so long as the fluid velocity
$\mathbf{v}=c\mathbf{E}\times\mathbf{B}/B^{2}$. Consequently, one expects the
Hall conductance of a two dimensional electron gas to vary linearly with
electron density, $n$, i.e. to exhibit the ‘classical’ Hall effect:
$\sigma_{xy}=nec/B=(e^{2}/h)\nu$, where the ‘filling factor’ $\nu\equiv
n\phi_{0}/B$ and $\phi_{0}=hc/e$ is the quantum of flux. In contrast, where
the quantum Hall effect arises, the Hall conductance remains constant,
$\sigma_{xy}=(e^{2}/h)\nu^{\star}$, for a finite range of $\nu$ on either side
of $\nu=\nu^{\star}$, where $\nu^{\star}$ is a rational number corresponding
to a quantized filling factor. Combining these observations, it is generally
asserted that disorder is necessary for the quantum Hall effect. Instead, we
show that the occurrence of a quasi-particle Wigner Crystal (QPWC) [1, 2, 3,
4, 5] for sufficiently small $|\nu-\nu^{\star}|$ results in a quantized Hall
plateau, even in the absence of disorder. To make the considerations concrete,
we derive the quantized Hall conductance both in a system with disorder,
taking the limit as the disorder strength tends to zero and as the system size
to infinity, and in a finite size system with no disorder but with a finite
pinning potential at the edges. We also discuss the finite $T$ phase diagram
and how thermal melting of the QPWC restores the classical behavior.

Classical Hall effect.— The derivation of the ‘classical’ Hall effect
typically invokes Galilean invariance [6, 7]. Although higher-order
corrections to effective mass theory do not have Galilean symmetry [8], it is
straightforward to show that translation invariance alone is sufficient for
the Hall conductivity to take its classical value (See Appendix). The
observation of quantized Hall plateau, on the other hand, is commonly
associated with the explicit breaking of translational symmetry by disorders
[9, 10, 11, 12].

Figure 1: Finite $T$ phase diagram of ‘ideal’ quantum Hall effect
a) Schematic phase diagram in the neighborhood of $\nu=\nu^{\star}$ (quantized
filling factor) in the limit of vanishing disorder. b) Resistivity tensor as a
function of $\nu$ at three fixed temperatures, as indicated in panel a. At
zero temperature (blue) the Hall resistivity is strictly quantized, whereas at
high temperature (black) the QPWC has melted and the Hall resistivity takes
its classical value. At intermediate temperatures (red), the Hall resistivity
shows sharp crossovers from the classical value to the quantized one as $\nu$
moves away from $\nu^{*}$. Moreover, the longitudinal resistivity $\rho_{xx}$
is very close to zero deep in the QPWC phase or near $\nu^{*}$ but becomes
finite away from those regions.

The occurrence of quasi-particle Wigner crystal.— In the absence of disorder,
however, there is one other possible source of a quantum Hall plateau:
spontaneous breaking of translational symmetry. Indeed, the presence of long-
range Coulomb interactions makes this inevitable close to a quantized filling
factor. Specifically, assuming the existence of a quantum Hall fluid at
$\nu=\nu^{\star}$, for $\delta\nu\equiv(\nu-\nu^{\star})$ small, the system
consists of a small density

$n_{\textrm{qp}}=\left(\frac{B}{\phi_{0}}\right)\left|\frac{e}{e^{\star}}\right||\delta\nu|$
(1)

of quasi-particles or quasi-holes (depending on the sign of $\delta\nu$),
where $e^{*}$ is the charge of a quasi-particle (i.e.
$\left|e^{\star}/e\right|=1$ for the integer quantum Hall effect and a
suitable rational fraction for the fractional quantum Hall effect). Thus, for
$|\delta\nu|$ sufficiently small, the spacing between quasi-particles is large
compared to the magnetic length, $\ell=\sqrt{\phi_{0}/2\pi B}$, and they
behave as classical particles with fixed guiding-centers. The result is that
the quasi-particles form a Wigner crystal below a transition temperature,

