text:
Prediction Skill of GEFSv12 in Depicting Monthly Rainfall and Associated Extreme Events over Taiwan during the Summer Monsoon
M. M. Nageswararao
Yuejian Zhu
Meng-Shih Chen
The skillful prediction of monthly scale rainfall in small regions like Taiwan is one of the challenges of the meteorological scientific community. Taiwan is one of the subtropical islands in Asia. It experiences rainfall extremes regularly, leading to landslides and flash floods in/near the mountains and flooding over low-lying plains, particularly during the summer monsoon season [June–September (JJAS)]. In September 2020, NOAA/NCEP implemented Global Ensemble Forecast System, version 12 (GEFSv12), to support stakeholders for subseasonal forecasts and hydrological applications. In the present study, the performance evaluation of GEFSv12 for monthly rainfall and associated extreme rainfall (ER) events over Taiwan during JJAS against CMORPH has been done. There is a marginal improvement of GEFSv12 in depicting the East Asian summer monsoon index (EASMI) as compared to GEFS-SubX. The GEFSv12 rainfall raw products have been calibrated with a quantile–quantile (QQ) mapping technique for further prediction skill improvement. The results reveal that the spatial patterns of climatological features (mean, interannual variability, and coefficient of variation) of summer monsoon monthly rainfall over Taiwan from QQ-GEFSv12 are very similar to CMORPH than Raw-GEFSv12. Raw-GEFSv12 has an enormous wet bias and overforecast wet days, while QQ-GEFSv12 is close to reality. The prediction skill (correlation coefficient and index of agreement) of GEFSv12 in depicting the summer monsoon monthly rainfall over Taiwan is significantly high (>0.5) in most parts of Taiwan and particularly more during peak monsoon months, September, and August, followed by June and July. The calibration method significantly reduces the overestimation (underestimation) of wet (ER) events from the ensemble mean and probabilistic ensemble forecasts. The predictability of extreme rainfall events (>50 mm day−1) has also improved significantly.
A Decade after Typhoon Morakot (2009): What Have We Learned about Its Physics and Predictability?
Chung-Chieh Wang
Hung-Chi Kuo
Yu-Han Chen
Shin-Hau Chen
Kazuhisa Tsuboki
Typhoon Morakot struck Taiwan during 7–9 August 2009 and became the deadliest tropical cyclone (TC) in five decades by producing up to 2635 mm of rain in 48 h, breaking the world record. The extreme rainfall of Morakot resulted from the strong interaction among several favorable factors that occurred simultaneously. These factors from large scale to small scale include the following: 1) weak environmental steering flow linked to the evolution of the monsoon gyre and consequently slow TC motion; 2) a strong moisture surge due to low-level southwesterly flow; 3) asymmetric rainfall and latent heating near southern Taiwan to further reduce the TC's forward motion as its center began moving away from Taiwan; 4) enhanced rainfall due to steep topography; 5) atypical structure with a weak inner core, enhancing its susceptibility to the latent heating effect; and 6) cell merger and back building inside the rainbands associated with the interaction between the low-level jet and convective updrafts. From a forecasting standpoint, the present-day convective-permitting or cloud-resolving regional models are capable of short-range predictions of the Morakot event starting from 6 August. At longer ranges beyond 3 days, larger uncertainty exists in the track forecast and an ensemble approach is necessary. Due to the large computational demand at the required high resolution, the time-lagged strategy is shown to be a feasible option to produce useful information on rainfall probabilities of the event.
Role of a Pacific Easterly Wave in the Genesis of Hagupit (2008)
Chen Zhao
Tim Li
Mingyu Bi
The Advanced version of the Weather Research and Forecasting (WRF-ARW) Model is used to investigate the influence of an easterly wave (EW) on the genesis of Typhoon Hagupit (2008) in the western North Pacific. Observational analysis indicates that the precursor disturbance of Typhoon Hagupit (2008) is an easterly wave (EW) in the western North Pacific, which can be detected at least 7 days prior to the typhoon genesis. In the control experiment, the genesis of the typhoon is well captured. A sensitivity experiment is conducted by filtering out the synoptic-scale (3–8-day) signals associated with the EW. The absence of the EW eliminates the typhoon genesis. Two mechanisms are proposed regarding the effect of the EW on the genesis of Hagupit. First, the background cyclonic vorticity of the EW could induce the small-scale cyclonic vorticities to merge and develop into a system-scale vortex. Second, the EW provides a favorable environment in situ for the rapid development of the typhoon disturbance through a positive moisture–convection feedback.
The National Hurricane Center Tropical Cyclone Model Guidance Suite
Mark DeMaria
James L. Franklin
Rachel Zelinsky
David A. Zelinsky
Matthew J. Onderlinde
John A. Knaff
Stephanie N. Stevenson
John Kaplan
Kate D. Musgrave
Galina Chirokova
Charles R. Sampson
The National Hurricane Center (NHC) uses a variety of guidance models for its operational tropical cyclone track, intensity, and wind structure forecasts, and as baselines for the evaluation of forecast skill. A set of the simpler models, collectively known as the NHC guidance suite, is maintained by NHC. The models comprising the guidance suite are briefly described and evaluated, with details provided for those that have not been documented previously. Decay-SHIFOR is a modified version of the Statistical Hurricane Intensity Forecast (SHIFOR) model that includes decay over land; this modification improves the SHIFOR forecasts through about 96 h. T-CLIPER, a climatology and persistence model that predicts track and intensity using a trajectory approach, has error characteristics similar to those of CLIPER and D-SHIFOR but can be run to any forecast length. The Trajectory and Beta model (TAB), another trajectory track model, applies a gridpoint spatial filter to smooth winds from the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS) model. TAB model errors were 10%–15% lower than those of the Beta and Advection model (BAM), the model it replaced in 2017. Optimizing TAB's vertical weights shows that the lower troposphere's environmental flow provides a better match to observed tropical cyclone motion than does the upper troposphere's, and that the optimal steering layer is shallower for higher-latitude and weaker tropical cyclones. The advantages and disadvantages of the D-SHIFOR, T-CLIPER, and TAB models relative to their earlier counterparts are discussed.
This paper provides a comprehensive summary and evaluation of a set of simpler forecast models used as guidance for NHC's operational tropical cyclone forecasts, and as baselines for the evaluation of forecast skill; these include newer techniques that extend forecasts to 7 days and beyond.
An Objective Scoring Method for Evaluating the Comparative Performance of Automated Storm Identification and Tracking Algorithms
Clarice N. Satrio
Kristin M. Calhoun
P. Adrian Campbell
Rebecca Steeves
Travis M. Smith
While storm identification and tracking algorithms are used both operationally and in research, there exists no single standard technique to objectively determine performance of such algorithms. Thus, a comparative skill score is developed herein that consists of four parameters, three of which constitute the quantification of storm attributes—size consistency, linearity of tracks, and mean track duration—and the fourth that correlates performance to an optimal postevent reanalysis. The skill score is a cumulative sum of each of the parameters normalized from zero to one among the compared algorithms, such that a maximum skill score of four can be obtained. The skill score is intended to favor algorithms that are efficient at severe storm detection, i.e., high-scoring algorithms should detect storms that have higher current or future severe threat and minimize detection of weaker, short-lived storms with low severe potential. The skill score is shown to be capable of successfully ranking a large number of algorithms, both between varying settings within the same base algorithm and between distinct base algorithms. Through a comparison with manually created user datasets, high-scoring algorithms are verified to match well with hand analyses, demonstrating appropriate calibration of skill score parameters.
With the growing number of options for storm identification and tracking techniques, it is necessary to devise an objective approach to quantify performance of different techniques. This study introduces a comparative skill score that assesses size consistency, linearity of tracks, mean track duration, and correlation to an optimal postevent reanalysis to rank diverse algorithms. This paper will show the capability of the skill score at highlighting algorithms that are efficient at detecting storms with higher severe potential, as well as those that closely resemble human-perceived storms through a comparison with manually created user datasets. The novel methodology will be useful in improving systems that rely on such algorithms, for both operational and research purposes focusing on severe storm detection.
Quantification of QLCS Tornadogenesis, Associated Characteristics, and Environments across a Large Sample
James S. Goodnight
Devin A. Chehak
Robert J. Trapp
The skillful anticipation of tornadoes produced by quasi-linear convective systems (QLCSs) is a well-known forecasting challenge. This study was motivated by the possibility that warning accuracy of QLCS tornadoes depends on the processes leading to tornadogenesis, namely, one that is dominated by an apparent release of horizontal shearing instability [shearing instability dominant (SID)] and one by a pre-tornadic mesocyclone [pre-tornadic mesocyclone dominant (PMD)] and its associated generative mechanisms. The manual classification of the genesis of 530 QLCS tornadoes as either SID or PMD was performed using heuristic, yet process-driven criteria based on single-Doppler radar (WSR-88D) data. This included 214, 213, and 103 tornadoes that occurred during 2019, 2017, and 2016, respectively. As a function of tornadogenesis process, 36% were classified as SID, and 60% were classified as PMD; the remaining 4% could not be classified. Approximately 30% of the SID cases were operationally warned prior to tornadogenesis, compared to 44% of the PMD cases. PMD tornadoes were also more common during the warm season and displayed a diurnal, midafternoon peak in frequency. Finally, SID cases were more likely to be associated with QLCS tornado outbreaks but tended to be slightly shorter lived. A complementary effort to investigate environmental characteristics of QLCS tornadogenesis revealed differences between SID and PMD cases. MLCAPE was relatively larger for warm-season SID cases, and 0–3-km SRH was relatively larger in warm-season PMD cases. Additionally, pre-tornadic frontogenesis was more prominent for cool-season SID cases, suggestive of a more significant role of the larger-scale meteorological forcing in vertical vorticity that fosters tornadogenesis through SID processes.
Randomized Subensembles: An Approach to Reduce the Risk of Divergence in an Ensemble Kalman Filter Using Cross Validation
Jean-François Caron
Ron McTaggart-Cowan
Mark Buehner
Pieter L. Houtekamer
Ervig Lapalme
In an ensemble Kalman filter, when the analysis update of an ensemble member is computed using error statistics estimated from an ensemble that includes the background of the member being updated, the spread of the resulting ensemble systematically underestimates the uncertainty of the ensemble mean analysis. This problem can largely be avoided by applying cross validation: using an independent subset of ensemble members for updating each member. However, in some circumstances cross validation can lead to the divergence of one or more ensemble members from observations. This can culminate in catastrophic filter divergence in which the analyzed or forecast states become unrealistic in the diverging members. So far, such instabilities have been reported only in the context of highly nonlinear low-dimensional models. The first known manifestation of catastrophic filter divergence caused by the use of cross validation in an NWP context is reported here. To reduce the risk of such filter divergence, a modification to the traditional cross-validation approach is proposed. Instead of always assigning the ensemble members to the same subensembles, the members forming each subensemble are randomly chosen at every analysis step. It is shown that this new approach can prevent filter divergence and also brings a cycling ensemble data assimilation system containing divergent members back to a state consistent with Gaussianity. The randomized subensemble approach was implemented in the operational global ensemble prediction system at Environment and Climate Change Canada on 1 December 2021.
Attribution of North American Subseasonal Precipitation Prediction Skill
Lantao Sun
Martin P. Hoerling
Jadwiga H. Richter
Andrew Hoell
James W. Hurrell
The skill of NOAA's official monthly U.S. precipitation forecasts (issued in the middle of the prior month) has historically been low, having shown modest skill over the southern United States, but little or no skill over large portions of the central United States. The goal of this study is to explain the seasonal and regional variations of the North American subseasonal (weeks 3–6) precipitation skill, specifically the reasons for its successes and its limitations. The performances of multiple recent-generation model reforecasts over 1999–2015 in predicting precipitation are compared to uninitialized simulation skill using the atmospheric component of the forecast systems. This parallel analysis permits attribution of precipitation skill to two distinct sources: one due to slowly evolving ocean surface boundary states and the other to faster time-scale initial atmospheric weather states. A strong regionality and seasonality in precipitation forecast performance is shown to be analogous to skill patterns dictated by boundary forcing constraints alone. The correspondence is found to be especially high for the North American pattern of the maximum monthly skill that is achieved in the reforecast. The boundary forcing of most importance originates from tropical Pacific SST influences, especially those related to El Niño–Southern Oscillation. We discuss physical constraints that may limit monthly precipitation skill and interpret the performance of existing models in the context of plausible upper limits.
Skillful subseasonal precipitation predictions have societal benefits. Over the United States, however, NOAA's official U.S. monthly precipitation forecast skill has been historically low. Here we explore origins for skill of North American week-3 to week-6 precipitation predictions. Skill arising from initial weather states is compared to that arising from ocean surface boundary states alone. The monthly and seasonally varying pattern of U.S. monthly precipitation skill is appreciably derived from boundary constraints, linked especially with El Niño–Southern Oscillation. Forecasts of opportunity are identified, despite the low skill of monthly precipitation forecasts on average. Potential limits of monthly precipitation skill are explored that provide insight on the juxtaposition of "skill deserts" over the central United States with high skill regions over western North America.
Forecast Dropouts in the NAVGEM Model: Characterization with Respect to Other Models, Large-Scale Indices, and Ensemble Forecasts
Justin G. McLay
Elizabeth Satterfield
A forecast "bust" or "dropout" can be defined as an intermittent but significant loss of model forecast performance. Deterministic forecast dropouts are typically defined in terms of the 500-hPa geopotential height (Φ500) anomaly correlation coefficient (ACC) in the Northern Hemisphere (NH) dropping below a predefined threshold. This study first presents a multimodel comparison of dropouts in the Navy Global Environmental Model (NAVGEM) deterministic forecast with the ensemble control members from the Environment and Climate Change Canada (ECCC) Global Ensemble Prediction System (GEPS) and the National Centers for Environmental Prediction (NCEP) Global Ensemble Forecast System (GEFS). Then, the relationship between dropouts and large-scale pattern variability is investigated, focusing on the temporal variability and correlation of flow indices surrounding dropout events. Finally, three severe dropout events are examined from an ensemble perspective. The main findings of this work are the following: 1) forecast dropouts exhibit some relation between models; 2) although forecast dropouts do not have a single cause, the most severe dropouts in NAVGEM can be linked to specific behavior of the large-scale flow indices, that is, they tend to follow periods of rapidly escalating volatility of the flow indices, and they tend to occur during intervals where the AO and Pacific North American (PNA) indices are exhibiting unusually strong interdependence; and 3) for the dropout events examined from an ensemble perspective, the NAVGEM ensemble spread does not provide a strong signal of elevated potential for very large forecast errors.
Short-Term Weather Forecast Skill of Artificial Neural Networks
Christopher C. Hennon
Andrew Coleman
We evaluate the short-term weather forecast performance of three flavors of artificial neural networks (NNs): feed forward back propagation, radial basis function, and generalized regression. To prepare the application of the NNs to an operational setting, we tune NN hyperparameters using over two years of historical data. Five objective guidance products serve as predictors to the NNs: North American Mesoscale and Global Forecast System model output statistics (MOS) forecasts, the High-Resolution Rapid Refresh (HRRR) model, National Weather Service forecasts, and the National Blend of Models product. We independently test NN performance using 96 real-time forecasts of temperature, wind, and precipitation across 11 U.S. cities made during the WxChallenge, a weather forecasting competition. We demonstrate that all NNs significantly improve short-range weather forecasts relative to the traditional objective guidance aids used to train the networks. For example, 1-day maximum and minimum temperature forecast error is 20%–30% lower than MOS. However, NN improvement over multiple linear regression for short-term forecasts is not significant. We suggest this may be attributed to the small number of training samples, the operational nature of the experiment, and the short forecast lead times. Regardless, our results are consistent with previous work suggesting that applying NNs to model forecasts can have a positive impact on operational forecast skill and will become valuable tools when integrated into the forecast enterprise.
We used approximately two years of historical weather data and objective forecasts for a number of cities to tune a series of artificial neural networks (NNs) to forecast 1-day values of maximum and minimum temperature, maximum sustained wind speed, and quantitative precipitation. We compare forecast error against common objective guidance and multiple linear regression. We found that the NNs exhibit about 25% lower error than common objective guidance for temperature forecasting and 50% lower error for wind speed. Our results suggest that NNs will be a valuable contributor to improving weather forecast skill when adopted into the existing forecast enterprise.


