Emergence, Stability and Decay of Skyrmions in Chiral Magnets

animation C. Schütte Emergence, Stability and Decay of Skyrmions in Chiral Magnets

Christian Pﬂeiderer

Physik-Department Technische Universität München Vortices in Daily Life... Kelvin‘s Vortex Model of Atoms

Lord Kelvin William Thomson perhaps the ﬁrst suggestion of (1824-1907) „topological solitons“ Towards a Uniﬁed Field Theory

key idea: Are bosons non-linear excitations of fermion ﬁelds?

(Are bosons topological solitons of fermion ﬁelds?)

Werner Heisenberg (1901-1976) NP 1932

RMP 29 296 (1957) From Tony Hilton Royle Skyrme to Skyrmions

key idea: Are fermions non-linear excitations of boson ﬁelds?

(Are fermions topological solitons of boson ﬁelds? winding number → baryon number) Tony Skyrme (1922-1987)

Proc. Royal Society London, Series A 260, 130 (1961) Proc. Royal Society London, Series A 262, 237 (1961) Nuclear Physics 31 556 (1962) From Tony Hilton Royle Skyrme to Skyrmions

Tony Skyrme (1922-1987)

Proc. Royal Society London, Series A 260, 130 (1961) Proc. Royal Society London, Series A 262, 237 (1961) Nuclear Physics 31 556 (1962) from Wikipedia from Wikipedia

winding number: +2 On the Winding Number in Spin Systems On the Winding Number in Spin Systems

real space

OP space On the Winding Number in Spin Systems

plot M. Rahn Winding in Two-Dimensional OP-Space

skyrmion (trivial) vortex Discovery of the Skyrmion citations per year 250 PRSL - Ser. A 260, 130 (1961) PRSL - Ser. A 262, 237 (1961) Nuclear Physics 31 556 (1962)

ISI: >3006 citations (170 before ’83)

0 1962 1983 2011

„Static properties of nucleons in the Skyrme model“ Adkins, Nappi, Witten, Nucl. Phys. B 228 552 (1983) ISI: 1484 citations

Ed Witten (ISI 68.800; h=124) 1994

2010 Magnetic Phase Diagram of MnSi

(spin-ﬂop phase)

Mühlbauer, et al. Science 323, 915 (2009) Neubauer, et al. PRL 102 186602 (2009) Magnetic Phase Diagram of B20 Compounds

Fe1-xCoxSi x=20% Fe1-xCoxSi x=20% Manyala et al, Nature Physics (2004) Physics et Nature al, Manyala

P213 insulators Mühlbauer, et al. Science 323, 915 (2009) Cu2OSeO3 Neubauer, et al. PRL 102 186602 (2009) Münzer, et al. PRB(R) 81 041203 (2010) Seki al., Science 336 198 (2012) Adams, et al. J. Phys. Conf. Series (2010) Adams et al., PRL, 108 237204 (2012) Bauer, et al. PRB(R) 82 064404 (2010) illustration from Bos et al. (2008) Real Space Observation with Lorentz Force Microscopy

at room temperature semiconductor insulator

FeGe Fe0.5Co0.5Si Cu2OSeO3

Münzer, et al. PRB(R) 81 041203 (2010) Yu et al., Nature 465 901 (2010)

Yu et al., Nat. Mater. 10 106 (2010) Seki al., Science 336 198 (2012) CP & Rosch, Nature (N&V) 465 880 (2010) Adams et al., PRL, 108 237204 (2012) Outline

● Introductory Remarks on Skyrmions ● Emergence of Skyrmions in Chiral Magnets ➢ Fluctuation-Induced First Order Transition ➢ Magnetic Phase Diagram ➢ Nature of the A-Phase ➢ Poor Man‘s Probe of Topology ● Topological Unwinding of Skyrmions ● Formation of a Topological Non-Fermi Liquid (?) ➢ Non-Fermi Liquid Puzze in MnSi ➢ Hall Effect under Pressure ● List of Untold Stories Challenges in NotSpintronics in this talk

typical current density 1012 A/m2

animation S. Maekawa

Emergent Electrodynamics of Skyrmions

current density: 106 A/m2 !!!

