Theory of the Strange Metal Sr3ru2o7

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Sean and , skonfo photoemis- from known is aeas enreported. been also have T T lna eitvt n a and resistivity -linear 2 sepce,wt each with expected, is T | T transport 2 tr cln u to due scaling bv which above T tr b T n that and eateto odne atrPyis h ezanIsiueo cec,Rehovot Science, of Institute Weizmann The Physics, Matter Condensed of Department -linear → a,c ch doi:10.1073/pnas.1915224117/-/DCSupplemental at online information supporting contains article This 1 the under Published Submission.y Direct PNAS a is article This interest.y competing no paper.y declare research, the authors wrote performed The and research, data, analyzed designed tools, S.A.H. reagents/analytic and new contributed E.B., C.H.M., contributions: Author iga ssece nFg .I oedetail: more In 1. Fig. phase in thermodynamic sketched and is transport diagram The criticality. quantum lic H in Criticality of mystery in Criticality the Quantum unraveling for point broadly. starting more metals solid strange a gives bands.” here “hot in transport or of frac- understanding regions” small conceptual “hot The in relatively localized a electrons, critical by the quantum of only tion strong experienced metals directly strange is of scattering families many in ates, cc Sr same anomalous the that the show explains will We contexts. other in approached. is field violation Here a to leads instead approached is ρ relation field Kadowaki–Woods critical the of the of as (11) tering conductivity below optical temperatures and At (10) heat specific temperature the above ing o cteigit h o lcrnpce n hence and pocket suppression electron space hot phase the no into is scattering there for classical so are scale, potential low-energy chemical (6, the a to cuprates above close of peak by context a the in scattered electrons the in 7), by noted is transport previously metal As electron strange electrons. to cold cold leads pocket, one hot a which into another in process, scattering owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To Sr ∼ rnil.Ti ati ent nepntesrnemetallicity strange the strong, underpin ruthenate. to unusually strontium seen of is is exclusion Pauli fact the electron This to subject principle. cold not are electrons second hot after the because hot a becomes A electron with electrons. cold colliding “cold” a which other are in the process that scattering than electrons mechanical “hot” quantum contains less Specifi- ruthenate properties. strontium electronic cally, well-established well- follows from metal, of detail of strange behavior studied in variant the widely forms how a unidentified show ruthenate, We novel strontium some work. at unclear whether is fundamentally theoreti- physics remained understood or are eluded has matter metals It has of time. strange metals” some whether “strange for understanding of cal behavior The Significance c 2 h cteigmcaimw aeietfidmyb relevant be may identified have we mechanism scattering The nadto othe to addition In ≈ RuO T A T 7.9 ∆ 3 A γ 2 4 Ru si rse isiztasto 1) eodruthen- Beyond (13). transition Lifshitz a crosses it as , n pcfi etcoefficient heat specific and ∼ steehneeto h pcfi eta h critical the as heat specific the of enhancement the is 2,addsly eea aoia etrso metal- of features canonical several displays and (2), ∆ Sr γ 3 NSlicense.y PNAS 2 Ru for O 2 Sr O T 3 7 lna eitvt 2 ,9,cc 9), 8, (2, resistivity -linear 7 Ru Sr sfil ue,wt critical a with tuned, field is T ρ 3 2 T ∼ tr Ru O tr h oiac fcc of dominance the , T 7 2 rdcsteosre anomalous observed the produces O 2 0 2,a u oe predicts. model our as 12), 10, (2, 2 log 7 A . y T ∼ γ https://www.pnas.org/lookup/suppl/ γ fadfeetruthenate, different a of ≡ 2 NSLts Articles Latest PNAS ewe h resistivity the between c Sr /T 3 Ru i.2sosthat shows 2 Fig. . → 2 O → c hscattering ch ai edof field -axis 7 → Sr ρ hscatter- ch developed ∼ 3 hscat- ch Ru T | . 2 f6 of 1 O 7 .