text:
NewsWorldAmericasUS politics
Don Jr is selling $500 video messages to fans after complaining about 'millions' in legal bills in from New York AG probe
'Thanks for helping us out and support us in going after liberals and crazies on CNN' Don Jr tells supporter in Australia in Cameo video
Gustaf Kilander
Wednesday 02 June 2021 21:02
Related video: Trump Investigation Evidence Will Be Heard by New Grand Jury
Donald Trump Jr is selling $500 video messages to fans on Cameo after complaining about "millions" in legal bills from the probe into the Trump Organization being carried out by the New York attorney general.
Cameo is a platform that allows celebrities to sell short, personalized video messages to fans and supporters.
"Don't worry about it if your wife's mad at you for saying that election night 2016 was the happiest night of your life... there's millions of people just like you, you can tell her I said that," Mr Trump says in a video to a supporter in Australia. "Thanks for helping us out and support us in going after the liberals and the crazies on CNN."
A video from Mr Trump delivered in two to seven days costs $525, a video sent to a supporter in less than 24 hours will set them back $787.
In his profile on Cameo, Mr Trump describes himself as a "Father, Patriot, Outdoorsman, Businessman, Political Commentator and #1 NYT Bestselling author".
National Enquirer owner fined for illegal Trump campaign aid
'I've had kidney stones that lasted longer': Trump mocked after he closes his blog after one month
Trump official suggested to Fauci that they use 'doggie cones' to solve coronavirus PPE crisis
"A portion of proceeds will be donated to Shadow Warriors Project," the profile then states.
The project supports private US military contractors and is led by Mark Geist. He was a private security contractor in Benghazi during the terror attack on the US embassy in 2012.
Donald Trump Jr is the executive vice president of the Trump Organization, which is currently under investigation by New York Attorney General Letitia James. Mr Trump went after Ms James during an appearance on Tucker Carlson Tonight on Fox News.
He said that she's "searching for a crime" and persecuting her political enemies.
"I think it's political persecution and I know that because she literally campaigned on it. She was going to investigate the crimes. The problem is it wasn't as though she was a part of this office. She had no idea but, in New York, it's okay to try to persecute your political enemies, to try to target them, to try to hurt them and they've been doing that for over five years," Mr Trump told Mr Carlson.
"So five years, no actual crime, they are in search of a crime," he added. "They've been trying to investigate to create a crime because that's what you're doing in New York… If this was happening in a banana republic, we'd be outraged. When it's going on in one of the largest states in America everyone seems to be silent on it."
"In New York, crime is going through the roof, taxes are going to go through the roof, property values are going down, murder rates are skyrocketing and, yet, rather than using taxpayer dollars to actually go and solve those issues to prevent the degeneration of New York even further, they're going to continue to spend those taxpayer dollars targeting their enemies and that's pretty disgusting," Mr Trump said.
"Even if they don't get anything, they get you to spend millions of dollars and that's sort of a win in of itself," he added.
Twitter users mocked Mr Trump for joining Cameo, with one tweeting: "This criminal enterprise is still raking in $$ from their loyal Id*ot following. What a P*G."
"I smell grift!" another added.
"Jr must be desperate for money!" A third said.
One account holder asked: "Why isn't all the money going to this cause? Is it because it's merely a front for his thieving self-interests."
"LOL better to be a crazy watching CNN than a [hypocrite] and idiot watching fox fake news. Who in their right mind would spend money to watch this fool??????? Oh yeah, Donnie Sr would," another said.
The Independent has reached out to the Trump Organization for comment.
New YorkDonald Trump JrcameoLetitia James
1/1Don Jr is selling $500 videos to fans after moaning about legal bills
Don Jr is selling $500 videos to fans after moaning about legal bills
Donald Trump Jr, Kimberly Guilfoyle, and Rep Doug Collins during a Trump Campaign press conference in Atlanta, Georgia, USA, on 5 November 2020.

