Theory of the Strange Metal Sr3Ru2O7
Connie Mousatova, Erez Berg1,b, and Sean A. Hartnolla,c
aDepartment of Physics, Stanford University, Stanford, CA 94305, USA; bDepartment of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot, 76100, Israel; cStanford Institute for Materials and Energy Science, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA
This manuscript was compiled on April 14, 2020
The bilayer perovskite Sr3Ru2O7 has been widely studied as a canon- Furthermore, it is the ‘hot’ electrons in these pockets that go ical strange metal. It exhibits T -linear resistivity and a T log(1/T ) critical as the field is tuned. These facts suggest that the ‘hot’ electronic specific heat in a field-tuned quantum critical fan. Critical- (h) electron pockets should be responsible for the anomalous ity is known to occur in ‘hot’ Fermi pockets with a high density of transport, but a suitable mechanism has been lacking. The hot states close to the Fermi energy. We show that while these hot pock- electrons are irrelevant as carriers of charge, because they are ets occupy a small fraction of the Brillouin zone, they are respon- short circuited by the faster ‘cold’ (c) electrons (4, 5). Instead, sible for the anomalous transport and thermodynamics of the mate- a distinctive cc → ch scattering process, in which one cold rial. Specifically, a scattering process in which two electrons from electron is scattered by another into a hot pocket, leads to the large, ‘cold’ Fermi surfaces scatter into one hot and one cold strange metal transport by the cold electrons. As previously electron renders the ostensibly non-critical cold fermions a marginal noted in the context of cuprates (6, 7), electrons in a peak Fermi liquid. From this fact the transport and thermodynamic phase close to the chemical potential are classical above a low energy diagram is reproduced in detail. Finally, we show that the same scat- scale, so there is no phase space suppression for scattering into tering mechanism into hot electrons that are instead localized near a the hot electron pocket and hence ρ ∼ T . two-dimensional van Hove singularity explains the anomalous trans- In addition to the T -linear resistivity (2, 8, 9), cc → ch port observed in strained Sr2RuO4. scattering above the temperature Ttr produces the observed anomalous specific heat (10) and optical conductivity (11) of Sr3Ru2O7. At temperatures below Ttr, the dominance of cc → ch scattering as the critical field is approached leads to a The conventional theory of metallic transport predicts that violation of the Kadowaki-Woods relation A ∼ γ2 between the the approach to the residual resistivity at low temperatures resistivity ρ ∼ AT 2 and specific heat coefficient γ ≡ c/T . Fig. should follow one of several simple power laws. If electron- 2 below shows that instead A ∼ ∆γ for Sr3Ru2O7 (2, 10, 12), electron scattering dominates then ρ ∼ T 2 is expected, with as our model predicts. Here ∆γ is the enhancement of the each factor of T coming from the suppression of final electron specific heat as the critical field is approached. states due to Pauli exclusion. Dominant electron-phonon scat- The scattering mechanism we have identified may be rele- 5 tering instead leads to ρ ∼ T in three dimensions. ‘Strange vant in other contexts. We will show that the same cc → ch metals’ can be defined as metals exhibiting an anomalous scattering explains the anomalous ρ ∼ T 2 log T of a different temperature dependence of the low temperature resistivity. ruthenate, Sr2RuO4, as it crosses a Lifshitz transition (13). The T -linear behavior ρ ∼ T has been widely observed, but Beyond ruthenates, in many families of strange metals strong 3/2 2 other scalings including ρ ∼ T and ρ ∼ T log T have also quantum critical scattering is only directly experienced by been reported. These are all stronger than the conventional a relatively small fraction of the electrons, localized in ‘hot 2 T scaling due to electronic scattering. A common scenario regions’ or in ‘hot bands’. The conceptual understanding of is that the anomalous scaling is observed above a ‘transport transport Sr3Ru2O7 developed here gives a solid starting point temperature’ Ttr, and that Ttr → 0 at some point in the phase for unravelling the mystery of strange metals more broadly. diagram, often associated with quantum criticality (1). A compelling theory of a strange metal must connect the anomalous transport behavior to other measurements, includ- Significance Statement ing thermodynamic and spectroscopic probes. Given the sim- ilarity of the strange metal phenomonon across materials, a The behavior of ‘strange metals’ has eluded theoretical un- thorough grounding of anomalous transport in specific mi- derstanding for some time. It has remained unclear whether croscopic electronic properties of a single material has the strange metals are fundamentally novel forms of matter or potential to uncover mechanisms that may be at work more whether some unidentified variant of well-understood physics is at work. We show how the behavior of strontium ruthen-
arXiv:1810.12945v2 [cond-mat.str-el] 12 Apr 2020 broadly. Here we show how such a comprehensive theory ate, a widely studied strange metal, follows in detail from well- can be achieved for the bilayer perovskite Sr3Ru2O7. This material has been widely studied as a prototypical strange established electronic properties. Specifically, strontium ruthen- metal (2), with T -linear resistivity in a quantum critical fan ate contains ‘hot’ electrons that are less quantum mechanical and diverging effective electron masses as the critical field is than the other ‘cold’ electrons. A scattering process in which approached at low temperature. a cold electron becomes hot after colliding with a second cold electron is unusually strong, because the hot electrons are not The key fact will be that Sr Ru O is known from photoe- 3 2 7 subject to the Pauli exclusion principle. This fact is seen to mission measurements to exhibit very shallow, small Fermi underpin the strange metallicity of strontium ruthenate. pockets. These lead to a sharp peak in the density of states close to the chemical potential (3). The distance of the peak from the chemical potential is comparable to the scale Ttr above which strange metal transport is observed at zero field. 1To whom correspondence should be addressed. E-mail: [email protected]
PNAS | April 14, 2020 | vol. XXX | no. XX | 1–10 Quantum criticality in Sr3Ru2O7. Criticality in Sr3Ru2O7 is Hot pockets and cc → ch scattering. Quantum criticality in field-tuned, with a critical c axis field of Hc ≈ 7.9T (2), and Sr3Ru2O7 involves a dichotomy between cold and hot electrons. displays several canonical features of metallic quantum criti- The hot electrons have a large density of states already in zero field, and only the hot electrons go critical as a function of T field. Specifically:
1. Angle-resolved photoemission (ARPES) reveals a compli- cated fermiology with multiple bands (3, 17). Quantum ⇢ T ⇠ oscillations show that all except one set of bands — the Ttr log T ⇠ γ2 pockets, which are not seen in Shubnikov-de Haas oscil- lations — 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 2 ⇢ m?hT meV from the chemical potential at zero field (3, 17). ⇠
2
T/v
Wh ~q
2 Fig. 3. Left: Illustration of cc → ch scattering, in which one of the fermions is scattered Fig. 2. As H → Hc at low temperatures, the coefficient A of the T resistivity increases by a factor of ∼ 7 (2), and is linearly proportional to the specific heat from the cold Fermi surface (blue) into the hot region (red). In general, the hot regions enhancement ∆γ (10, 12). could also be on the same Fermi sheet as the cold fermions. Right: a sharp peak in the density of states ν() close to the chemical potential, due to the hot fermions.