PHYSICS Fig. 3 illustrates the interplay of hot and cold electrons. The Fermi surface mostly consists of regions with a high Fermi velocity, as well as a set of “hot pockets” with a large density of states peaked near or at the chemical potential. The essential feature for the present discussion is that the high density of states is localized in a small part of the Brillouin zone, whereas the light portions of the Fermi surface are much more extended. The density of states of the “hot electrons” is high, whereas the “cold electrons” away from the hot spots dominate the current due to their higher Fermi velocity. It is therefore natural for the transport lifetime to be determined by scattering processes where both cold and hot electrons are involved. Several such processes are possible: A cold electron can scatter off a hot one, denoted ch → ch, and a pair of cold electrons can scatter into a cold and a hot electron (cc → ch), etc. The ch → cc or cc → ch processes involve large momentum transfer and contribute significantly to transport. Moreover, for Fig. 1. Schematic phase diagram of Sr3Ru2O7 as a function of field and a large enough cold Fermi surface, all cold electrons can par- temperature. There is a marginal Fermi liquid of cold electrons above Ttr. ticipate in cc → ch processes and no “short circuiting” occurs. In the conventional Fermi liquid below Ttr, transport and specific heat are Other types of scattering are either small angle, and do not effi- dominated by the diverging hot electron mass m?h as H → Hc. ciently degrade current, or not active across the entire Fermi surface. We discuss this further in SI Appendix. In the following 1) At H = 0 the resistivity is T linear above Ttr ∼ 20 K and goes we show that cc → ch scattering reproduces the phase diagram 2 as T at the lowest temperatures. As H → Hc , the scale Ttr in Fig. 1. apparently collapses toward zero (2, 8, 9). 2) As H → Hc at low temperatures, an effective mass m? is A Marginal Fermi Liquid from Classical Fermions strongly enhanced. The growth of m? is seen in the coefficient The peak in the density of states due to the hot γ2 pock- 2 A of the low-temperature resistivity ρ ∼ AT (2), NMR spin ets (at H = 0, from ARPES) occurs at a distance from the relaxation (14), and the low-temperature specific heat coef- chemical potential that roughly coincides with the temperature ficient γ ≡ c/T (10, 12). However, we see in Fig. 2 that the above which T -linear resistivity is observed. Let the distance specific heat enhancement ∆γ follows the scaling A ∼ ∆γ, of the peak from the chemical potential be h and its width be distinct from the conventional Kadowaki–Woods relation Wh (Fig. 3). At zero field, Wh |h|. At temperatures above 2 . A ∼ γ . |h| + Wh the γ2 fermions are classical and nondegenerate. A 3) At H = Hc a low-temperature divergence γ ∼ log T is cc → ch scattering process can therefore be expected to be sup- observed (10, 15), until cut off at a spin-ordering transition pressed by T rather than T 2. We now verify that this is the case at a temperature of about 1 K (16). by showing that this scattering leads to marginal Fermi liquid 4) An extended Drude fit to σ(ω) in the T -linear regime reveals (MFL) (21) behavior of the cold fermions, as has been previously a scattering rate Γ(ω) ∼ ω for ω & T (11). noted (6, 7). To connect with the MFL cleanly, we write the scattering pro- Hot Pockets and cc → ch Scattering cess as the interaction of a cold fermion with a bosonic mode Quantum criticality in Sr3Ru2O7 involves a dichotomy between cold and hot electrons. The hot electrons have a large density of states already in zero field, and only the hot electrons go critical as a function of field. Specifically: 1) Angle-resolved photoemission (ARPES) reveals a compli- cated fermiology with multiple bands (3, 17). Quantum oscillations show that all except one set of bands—the γ2 pockets, which are not seen in Shubnikov–de Haas oscillations—do not exhibit a strongly divergent mass at the critical field (18). The γ2 pockets are remarkably shallow and give rise to a peak in the density of states at a few millielectronvolts from the chemical potential at zero field (3, 17). 2) The coefficient γ as a function of temperature shows a maximum at Tpk ≈ 7K at zero field (10). A similar peak is seen at 0 Tpk ≈ 17K in the magnetic susceptibility (19, 20). As H → Hc , Tpk collapses to zero. At zero field the hot electron peak in the density of states seen in photoemission quantitatively explains the peaks seen in the specific heat coefficient γ and magnetic susceptibility χ. Using the measured density of states of the γ2 pockets (3) in standard free-electron formulas for γ and χ leads to peaks at temperatures of 9 K and 19 K, respectively. These are in good agreement with the zero-field experimental values mentioned above. This is the first imprint of the hot electrons on physical observables. In the 2 remainder we show that cc → ch scattering into the hot electron Fig. 2. As H → Hc at low temperatures, the coefficient A of the T resistivity pockets fundamentally determines the transport behavior both increases by a factor of ∼ 7 (2), and is linearly proportional to the specific above and below Ttr. heat enhancement ∆γ (10, 12).