redpajama_set_name: RedPajamaCommonCrawl


text:
African Lit, Book Review
Review: Things Fall Apart
December 1, 2017 December 9, 2018 by Mini
Title: Things Fall Apart
Author: Chinua Achebe
Rating: 3/5 ★★★☆☆
Set against the backdrop of colonial times, this book is written with a political agenda of showcasing that impact of modernism on the lives of Nigerian communities. This story permeated within me a sense of empathy towards the lost and forgotten cultures who fell victim to the impending reformation. I dealt with a sense of epiphany throughout the book making me fearsome of the realities faced by so many people who voice had been dampened by the foreign intrusion.
This is a modern African literature presenting the ideological contestation of the traditional and modern ideas within the time frame of pre and post-colonial life in Nigeria. The style of writing is a conscious blend of his native language and English. Achebe has received criticism for choosing English as a medium to voice the adversities of people instead of their native language. However, in anticipation of an expansive reader base, he has written the book in English but has kept some of the local words as it is making sure that his native language doesn't fall victim of globalization. The major portion of the story dealt with a constant tussle between the old and the new, modernism and traditionalism. In the beginning, the novel had shed light on the Nigerian ways of doing things with storytelling being a prominent part of their culture. The invasion of white missionaries and colonial government not only disintegrated the Igbo society but also imposed English language attacking the customs of a collapsible culture. This had a deeper impact on the lives of native communities as they had to let go of their values and culture system which led to a systematic destruction of individual identities.
The protagonist Okonkwo was a strong and cruel man who earned fame in his early years as an accomplished wrestler as opposed to his drunkard and incompetent father. Okonkwo believed in the power of his dominant masculinity and considered emotions a sign of weakness. He was determined to become wealthy and had three wives to command. The story is a struggle of his resistance towards missionary ideologies and the ambiguities of cultural clash consequential of power politics. Did Okonkwo really fight for protecting his traditions or it was the inability to accept the flaws in his culture? Was he justified?
For me, Okonkwo remained a ruthless and remorseless person despite being portrayed as a tragic hero by many. I was cringing away from the violence and killings. Often, I found myself quickly turning the pages to avoid reading the Okonkwo's beating his women and children and killing one which is why this remains a 3 star for me. I wouldn't recommend it as a must read but this book is a good reminder of the blessings and privileges we tend to ignore sometimes.
African Lit blog Book Review Bookish Daily books related november reviews
Review: Chronicle of a Death Foretold
Review: Murder on the Orient Express
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text:
\section{Introduction\label{sec:Intro}}

Multipolar interactions and related ordering phenomena have attracted
great interest because, unlike commonly-known long-range orders of
electric or magnetic dipole moments, multipoles are often related to
more exotic phases which are difficult to probe directly by
conventional methods~\cite{Kuramoto2009,Santini2009}.
For systematic investigation of the collective behavior of multipole
moments, $f$-electron systems are suitable choices since the strong
coupling between spin and orbital degrees of freedom of $f$-electrons
facilitates multipole formation.
Indeed, the actinide dioxides with $5f$-electrons exhibit a variety
of multipolar ordering phenomena~\cite{Santini2009}.
For lanthanides with $4f$-electrons, multipolar, and especially
quadrupolar orders, have been discovered for different
systems~\cite{Tm2000,Dy2000,Np2002,Pr2006,Cameron2016}.

YbRu$_2$Ge$_2$, a heavy-fermion metal with tetragonal structure
(space group I4/mmm, No. 139; point group D$_{4h}$), has been
suggested to hold a ferroquadrupolar (FQ) order at low
temperature~\cite{Jeevan2006,Takimoto2008,Thalmeier2008,Nicklas2010,Prasad2010,Jeevan2011,Fisher2018,ZXShen2019}.
It undergoes a second-order phase transition at T$_Q$=10\,K, before
entering into an antiferromagnetic (AFM) phase below T$_{N1}$=6.5\,K~\cite{Jeevan2006,Jeevan2011}.
At T$_{N2}$=5.5\,K, there may exist a small change in the magnetic
structure~\cite{Jeevan2011,Nicklas2010}. Early studies show that the
transition at T$_Q$ is not magnetic, and T$_Q$ increases when
magnetic field is applied along the easy
direction~\cite{Jeevan2006,Jeevan2011}. This behavior at T$_Q$ is
similar to that of TmAu$_2$ at its FQ ordering
temperature~\cite{Kosaka1998}, hence suggesting a FQ phase in
YbRu$_2$Ge$_2$. The existence of a FQ order is further supported by
recent elastoresistivity studies where above T$_Q$ the
elastoresistivity in the quadrupolar symmetry channels displays a
Curie-Weiss behavior~\cite{Fisher2018}. Below T$_Q$, an orthorhombic
structural distortion is observed by X-ray diffraction, which
confirms that the FQ state breaks B$_{1g}$ ($x^{2}-y^{2}$)
symmetry~\cite{Fisher2018}.