Jonietz et al, Science 330, 1648 (2010) Schulz et al. Nature Physics 8 301 (2012) Everschor et al., PRB 86 054432 (2012) animation A. Rosch

Collaborations

samples bulk properties neutron scattering A. Bauer R. Ritz T. Adams S. Mühlbauer A. Neubauer C. Schnarr A. Chacon R. Georgii W. Münzer M. Halder J. Kindervater W. Häußler S. Gottlieb C. Franz G. Brandl P. Link M. Wagner B. Pedersen F. Rucker S. Dunsiger T. Keller (MPI) M. Janoschek2 P. Böni M. Hirschberger1 F. Jonietz P. Niklowitz3 F. Bernlochner T. Schulz4 A. Tischendorf M. Rahn5 F. Birkelbach

1Princeton 3London TOPFIT 2Los Alamos 4Mainz 5Oxford Collaborations

Köln Berkeley Dresden R. Bamler J. Koralek P. Milde S. Buhrandt D. Maier D. Köhler J. Waizner J. Orenstein J. Seidel C. Schütte A. Vishwanath L. Eng K. Everschor M. Garst Lausanne München B. Binz H. Berger T. Schwarze A. Rosch D. Grundler Braunschweig Utrecht R. Hackl (WMI) P. Lemmens R. Duine

TOPFIT Hierarchical Energy Scales in B20 Compounds and the Fluctuation-Induced First Order Transition Hierarchical Energy Scales in B20 compounds Landau-Lifshitz vol. 8, §52

(1) ferromagnetism (2) Dzyaloshinsky-Moriya

(3) crystal ﬁeld (P213):

Si,Ge Si,Ge locked to <111> or <100> TM TM

B20: no inversion center center inversion no B20: 

TN (K)  (Å) MnGe 170 30 to 60 Mn1-xFexSi < 28 180 to 120 Fe1-xCoxSi < 45 > 300 FeGe 280 700

left-handed right-handed Cu2OSeO3 54 620 Breakthrough for „Itinerant Spin Fluctuations“ @RESEDA (FRM II)

MnSi Ishikawa et al PRB (1985)

Lonzarich JMMM 45 43 (1984) Lonzarich, Taileffer. J. Phys. Cond. Matter 18 4339 (1985) Prediction of a Spontaneous Skyrmion Phase

claim of a skyrmion liquid

Pappas et al., PRL 102, 197202 (2009)

Rößler, Bogdanov, CP, Nature 442, 797 (2006) cf Hamann et al., PRL 107, 037207 (2011) Fluctuation-Induced First Order Transition a helimagnetic Brazovskii transition

FM exchange J a = 11meV

DM-interaction D a2 = J Q a2 = 1.7meV

cubic anisotropy J1 a = 0.37meV

(a: lattice constant)

experiments @ MIRA, FRM II cf. claim of a Bak-Jensen M. Janoschek, M. Garst et al. 1st order transition arXiv/1205.4780 S. Grigoriev et al., PRB 81, 144413 (2010) Fluctuation-Induced First Order Transition a helimagnetic Brazovskii transition

M. Janoschek, M. Garst et al. arXiv/1205.4780 Magnetic Phase Diagram of B20 Compounds Magnetic Phase Diagram of B20 Compounds

mostly a spin-ﬂop phase

(spin-ﬂop phase)

B Ishikawa & Arai JPSJ 53, 2726 (1984)

spin-flop

T Magnetic Phase Diagram of B20 Compounds

(spin-ﬂop phase)

Ishikawa & Arai JPSJ 53, 2726 (1984) ???? helical phase

Kadowaki JPSJ 51, 2433 (1982) Magnetic Phase Diagram of B20 Compounds

(spin-ﬂop phase)

Ishikawa & Arai JPSJ 53, 2726 (1984)

Wilhelm et al., PRL 107, 127203 (2011)

purly based on ac susceptibility @ 1 kHz ill-deﬁned sample shape Magnetic Phase Diagram of B20 Compounds