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MFL Fermi dis- explicitly the the fermion about obtain cold linearized the be to allows persion which respectively), and velocity, energy, Fermi Fermi momentum, Fermi fermions fermions’ cold cold the The denote energies [2]. in at simplifications are to leads This nonzero. | we above, f to identity the Similarly through formulations general. mon keep Eq. written can have we which fermions, Here are distributions Bose– and Fermi–Dirac Einstein the and constant coupling four-fermion Eq. form, this In Σ in work (we self-energy the generality) acquire for dimensions fermions spatial cold the tion, aigtecl em ufc ob on o simplicity, for round be to surface Fermi cold the Taking Π b identity the using differently, 1/(e with the fermions weight cold treat spectral free We from polarizability. starting fermion perturbatively, interaction cold–hot the by generated cold the fermions. as hot sheet the Fermi to In due same (red). potential, the region on hot be the also ( into fermions. could (blue) regions surface hot Fermi the cold general, the from scattered is 3. Fig. ostve al. et Mousatov AB ω (Ω h oaiaiiyis polarizability The (Ω). c 00 ch 00 | ∼ | ttmeaue rae than greater temperatures At (ω (Ω, 0 ) Ω/T , − Im | ∼ Ω| ~ k lutaino cc of Illustration (A) ~ q = ) f − hr eki h est fstates of density the in peak sharp A B) F = ) G Ω+Ω + (Ω ( h scmol expressed commonly is [1] (22), textbooks In 1). R λ × ~ q Ω × Z 2 (Ω ) 0 Im | | Z ≈ Im 0 Im (2π d 0 Ω, + T G ). 2 × 1 (2π π 2 Π d G d c Ω G k R namnfsl hsclwy eae ocom- to related way, physical manifestly a in stasaetyFrisgle rule: golden Fermi’s transparently is ch 00 ) Z Z d c R d 0 (ω 0 c R q ) (Ω, ~ k Ω + (Ω v d Z (ω d (2π F 0 d − π Z Ω + → c 0 d , ~ q , k d | Ω, 0 ~ k k ~ k ~ ) q F π ) Ω hsatrn,i hc n ftefermions the of one which in scattering, ch d v d 0 = ) wherever 0 c π ) ≈ F 1 Im ) ~ Ω k 0 f δ = c f f (| Ω 1/(e = (Ω) steseta egto h hot the of weight spectral the is − F f Im πδ (ω (Ω ~ E k (ω ( ~ q G 0 F ~ q (ω − | 0 − G Π ) c h Ω) + R − tanh ) f h b R Ω)b (Ω − (where | (ω k Ω)b ch 00 (Ω) (Ω Im F f h 0 ) (Ω (Ω, | c c, f Ω, + (Ω)/f 0 ) + Ω/T ~ (−Ω k (Ω) G Ω, + 2T 0 u oteinterac- the to Due ). ν Ω W ~ q h Ω) + R ls otechemical the to close () k (Ω ). h ~ 1) + k . 0 F (ω ) ehave we , ~ k 0 c 0 + 0 f , Ω, + = ) + (−Ω v ~ q F ~ and q ). c f ). and , ~ k (Ω 0 )/b 0 + λ b − Ω = (Ω) Ω = (Ω) ~ sthe is q ω T E ) + ) [2] [3] [1] [4] F ∼ is d c odfrincnsatrit scntandb h ieo the of size rate the scattering the by finds one Γ constrained integrals is the evaluating into regions; scatter hot can Fermi that fermion cold so the cold than flat, that smaller be so are to regions sea, hot band The hot zone. Brillouin the taking by Im [4] from rate tering bv h hrceitceeg clso h o emos(such fermions hot the of scales as is energy temperature characteristic the the that is above single-particle assumption only the The function. for Green’s result lifetime. this transport the angle, determines also large rate scattering is that scattering observed shows ch the [2], precisely the above evaluate in found to states of (3) density pockets determined experimentally the cc using A the above. of noted evaluation conductivity careful optical and resistivity experimental becomes that polarization MFL h ieaislaigt hsrsl in result this to to leading kinematics the to Eq. emt eitvt.Hwvr because However, temperature-independent resistivity. a to contributes to term It region). proportional hot the rate as scattering size a to leading λ disorder, elastic as small For h aiu ntedniyo ttsi i.3i