The FQ order, namely the ordering of Yb$^{3+}$ 4f-electron charge
distribution at zero wavevector, can be probed indirectly by studying
the lattice dynamics and crystal-field (CF) excitations. In a FQ
arrangement, aligned charge quadrupoles uniformly distort the lattice
via a coupling between the quadrupole moment and the strain field
with the same symmetry. The induced distortion reduces the
point-group symmetry of the lattice system, splitting degenerate
phonon modes; the distortion also modifies the energy and lifetime of
the phonon modes of the same symmetry. Such anomalies can be revealed
by investigating the phonon spectra. Besides, the quadrupolar moments
are carried by the CF ground state of Yb$^{3+}$. The tetragonal CF
potential splits the $^2F_{7/2}$ ground multiplet into two $\Gamma_6$
and two $\Gamma_7$ Kramers doublets.
The magnetic entropy right above T$_Q$ is nearly
R\,ln\,4~\cite{Jeevan2006}, suggesting that the CF ground state is a
quasi-quartet consisting of two quasi-degenerate Kramers doublets.
The quasi-quartet ground state was recently confirmed by
angle-resolved photo-emission spectroscopy
studies~\cite{ZXShen2019}. This quasi-quartet near degeneracy is
essential for forming a quadrupolar ground state and deserves a
detailed study.

Raman spectroscopy is a conventional tool for studying phonon modes~\cite{Cardona1982} and CF excitations~\cite{Cardona2000}. Here we study the lattice dynamics, low-energy quadrupole fluctuations, and CF excitations in YbRu$_2$Ge$_2$. We assign four Raman-active phonon modes, and reveal an anomalous intensity enhancement of two phonon modes on cooling. The three CF transitions within the $^2F_{7/2}$ ground multiplet are identified and a CF level scheme is in turn established. We demonstrate that low-energy Raman response undergoes remarkable enhancement on cooling towards T$_Q$ and that the static electronic Raman susceptibility in the corresponding quadrupole channels follows nearly perfect Curie behavior, signifying that the relatively strong coupling to the lattice in the B$_{1g}$-symmetry channel enhances by about 10\,K the vanishingly small electronic Weiss temperature to the FQ transition temperature T$_Q$.

\section{Experimental\label{sec:Exp}}

Single crystals of YbRu$_2$Ge$_2$ were grown by flux method; details of the growth can be found in Ref~\cite{Fisher2018}. Two samples were used in this study: one was cleaved in ambient condition to expose its xy crystallographic plane, the other had a clean as-grown xz crystallographic plane. The xy crystallographic plane was examined under a Nomarski microscope to find about 200$\times$200\,$\mu$m$^{2}$ strain-free area.

Raman scattering measurements were performed in a quasi-back scattering geometry from sample placed in a continuous helium-gas-flow cryostat. We used 476.2, 647.1 and 752.5\,nm lines from a Kr$^+$ ion laser for excitation. Incident light with no more than 14\,mW power was focused to a 50$\times$100\,$\mu$m$^{2}$ spot. Particularly, for measurements below 10\,K, the power of the incident light was reduced to 2\,mW. The temperatures reported in this paper were corrected for laser heating, which was estimated to be \,0.75\,$\pm$\,0.25\,K/mW~\footnote{Optical absorption coefficient and thermal conductivity data for YbRu$_2$Ge$_2$, which are required for a model estimation of the laser heating~\cite{Maksimov1992}, are currently unavailable. Hence, we base the estimate of laser heating at low temperatures on comparison with other heavy-fermion metals with comparable electrical conductivity~\cite{Fisher2018}: CeB$_{6}$~\cite{Takase1980} and URu$_{2}$Si$_{2}$~\cite{Palstra1986}, for which laser heating for the same experimental set-up was established in the range of 0.5$\sim$1.0\,K/mW.}.

Seven polarization configurations were employed to probe excitations in different symmetry channels. The relationship between the scattering geometries and the symmetry channels~\cite{Hayes2004} is given in Table~\ref{tab:Exp1}.

\begin{table}[t]
\caption{\label{tab:Exp1}The relationship between the scattering geometries and the symmetry channels. For scattering geometry E$_{i}$E$_{s}$, E$_{i}$ and E$_{s}$ are the polarizations of incident and scattered light; X, Y, X', Y' and Z are the [100], [010], [110], [1$\overline{1}$0] and [001] crystallographic directions; R and L are right and left circular polarizations. A$_{1g}$, A$_{2g}$, B$_{1g}$, B$_{2g}$ and E$_{g}$ are the irreducible representations of the D$_{4h}$ group.}
\begin{ruledtabular}
\begin{tabular}{ll}
Scattering Geometry&Symmetry Channel\\
\hline
XX&A$_{1g}$+B$_{1g}$\\
XY&A$_{2g}$+B$_{2g}$\\
X'X'&A$_{1g}$+B$_{2g}$\\
X'Y'&A$_{2g}$+B$_{1g}$\\
XZ&E$_{g}$\\
RR&A$_{1g}$+A$_{2g}$\\
RL&B$_{1g}$+B$_{2g}$\\
\end{tabular}
\end{ruledtabular}
\end{table}

We used a custom triple-grating spectrometer with a
liquid-nitrogen-cooled charge-coupled device (CCD) detector for
analysis and collection of the scattered light. The data were corrected for the spectral response of the system. The measured secondary-emission intensity $I(\omega,T)$ is related to the Raman response $\chi''(\omega,T)$ by $I(\omega,T)=[1+n(\omega,T)]\chi''(\omega,T)+L(\omega,T)$, where $n$ is the Bose factor, $\omega$ is energy, $T$ is temperature. $L(\omega,T)$ represents the far tail of photo-luminescence, which in the narrow spectral window of interest was approximated by a linear frequency dependence.

\section{Results and Discussion\label{sec:Res}}

\subsection{Lattice Dynamics\label{subsec:LD}}

The spectra of phonon modes are presented in Fig.~\ref{fig:P}. By
group theory, four Raman-active optical phonon modes are expected for
YbRu$_2$Ge$_2$ structure: $A_{1g}\oplus B_{1g}\oplus 2E_{g}$.
A$_{1g}$ and B$_{1g}$ modes are accessible in XX geometry and E$_{g}$ modes in XZ geometry.
The phonon energies at 13\,K are tabulated in Table~\ref{table:ModeSummary}.

\begin{figure}
\includegraphics[width=0.48\textwidth]{Fig1.pdf}
\caption{\label{fig:P}Raman response $\chi''(\omega,T)$ of four Raman-active optical phonon modes at low temperature measured with the 647.1\,nm excitation. The XX and XZ spectra are offset by 1.5 and 3 arbitrary units (arb. units), respectively. The spectral resolution is 1.0\,cm$^{-1}$.}
\end{figure}

\begin{table}[b]
\caption{\label{table:ModeSummary}Summary of the energy of the phonon
and crystal-field (CF) modes. The coupled CF and phonon modes are labeled by ``(c)". Results of this work are compared to inelastic neutron scattering (INS) study~\cite{Jeevan2010}. Units are cm$^{-1}$.}
\begin{ruledtabular}
\begin{tabular}{lll}
Mode&This work&INS\\
\hline
$\Gamma_6^{(1)}\rightarrow\Gamma_7^{(1)}$&2&--\\
$\Gamma_6^{(1)}\rightarrow\Gamma_7^{(2)}$&95&89\\
$\Gamma_6^{(1)}\rightarrow\Gamma_6^{(2)}$ (c)&239&--\\
\hline
A$_{1g}$ (c)&268&260\\
B$_{1g}$&168&170\\
E$_{g}^{(1)}$&138&--\\
E$_{g}^{(2)}$ (c)&301&--\\
\end{tabular}
\end{ruledtabular}
\end{table}