(spin-ﬂop phase)

Ishikawa & Arai JPSJ 53, 2726 (1984) ???? helical phase

Kadowaki JPSJ 51, 2433 (1982) Magnetic Phase Diagram of B20 Compounds

(spin-ﬂop phase)

Ishikawa & Arai JPSJ 53, 2726 (1984)

Kadowaki JPSJ 51, 2433 (1982) Magnetic Phase Diagram of MnSi Revisited

Brazovskii regime

Bauer, CP, PRB 85, 214418 (2013) Speciﬁc Heat of MnSi

T B (mT) MnSi c B || <110> 3.5 0 ) 1 - +10 K 1

- 3.0 l 120

o T A1 +5 m

J 2.5 (

250 C T +10 2.0 2 700 26 27 28 29 30 31 32 T (K)

Bauer, Garst, CP, PRL 110, 177207 (2013) Speciﬁc Heat of MnSi

ﬁrst order transition

) 100 T 2 (a) 0 mT (b) 50 mT (c) 100 mT (d) T 120 mT (e) 140 mT (f) 145 mT - T c c T c T c

K T c c 1 - l o m

J 80 m (

l e 60 C

) 100 2 (g) 155 mT (h) 165 mT (i) 180 mT (j) 210 mT (k) 220 mT (l) 230 mT - T T A2 T K T T T A2 c 1

- A2 A2 A2 l

o T A1 m

T A1

J 80 T m A1 (

T T A1 A1 T

l e 60 C

) 100 ) 2 (m) 280 mT (n) 340 mT (o) 370 mT (p) 390 mT (q) 410 mT (r) 430 mT 2 (s) - - 70 410 mT K K 1 1

- T - l c l o T o m m c T J 80 c J 60 T

m c m

( T ( c T T

/ T /

l c l e 60 e 50 570 mT C C 28 29 30 28 29 30 28 29 30 28 29 30 28 29 30 28 29 30 26 28 30 T (K) T (K) T (K) T (K) T (K) T (K) T ( K) second order transition

Bauer, Garst, CP, PRL 110, 177207 (2013) Magnetic Phase Diagram of MnSi Revisited

MnSi, B || <110> (a) B (a) C(T) Re Im c1 χac χac ) ΔS 0.5 1 tot - 6 0.4

K TCP 1

- ΔS FM

l A2 0.4 ) o

) 0.3 TCP S C T m Δ ( T 4

A1 0.3 ( t J

n i B

m 0.2 B ( 0.2 S B S 2 A1 VI

Δ 0.1 B 0.1 A2 H FD PM 0 0 0 0 0.1 0.2 0.3 0.4 0.5 27 28 29 30 31 32 B (T) T ( K) (b) (b) TCP

) ΔS 0.4 1 tot

- 5

K C 0.3 1

- ΔS l 4 A2 0.3 ) o ) S T ( m Δ T

A1 0.2 ( 3 t J

n 0.2 S i B m

B B (

S 0.1 0.1 Δ 1 B A2 H FD 0 0 0 0.12 0.16 0.20 0.24 28.0 28.5 29.0 B (T) T (K)

Bauer, Garst, CP, PRL 110, 177207 (2013) Nature of the A-Phase Neutron Scattering Pattern in the A-Phase previous work

cf. Lebech, Bernhoeft (1993) Grigoriev, et al. (2006)

new work

Mühlbauer et al, Science 323, 915 (2009) Very Weak Pinning to Crystal Lattice: Neutrons all measurements at MIRA (FRM II)

(110) (111) (110)

(100)

dependence on orientation Stabilization through anisotropy? = Bogdanov & Yablonskii JETP 68 101 (1989)

spin order not sensitive to orientation!