tadistance a at is 3 Fig. in states of density | the in maximum The Field Critical the to Approach Low-Temperature additional an the by suppressed of Therefore, factor is electrons. rate cold scattering the transport effective for scattering small-angle ily elec- hot the the of finds in One states electron suppression. final λ Fermi–Dirac an and no off initial have scatters the tron electron Both cold region. a hot which in cess, 23). (9, rate scattering “Planckian” observed estimate we bands 0.1~/m over averaging Again mass. phase estimate available we the bands 0.5 controls over [5] are Averaging in scattering. There surfaces for Fermi space 9. cold ref. the of from ity with numbers pockets hot using eight prefactor the For controlled. estimate is computation perturbative neddmntsi the in the dominates indeed than greater 3. Appendix is Fig. in region shown despite annulus hot that cold the requirement, broadened This thermally of the area of area the when inates that simplicity for ing utb mle hnta o cteigit o fermions: hot into scattering for that fermions cold than into k smaller scattering for be space must phase the dominate, to cc from ity surface). F 2 2 c h h result The iia optto a edn o h ch the for done be can computation similar A hr sas conventional a also is There k k c Σ d | ≡ W G × k F F 3 2 rmteceia oeta.A eprtrswl eo this below well temperatures At potential. chemical the from 3 c 00 2 = B h h h Σ R h 10 a oe nrf .I ste elkonta netn the inserting that known well then is It 6. ref. in noted was ∼ T /v R (ω c 00 e k and −4 (ω /v max(T F hs ubr edto lead numbers These . n etrn units: restoring and F (2π (k , d c evrf httepaesaefrcc for space phase the that verify we h ~ k F k 2 0) = F h irsoi coupling microscopic The s/m. (the = ) q ) F c en ml because small being → 2 h k h h /k δ F 3 h rfco of prefactor the , ω .W a siaetemgiueo h scat- the of magnitude the estimate can We ). (ω πδ h csatrn.FrteMFL the For scattering. cc , ∼ k F sotie o n omo h o fermion hot the of form any for obtained is q ω iera ihrfeunis hsepan the explains This frequencies. higher at linear F 2 c − (ω 3 ) ) h nerli etitdt eino h same the of region a to restricted is integral λ 2 noteexpression the into emtial,cc Geometrically, . k 2) hs hr sa is there Thus, (21). othat so , ) 2 F c, k o oergo farea/volume of region some for c ~ λ k F d Γ nti ae h hs pc htthe that space phase the case, this In . P − k −2 c ∼ c T ~ B q i ∼ hssatrn hrfr behaves therefore scattering This ). T v lna eieof regime -linear k k F F → d F λ 2 Γ c h h,i 2 /k /v tr hsatrn aein rate scattering ch k v k F T F E 2 ≈ F 2 ∼ F Γ F 2 k 2 h c c F c B 2 h 3A 0.03 ∼ k k c /k sstb igecl Fermi cold single a by set is T F T otiuint h resistiv- the to contribution 4 B sahg-nrysae In scale. high-energy a is T ~ k k h /~ F T 2 egv oedtisof details some give We . F B /k tr 1 Specializing Appendix. SI c h T NSLts Articles Latest PNAS . o h efeeg leads self-energy the for × T ≈ → F ssaladthe and small is [5] in 4 −2 ossetwt the with consistent /~, T c lna cteigrate scattering -linear E 0K 20 × λ Sr k lna cteigis scattering -linear h nes veloc- inverse The . hsatrn dom- scattering ch F T F Sr c k a nt of units has 3 c lna scattering -linear a esatisfied be can , F Ru hsi necessar- is this , λ 3 → eas cc Because . h Ru hv ∼ v 2 F hscattering ch O IAppendix, SI 2 c Σ → O /~ 7 F k . c 00 7 F c d (ω hv /k (assum- hpro- ch h ecan we | F −1 nthe in , F ~ ~ c k f6 of 3 c per ≈ i ) ≈ i [5] → γ SI ∼ 2