In Fig.~\ref{fig:P_Para} we show the temperature dependence of the spectral parameters (energy, FWHM, and integrated intensity) of the phonon modes. The spectral parameters were obtained by fitting the measured spectral peaks with Lorentzian lineshapes.

\begin{figure}
\includegraphics[width=0.44\textwidth]{Fig2.pdf}
\caption{\label{fig:P_Para}
Temperature dependence of (a) the energy, (b) the FWHM and (c) the
integrated intensity of the Raman-active optical phonon modes. The
energy and integrated intensity are normalized to their respective
value at 304\,K. The integrated intensity of the very weak
E$_{g}^{(1)}$ phonon mode is not shown. The solid lines in (a) and
(b) represent the fits to anharmonic decay
model~\cite{Klemens1966,Cardona1984}, while the solid lines in (c)
represent the fits to Eq.~(\ref{eq:Int}). Insets: zoom-in of the
low-temperature data points showing how the physical properties
change across the phase-transition temperature T$_Q$=10\,K. The
dashed line in the insets indicate T$_Q$. The vertical error bars are derived from Lorentzian fits and represent one standard deviation; the horizontal error bars are derived from the uncertainty of laser heating estimation.}
\end{figure}

The temperature dependence of both frequency and FWHM of the phonon
modes [Fig.~\ref{fig:P_Para}(a) and (b)] is in accordance with a
simple model assuming anharmonic decay into two phonons with
identical frequencies and opposite
momenta~\cite{Klemens1966,Cardona1984}:
\begin{equation}
\omega(T)=\omega_0-\omega_2[1+\frac{2}{e^{\hbar\omega_0/2k_BT}-1}],
\label{eq:energyTwo}
\end{equation}
and
\begin{equation}
\Gamma(T)=\Gamma_0+\Gamma_2[1+\frac{2}{e^{\hbar\omega_0/2k_BT}-1}].
\label{eq:gammaTwo}
\end{equation}

The fitting results are summarized in Table~\ref{tab:P}. Small
deviations between the measured energy and the fitting curve for the
B$_{1g}$ mode could be due to an additional decay channels, for example decay into one acoustic and one optical mode.

\begin{table}
\caption{\label{tab:P}The fitting parameters for the energy and FWHM of the four Raman-active optical phonon modes.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
Mode&$\omega_0$&$\omega_2$&$\Gamma_0$&$\Gamma_2$\\
\hline
E$_{g}^{(1)}$&138.4$\pm$0.1&0.70$\pm$0.03&1.7$\pm$0.5&0.3$\pm$0.1\\
B$_{1g}$&167.92$\pm$0.01&0.212$\pm$0.002&6.08$\pm$0.02&0.094$\pm$0.005\\
A$_{1g}$&270.15$\pm$0.04&2.27$\pm$0.02&3.6$\pm$0.1&1.71$\pm$0.07\\
E$_{g}^{(2)}$&303.32$\pm$0.02&2.67$\pm$0.01&2.98$\pm$0.05&0.75$\pm$0.03\\
\end{tabular}
\end{ruledtabular}
\end{table}

The integrated intensity of the A$_{1g}$ and E$_{g}^{(2)}$ phonon modes has more than 50\% increase on cooling, in contrast to the behavior of the B$_{1g}$ phonon mode, whose integrated intensity is nearly temperature-independent [Fig.~\ref{fig:P_Para}(c)]. The increase of the integrated intensity on cooling suggests a coupling of a CF transition to these two phonon modes~\cite{Cooper1987}. This coupling is enhanced when the energies of the CF splitting and the phonon modes are close. Indeed, such a CF excitation, 239\,cm$^{-1}$ at 13\,K, exists. The mechanism of this coupling will be discussed in Subsection.~\ref{subsubsec:Coupling}.

Because the FQ order parameter is of B$_{1g}$ symmetry~\cite{Fisher2018}, the energy and lifetime of the B$_{1g}$ phonon mode are expected to exhibit anomalies across T$_Q$ due to electron-phonon coupling. Moreover, breaking of the four-fold rotational symmetry should split the two E$_{g}$ phonon modes~\cite{Zhang2016}. However, as shown in the insets of Fig.~\ref{fig:P_Para}, B$_{1g}$ and E$_{g}$ phonon modes do not exhibit significant anomaly across T$_Q$. E$_{g}$ phonon modes do not show notable splitting at 4.5\,K, either [Fig.~\ref{fig:P}]. The splitting of the E$_{g}^{(1)}$ phonon mode is challenging to observe due to its weak intensity. Because the FWHM of the E$_{g}^{(2)}$ phonon mode is 4\,cm$^{-1}$ at 4.5\,K, we set the upper limit of the splitting of the E$_{g}$ phonon modes to be about 4\,cm$^{-1}$ at 4.5\,K.

\subsection{Quadrupolar fluctuations\label{subsec:Q}}

In the tetragonal phase above $T_{Q}$, the four-fold rotational symmetry along the z-axis is preserved and the CF ground state supports no static xy-plane quadrupole moment.
However, dynamical quadrupolar fluctuations with zero time average quadrupolar moment are allowed~\footnote{Detailed discussion of the symmetry
breaking and the wavefunction mixing can be found in
Subsection.\ref{subsec:CF}}.

In Fig.~\ref{fig:CF1} we show the spectra of low-energy quadrupolar fluctuations.
They are present in RL geometry yet absent in RR geometry [Fig.~\ref{fig:CF1}(a)].
By group theory, the absence of A$_{1g}$ and A$_{2g}$ components indicates that the CF ground state is a quasi-quartet composed of one $\Gamma_6$ and one $\Gamma_7$ doublets.

\begin{figure}
\includegraphics[width=0.44\textwidth]{Fig3.pdf}
\caption{\label{fig:CF1}
Raman response $\chi''(\omega,T)$ for (a) RR and RL scattering geometries with the 647.1\,nm excitation, (b) X'Y' geometry with the 752.5\,nm excitation, and (c) XY geometry with the 752.5\,nm excitation. The solid lines are Drude fits, Eq.~(\ref{eq:Drude}). The error bars represent one standard deviation.}
\end{figure}

The static Raman susceptibility $\chi_{\mu}(0,T)$ in the symmetry channel $\mu$ ($\mu$\,=\,B$_{1g}$ or B$_{2g}$) can be obtained from the Raman response $\chi_{\mu}^{\prime\prime}(\omega,T)$ by virtue of the Kramers-Kronig relation~\cite{Thorsmolle2016,Gallais2016}:
\begin{equation}
\chi_{\mu}(0,T)=\frac{2}{\pi}\int_{0}^{\omega_{max}} \frac{\chi_{\mu}^{\prime\prime}(\omega,T)}{\omega} d\omega~,
\label{eq:KK}
\end{equation}
in which we choose the upper cutoff for the spectra of fluctuations at $\omega_{max}$\,=\,40\,cm$^{-1}$, see Fig.~\ref{fig:CF1}.

We use Drude lineshape
\begin{equation}
\chi_{\mu}^{\prime\prime}(\omega,T)\propto\frac{Q_{\mu}^2\omega}{\omega^2+\gamma_{\mu}^2}
\label{eq:Drude}
\end{equation}
to extrapolate the Raman response below the instrumental cutoff
5\,cm$^{-1}$.
In Eq.~(\ref{eq:Drude}), $Q_{\mu}$ is the magnitude of
the quadrupolar moment, and $\gamma_{\mu}$ reflects the decay rate.
In the Raman scattering process light couples to the system's charge
quadrupole moment.

\begin{figure}[b]
\includegraphics[width=0.48\textwidth]{Fig4.pdf}
\caption{\label{fig:CF1_Para}Temperature dependence of the static electronic Raman susceptibility $\chi(0,T)$ for B$_{1g}$ (red) and B$_{2g}$ (blue) quadrupole channels derived from Raman response shown in Fig.~\ref{fig:CF1}. Inset: temperature dependence of $1/\chi(0,T)$. The solid lines are Curie-Weiss fits, Eq.~(\ref{eq:SRS}). The vertical error bars represent one standard deviation; the horizontal ones are derived from the uncertainty of laser heating estimation.}
\end{figure}

Theoretically, the low-energy Raman response in the quadrupolar channels contains both the lattice and the electronic contributions~\cite{Thorsmolle2016,Gallais2016}. However, the energy of lattice fluctuations is much lower than the instrumental cutoff (5\,cm$^{-1}$), and Eq.~(\ref{eq:Drude}) only takes into account of the electronic contribution. Thus, only electronic quadrupole fluctuations are included in the derived susceptibility $\chi_{\mu}(0,T)$.