$T_{\textrm{qpwc}}=A_{\textrm{wc}}\ \frac{(e^{*})^{2}}{4\pi\epsilon}\left[\pi
n_{qp}\right]^{1/2}\sim|\delta\nu|^{1/2}\ ,$ (2)

where $\epsilon$ is the dielectric constant and $A_{\textrm{wc}}\approx 1/128$
[13, 14, 15]. For $T<T_{\textrm{qpwc}}$ the quasi-particle Wigner crystal
(QPWC) has quasi-long-range order and a finite shear modulus; it only has true
long-range order at $T=0$. The resulting phase diagram for the ‘ideal’ (i.e.
zero disorder) quantum Hall system in the neighborhood of $\nu^{\star}$ is
thus as shown in Fig. 1(a). Importantly, at low $T$ in the crystalline state,
the lowest lying current carrying excitations are interstitials, whose density
is exponentially activated as

$n_{\textrm{int}}\sim\exp[-\Delta/T]\ \ {\rm with}\ \ \Delta=A_{\textrm{int}}\
\frac{(e^{\star})^{2}}{4\pi\epsilon}\left[n_{\textrm{qp}}\right]^{1/2},$ (3)

where $A_{\textrm{int}}\approx 0.136$ [16, 17].

The width of the QPWC phase can be estimated using the composite fermion
approach[18]. In the absence of Landau level mixing, it was estimated in
Ref.[19] that an electron WC has lower energy than any quantum Hall liquid for
$\nu<\nu_{0}$ where $1/7<\nu_{0}<1/5$; Landau level mixing will generally tend
to lead to a larger value of $\nu_{0}$. In the mean-field treatment of
composite fermions with $2m$ flux quanta attached to each electron, an
electron state with $\nu=(p+f)/[1+2m(p+f)]$ corresponds to a state with $p$
fully filled and one $f$-filled Landau level of composite fermions. ($m$ and
$p$ are integers and $0<f<1$.) This reasoning leads to the conclusion that if
there is a stable FQH liquid at $\nu^{\star}=p/[1+2mp]$, there should be a
stable QPWC for

$\frac{p-\nu_{0}}{1+2m(p-\nu_{0})}<\nu<\frac{p+\nu_{0}}{1+2m(p+\nu_{0})}.$ (4)

For instance, taking $\nu_{0}\approx 1/7$ [19] the QPWC associated with the
$\nu=1/3$ plateau would extend over the range $1/3-0.018<\nu<1/3+0.014$ in the
absence of disorder.

The pinning of QPWC and quantum Hall effect.— Because a Wigner crystal of
electrons is insulating in the pinning regime [20, 21, 22, 23, 24], the DC
conductivity of the QPWC must come exclusively from the quantum Hall
condensate. As in the case of any charge-density wave (CDW) [25, 26, 27, 28,
29, 30], the one subtlety concerns the possible contribution of a sliding
state of the QPWC. We analyze this problem in two distinct ways:

Firstly, we consider a sequence of ‘real’ systems with disorder strength, $D$,
as $D\to 0$. As is well known, disorder in two dimension both destroys long-
range CDW order [31]– i.e. it rounds the transition at $T_{\textrm{qpwc}}$ –
and pins the CDW [25, 26]. This means that for a QPWC with $D\neq 0$, there is
a quantized Hall response in a finite range of filling factors:
$\sigma_{xy}\to(e^{2}/h)\ \nu^{\star}$ as $T\to 0$ for
$\nu\in[\nu^{*}-\delta\nu_{-},\nu^{*}+\delta\nu_{+}]$ where
$\delta\nu_{\pm}(D)$ approach finite positive values deduced from Eq. 4 as
$D\to 0$. On the other hand, for field strengths greater than a critical
field, $E^{\star}$, the QPWC depins (slides) and contributes to the current.
The depinning field in the weak disorder limit can be estimated using the
arguments of Lee and Rice [25, 27]:

$E^{\star}(D)\approx\frac{\epsilon}{(e^{\star})^{3}}\ D^{2},\ $ (5)

where $D^{2}$ is the mean square pinning energy per unit cell of the QPWC. In
other words, for ever smaller $D$, in order to remain in the linear response
regime, we must restrict measurements of the conductivity to an increasingly
smaller range of field strengths, $V_{H}/L=E\ll E^{\star}(D)$, where $V_{H}$
is the Hall voltage and $L$ is the width of the device as in Fig. 2. We note
that even though the critical current density $j_{x}^{*}=\rho_{xy}E^{*}$ goes
to $0$ as $D\rightarrow 0$, the critical current $I_{x}^{*}\sim LD^{2}$ could
remain finite if we simultaneously take $L\to\infty$ in an appropriate manner.
To obtain a sense of the expected magnitude of the critical current, we make a
crude estimate
$D^{2}\sim\left(\frac{ee^{*}}{4\pi\epsilon\ell}\right)^{2}\times\left({n_{\textrm{i}}\ell^{2}}\right)$
where $n_{\textrm{i}}$ is the density of impurities. In the cleanest two
dimensional electron system as of today [32] (in a $d=30$nm width GaAs quantum
well) $n_{\textrm{i}}=3\times 10^{7}\textrm{cm}^{-2}$; for a 2D electron
density $n=1\times 10^{11}\textrm{cm}^{-2}$ and $B=12$T, which corresponds to
$\nu=1/3,$ this estimate yields a critical Hall voltage $V_{H}^{*}\sim
34\textrm{mV}$ and correspondingly, for a sample with linear size
$L_{x}=0.4\textrm{cm}$, to a critical current $I_{x}\sim 0.43\mu\textrm{A}$.

Figure 2: Schematic of a Hall bar
As the current flows through the current leads, the longitudinal and Hall
voltages are measured at the voltage leads. $L$ is the linear size of the
sample.

Alternatively, consider measuring the Hall response in an ‘ideal’ ($D=0$)
sample of finite size as in Fig. 2. Here the current enters and leaves through
current leads at either end, and the longitudinal and Hall voltages are
measured at the voltage leads on the sides of the device. Because the QPWC has
a finite shear modulus, it cannot smoothly flow through the current leads at
either end of the device – it is pinned by the edges of the device. On the
other hand, if the current exceeds a critical value, the crystal will be
sheared, and a portion of the QPWC will begin to flow through the device. As
in the disordered case, this means that the QPWC is pinned for small enough
$E$. Now, assuming finite pinning potential at the edges, the critical field
scales to 0 as $E^{*}\sim 1/L$ in the thermodynamic limit, but the critical
Hall voltage and current remain finite even in the limit $L\rightarrow\infty$:
$V_{H}^{*}=\rho_{xy}I_{x}^{*}=E^{*}L$. Again, so long as measurements are
carried out with $I_{x}\ll I_{x}^{*}$, a quantized Hall response will be
observed (to an exponential accuracy with small corrections,
$\delta\sigma_{xy}\sim\exp[-L/\ell]$). Importantly, the width of the plateau,
$[\delta\nu_{-}+\delta\nu_{+}]$, approaches a constant as $L\to\infty$.

The melting of QPWC.— The role of the QPWC can (in principle) be unambiguously
inferred from the thermal evolution even in the presence of weak disorder. For
weak enough disorder, one expects a sharp crossover in conductivity at
$T=T_{\textrm{qpwc}}(\delta\nu)$: for $T>T_{\textrm{qpwc}}$, the Hall response
should be close to its ‘classical’ value, while for $T<T_{\textrm{qpwc}}$ the
quantum Hall effect should be seen up to exponentially small corrections [33]
proportional to the concentration of interstitials, $n_{\textrm{int}}$. Since
$T_{\textrm{qpwc}}(\delta\nu)$ vanishes as $\delta\nu\to 0$ as given in Eq. 2,
the finite $T$ Hall response should exhibit a peculiar reentrant behavior as a
function of $\nu$ as shown in Fig. 1b. The temperature below which this
reentrant behavior can be seen is evaluated to be $T_{\textrm{qpwc}}\approx
0.02\textrm{K}$, where we take $\nu=1/3-0.009\ (=1/3-\delta\nu_{-}/2)$ and
$n=10^{11}\textrm{cm}^{-2}$ in Eq. 2.