PHYSICS 2 2 scale all of the fermions are degenerate, and electron–electron kFh/kFc · log(Λ/T ). Here m?c is the unrenormalized cold elec- 2 scattering leads to the usual T resistivity. If |h| < Wh, how- tron mass. The energy cutoff scale Λ ∼ vFckFh because MFL ever, there remains a degenerate Fermi pocket of hot fermions scattering Γc ∼ ω requires ω/vFc . kFh, for the scattering phase that makes a large contribution to the density of states and cc space to be determined by the hot fermion pocket (SI Appendix). → ch scattering is still important. In this regime ω ∼ Ω ∼ Ω0 ∼ Precisely such a logarithmic temperature dependence is seen Ω + Ω0 ∼ T , and the formulas above give the scattering rate cleanly at low temperatures at the critical field, as we recalled (reinstating factors of kB and ~ in the final term) above. The onset of logarithmic behavior is also visible in the data away from the critical field, at temperatures above Ttr. At 2 T 2 kFh 2 m?h (kB T ) temperatures below tr, within the hot electron peak in the den- Γc ∼ λ 2 T ∼ , [6] sity of states, the hot fermions become degenerate and the cold vFhvFc m?c ~EFc 2 2 2 fermion mass enhancement is cut off as γcold ∼ kB m?c · kFh/kFc · log (Λ/(| | + W )). The relative contribution of hot and cold where we define the mass of the heavy electrons through their h h electrons to the low-temperature γ at the H ≈ Hc is discussed density of states at the Fermi level, m = ν (0)/2π, and sim- ?h h further in SI Appendix. ilarly for m?c. In the final term in [6] we again assumed for The observed behaviors of the specific heat and transport data simplicity that λ ∼ /m?c is set by a single cold Fermi surface. ~ shown in Fig. 1 are entirely reproduced by the model of dominant See SI Appendix for details of the kinematics leading to [6]. cc → ch scattering if Ttr ∼ |h| + Wh collapses toward zero at The important factor is the ratio m /m?c that, we will posit, ?h the critical field. This collapse is cut off near the critical field becomes large as the critical field is approached. The scatter- by the onset of spin ordering at around 1 K, as shown in Fig. 1. ing rate Γc determines the resistivity through the Drude formula 2 Therefore the width of the peak in the density of states need not ρ = m?cΓc/(nce ). The resulting scaling of the resistivity with the strictly go to zero. large hot mass is therefore ρ ∝ m T 2. In contrast, if all fermions ?h The data suggest that it is predominantly the scale | | that have a strongly enhanced mass (become hot) or if ch → ch scat- h drives the collapse of Ttr toward the critical field. We recalled tering dominates (for example, if the hot band is large in the above that a peak in the specific heat is seen at a temperature Brillouin zone), then ρ ∝ m2 T 2. ?h T . This Schottky anomaly-like peak in γ(T ) is quantitatively At the same low temperatures, the electronic specific heat is pk explained at zero field by the peak in the density of states seen also given by the conventional Fermi liquid formula. If m ?h in ARPES (3). As the temperature is raised more states in the m?c, then hot electrons dominate the specific heat, even while hot region become accessible and hence the specific heat coef- cold electrons dominate transport. The enhancement of the spe- ficient increases until the temperature crosses ∼ max(| |, W ), cific heat coefficient γ due to the increasing mass of the hot h h beyond which the specific heat starts to decrease (and become fermions is ∆γ ∼ k 2 m . Together with the results in the pre- B ?h dominated by the cold electrons). The simultaneous collapse of vious paragraph, this leads to A ∼ ∆γ, where A is the coefficient T and Ttr as the critical field is approached is therefore nat- in the resistivity ρ ∼ AT 2. The notation ∆γ refers to the fact pk urally explained by | | → 0. Such behavior is consistent with a that the specific heat of the cold fermions has been subtracted h recent band structure analysis (28). As noted above, the essen- out.