MnSi measurements @ MIRA, FRM II Fluctuation-Stabilized Multi-q Structure

triple-q + uniform M Binz, Vishwanath, Aji PRL (2006)

perp. single-q: metastable triple-q: metastable

Mühlbauer, et al. Science 323, 915 (2009)

Monte-Carlo (includes ﬂuctuations) B=0

Buhrandt & Fritz, arXiv/1304.6580 Fluctuation-Stabilized Multi-q Structure

triple-q + uniform M Binz, Vishwanath, Aji PRL (2006)

perp. single-q: metastable triple-q: metastable

Mühlbauer, et al. Science 323, 915 (2009)

Monte-Carlo (includes ﬂuctuations)

Buhrandt & Fritz, arXiv/1304.6580 Topological Properties of the Rigorous Solution phase btw fundamental modes projection from above φ 0.5

!0.5

!1.

!1.5

!2.

center: M antiparallel B

winding number per unit cell:

lattice of topological knots skyrmions

Adams et al., PRL 107, 217206 (2011) Poor Man‘s Experimental Probe of Topology (emergent magnetic ﬁeld) From Topological Winding to Berry‘s Phase Hall Effects in Magnetic Materials

normal HE topological HE ρxy anomalous HE electron-like

B hole-like

ρxy = R0B Berry phase (real space) Berry phase (momentum space) cf Nagaosa et al., RMP 82, 1539 (2010) Anomalous Hall Effect in MnSi

A phase (arb. units) χ

p=0

0 0.1 0.2 0.3 0.4 0.5 0.6 B(T) Lee et al. PRB 75, 172403 (2007) see also Neubauer et al., Physica B (2009) Anomalous Hall Effect in MnSi

A phase (arb. units) χ

p=0 Neubauer et al., PRL 102 186602 (2009) 0 0.1 0.2 0.3 0.4 0.5 0.6 B(T) Emergent Magnetic Field of Skyrmions

Pﬂeiderer, Rosch Nature (N&V) 465 880 (2010) conduction electron tracks spin structure: collect Berry phase express as Aharonov-Bohm phase represents effective ﬁeld 1

1 trivial topology: Φ =0 (1) non-trivial topology: B! -132.5T (1) eﬀ ≈− Binz, Vishwanath ∆PhysicaC B 403 1336 (2008) Neubauer, et al. PRL 102 186602 (2009)ln( T ) (2) T ∝− cf giant emergentΦ =0 ﬁelds in MnGe (2) Kanazawa et al., PRL 106 156603 (2011)

χ χ χ T 3/4 (3) ∆C 0 1 ∝ ln(T−) (3) T ∝−

5/3 χ χ ρ χρT03/4 T (4) (4) ∝ 0 −− 1 ∝

5/3 ρ ρ0 T 2µB (5) − ∝ωL = ! (5)

2µB ωL = (6) m ! 0.012µ (6) exp ≈ B

m 0.012µ (7) exp ≈ B m 0.015(5)µ (7) calc ≈ B m 0.015(5)µ (8) calc ≈ B ∆a =4.3 10−4 (8) ∆a a · =4.3 10−4 (9) a ·

∆c −4 ∆c =2.1 10 (9) =2c.1 10−4 · (10) c ·

∆η ∆η −4 =5 10=5−4 10 (11) (10) η c η ·c · ! ! ! ! ! ! ∆η −4 f ∆η =3.8 10 −4 (12) fη FWHM · =3.8 10 (11) " # η FWHM · " # t (13) t (12)

B/B0 (14)

B/B0 (13)

U 3 B (15) $ JQ2 " # U 3 B (14) $ JQ2 " #

B =150mT (15) | | Topological Unwinding of a Skyrmion Lattice by Magnetic Monopoles Skyrmion Lattices in B20 Compounds

MnSi pure metal Mühlbauer, et al. Science 323, 915 (2009) Neubauer, et al. PRL 102 186602 (2009) Janoschek, et al. J. Phys. Conf. Series (2010)

Mn1-xFexSi, Mn1-xCoxSi quantum criticality Bauer, et al. PRB 82 064404 (2010) plot taken from Manyala et al., Nature Materials (2004) et Materials al., Nature from taken Manyala plot

Fe1-xCoxSi semiconductor Münzer, et al. PRB(R) 81 041203 (2010) Adams, et al. J. Phys. Conf. Series (2010) CP et al., J. Phys.: Cond. Matter 82 064404 (2010) Metastable Skyrmion Lattice in Fe1-xCoxSi