## Appendix

Here we prove that the Hall conductivity takes the classical value
$\sigma_{xy}=nec/B=(e^{2}/h)\nu$ as discussed in the main article. Consider a
gauge-invariant many-body Hamiltonian

$H=\sum_{i}K\left[\mathbf{p}_{i}+\frac{e}{c}\mathbf{A}(\mathbf{r}_{i},t)\right]+V\left[\left\\{\
\mathbf{r}_{j}\right\\}\right]$ (6)

in which $V$ is invariant under (continuous) translation,
$\mathbf{r}_{i}\to\mathbf{r}_{i}+\mathbf{a}$, where $\mathbf{r}_{i}$ is the
(2d) position of the $i$-th electron and $\mathbf{p}_{i}$ the conjugate
momentum. We consider an infinite cylinder with periodic boundary conditions
in the $\hat{y}$ direction, a uniform perpendicular magnetic field,
$\mathbf{B}=B\hat{z}$, and an electric field $\mathbf{E}=E\hat{y}$, which can
be represented, in an appropriate gauge, as
$\mathbf{A}(\mathbf{r},t)=\left[-cEt+Bx\right]\hat{y}$. The electric field
dependent term in $H$ can be removed by a suitable transformation to a co-
moving frame with velocity $\mathbf{v}=(E/B)\hat{x}$. Specifically, let
$\mathbf{r}_{j}^{\prime}=\mathbf{r}_{j}-c(E/B)t\hat{x}$ and correspondingly
$\mathbf{p}_{i}^{\prime}=\mathbf{p}_{i}$, then $i\hbar\frac{\partial}{\partial
t}=i\hbar\frac{\partial}{\partial
t^{\prime}}+c\frac{E}{B}\sum_{i}p_{i,x}^{\prime}$ and the Hamiltonian
transforms to:

$H^{\prime}=\sum_{j}K\left[\mathbf{p}_{j}^{\prime}+\frac{e}{c}Bx_{j}^{\prime}\hat{y}\right]+V\left[\left\\{\mathbf{r}_{j}^{\prime}\right\\}\right]-c\frac{E}{B}\sum_{j}p_{j,x}^{\prime},$
(7)

which is time-independent. The current density in the $x$-direction is simply
proportional to the center of mass velocity,
$j_{x}^{\prime}=-ne\langle\dot{X}\rangle$ (where $n$ is the electron density),
which must likewise be time-independent. The total momentum in $y$-direction,
$P_{y}^{\prime}=P_{y}$, is conserved, while for fixed $P_{y}^{\prime}$,
$\langle K^{\prime}\rangle$ would diverge with time as $|X^{\prime}|\to\infty$
for any $\left<\dot{X}^{\prime}\right>\neq 0$. Consequently,
$\left<\dot{X}^{\prime}\right>=0$ and $j_{x}^{\prime}=0$. However, the current
density transforms as
$\mathbf{j}^{\prime}(\mathbf{r}^{\prime})=\mathbf{j}(\mathbf{r})+ec\frac{E}{B}n(\mathbf{r})\hat{x}$,
and hence, $j_{x}=-necE/B$, which gives classical Hall conductivity.

## Acknowledgement

We thank S. Hartnoll and C. Murthy for constructive criticism. This work was
supported, in part, by the U. S. Department of Energy (DOE) Office of Basic
Energy Science, Division of Materials Science and Engineering at Stanford
under contract No. DE-AC02-76SF00515.

## Contributions

Both authors contributed essentially to all aspects of this paper.

## Competing Interests

The authors declare no competing financial or non-financial interests.

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