3. At H = H a low temperature divergence γ ∼ log T is c The density of states of the ‘hot electrons’ is high, whereas observed (10, 15), until cut off at a spin ordering transition the ‘cold electrons’ away from the hot spots dominate the cur- at a temperature of about 1K(16). rent due to their higher Fermi velocity. It is therefore natural 4. An extended Drude fit to σ(ω) in the T -linear regime for the transport lifetime to be determined by scattering pro- reveals a scattering rate Γ(ω) ∼ ω for ω & T (11). cesses where both cold and hot electrons are involved. Several
2 | Mousatov et al. such processes are possible: a cold electron can scatter off a the explicitly MFL form hot one, denoted ch → ch; a pair of cold electrons can scatter into a cold and a hot electron (cc → ch); etc. 00 Ω Πch(Ω, ~q ) ≈ F (~q ) tanh . [3] The ch → cc or cc → ch processes involve large momentum 2T transfer, and contribute significantly to transport. Moreover, Taking the cold Fermi surface to be round for simplicity, for a large enough cold Fermi surface, all cold electrons can Z d 0 Z 0 participate in cc → ch processes and no ‘short circuiting’ occurs. π d k ~ 0 dΩ 1 R 0 ~ 0 F (~q ) ≈ d δ(|k |−kF c) Im Gh (Ω +Ω, k +~q ) . Other types of scattering are either small angle, and do not 2 (2π) π vF c efficiently degrade current, or are not active across the entire [4] Fermi surface. We discuss this further in the Supplementary Equation [3] was noted in Ref. (6). It is then well known that Information (SI). In the following we show that cc → ch inserting the MFL polarization [3] into the expression [1] for 00 scattering reproduces the phase diagram in Fig.1. the self-energy leads to Σc ∼ max(T, ω) (21). Thus, there is a T -linear scattering rate that becomes ω-linear at higher A Marginal Fermi Liquid from classical fermions. The peak in frequencies. This explains the experimental resistivity and the density of states due to the ‘hot’ γ2 pockets (at H = 0, optical conductivity noted above. A careful evaluation of the from ARPES) occurs at a distance from the chemical potential cc → ch scattering rate in the SI, using the experimentally that roughly coincides with the temperature above which T - determined density of states in the γ2 pockets (3) to evaluate linear resistivity is observed. Let the distance of the peak [2], shows that T -linear scattering is found above precisely from the chemical potential be h, and its width be Wh (see the observed Ttr ≈ 20K. Because cc → ch scattering is large Fig.3). At zero field, Wh . |h|. At temperatures above angle, this result for the single particle scattering rate also |h| + Wh the γ2 fermions are classical and non-degenerate. determines the transport lifetime. A cc → ch scattering process can therefore be expected to The result [3] is obtained for any form of the hot fermion be suppressed by T rather than T 2. We now verify that this Green’s function. The only assumption is that the temperature is the case by showing that this scattering leads to Marginal is above the characteristic energy scales of the hot fermions Fermi Liquid (MFL) (21) behavior of the cold fermions, as (such as Wh and h). We can estimate the magnitude of the has been previously noted (6,7). scattering rate from [4] by taking the hot band to be flat, so R ~ d To connect with the MFL cleanly, we write the scattering that Im Gh (ω, k) = πδ(ω) for some region of area/volume kF h process as the interaction of a cold fermion with a bosonic in the Brillouin zone. The hot regions are smaller than the mode generated by the cold-hot fermion polarizability. We cold Fermi sea, so that kF h kF c. In this case, the phase treat the interaction perturbatively, starting from free cold space that the cold fermion can scatter into is constrained by R ~ fermions with spectral weight Im Gc (ω, k) = πδ(ω−c,~k). Due the size of the hot regions; evaluating the integrals one finds to the interaction, the cold fermions acquire the self energy 00 2 d−2 d 2 the scattering rate Γc ≡ Σc (ω = 0) ∼ λ kF c kF h/vF c T . (we work in d spatial dimensions for generality) We give some details of the kinematics leading to this result in the SI. Specializing to d = 2 and restoring units: Z ddq Z dΩ f(ω − Ω)b(Ω) Σ00(ω,~k) = λ2 c d k2 k T (2π) π f(ω) Γ ∼ λ2 F h B . [5] c 2 R ~ 00 vF c ~ ×Im Gc (ω − Ω, k − ~q )Πch(Ω, ~q ) . [1]