The obtained temperature dependence of the static electronic Raman susceptibilities for both B$_{1g}$ and B$_{2g}$ quadrupole channels are shown in Fig.~\ref{fig:CF1_Para}. The static Raman susceptibility $\chi_{\mu}(0,T)$ obeys Curie-Weiss temperature dependence
\begin{equation}
\chi_{\mu}(0,T)\propto\frac{Q_{\mu}^2}{T-T^{\mu}_{W}}~,
\label{eq:SRS}
\end{equation}
where $T^{\mu}_{W}$ is the Weiss temperature:
\begin{equation}
T^{\mu}_{W}=\kappa_{\mu}Q_{\mu}^2~,
\label{eq:Tw}
\end{equation}
in which $\kappa_{\mu}$ measures the strength of the electronic intersite quadrupolar interactions.

By fitting the data, the ratio of $Q_{B_{1g}}$ to $Q_{B_{2g}}$ is
determined to be nearly 1.4. The derived Weiss temperatures,
T$^{B_{1g}}_{W}$\,$\sim$\,$-2$ and
T$^{B_{2g}}_{W}$\,$\sim$\,$+2$\,K
\footnote{We note that in
the elastoresistivity studies, the Weiss temperature obtained from a
fit to the static quadrupole-strain susceptibility is also a measure
of an extrapolated bare electronic quadrupolar transition temperature
because of clamped lattice~\cite{Fisher2018}. This is the reason why
the Weiss temperatures extrapolated from the Raman susceptibility and
from quadrupole-strain susceptibility agree.}. The nearly exact Curie
law is not surprising because both direct-exchange and super-exchange
between local quadrupolar moments are expected to be vanishingly weak
due to compactness of the $f$-orbitals. Although itinerant electrons
provide effective coupling between the local magnetic dipole moments
at Yb$^{3+}$ sites, and the resulting RKKY interaction~\footnote{See
Ref.~\cite{RKKY} and the references therein.} leads to AFM order
below T$_{N1}$=6.5\,K, these itinerant electrons do not provide a
significant effective coupling between the local electric quadrupole moments at
Yb$^{3+}$ sites.

The true B$_{1g}$-symmetry FQ order develops at
T$_{Q}$\,=\,10\,K~\cite{Fisher2018}, about 10\,K above the Weiss
temperature T$^{B_{1g}}_{W}$. Because YRu$_2$Ge$_2$, the non-magnetic
analog of the same structure, has no orthorhombic
transition~\cite{Bouvier1996,Fisher2018}, the quadrupolar
fluctuations of YbRu$_2$Ge$_2$ lattice themselves should have
little tendency towards a structural instability.
Nevertheless, coupling between the lattice strain fields and the local electronic quadrupole moments of the same symmetry enhances the transition temperature~\cite{Bohmer2015,Bohmer2016,Thorsmolle2016,Gallais2016}:
\begin{equation}
T^{\mu}_{Q} = T^{\mu}_{W}+(\lambda_{\mu}^2/C_{\mu})Q_{\mu}^2 = (\kappa_{\mu}+\lambda_{\mu}^2/C_{\mu})Q_{\mu}^2~,
\label{eq:Diff}
\end{equation}
where $\lambda_{\mu}$ measures the coupling between the local
charge quadrupole moments on Yb$^{3+}$ sites and the lattice strain
fields, and $C_{\mu}$ is the symmetrized elastic constant, which is
(C$_{11}$-C$_{12}$)/2 for the B$_{1g}$ channel or C$_{66}$ for the
B$_{2g}$ channel~\cite{Mitsumoto2013}. The true quadrupolar
transition temperature T$_Q$ equals to the largest of two T$^{\mu}_{Q}$.
Because the FQ order in YbRu$_2$Ge$_2$ has B$_{1g}$ symmetry, the
T$^{B_{1g}}_{Q}$\,=\,T$_Q$ and non-realized T$^{B_{2g}}_{Q}$\,\textless\,T$_Q$.

Tuning an additional parameter (magnetic field, pressure or doping, for instance) may affect the electron-lattice coupling and induce a transition from B$_{1g}$ FQ ordering to B$_{2g}$ FQ ordering. Indeed, although T$_Q$ stays constant up to application 9\,GPa pressure with zero magnetic field~\cite{Nicklas2010} and increases with in-plane magnetic field at ambient pressure~\cite{Jeevan2006}, experimental results do show suppression of T$_Q$ by Si~\cite{Prasad2010} or Rh~\cite{ZXShen2019} doping, and by applying magnetic field under 1.23\,GPa pressure~\cite{Nicklas2010}. These results suggest a competition between B$_{1g}$- and B$_{2g}$-symmetry FQ order.

\subsection{Crystal-Field Excitations\label{subsec:CF}}

Within the $^2F_{7/2}$ multiplet, there are three CF excitations
corresponding to transitions from the CF ground state to the three CF
excited states.
From group theoretical considerations~\cite{Koster1963}, the CF transitions between levels of the same symmetry (i.e. $\Gamma_6\rightarrow\Gamma_6$ or $\Gamma_7\rightarrow\Gamma_7$) contain A$_{1g}$, A$_{2g}$ and E$_{g}$ representations, whereas those between levels of different symmetry (i.e. $\Gamma_6\rightarrow\Gamma_7$ or $\Gamma_7\rightarrow\Gamma_6$) contain B$_{1g}$, B$_{2g}$ and E$_{g}$ symmetry representations. The Raman intensities in different symmetry channels may vary due to matrix element effect.

The lowest-energy CF transition, namely the transition between the two quasi-degenerate Kramers doublet does not clearly exhibit itself in the low-energy Raman spectra [Fig.~\ref{fig:CF1}].
The CF excitations from the ground state to the remaining two higher
energy states are shown in Fig.~\ref{fig:CF23} at 95\,cm$^{-1}$ and
239\,cm$^{-1}$.
These two transitions are expected to appear in all Raman-active symmetry channels,
because the two low-lying doublets within the quasi-quartet have
roughly the same population at 11\,K.
With the 476.2\,nm excitation, the 95\,cm$^{-1}$ transition indeed appears as a weak peak for four linear polarizations, while the 239\,cm$^{-1}$ transition overlaps with the strong A$_{1g}$ phonon mode. With the 647.1\,nm excitation, instead, the 95\,cm$^{-1}$ transition becomes weaker, but the 239\,cm$^{-1}$ transition is identifiable, manifesting itself as a peak in the RL spectrum and a shoulder in the RR spectrum.

\begin{figure}
\includegraphics[width=0.44\textwidth]{Fig5.pdf}
\caption{\label{fig:CF23}Raman response $\chi''(\omega)$ of the CF excitations from the quasi-quartet to the remaining two CF levels at low temperature. The dashed lines indicate the position of the two CF transitions. (a) The spectra for four linear polarizations measured at 11\,K with the 476.2\,nm excitation. The XX and X'X' spectra are offset by 0.5 arbitrary units (arb. units). The spectral resolution is 3.5\,cm$^{-1}$. (b) The spectra for two circular polarizations measured at 15\,K with the 647.1\,nm excitation. The RR spectrum is offset by 0.5 arbitrary unit. The spectral resolution is 2.5\,cm$^{-1}$.}
\vspace{-2mm}
\end{figure}

The symmetry of the CF levels are assigned by the following argument: because YbRh$_2$Si$_2$ and YbIr$_2$Si$_2$, which have the same lattice structure as YbRu$_2$Ge$_2$, both have a $\Gamma_6$ CF ground state~\cite{Kutuzov2011,Leushin2008}, we suggest that the CF ground state of YbRu$_2$Ge$_2$ is also of $\Gamma_6$ symmetry (denoted as $\Gamma_6^{(1)}$). The other Kramers doublet within the quasi-quartet, in turn, is of $\Gamma_7$ symmetry (denoted as $\Gamma_7^{(1)}$).