Münzer, Neubauer, Mühlbauer, Franz, Adams, Jonietz, Georgii, Böni, Pedersen, Schmidt, Rosch, Pﬂeiderer, PRB(R) 81 041203 (2010) A1 20 mT B1 20 mT Magnetic Force Microscopy

⊗" A2 10 mT B2 10 mT

⊗" A3 0 mT B3 0 mT

⊗" A4 -5 mT B4 -5 mT

⊗" A5 -10 mT B5 -10 mT

⊗" [110] [100] Milde et al, Science 340, 1076 (2013) 20 µm-1 1 µm 200 nm Magnetic Force MFM SANS Microscopy

Milde et al, Science 340, 1076 (2013) Monte Carlo Simulations in the Metastable Regime

(a) (b) 1.2 1 2 h 0.8 ×λ 0.6

W/A 0.4 0.023 0.2 surface 1 surface 2 MP 0 0 0.02 0.04 0.06 0.08 0.1 B/J (c) 0.05 MP

3 AMP h 0.04 ×λ 0.03

0.02

0.01 #monopoles/V

0 0 0.02 0.04 0.06 0.08 0.1 B/J animation C. Schütte Topological Unwinding of Skyrmions by means of Magnetic Monopoles

1 FQ 1 FQ

1 FQ

Paul Dirac

1 FQ prediction of magnetic monopoles to explain quantized electric charge

defects in (emergent) B-ﬁeld with quantized charge Formation of a Topological Non-Fermi Liquid Non-Fermi Liquid Puzzle in MnSi Revisited Magnetic Quantum Phase Transition in MnSi

40 MnSi T (ρ, χ) T (Larmor) c c,L T (ENS) T (Larmor) 30 0 TE

20 T (K)

10 FL NFL

0 0 5 10 15 20 25 p (kbar)

CP, McMullan, Lonzarich Physica B 199-200, 634 (1994) CP, McMullan, Julian, Lonzarich PRB 55, 8330 (1997) Thessieu, CP, Stepanov, Flouquet, JPCM 9, 6677 (1997) CP, Julian, Lonzarich Nature 414, 427 (2001) Doiron-Eyraud et al, Nature (2003) 1

χ γ (1) NFL-Resistivity0 ∝ without Quantum Criticality

40 MnSi T (ρ, χ) 2 T (Larmor) c A γ c,L (2) T (ENS) ∝ T (Larmor) 30 0 TE

20 χ0 (3) T (K)

10 FL NFL C(T ) = γT + . . . 1 (4)

0 ) 0 5 10 15 20 25 2 0.1

p (kbar) cm/K µ Ω (

α 2 0.01 expt. ρ(T ) = ρ0 + AT + . . . /T (5) calc. Δρ

singular für T→0 0.001 0.01 0.1 1 10 100 CP, McMullan, Lonzarich Physica B 199-200, 634 (1994) T (K) CP, McMullan,χ(T Julian,) Lonzarichχ0 PRBc 55on, 8330st (1997) (6) Thessieu, CP, Stepanov,≈ Flouquet,≈ JPCM 9, 6677 (1997) CP, Julian, Lonzarich Nature 414, 427 (2001) Doiron-Eyraud et al, Nature (2003)

− WΩ L = 2.45 10 8 (7) · K2

κ = L (8) σT

2 p0 1 + 3 g(cos θ)Pl(cos θ)do > 0 (9) V0(2π¯h) !