For small kF h, the prefactor of kB T/~ in [5] is small and the In this form, the above equation is transparently Fermi’s perturbative computation is controlled. For Sr3Ru2O7 we Golden rule: λ is the four-fermion coupling constant and can estimate the prefactor using numbers from (9). There the Fermi-Dirac and Bose-Einstein distributions are f(Ω) = are eight hot pockets with P k2 ≈ 0.03A−2. The inverse Ω/T Ω/T i F h,i 1/(e + 1) and b(Ω) = 1/(e − 1). In textbooks (22), [1] velocity of the cold Fermi surfaces in [5] controls the available is commonly expressed differently, using the identity f(ω − phase space for scattering. Averaging over bands we estimate Ω)b(Ω)/f(ω) = f(Ω − ω) + b(Ω). The polarizability is −1 −4 hvF c i ≈ 0.5 × 10 s/m. The microscopic coupling λ has Z d 0 Z 0 0 0 units of ~/mass. Again averaging over bands we estimate 00 d k dΩ f(Ω + Ω)f(−Ω ) λ ∼ hvF /kF i ≈ 0.1 /m . These numbers lead to Γ ∼ kB T/ , Πch(Ω, ~q ) = d c c ~ e ~ (2π) π b(Ω) consistent with the observed ‘Planckian’ scattering rate (9, 23). R 0 ~ 0 R 0 ~ 0 ×Im Gc (Ω , k ) Im Gh (Ω + Ω, k + ~q ) . [2] A similar computation can be done for the ch → ch process, in which a cold electron scatters off an electron R 0 ~ 0 Here Im Gh (Ω + Ω, k + ~q ) is the spectral weight of the hot in the hot region. Both the initial and final states of the fermions, which we can keep general. Similarly to above, hot electron have no Fermi-Dirac suppression. One finds 2 we have written equation [2] in a manifestly physical way, Σ00(ω,~k) ∼ λ2k2 R d q δ(ω − ). This scattering there- c F h (2π)2 c,~k−~q related to common formulations though the identify f(Ω0 + fore behaves as elastic disorder, leading to a scattering rate Ω)f(−Ω0)/b(Ω) = f(Ω0) − f(Ω + Ω0). 2 3 proportional to λ kF h/vF c (the q integral is restricted to a At temperatures greater than |h| + Wh, we have T ∼ |ω| ∼ region of the same size as the hot region). It contributes a 0 0 R 0 ~ 0 |Ω| ∼ |Ω | |Ω +Ω| wherever Im Gh (Ω +Ω, k +~q ) is nonzero. temperature-independent term to resistivity. However, be- This leads to simplifications in [2]. The cold fermions are at cause kF h kF c, this is necessarily small angle scattering for energies T vF ckF c = EF c (where kF c, vF c, EF c denote the the cold electrons. Therefore, the effective transport scattering 2 cold fermions’ Fermi momentum, Fermi velocity, and Fermi rate is suppressed by an additional factor of (kF h/kF c) , so 4 4 energy, respectively), which allows the cold fermion dispersion that Γtr ∼ kF h/kF c × kF hvF c/~ (assuming for simplicity that to be linearized about the Fermi energy. In this way we obtain λ ∼ vF c/kF c is set by a single cold Fermi surface).
Mousatov et al. PNAS | April 14, 2020 | vol. XXX | no. XX | 3 There is also a conventional T 2 contribution to the resistiv- in a small region of the Brillouin zone. However, in using ity from cc → cc scattering. For the MFL T -linear scattering [4] we are also assuming that the integrated single-particle to dominate, the phase space for scattering into cold fermions spectral weight of the hot electrons remains finite as their must be smaller than that for scattering into hot fermions: mass diverges. This leads to a picture in which the growth of 2 kF ckB T/vF c kF h. Geometrically, cc → ch scattering domi- the mass is primarily a band structure effect associated with nates when the area of the hot region is greater than the area a strongly enhanced density of states at the Fermi energy as of the thermally broadened cold annulus shown in Fig.3. This H → Hc (28). This spectral weight must survive the presence 2 2 requirement, that kB T kF h/kF c × EF c, can be satisfied of any quantum critical scattering. despite kF h being small because EF c is a high energy scale. In the SI we verify that the phase space for cc → ch scattering The collapse of Ttr at the critical field. The hot electrons dom- indeed dominates in the T -linear regime of Sr3Ru2O7. inate the specific heat at low temperatures, especially as the critical field is approached. At temperatures above the peak Low temperature approach to the critical field. The maximum in the density of states their contribution becomes small: in the density of states in Fig.3 is at a distance |h| from the 2 2 2 3 γhot ∼ kF h max Wh , h /(kB T ). In computing γhot here the chemical potential. At temperatures well below this scale all of chemical potential is kept fixed and constant, controlled by the the fermions are degenerate, and electron-electron scattering large total number of cold electrons. The cold fermions domi- 2 leads to the usual T resistivity. If |h| < Wh, however, there nate the specific heat in this regime. The MFL scattering rate remains a degenerate Fermi pocket of hot fermions that makes 00 Σc ∼ max(T, ω) implies a logarithmic effective mass enhance- a large contribution to the density of states and cc → ch 2 2 2 2 ment so that ∆γcold ∼ kB ∆m?c ∼ kB m?c · kF /kF · log(Λ/T ). scattering is still important. In this regime ω ∼ Ω ∼ Ω0 ∼ h c Here m?c is the unrenormalized cold electron mass. The en- Ω + Ω0 ∼ T , and the formulae above give the scattering rate ergy cutoff scale Λ ∼ vF ckF h because MFL scattering Γc ∼ ω (re-instating factors of kB and ~ in the final term) requires ω/vF c . kF h, in order for the scattering phase space to be determined by the hot fermion pocket (see SI). Pre- k m (k T )2 Γ ∼ λ2 F h T 2 ∼ ?h B , [6] cisely such a logarithmic temperature dependence is seen c v v2 m E F h F c ?c ~ F c cleanly at low temperatures at the critical field, as we re- where we define the mass of the heavy electrons through their called above. The onset of logarithmic behavior is also visible in the data away from the critical field, at temperatures above density of states at the Fermi level: m?h = νh(0)/2π, and T . At temperatures below T , within the hot electron peak similarly for m?c. In the final term in [6] we again assumed tr tr in the density of states, the hot fermions become degener- for simplicity that λ ∼ ~/m?c is set by a single cold Fermi surface. See the SI for details of the kinematics leading to [6]. ate and the cold fermion mass enhancement is cut off as 2 2 2 The important factor is the ratio m?h/m?c that, we will posit, γcold ∼ kB m?c · kF h/kF c · log Λ/(|h| + Wh) . The relative becomes large as the critical field is approached. The scattering contribution of hot and cold electrons to the low temperature rate Γc determines the resistivity through the Drude formula γ at the H ≈ Hc is discussed further in the SI. 2 ρ = m?cΓc/(nce ). The resulting scaling of the resistivity with The observed behaviors of the specific heat and transport 2 the large hot mass is therefore ρ ∝ m?hT . In contrast, if all data shown in Fig.1 are entirely reproduced by the model fermions have a strongly enhanced mass (become hot) or if of dominant cc → ch scattering if Ttr ∼ |h| + Wh collapses ch → ch scattering dominates (for example, if the hot band is towards zero at the critical field. This collapse is cut off near 2 2 large in the Brillouin zone), then ρ ∝ m?hT . the critical field by the onset of spin ordering at around 1K, At the same low temperatures, the electronic specific heat as shown in Fig.1. Therefore the width of the peak in the is also given by the conventional Fermi liquid formula. If density of states need not strictly go to zero. m?h m?c then hot electrons dominate the specific heat, even The data suggests that it is predominantly the scale |h| while cold electrons dominate transport. The enhancement of that drives the collapse of Ttr toward critical field. We recalled the specific heat coefficient γ due to the increasing mass of above that a peak in the specific heat is seen at a temperature 2 the hot fermions is ∆γ ∼ kB m?h. Together with the results Tpk. This Schottky anomaly-like peak in γ(T ) is quantitatively in the previous paragraph, this leads to A ∼ ∆γ, where A is explained at zero field by the peak in the density of states seen 2 the coefficient in the resistivity ρ ∼ AT . The notation ∆γ in ARPES (3). As the temperature is raised more states in the refers to the fact that the specific heat of the cold fermions hot region become accessible and hence the specific heat coeffi- has been subtracted out.∗ We have recovered precisely the cient increases until the temperature crosses ∼ max(|h|,Wh), observed linear scaling shown in Fig.2. The key input is beyond which the specific heat starts to decrease (and become the hypothesis that m?h becomes large as H → Hc, which dominated by the cold electrons). The simultaneous collapse leads to the dominance of cc → ch scattering. This scaling of T and T as the critical field is approached is therefore 2 pk tr is distinct from the Kadowaki-Woods relation A ∼ γ (24), naturally explained by | | → 0. Such behavior is consistent 2 2 h which instead follows from the resistivity ρ ∝ m?hT . with a recent band structure analysis (28). As noted above, Microscopically, the mass enhancement has been assumed the essential requirement on the width Wh of the peak is that to be tied up with metamagnetic quantum criticality of the it should be of order 1 kelvin at the critical field, so that the hot electrons (2, 25–27). The results above are not sensitive MFL behavior can persist all the way down to the onset of to the details of this physics beyond requiring a divergent m?h the low temperature ordered phase. ∗ As m?h diverges, m?c also becomes enhanced due to cc→ch scattering. We will see shortly As h → 0, the density of states at the chemical potential that the enhancement is only logarithmic: m ∝ k2 m v /k . Close to the ∆ ?c F h log ( ?h F c F h) increases. This leads to an increase in the specific heat coeffi- 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 of the specific heat data in the cient at low temperature, tying the collapse of Ttr to the low SI suggests that ∆m?c ∼ m?c is not negligible. temperature mass enhancement we described above.
