Metamagnetism and Quantum Criticality in Sr3ru2o7

Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Metamagnetism and Quantum Criticality in Sr3Ru2O7

Ben Simons (Cavendish Laboratory)

Collaborators (at St. Andrews):

Andrew Green (theory) Santiago Grigera (experiment & theory) Andrew Mackenzie’s group (experiment)

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Quantum Critical Phenomena — the hype! Continuous phase transition can be driven by quantum dynamical as well as thermal ﬂuctuations

Classical critical region: system static on thermal timescales Quantum critical region: properties controlled by critical ﬂuctuations of soft mode associated with ordering

Quantum critical scaling captured by generalization of Ginzburg-Landau theory due to Hertz (’76) and Millis (’93)

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Quantum Critical Phenomena — the current reality! Continuous phase transitions can be driven by quantum dynamical as well as thermal ﬂuctuations

Classical critical region: system static on thermal timescales Quantum critical region: properties controlled by critical ﬂuctuations of soft mode associated with ordering

Yet, despite appeal of universal classiﬁcation, precious few examples of Hertz-Millis scaling are actually observed!

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Motivation Apparent failure of Hertz-Millis scaling ascribed to eﬀect of non-critical slow modes, disorder, Kondo physics, development of new phase behaviour, etc.

It has been argued that metamagnetic bilayer ruthenate Sr3Ru2O7 provides a benchmark system to test conventional framework of QCP in metals...

However, despite apparent simplicity, Sr3Ru2O7 is found to exhibit remarkable phase behaviour...

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Outline

1 Experiment: metamagnetism in Sr3Ru2O7

2 Theory: from phenomenology to microscopics

3 Summary

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Metamagnetism and quantum criticality

In many weakly ferromagnetic metals (e.g. MnSi, UGe2, ZrZn2, etc.) magnetic transition can be tuned, e.g. by pressure In many (perhaps all?) cases, transition becomes weakly first order before the QCP reached Continued in field, critical line bifurcates at tricritical point to H reveal planes of first order metamagnetic transitions

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Ruthenate family: the Ruddlesten-Popper series Amongst this general class, layered perovskite strontium ruthenate family Srn+1RunO2n+3 assumes distinguished role:

n = 1 n = 2 n = ∞

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Ruthenate family: the Ruddlesten-Popper series Amongst this general class, layered perovskite strontium ruthenate family Srn+1RunO2n+3 assumes distinguished role:

n=2: Sr3Ru2O7 bilayer compound found to be strongly enhanced paramagnetic metal on verge of ferromagnetism with anisotropic Fermi liquid-like correlations a (ρc/ρab ≈ 100)

aIkeda et al., PRB ’00

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Metamagnetic quantum criticality in Sr3Ru2O7

Field angle provides “tuning parameter” to realise QCP:a critical line from maxima in Re χ(ω); ﬁrst order plane from maxima in Im χ(ω)

aGrigera et al., PRB ’03

Re χ Im χ

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Metamagnetic quantum criticality in Sr3Ru2O7

However, ﬁt to Hertz-Millis 5/3 scaling ρ(T ) = ρ0 + AT unconvincinga... while new generation of ultra pure samples expose region of anomalous transportb... with “saturation” of resistivity inside peakc aGrigera et al, Science ’01 bPerry et al., PRL ’04 cGrigera et al., Science ’04 d ln(ρ − ρ ) α = 0 d ln T

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Metamagnetic quantum criticality in Sr3Ru2O7

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Metamagnetic quantum criticality in Sr3Ru2O7

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Phase diagram (c-axis) — combining measurements of:

2 resistivity ∂H ρ ∂T ρ ac susceptibility magnetostriction λ(B) thermal expansion dc magnetisation •

Im χ

Grigera et al., Science ’04

Consistent with new phase?

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Studies of ﬁeld-angle dependence...

T = 100mK

...suggest anomalous region better interpreted as bifurcation of parent metamagnetic transition (Green et al., PRL ’05)

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Landau phenomenology Experiment: metamagnetism in Sr Ru O 3 2 7 Quantum tricriticality Theory: from phenomenology to microscopics Beyond Landau mean-ﬁeld theory Summary Lessons from microscopics

Questions: What is origin of phase bifurcation? Why acute sensitivity to sample purity? Are magnetic/resistance anomalies signatures of new phase? Is the bifurcation phenomenon linked to quantum criticality?

Questions: What is origin of phase bifurcation? Why acute sensitivity to sample purity Are magnetic/resistance anomalies signatures of new phase? Is the bifurcation phenomenon linked to quantum criticality?