∗ We have recovered precisely the observed linear scaling tial requirement on the width W of the peak is that it should be shown in Fig. 2. The key input is the hypothesis that m becomes h ?h of order 1 K at the critical field, so that the MFL behavior can large as H → Hc , which leads to the dominance of cc → ch persist all of the way down to the onset of the low-temperature scattering. This scaling is distinct from the Kadowaki–Woods ordered phase. relation A ∼ γ2 (24), which instead follows from the resistivity As → 0, the density of states at the chemical potential ρ ∝ m2 T 2. h ?h increases. This leads to an increase in the specific heat coef- Microscopically, the mass enhancement has been assumed to ficient at low temperature, tying the collapse of Ttr to the be tied up with metamagnetic quantum criticality of the hot elec- low-temperature mass enhancement we described above. trons (2, 25–27). The results above are not sensitive to the details In the scenario we have outlined, nondegenerate hot fermions of these physics beyond requiring a divergent m in a small ?h are important even at low temperatures at the critical field. In region of the Brillouin zone. However, in using [4] we are also this case, as noted above, elastic ch → ch scattering gives an addi- assuming that the integrated single-particle spectral weight of the tional contribution to the residual, temperature-independent hot electrons remains finite as their mass diverges. This leads to a resistivity. A strong peak in the residual resistivity at the criti- picture in which the growth of the mass is primarily a band struc- cal field is indeed seen in the data (2). Furthermore, the Lorenz ture effect associated with a strongly enhanced density of states ratio at low temperatures shows increased elastic scattering at at the Fermi energy as H → Hc (28). This spectral weight must the critical field (29). survive the presence of any quantum critical scattering. Our discussion has been purely in terms of the fermionic band structure. Direct contributions to transport and thermodynam- The Collapse of Ttr at the Critical Field ics from quantum critical collective modes are not necessary The hot electrons dominate the specific heat at low temper- to explain the data, as considered in refs. 30 and 31. Nonethe- atures, especially as the critical field is approached. At tem- less, we mention two ways in which such bosonic physics could peratures above the peak in the density of states their con- 2 2 2 3 additionally be present. First, quantum critical fluctuations of tribution becomes small: γhot ∼ kFh max Wh , h /(kB T ). In an overdamped bosonic order parameter with zB = 2 in two computing γ here the chemical potential is kept fixed and hot dimensions would contribute to a logarithmic specific heat at low constant, controlled by the large total number of cold electrons. temperatures (32). Second, order parameter fluctuations provide The cold fermions dominate the specific heat in this regime. 00 an additional scattering mechanism. A different source of T - The MFL scattering rate Σc ∼ max(T , ω) implies a logarithmic 2 2 linear scattering will be called for at the lowest temperatures if effective mass enhancement so that ∆γcold ∼ kB ∆m?c ∼ kB m?c · Wh does not in fact collapse down to around 1 K at the critical field (as we assumed above).