4 | Mousatov et al. In the scenario we have outlined, non-degenerate hot ~k0 must be near the Fermi surface means this latter integral is d−2 fermions are important even at low temperatures at the critical over a region of size kF c . All told, we obtain the decay rate field. In this case, as noted above, elastic ch → ch scattering gives an additional contribution to the residual, temperature- kd−2 Z T kd kd−2 k T k T d/z−η Γ ∼ λ2 F c T d ν () ∼ λ2 F h F c B B . independent resistivity. A strong peak in the residual re- c 2 h 2 vF c −T vF c ~ Wh sistivity at the critical field is indeed seen in the data (2). [9] Furthermore, the Lorenz ratio at low temperatures shows The scattering rate [9] is the number of states in the peak increased elastic scattering at the critical field (29). that are thermally accessible. In the final term we re-inserted Our discussion has been purely in terms of the fermionic factors of ~ and kB . band structure. Direct contributions to transport and thermo- There can be logarithmic corrections to [9]. For example, a dynamics from quantum critical collective modes is not neces- van Hove point in two dimensions has z = 2 with the dispersion 2 2 2 sary to explain the data, as considered in (30, 31). Nonetheless, ω ∼ kx − ky = k cos(2θ). This leads to a logarithmic diver- we mention two ways in which such bosonic physics could ad- gence in the integrals over q0 in [2] at θ ≈ ±π/4. Assuming the 2 ditionally be present. Firstly, quantum critical fluctuations anomalous dimension η is small, we obtain Γc ∼ T log(Wh/T ). of an overdamped bosonic order parameter with zB = 2 in Another interesting general case is a multi-critical point with two dimensions would contribute to a logarithmic specific heat z = 4 in two dimensions (28, 33). As long as the anomalous 3/2 at low temperatures (32). Secondly, order parameter fluctua- dimension η is small, this gives Γc ∼ T . Both of these two tions provide an additional scattering mechanism. A different scalings of Γc with temperature have been obtained previously source of T -linear scattering will be called for at the lowest in (34) from a variational Boltzmann equation computation. temperatures if Wh does not in fact collapse down to around Band structure effects on transport are also considered in 1K at the critical field (as we assumed above). (35, 36). In our discussion above, the scattering mechanism responsible for the resistivity is physically transparent. Scattering into a divergent density of states. Taking a step back from Sr3Ru2O7, nondegenerate fermions will be present Scattering into the van Hove point in Sr2RuO4. Under uniaxial in a localized region of the Brillouin zone whenever a divergence pressure, Sr2RuO4 undergoes a Lifshitz transition, traversing in the density of states occurs at the chemical potential, such a van Hove singularity at a critical compressive strain εvH as at a van Hove point. In general, a dispersion with the (37). Resistivity measurements on very pure samples show z scaling h,~k ∼ k (this need not be a minimum in k, e.g. at a a characteristic strange metal ‘fan’ emanating from a crit- 2 2 ical strain at zero temperature (13). Similar behavior has van Hove point h,~k ∼ kx − ky) leads to a density of states been found upon traversing the van Hove point by chemical d/z−1 νh() ∼ , [7] doping (38, 39) or by epitaxial strain (40). Outside the fan, the resistivity ρ ∼ T 2. In the uniaxial strain experiment the 2 which is divergent for z > d (for z = d there can be a loga- resistivity near the critical strain fits ρ ∼ T log(TvH/T ), with rithmic divergence). While this section shares with our earlier TvH = 230K, to excellent accuracy over a range of 40 times discussion the importance of nondegenerate fermions, it de- the residual resistivity (13). This is precisely the behavior scribes a different scenario. The MFL discussed above arises at obtained above for cc → ch scattering near a van Hove singu- temperatures above a sharp peak in the density of states. The larity. The single-particle scattering rate scales as T due to physics discussed here instead arises at temperature within ch→ch proceeses (41). a broader, power law or logarithmic, peak that is located at At high temperatures, the resistivity of unstrained Sr2RuO4 the Fermi energy. The density of states of the hot electrons gradually crosses over from quadratic to linear temperature diverges while Wh remains finite. dependence. The linear regime onsets about 600K(42). It is The lifetime of cold fermions due to cc → ch scattering interesting to note that this temperature is roughly comparable can again be computed from [1] and [2] above. The energies to the scale TvH mentioned above. This conceivably suggests 0 are now ω ∼ Ω ∼ Ω ∼ T EF c. The dispersion of the that the high temperature T -linear behavior might also be cold electrons can be linearized about their Fermi surfaces, as due to cc → ch scattering into the band containing the van previously. The hot electrons obey Hove point, although the ‘hot’ band is not small in this case. η z R ~ 1 Wh ω kF h Outlook. We have shown that the strange metal phenomenol- Im Gh (ω, k) = G z , [8] ω ω k Wh ogy of two strontium ruthenates can be understood from the for some scaling function G and, for generality, allowing for cc → ch scattering of cold electrons into small regions of hot, non-degenerate electrons. This simple process is a fermionic an anomalous dimension η for the hot electrons. Here kF h is a characteristic momentum scale of the hot fermions. This analogue of the well-established T -linear scattering rate that anomalous dimension further shifts the exponent in the density arises from scattering off classical bosons. It is distinct, how- of states [7], so that ν () ∼ R ddk Im GR(,~k) ∼ d/z−1−η. ever, from scattering off a classical fermion, which would h h correspond to ch → ch.† The simplest case of free electrons is Im GR(ω,~k) = πδ(ω − h In Sr Ru O in particular, the proposed dominance of cc kz W /kz ). The scattering kinematics is similar to the cases 3 2 7 h F h → ch scattering successfully explains quantitative properties considered previously, and is described in detail in the SI. of dc and optical charge transport and specific heat. It will be The resulting scattering rate follows from scaling: of the two momentum integrals in [1] and [2], one is over the hot region †In SDW transitions, scattering off composite operators at a hot spot has been argued to lead to in the Brillouin zone, with |~k0 + ~q| ∼ T 1/z, while the other is ‘lukewarm’ fermions over the entire Fermi surface (43–45). This is analogous to ch → ch scattering in our framework. In particular, the fact that the scattering is small angle means that it does not over the cold Fermi surface. The constraint that ~k,~k − ~q and strongly impact transport. It may be interesting to look at cc → ch scattering in those models.