The small difference of the excitation energy measured in RR and RL
geometries near 239\,cm$^{-1}$ [Fig.~\ref{fig:CF23}(b)] serves as an
estimation of the quasi-quartet splitting. Using the Lorentzian fits,
we find that the excitation energy measured in RR geometry is higher
by 2\,$\pm$\,1\,cm$^{-1}$ than that in RL geometry at 25\,K and
15\,K. Therefore, the symmetry of the CF state at 239\,cm$^{-1}$ is
defined to be $\Gamma_6$ (denoted as $\Gamma_6^{(2)}$), and the quasi-quartet splitting is estimated to be 2\,$\pm$\,1\,cm$^{-1}$. Because there are only two $\Gamma_6$ and two $\Gamma_7$ states within the $^2F_{7/2}$ multiplet, the CF state at 95\,cm$^{-1}$ can only be of $\Gamma_7$ symmetry (denoted as $\Gamma_7^{(2)}$).

The energies of the CF excitations at 15\,K are summarized in
Table~\ref{table:ModeSummary}
\footnote{We note that would the CF ground state be $\Gamma_7$
symmetry, the symmetry of the CF states at 2\,cm$^{-1}$,
95\,cm$^{-1}$ and 239\,cm$^{-1}$ should instead be $\Gamma_6$,
$\Gamma_6$ and $\Gamma_7$, respectively.}.

In an inelastic neutron scattering study of YbRu$_2$Ge$_2$, excitations at 89\,cm$^{-1}$, 170\,cm$^{-1}$ and 260\,cm$^{-1}$ are resolved at 5\,K with the magnitude of momentum transfer being $\sim$1.9\AA$^{-1}$ (Ref.~\cite{Jeevan2010}).
Their data well match our assignments; the comparison is shown in Table~\ref{table:ModeSummary}.
This consistency not only supports our assignments, but also suggests that the CF excitations and optical phonon modes have little dispersion.

\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{Fig6.pdf}
\caption{\label{fig:EP} Schematic energy diagram of the CF states (red horizontal lines) and the phonon modes (blue horizontal lines). The coupled CF transition and phonon modes are circled in purple. On the left are the angular electron-cloud distributions of the four CF states; on the right are the vibration patterns of the phonon modes.}
\end{figure}

\subsubsection{Angular Electron-Cloud Distribution of the Crystal-Field States\label{subsubsec:Model}}

We use the following single-ion Hamiltonian to calculate the angular electron-cloud distribution at Yb sites:
\begin{equation}
H=H_{CF}+H_{B_{1g}}~.
\label{eq:H}
\end{equation}

The first term in Eq.~(\ref{eq:H})
\begin{equation}
H_{CF}=B_2^0\hat{O}_2^0+B_4^0\hat{O}_4^0+B_4^4\hat{O}_4^4+B_6^0\hat{O}_6^0+B_6^4\hat{O}_6^4
\label{eq:HCF}
\end{equation}
is the general expression for a CF potential of tetragonal site symmetry~\cite{Fischer1987}. The $\hat{O}^0_2$, $\hat{O}^0_4$, $\hat{O}^4_4$, $\hat{O}^0_6$, and $\hat{O}^4_6$ operators are Stevens operators~\cite{Stevens1952}. The five $B$'s are the CF coefficients.

From the CF level diagram, we cannot uniquely determine the CF
Hamiltonian and wavefunctions if we assume tetragonal site symmetry.
Hence, we approximate the real tetragonal CF potential with a dominating cubic CF potential~\cite{Lea1962} plus a small $\hat{O}^0_2$ axial term:
\begin{equation}
H_{Tetra}=B_2^0\hat{O}_2^0+B_4(\hat{O}^0_4+5\hat{O}^4_4)+B_6(\hat{O}^0_6-21\hat{O}^4_6)~.
\label{eq:HT}
\end{equation}

A cubic CF potential would split the $^2F_{7/2}$ multiplet into one
quartet $\Gamma_8$, one doublet $\Gamma_7$, and one doublet
$\Gamma_6$ states of O$_{h}$ group. Reducing the cubic symmetry to
the tetragonal symmetry, the quartet $\Gamma_8$ state of O$_{h}$
group would be split into one $\Gamma_7$ and one $\Gamma_6$ states of
D$_{4h}$ group. Because YbRu$_2$Ge$_2$ has a quasi-quartet CF ground
state, it is possible that this quasi-quartet is induced by a small
tetragonal perturbation to a large cubic CF potential. This small
perturbation is represented by the first term in Eq.~(\ref{eq:HT}).
We cannot rule out an alternative scenario that the quasi-quartet CF ground state of YbRu$_2$Ge$_2$ is of accidental degeneracy, rather than derived from the quartet $\Gamma_8$ state of cubic symmetry.
Nevertheless, the Hamiltonian $H_{Tetra}$ preserves the 4-fold
rotational symmetry along z-axis, and is sufficient to provide
qualitative insights.
In Appendix~\ref{sec:AB2g} we show that based on our assumption, the
ratio of $Q_{B_{1g}}$ to $Q_{B_{2g}}$ is calculated to be 1.34, close
to the experimentally determined ratio of 1.4
[Subsection.\ref{subsec:Q}]. This consistency supports our choice of
Eq.~(\ref{eq:HT}).
Experimentally, the wavefunction of the CF ground
state could be determined by core-level non-resonant inelastic X-ray
scattering, which has been used for Ce-based heavy
fermion systems~\cite{Sundermann2019}.

The second term in Eq.~(\ref{eq:H})
\begin{equation}
H_{B_{1g}}=\frac{V}{2}(\hat{J}_x^2-\hat{J}_y^2)=\frac{V}{2}(\hat{J}_+^2+\hat{J}_-^2)
\label{eq:HB1g}
\end{equation}
represents the effective quadrupole-field (QF) potential of B$_{1g}$ symmetry. $V$ measures the strength of the QF potential.

Above T$_Q$, there is no static B$_{1g}$ QF potential and we define $H=H_{Tetra}$.
We diagonalize $H_{Tetra}$ in the basis of $|J,m_J\rangle$, where $J=7/2$ and $m_J$ are the quantum numbers of $\hat{\mathbf{J}}$ and $\hat{J}_z$, respectively. After diagonalization, the CF transition energies can be expressed in terms of $B_2^0$, $B_4$, and $B_6$. We fit the experimentally determined CF level diagram by these three adjustable parameters. There are four sets of parameters which reproduce the level diagram, and we choose the set with the smallest $B_2^0$ value. The fitting results thus are $B_2^0$\,=\,-0.164\,cm$^{-1}$, $B_4$=0.0518\,cm$^{-1}$, and $B_6$=-0.00442\,cm$^{-1}$.
The corresponding angular electron-cloud distribution of the CF states is plotted in Fig.~\ref{fig:EP}.

Below T$_Q$, there is a finite static B$_{1g}$ QF potential, here we
define $H=H_{Tetra}+H_{B_{1g}}$.
We assume that the values of $B_2^0$, $B_4$, and $B_6$ do not change.
We diagonalize $H$ in the basis of $|J,m_J\rangle$, and after
diagonalization, the CF transition energies can be expressed in terms
of $V$. We find that $V$=0.523\,cm$^{-1}$ renders an additional
2\,cm$^{-1}$ splitting of the ground quartet. In Fig.~\ref{fig:FQ},
we plot the angular electron-cloud distribution of the ground quartet
for $V$=0.523\,cm$^{-1}$. The charge distribution looks different
from [100] and  [010] directions because the $\Gamma_5^{(1)}$ and
$\Gamma_5^{(2)}$ doublets carry B$_{1g}$ quadrupole moment.
Furthermore, the quadrupole moment carried by $\Gamma_5^{(1)}$ state
and that carried by $\Gamma_5^{(2)}$ state have approximately same
magnitude but an opposite sign~\footnote{Strictly speaking, the
quadrupole moments have the same magnitude only if we ignore the
contribution by higher energy CF states.}.