* As m?h diverges, m?c also becomes enhanced due to cc → ch scattering. We will see Scattering into a Divergent Density of States 2 shortly that the enhancement is only logarithmic: ∆m?c ∝ k log (m v c/k ). Close Fh ?h F Fh Taking a step back from Sr3Ru2O7, nondegenerate fermions to the critical ﬁeld this growth is parametrically weaker than that of m?h, plausibly explaining why it is not seen in quantum oscillations (18). That said, our interpretation will be present in a localized region of the Brillouin zone when- of the speciﬁc heat data in SI Appendix suggests that ∆m?c ∼ m?c is not negligible. ever a divergence in the density of states occurs at the chemical

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[2] and ω [1] from computed be again while diverges electrons hot density The the energy. of Fermi states or the of law at located power is broader, that a discussed peak within physics logarithmic, temperature The at states. temperatures arise of at instead density the here arises in peak above sharp discussed a above MFL differ- a The describes scenario. it ent fermions, nondegenerate discussion earlier of our importance with the shares section this While divergence). dispersion a general, In point. Hove van scaling the a with at as such potential, ostve al. et Mousatov rate decay the obtain we told, for divergent is which point Hove van a at Γ ~ k x 2 c ∼ 0 .Freape van a example, For [9]. to corrections logarithmic be can There h ieieo odfrin u occ to due fermions cold of lifetime The − + ∼ Ω k k ~ q B λ y ε ∼ 2 ∼ | 2 . ν vH = k h Ω v F () d T F k 0 2 −2 3) eitvt esrmnso eypr samples pure very on measurements Resistivity (37). c η ∼ c 2 Im 1/z ∼ cos(2θ ssal eobtain we small, is T T R hl h te soe h odFrisurface. Fermi cold the over is other the while , G Z h, h d R −T Γ q T Im ~ d k (ω E 0 c 9 hslast oaihi iegnein divergence logarithmic a to leads This ). k ~ k ∼ at [2] in ∼ F d h, , stenme fsae ntepa htare that peak the in states of number the is , Im G z c ~ η k ~ ν k k ~ T k h iprino h odeetoscan electrons cold the of dispersion The . > h = ) z R o h o lcrn.Here electrons. hot the for − ∼ ν 3/2 G h (ω ti edntb iiu in minimum a be not need (this h d Sr () G ~ q k h () R ω x oho hs w clnsof scalings two these of Both . 1 , (for 2 2 θ (, and ~ k n,frgnrlt,alwn o an for allowing generality, for and, RuO ∼ − h eutn scatter- resulting The Appendix. SI ± ≈ ∼ = ) ~ k W λ k z ω ) y ~ k 2 2 d Γ πδ = h ∼ suigteanomalous the Assuming 4. π/ ed oadniyo states of density a to leads ) 4 0 /z k z c F utb erteFrisur- Fermi the near be must d nege isiztransi- Lifshitz a undergoes d −1 2 = ∼ (ω η h v d G k hr a ealogarithmic a be can there F /z 2 T , F Sr − d c −1η ihtedispersion the with −2 c 2 2 k W log( RuO k z ω W → k z W B h ipetcase simplest The . ~ k W h T F hsatrn can scattering ch h z 4 ean finite. remains /k h h h /T F z k k , W h z B F h scat- The ). Another ). h T 4 = h k sachar- a is F d −2 c d k ntwo in /z e.g., , All . ω η −η [7] [9] [8] Γ ∼ 7, is ~ c . f lsia oos ti itnt oee,fo cteigofa ch off to scattering correspond from would however, which distinct, fermion, classical is It bosons. classical off well- the of analogue fermionic nondegenerate established a hot, is of process regions simple This small electrons. two into cc electrons of the cold from phenomenology of understood tering metal be strange can the ruthenates strontium that shown have We case. this Outlook in small not is band hot the although cc hrooe sas oetal estv rb ftepresence the electrons. of nondegenerate probe sensitive of across potentially a behavior also is in Hall thermopower change the dramatic as inter- such a T be observables shows will subtle which It more coefficient, heat. whether specific see and to dc esting transport of charge properties optical quantitative and explains successfully scattering ch compara- It high-temperature roughly the (42). is that K temperature scale 600 the this temperature to that about ble linear note at to onsets to interesting regime is quadratic linear from The single-particle dependence. over The crosses gradually singularity. Hove van as cc scales for rate a above scattering obtained near behavior the scattering resistivity precisely residual is the times This 40 (13). of range a over accuracy excellent