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B 62(10):R6089–R6092. peak moving towards the Fermi energy (48) and is therefore 21. Varma CM, Littlewood PB, Schmitt-Rink S, Abrahams E, Ruckenstein AE (1989) Phe- difficult to square with several aspects of the phenomenology. nomenology of the normal state of Cu-O high-temperature superconductors. Phys. Rev. Lett. Turning to other families of strange metals, heavy fermion 63(18):1996–1999. 22. Mahan G (2000) Many-body physics. Springer. systems contain narrow peaks in the density of states. A 23. Zaanen J (2004) Why the temperature is high. Nature 430:512. dichotomy between hot and cold fermions may be at work 24. Kadowaki K, Woods S (1986) Universal relationship of the resistivity and specific heat in in those cases also, and hence they are candidates for the heavy-fermion compounds. Solid State Communications 58(8):507 – 509. 25. Perry RS, et al. (2001) Metamagnetism and Critical Fluctuations in High Quality Single Crys- approach we have taken here. For more general strange metals, tals of the Bilayer Ruthenate Sr3Ru2O7. Phys. Rev. Lett. 86(12):2661–2664. however, the following caveat should be kept in mind. While 26. Millis AJ, Schofield AJ, Lonzarich GG, Grigera SA (2002) Metamagnetic quantum criticality in metals. Phys. Rev. Lett. 88(21):217204. controlled quantum Monte Carlo modelling of e.g. density 27. Tokiwa Y, Mchalwat M, Perry RS, Gegenwart P (2016) Multiple Metamagnetic Quantum Criti- wave or Ising nematic criticality in metals clearly shows distinct cality in Sr3Ru2O7. Phys. Rev. Lett. 116(22):226402. hot and cold regions of the Fermi surface (49), the hot regions 28. Efremov DV, et al. (2018) Multicritical fermi surface topological transitions. do not necessarily lead to peaks in the density of states because 29. Ronning F, et al. (2006) Thermal conductivity in the vicinity of the quantum critical end point in sr3ru2o7. Phys. Rev. Lett. 97(6):067005. the spectral weight of the hot fermions can vanish in tandem 30. Rost AW, et al. (2010) Power law specific heat divergence in Sr3Ru2O7. Physica Status with a divergent mass (50). It remains to be understood how Solidi (b) 247(3):513–515. 31. Mackenzie A, Bruin J, Borzi R, Rost A, Grigera S (2012) Quantum criticality and the formation variants of the cc → ch scattering mechanism identified in this of a putative electronic liquid crystal in Sr3Ru2O7. Physica C: Superconductivity 481:207 – work could underpin the strange metallic physics arising in 214. Stripes and Electronic Liquid Crystals in Strongly Correlated Materials. those cases. 32. Millis AJ (1993) Effect of a nonzero temperature on quantum critical points in itinerant fermion systems. Phys. Rev. B 48(10):7183–7196. 33. Yuan NFQ, Isobe H, Fu L (2019) Magic of high order van Hove singularity. Data availability. The data and codes used in this work are 34. Hlubina R (1996) Effect of impurities on the transport properties in the van hove scenario. available upon request. Phys. Rev. B 53(17):11344–11347. 35. Buhmann JM (2013) Unconventional scaling of resistivity in two-dimensional fermi liquids. Phys. Rev. B 88(24):245128. ACKNOWLEDGMENTS. We have had helpful discussions with 36. Herman Fcv, Buhmann J, Fischer MH, Sigrist M (2019) Deviation from fermi-liquid transport Chandra Varma, Andrew Mackenzie, Andrey Chubukov and Jan behavior in the vicinity of a van hove singularity. Phys. Rev. B 99(18):184107. Zaanen. This work is supported by the Department of Energy, Office 37. Steppke A, et al. (2017) Strong peak in Tc of Sr2RuO4 under uniaxial pressure. Science 355(6321):9398. of Basic Energy Sciences, under Contract No. DEAC02-76SF00515. 38. Kikugawa N, Bergemann C, Mackenzie AP, Maeno Y (2004) Band-selective modification EB acknowledges financial support from the European Research of the magnetic fluctuations in Sr2RuO4: A study of substitution effects. Phys. Rev. B Council (ERC) under grant HQMAT (grant no. 817799), the Israel- 70(13):134520. US Binational Science Foundation (BSF), and the hospitality of the 39. Shen KM, et al. (2007) Evolution of the Fermi Surface and Quasiparticle Renormalization Aspen Center for Physics, supported by NSF grant PHY-1607611, through a van Hove Singularity in Sr2−y Lay RuO4. Phys. Rev. Lett. 99(18):187001. where part of this work was done. CHM is supported by an NSF 40. Burganov B, et al. (2016) Strain Control of Fermiology and Many-Body Interactions in Two- graduate fellowship. Collaborative work for this paper was made Dimensional Ruthenates. Phys. Rev. Lett. 116(19):197003. possible by the Nordita program “Bounding Transport and Chaos in 41. Pattnaik PC, Kane CL, Newns DM, Tsuei CC (1992) Evidence for the van hove scenario in high-temperature superconductivity from quasiparticle-lifetime broadening. Phys. Rev. B Condensed Matter and Holography”, which was additionally funded 45(10):5714–5717. by ICAM and Vetenskapradet. 42. Tyler AW, Mackenzie AP, NishiZaki S, Maeno Y (1998) High-temperature resistivity of Bad metallic transport in a good metal. Phys. Rev. B 58(16):R10107–R10110. 