Some current proposals: Pomeranchuk instability;a nanoscale charge instabilities;b electron nematic;c magnetic mesophase formation;d ...

aGrigera et al., Science ’05; Nilsson and Castro Neto, ’05 bHonerkamp, PRB ’05 cKee and Kim, PRB ’05 dBinz, Braun, Rice and Sigrist, ’05

Landau theory of phase bifurcation

Vicinity of metamagnetic critical point described by Landau expansion (in m = M − M∗)

βH r u = m2 + m4 − hm V 2 4

However, if sign of u is reversed, must consider generalisation

βH0 r s u 1 = m2 + m3 + m4 + m6 − hm V 2 3 4 6 i.e. only quintic term removable by rescaling

Landau theory of phase bifurcation

s = 0 At tricritical point r = u = h = 0 critical line bifurcates; cf. Landau theory of weak ferromagnetisma

aWohlfarth and Rhodes, Phil. Mag. ’62

βH0 r s u 1 = m2 + m3 + m4 + m6 − hm V 2 3 4 6

Landau theory of phase bifurcation

s 6= 0 Tricritical point becomes “dislocated”

βH0 r s u 1 = m2 + m3 + m4 + m6 − hm V 2 3 4 6

Applied to Sr3Ru2O7...

Rotation in (r,u,h) ⇒ good qualitative ﬁt to experiment and suggests that, for T & 1K, ﬂuctuations controlled by quantum tricritical point...

Quantum tricriticality in Sr3Ru2O7 ...where critical ﬂuctuations controlled by Hertz-Millis action:

Z |ω| Z S = d3q dω r + q2 + |m(q)|2 + u m4 + v m6 0 v|q| 0 0

Self-consistent Renormalisation Conventional QCP r(0) = 0, u(0) > 0 r(T ) ∼ χ−1(T ) ∝ T 4/3 R T 3 5/3 ⇒ ρ(T ) ∝ Im Σ ∼ r(T ) ∝ T QTCP r(0) = u(0) = 0 r(T ) ∼ χ−1(T ) ∝ T 8/3 ⇒ ρ ∝ T 1/3 Data: Grigera et al., unpub.

Beyond Landau phenomenology?

Explanation of resistance anomaly in bifurcation region and sensitivity to sample purity calls for more microscopic theory...

Independence of ρ on T ⇒ domain formation Magnetostriction data ⇒ strong magnetoelastic coupling

Magnetoelastic coupling

Coupling of magnetization to strain ﬁeld ψ = ∇ · u provides mechanisma for phase bifurcation...

Z ∗ 2 βH = βH0[m] + g (M + m) ψ + βHψ[ψ]

ψ ﬂuctuations −ve contribution to u degeneracy of soft ﬂuctuations domain formation and resistance anomaly i.e. not a new phase a Rice ’54, Larkin & Pikin ’69, etc. but a crossover

Field-current anisotropy of resistivity

However, sensitivity of resistance j|| anomaly to relative orientation of current and ﬁeld casts doubt on strain induced domains...

j⊥

Grigera et al., unpublished

...and “secondary” metamagnetic transitions suggest: Bifurcation structure and resistance anomaly more generic...

ρT =100mK, j||

...with evidence for bifurcation of bifurcations!

...and “secondary” metamagnetic transitions suggest: Bifurcation structure and resistance anomaly more generic...

ρT =100mK, j⊥

...with evidence for bifurcation of bifurcations!

Magnetic textures? Is current-field anisotropy compatible with magnetic texture, e.g. spin spiral, oriented by field...? ...consistent with NMR,a which indicates spin fluctuations dominated by AF correlations in bifurcation region...

aKitagawa et al., PRL ’05

...hint provided by microsopic theory of parent transition...

Lessons from microscopic theory of weakly ferromagnetic metal:

Development of tricritical point accompanied by sign reversal of gradient term...a aBerridge, Green, Grigera and BDS, ’06

Z 2 3 βH = rm2 + um4 + vm6 − h · m + u(∇m)2 + v(∇2m)2 + ··· 3 5

...a property familiar from Fulde-Ferrel-Larkin-Ovchinnikov theory of superconductivity

Magnetic “FFLO phase”

For h = 0, low T transition to planar spiral magnetic phase preempts formation of ferromagnetic phase For h 6= 0, spin spiral orients around ﬁeld direction

Application to Sr3Ru2O7? Captures disorder sensitivity, ubiquity, resistance anomaly & sensitivity to orientation, but...

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

Summary Bilayer strontium ruthenate does not appear to be a simple paradigm for metamagnetic quantum criticality Ultra-high purity samples reveal anomalous phase behaviour near putative QCP...... captured by simple Landau phenomenology involving symmetry broken tricritical point structure Recent measurements indicate that phase bifurcation (and resistance anomalies) may be ubiquitous feature of metamagnetic system... while microscopic theory hints at development of magnetic texture in region of bifurcation, a magnetic counterpart of FFLO phase

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7 Experiment: metamagnetism in Sr3Ru2O7 Theory: from phenomenology to microscopics Summary

NbFe2

Grosche et al. ’05

Ben Simons (Cavendish Laboratory) Metamagnetism and Quantum Criticality in Sr3Ru2O7