The FQ phase transition reflects the competition between the entropy and energy terms in the Helmholtz free energy of the system. Above T$_Q$, the entropy term dominates and the system prefers a quasi-degenerate CF ground state. Below T$_Q$, instead, the system pursues lowest possible energy, and an orthorhombic quadrupolar field fulfills the goal: this field mixes the wavefunctions of the quasi-degenerate $\Gamma_6^{(1)}$ and $\Gamma_7^{(1)}$ states, increasing their separation and in turn reducing the ground state energy. In view of group-theoretical considerations, the $\Gamma_6$ and $\Gamma_7$ irreducible representations of the D$_{4h}$ group become the $\Gamma_5$ representation of the D$_{2h}$ group. Correspondingly, the $\Gamma_6^{(1)}$ and $\Gamma_7^{(1)}$ states of the D$_{4h}$ tetragonal phase are mixed by the Hamiltonian $H_{B_{1g}}$, and become the $\Gamma_5^{(1)}$ and $\Gamma_5^{(2)}$ states of the D$_{2h}$ orthorhombic phase.

\begin{figure}
\includegraphics[width=0.48\textwidth]{Fig7.pdf}
\caption{\label{fig:FQ} (a) The effect of the B$_{1g}$ quadrupole-field potential on the ground quasi-quartet. The $\Gamma_6^{(1)}$ and $\Gamma_7^{(1)}$ doublets of the D$_{4h}$ group are mixed to form the $\Gamma_5^{(1)}$ and $\Gamma_5^{(2)}$ doublets of the D$_{2h}$ group. The wavefunctions are expressed in the basis of $|J=7/2,m_J\rangle$. (b) The angular electron-cloud distribution of the $\Gamma_5^{(1)}$ and $\Gamma_5^{(2)}$ doublets viewed from three orthogonal directions.}
\end{figure}

There are two obvious choices for the macroscopic order parameter of
the B$_{1g}$-symmetry FQ phase.
First is the quadrupole moment per unit volume:
\begin{equation}
\Psi\propto(n_{\Gamma_5^{(1)}}-n_{\Gamma_5^{(2)}})Q_{B_{1g}}~,
\label{eq:MOP1}
\end{equation}
where $n_{\Gamma_5^{(1)}}$ and $n_{\Gamma_5^{(2)}}$ are the occupancy for the $\Gamma_5^{(1)}$ and $\Gamma_5^{(2)}$ states, respectively.
The second choice is the lattice orthorhombicity, which
couples to the quadrupolar order:
\begin{equation}
\Psi\propto\frac{a-b}{a+b}~,
\label{eq:MOP2}
\end{equation}
where $a$ and $b$ are the in-plane lattice constant.
The orthorhombicity as a function of temperature has been measured by
X-ray diffraction~\cite{Fisher2018}.

\subsubsection{Coupling between the Crystal-Field Transition and the Phonon Modes\label{subsubsec:Coupling}}

The coupling between the $\Gamma_6^{(1)}\rightarrow\Gamma_6^{(2)}$ CF transition and the A$_{1g}$ and E$_{g}^{(2)}$ phonon modes originates from the modulation of the electron-cloud distribution of CF states by lattice vibration [Fig.~\ref{fig:EP}]. Such coupling is allowed by group theory because $\Gamma_6\otimes\Gamma_6\,=\,A_{1g}\oplus A_{2g}\oplus E_{g}$. We note that the phonon energy and linewidth can be well accounted for by the anharmonic decay model [Fig.~\ref{fig:P_Para}(a) and (b)], suggesting that renormalization due to electron-phonon coupling is small. In the Appendix~\ref{sec:ADerivation} we show that for small coupling strength, the temperature dependence of the integrated intensity of the phonon modes, $I.I.(T)$, has the following phenomenological expression:
\begin{equation}
I.I.(T)=Af_{(1)}(T)[1-f_{(2)}(T)]+B~,
\label{eq:Int}
\end{equation}
where $A$ and $B$ are two constants; $f_{(1)}(T)=2/Z(T)$ measures the occupancy of the $\Gamma_6^{(1)}$ CF state, and $f_{(2)}(T)=2e^{-E_4/k_BT}/Z(T)$ measures the occupancy of the $\Gamma_6^{(2)}$ CF state. $Z=2\sum_{i=1}^{4}e^{-E_i/k_BT}$ is the partition function; $E_1$=0\,cm$^{-1}$, $E_2$=2\,cm$^{-1}$, $E_3$=95\,cm$^{-1}$ and $E_4$=239\,cm$^{-1}$ are the energies of the CF levels [Table~\ref{table:ModeSummary}].

In Eq.~(\ref{eq:Int}), the constant $B$ represents the temperature-independent spectral weight of the phonon mode. Without the interaction $v$ and in the absence of a phase transition, the integrated intensity of the phonon modes is expected to be temperature-independent. The first term, which is temperature-dependent, can be interpreted as the spectral weight transferred from the CF mode to the phonon mode. This transferred spectral weight is proportional to the occupancy of the ground CF state $\Gamma_6^{(1)}$, and the un-occupancy of the excited CF state $\Gamma_6^{(2)}$. The constant $A$ is a measure of the transferred spectral weigh at zero-temperature~\footnote{We note that there is no general sum rule for Raman
spectroscopy.}.

Because of the phase transition at T$_Q$=10\,K, Eq.~(\ref{eq:Int}) is only valid above 10\,K. In addition, group theory allows the $\Gamma_7^{(1)}\rightarrow\Gamma_6^{(2)}$ CF mode to couple to the E$_{g}^{(2)}$ phonon mode, which is not considered by simplified Eq.~(\ref{eq:Int}). Because the splitting between the $\Gamma_6^{(1)}$ and $\Gamma_7^{(1)}$ states is only 2\,cm$^{-1}$, including the contribution from the $\Gamma_7^{(1)}\rightarrow\Gamma_6^{(2)}$ CF mode will only influences the fitting curve at temperature much lower than T$_Q$, a temperature range where Eq.~(\ref{eq:Int}) is invalid.

We use Eq.~(\ref{eq:Int}) to fit the phonon intensity data above 10\,K in Fig.~\ref{fig:P_Para}(c). For the A$_{1g}$ phonon mode, $A=3.14\pm0.08$ and $B=0.06\pm0.03$; for the E$_{g}^{(2)}$ phonon mode, $A=2.08\pm0.05$ and $B=0.35\pm0.02$. These values show that at low-temperature, the integrated intensity of the A$_{1g}$ and E$_{g}^{(2)}$ modes is mainly contributed by the transferred spectral weight. The fitting curves match the data well, which further supports our CF level scheme.

\section{Conclusion\label{sec:Conclusion}}
In summary, the Raman scattering study of YbRu$_2$Ge$_2$ focuses on the origin of the ferroquadrupolar transition, as well as on the spectroscopy of phonons and CF excitations within the $^2F_{7/2}$ ground multiplet of Yb$^{3+}$ ion.

The deduced CF level scheme verifies the proposed quasi-quartet ground state, and we estimate that the splitting between two quasi-degenerate Kramers doublets is about 2\,cm$^{-1}$. The static electronic Raman susceptibilities in both B$_{1g}$ and B$_{2g}$ quadrupole channels essentially exhibit Curie law, signifying relatively strong coupling to the lattice in the B$_{1g}$-symmetry channel that enhances the vanishingly small electronic Weiss temperature to the temperature of quadrupole phase transition at 10\,K.

The temperature dependence of the energy and FWHM of the observed phonon modes are described by anharmonic decay model. The integrated intensities of the A$_{1g}$ and E$_{g}^{(2)}$ phonon modes show more than 50\% enhancement on cooling, which is explained by strong coupling between these phonons and the CF transitions with similar energies.

\begin{acknowledgments}
We thank K. Haule and G. Khanal for discussions. The spectroscopic work at Rutgers (M.Y. and G.B.) was supported by NSF Grant No. DMR-1709161. The work at Stanford (E.W.R. and I.R.F.), including the crystal growth and characterization, was supported by the Gordon and Betty Moore Foundation Emergent Phenomena in Quantum Systems Initiative through Grant No. GBMF4414. G.B. also acknowledges the QuantEmX grant from ICAM and the Gordon and Betty Moore Foundation through Grant No. GBMF5305 allowing to make G.B. a collaborative visit to Stanford. Work at NICBP was supported by IUT23-3 grant.
\end{acknowledgments}

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