1. Sachdev S, Keimer B (2011) Quantum criticality. Physics Today 64:29. Sr2RuO4 : 43. Hartnoll SA, Hofman DM, Metlitski MA, Sachdev S (2011) Quantum critical response at the 2. Grigera SA, et al. (2001) Magnetic Field-Tuned Quantum Criticality in the Metallic Ruthenate onset of spin-density-wave order in two-dimensional metals. Phys. Rev. B 84(12):125115. Sr3Ru2O7. Science 294(5541):329–332. 44. Chubukov AV, Maslov DL, Yudson VI (2014) Optical conductivity of a two-dimensional metal 3. Tamai A, et al. (2008) Fermi Surface and van Hove Singularities in the Itinerant Metamagnet at the onset of spin-density-wave order. Phys. Rev. B 89(15):155126. Sr3Ru2O7. Phys. Rev. Lett. 101(2):026407. 45. Abrahams E, Schmalian J, Wölfle P (2014) Strong-coupling theory of heavy-fermion criticality. 4. Hlubina R, Rice TM (1995) Resistivity as a function of temperature for models with hot spots Phys. Rev. B 90(4):045105. on the fermi surface. Phys. Rev. B 51(14):9253–9260. 46. Perry R, et al. (2000) Hall effect of Sr Ru O . Physica B: Cond. Mat. 284-288:1469 – 1470. 5. Rosch A (1999) Interplay of disorder and spin fluctuations in the resistivity near a quantum 3 2 7 47. Liu Y, et al. (2001) Electrical transport properties of single-crystal The possible critical point. Phys. Rev. Lett. 82(21):4280–4283. Sr3Ru2O7 : existence of an antiferromagnetic instability at low temperatures. Phys. Rev. B 63(17):174435. 6. Varma CM (1994) Theoretical framework for the normal state of copper oxide metals in 48. Iwaya K, et al. (2007) Local Tunneling Spectroscopy across a Metamagnetic Critical Point in Strongly Correlated Electronic Systems, the Los Alamos Symposium 1991, eds. Hebard AF, the Bilayer Ruthenate . Phys. Rev. Lett. 99(5):057208. et al. (Addison-Wesley, Reading MA), pp. 573–603. Sr3Ru2O7 49. Berg E, Lederer S, Schattner Y, Trebst S (2019) Monte carlo studies of quantum critical metals. 7. Ruckenstein AE, Varma CM (1991) A theory of marginal fermi-liquids. Physica C: Supercon- Annual Review of Condensed Matter Physics 10(1):63–84. ductivity 185-189:134 – 140. 50. Schattner Y, Gerlach MH, Trebst S, Berg E (2016) Competing orders in a nearly antiferromag- 8. Capogna L, et al. (2002) Sensitivity to Disorder of the Metallic State in the Ruthenates. Phys. netic metal. Phys. Rev. Lett. 117(9):097002. Rev. Lett. 88(7):076602.
6 | Mousatov et al. Supplementary Material
4
focused on cc → ch scattering. Here we firstly discuss in a little 3
the relatively low density of states of the cold electrons, these 2
fer (if the h electron scatters within the same hot pocket), 1
or they involve very specific momenta corresponding to the 1
Mousatov et al. PNAS | April 14, 2020 | vol. XXX | no. XX | 7 ~
c
k
k
~ ~
~ ✓ k
~
Fig. S2. Kinematics for cc → ch scattering. Top right half shows the phase space available to c → h scattering for a fixed momentum transfer ~q and energy transfer ~ Ω = 0 as representative cases. The green arcs, connected by ~q, depict the phase k
In the lower left half of Fig. S2 the first restriction in the integral δq˜⊥ = δq⊥ cos θ + δqk sin θ, over ~q forces ~k − ~q to lie within momentum k of the blue dashed F h δq˜k = −δq⊥ sin θ + δqk cos θ. [S9] circle. The second restriction requires ~k − ~q to lie within T of the vF c cold Fermi surface. The two regions denoted by these restrictions Using [S8,S9], Eq. [S7] becomes T intersect in the green arcs of length ∼ kF h and width ∼ v , so Z 2 2 0 F c ~ 2 d δqd˜ δk 0 T Γc(k, T ) = 2πλ δ vF cδk + vF cδq˜⊥ − ~ ~0 that the phase space available to ~q goes like kF h. It follows that 4 ⊥ h,kh+δk +δ~q vF c (2π) T k T 0 0 00 2 F h × f(vF cδk⊥)[1 − f(−vF cδq˜⊥)][1 − f(vF cδk⊥ + vF cδq˜⊥)]. Σc (Ω, ~q ) ∼ λ min( , kF h) , [S5] vF h vF c vF c [S10] thereby reproducing [5] and [6] in the main text. Let us define the generalized hot density of states It is straightforward to generalize these geometric arguments to general dimension d. The momentum ~k0 has phase space ∼ 0 Z d δkk d δq˜k d−2 T kF min(kF h, ) because the one-dimensional arc in the upper ν˜ (ε) = δ ε − ~ ~0 . [S11] h vF h h 2 h,kh+δk +δ~q right of Fig. S2 becomes a (d − 1)-dimensional surface. Similarly, (2π) d−2 T the momentum ~q has phase space ∼ k kF h × . The self ‡ F c vF c There are in fact two distinct scattering processes that need to be summed over, related by an energy for general dimension d is therefore exchange of the two outgoing electrons, each with its own matrix element. The cc→ch scattering λ d−2 d−1 amplitude includes both processes. k k § 00 2 F c F h T For the total scattering rate out of ~k, we should add also the ch→cc process; the analysis of this Σ (~q, Ω) ∼ λ min(kF , ) T. [S6] c v2 h v process is similar to that of cc→ch, and gives the same parameteric temperature dependence. F c F h
8 | Mousatov et al. 3 80
2 60
( 0
k 20
↵
↵
-2
1000 -3
-3 -2 -1 0 1 2 3 800 kx (1/a)
Fig. S4. Regions of the Brillouin Zone occupied by cold fermions at a temperature of 600 20 K (in blue), as well as the approximate region occupied by hot fermions (in red). The cold regions are defined by energies between −20 K and 20 K, determined 400 from treating each band velocity (taken from Refs. (9, 18)) as isotropic. Fermi surface contours and hot regions from Ref. (3). The lattice parameter is taken to be a = 3.89 200 A.