ealcpyisaiigi hs cases. those strange in the arising underpin physics could (50). cc metallic work mass the this of in divergent identified variants a mechanism how ing the understood with of be tandem to to weight in remains lead spectral It vanish the necessarily can because not states fermions do of hot density regions the hot the in of the peaks regions (49), cold and surface hot distinct Fermi shows crit- clearly nematic metals Ising or in wave icality density quantum e.g., fol- controlled of, While have the modeling mind. Carlo however, we in Monte kept metals, cases approach be strange should those the caveat general in for lowing more work candidates For at are here. be they taken may hence dichotomy fermions and A cold also, states. of and density hot the between in peaks narrow contain tems of aspects energy several Fermi with square the to toward phenomenology. difficult the moving therefore peak are the and field (48) show of clearly function needed Existing a not is description. as our do energy data of Fermi (STM) aspect microscopy the this tunneling toward falsify) scanning moves (or justify it fully as to peak narrow- the the of of ing confirmation experimental neces- Direct is temperatures. This field. the how critical peak explain and the to this at framework of our K) within width 1 sary order the our (of that of small assumption aspect becomes the speculative been has most collapses discussion that The field experimentally. zero at as grounded states energy of Fermi density the the toward in peak a of ence fits strain critical the 3,3)o yeiailsri 4) usd h a,teresistiv- the fan, the Outside (40). strain epitaxial ity by or 39) (38, † nl en hti osntsrnl mattasot tmyb neetn olo at look to interesting be may It transport. impact strongly not cc does it to that analogous means is angle This (43–45). surface Fermi entire ch the over fermions “lukewarm” to lead nSWtastos cteigofcmoieoeaosa o pthsbe rudto argued been has spot hot a at operators composite off scattering transitions, SDW In tr In u ecito of description Our unn oohrfmle fsrnemtl,hayfrinsys- fermion heavy metals, strange of families other to Turning thg eprtrs h eitvt funstrained of resistivity the temperatures, high At → → → ρ 4,4) a lob nesodfo hsprpcie The perspective. this from understood be also can 47), (46, T ∼ hsatrn ntoemodels. those in scattering ch Sr hsatrn norfaeok npriua,tefc httesatrn ssmall is scattering the that fact the particular, In framework. our in scattering ch hsatrn notebn otiigtevnHv point, Hove van the containing band the into scattering ch o 1/T log T 3 Ru 2 nteuixa taneprmn h eitvt near resistivity the experiment strain uniaxial the In . 2 T O lna cteigrt htaie rmscattering from arises that rate scattering -linear 7 pcfi etaeal oetn ont h lowest the to down extend to able are heat specific T npriua,tepooe oiac fcc of dominance proposed the particular, in vH etoe bv.Ti ocial suggests conceivably This above. mentioned ρ Sr ∼ T 3 T Ru T u och to due 2 lna eairmgtas edeto due be also might behavior -linear H 2 T log( O → 7 ssrcue rudtepres- the around structured is H vH c → hs ttmnsaewell are statements These . /T hpoess(41). processes ch NSLts Articles Latest PNAS with ), T lna resistivity -linear T → vH → ch. 230 = → hscatter- ch † Sr hscat- ch 2 | RuO → ,to K, f6 of 5 → ch 4

PHYSICS Data Availability Quantum Matter (HQMAT) (Grant 817799); the Israel–US Binational Sci- ence Foundation; and the hospitality of the Aspen Center for Physics, The data and codes used in this work are available upon request. supported by NSF Grant PHY-1607611, where part of this work was done. ACKNOWLEDGMENTS. We have had helpful discussions with Chandra C.H.M. is supported by an NSF graduate fellowship. Collaborative work Varma, Andrew Mackenzie, Andrey Chubukov, and Jan Zaanen. This work for this paper was made possible by the Nordita program “Bounding is supported by the Department of Energy, Ofﬁce of Basic Energy Sci- Transport and Chaos in Condensed Matter and Holography,” which was ences, under Contract DEAC02-76SF00515. E.B. acknowledges ﬁnancial additionally funded by Institute for Complex Adaptive Matter (ICAM) and support from the European Research Council under Grant Horizons in Vetenskapradet.

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