0 0 10 20 30 40
0 Finally, we change variables from (δkk, δq˜k) to Fig. S5. (Left) observed density of states for the hot electron peak (from Ref. (3)). 0 (Right) Scattering rate [S15] calculated from hot density of states, due to cc → ch (k1, k2) = (−δq˜k sin θ, δq˜k cos θ +δkk)+(δq˜⊥ cos θ +δk⊥, δq˜⊥ sin θ). [S12] scattering in Sr3Ru2O7. This gives Z 2 dk1dk2 2 ν˜h(ε) = δ ε − ~ = νh(ε), | sin 2θ| (2π)2 h,kh+(k1,k2) | sin 2θ| density of states, which is at around 2 – 4 meV below the chemical [S13] potential. where νh(ε) is the hot density of states. The factor 2/| sin 2θ| originates from the Jacobian; we have assumed that sin 2θ 6= 0. At Accessible phase space for hot and cold scattering in Sr3Ru2O7. points where sin 2θ = 0, the cold dispersion cannot be linearized, The cc → ch scattering we have focused on competes with cc → cc and a more careful treatment is needed. However, since this occurs scattering. The latter leads to a conventional T 2 scattering rate. In only at a discrete set of points, the transport properties are not order for a strong T -linear scattering to be visible, the phase space expected to be affected. for cc → ch scattering should be comparable or greater than that Inserting [S13] in [S10], we arrive at the result: for cc → cc scattering (assuming that the various scatterings have comparable interaction strengths). In the main text we explained Z T that this condition can be met fairly naturally because the region ~ 2 T Γc(k, T ) ∼ λ d νh(), [S14] of the Brillouin Zone occupied by the hot fermions is of order k2, v2 h F c −T while a cold fermion occupies a region of size kcT/vF c and T/vF c −α is small for temperatures well below the Fermi energy. We consider the divergent hot density of states νh() ∼ ( − h) |−h| Sr3Ru2O7 has a complicated band structure, which in turn [or νh() ∼ ln , corresponding to the case of a van Hove Wh complicates the estimation of the available scattering phase space singularity in d = 2] within an energy range of Wh. At low tem- as a function of temperature. We will use the ARPES data from 2 peratures T < |h| where the peak in νh() is not excited, Γc ∼ T . Refs. (3, 17), together with Fermi velocities extracted from quantum At intermediate temperatures |h| < T < Wh, the integral gives oscillations in Refs. (9, 18). There are five cold bands at the chemical 1−α 2−α T [or T ln(T/Wh), in the van Hove case], so that Γc ∼ T [or potential. However, the bands have a nontrivial structure already 2 T ln(T/Wh)], respectively. The former is consistent with Eq. [9] in at low energies of a few meV from the chemical potential. This the main text, with 1 + η − d/z = α. At high temperatures Wh < T , means that the well-characterized band structure at the chemical 1 the universal Γc ∼ T is recovered. potential alone is insufficient to determine the scattering phase space at temperatures that are well within the observed T -linear Onset of T-linear scattering. We can evaluate [S10] more carefully regime. For example, the δ band has a depth of only about 8 meV to get a more precise version of Eq. [S14]. Integrating over the below the chemical potential. The Fermi velocity of this band at the momenta at fixed hot fermion energy gives chemical potential is small, only a factor of 2 greater than that of the hot fermions. At relatively low temperatures, then, it is possible λ2 Z ν () Γ ∝ d h . [S15] that this band should be considered as ‘hot’. c v2 sinh(/k T ) F c B We have therefore estimated the available phase space at T = 20 K, the lowest temperature showing T -linear transport in zero field. The hot electron density of states νh() for Sr3Ru2O7 has been de- termined from ARPES measurements in Ref. (3), and is reproduced At this temperature it should be reasonable to use the band structure in Fig. S5 (left). Using that data in the integral [S15] we obtain at the chemical potential. Fig. S4 shows the region of the Brillouin the temperature-dependent scattering rate shown in Fig. S5 (right). Zone occupied by thermally broadened cold bands (in blue) at this 2 temperature, together with the approximate area of the hot, almost The figure exhibits a crossover from Γc ∼ T to Γc ∼ T at the flat band (in red). The thickness of the blue lines is 2kB T , such scale Ttr ≈ 20K, in agreement with the transport data discussed vF in the main text. It is interesting to note that T -linear scattering that thicker cold bands have a lower Fermi velocity. The red regions onsets before the temperature reaches the center of the peak in the denote the approximate area of the Brillouin zone where there is
Mousatov et al. PNAS | April 14, 2020 | vol. XXX | no. XX | 9 a nearly-flat band (to within ±1meV) centered at an energy of the spin. The area of the hot region shown should be multiplied about 3meV below the Fermi energy (3) (this is the region that by a factor of 4 (including the spin multiplicity). The thermally gives rise to the peak in the DOS shown in Fig. S5 above). In broadened cold bands are found to occupy a phase space of roughly comparing the total area occupied in the Brillouin Zone, one must (1.3, 0.61, 0.84, 3.8, 2.0)/a2, summing up to 9.0/a2, while the hot account for the multiplicities discussed in Refs. (9, 18). For the regions have a total phase space of about 10/a2. Therefore, indeed, bands (δ, α1, α2, γ1, β) this involves multiplying the area shown in the hot electron phase competes with and slightly dominates over Fig. S4 by (2, 1, 1, 2, 1) in addition to an overall factor of 2 for the cold electron phase at this temperature.
10 | Mousatov et al.