J γ 1 (10) ≈ molK2

∆C(T ) γT (11) c ≈ c

φ1 (12)

φ2 (13)

h − Φ = = 2 10 15Vs (14) 0 2e ·

ˆ2 3 S) ψ = ψ (15) | # 4| #

m µ (16) − l B 1

χ γ (1) NFL-Resistivity0 ∝ without Quantum Criticality

40 MnSi T (ρ, χ) 2 T (Larmor) c A γ c,L (2) T (ENS) ∝ T (Larmor) 30 0 TE

20 χ0 (3) T (K)

10 FL NFL C(T ) = γT + . . . 2.2 (4) 0 2 0 5 10 15 20 25 p (kbar) 1.8

α 1.6 α ρ(T ) = ρ0 + AT + . . . 1.4 (5)

1.2 p = 15.5 kbar 0.5 K < T < 2.0 K singular für T→0 1 0 0.2 0.4 0.6 0.8 1 CP, McMullan, Lonzarich Physica B 199-200, 634 (1994) B (T) CP, McMullan,χ(T Julian,) Lonzarichχ0 PRBc 55on, 8330st (1997) (6) Thessieu, CP, Stepanov,≈ Flouquet,≈ JPCM 9, 6677 (1997) CP, Julian, Lonzarich Nature 414, 427 (2001) Doiron-Eyraud et al, Nature (2003)

− WΩ L = 2.45 10 8 (7) · K2

κ = L (8) σT

2 p0 1 + 3 g(cos θ)Pl(cos θ)do > 0 (9) V0(2π¯h) !

J γ 1 (10) ≈ molK2

∆C(T ) γT (11) c ≈ c

φ1 (12)

φ2 (13)

h − Φ = = 2 10 15Vs (14) 0 2e ·

ˆ2 3 S) ψ = ψ (15) | # 4| #

m µ (16) − l B Neutron Spin-Echo & Larmor Diffraction

sample 3.5x10-3 thermal expansion Co [111] -3 θ 3.0x10 dilatometry [1] B TRISP Larmor diffraction

-3 [1] F.R. Kroeger, C.A. Swenson, TC2 TC3 2.5x10 J. Appl. Phys. 48, 853 (1977) vp v G 2.0x10-3

-3 vt 1.5x10 L1 L2 L(T)/L(5K) - 1 1.0x10-3

TC1 TC4 5.0x10-4 α in out Rekveldt, Keller, Golub, EPL (2001) 0.0

1.0x10-5

5.0x10-6

0.0

-5.0x10-6 difference -1.0x10-5 dilatometry - TRISP 050100150200250300 T[K]

resolution 10-6 !!!!

measurements at TRISP (FRM II) Non-Fermi Liquid Metal without Quantum Criticality

2 40 1 2 7 0 p (kbar) the magnetic moment, but by the weakest energy scale: 0 the pinning potential of the helical order. However, T 6.2 T (ρ, χ) T (Larmor) 0 ) MnSi 1 0 p=0 c 5 12.5 (222)

c,L -

is not seen in the resistivity and susceptibility. More- 6.2 0 12.5 (224) 1

* 13.9 dn over, µ-SRTexp (ENS)eriments show phaTse se g(Larmor)regation below p 8 12.5 ( c 13.9 up

30 0 TE a

and suggest that the partial order may even be dynamic ) 14.2

5 6 p=0 14.5 - 16.1 [20]. The NFL behavior may hence be related to a subtle 0 0 4 1 0 2 10 16.3 * 2 2 quantum critical point. As the experimental probes used ( 4 13.9 T (K ) 2 1

up/dn so far, notably resistivity, susceptibility, neutron scatter- 2 14.2 20 a ing and µ-SR show diﬀerent cross-over scales the possible 2 T (K) existence of such a quantum critcal point is the outstand- 0 ing challenge. This issue may be resolved with precision 14.5 -2 16.1 measurements of the lattice constant as the conjugate 16.3 10 2 1 T variaFLble of the control parameter. -4 TE For meaningful experiments in MNFLnSi a resolution of 0 1 0 2 0 3 0 4 0 5 0 the lattice constant better than 10−5 is necessary. Con- T (K) 0 ventional capacitive bulk methods cannot be used in the pressure and tempp*erature rapnge of interest [21, 22]. A FIG. 1: Temperature dependence of magnetic and electronic very elegant method are scattecring experiments, but the contributions, a2, of the lattice constant of MnSi at various 8 pressures. The inset displays the lattice constant at ambient resolution of both, synchrotron radiation and neutron 2 pressure versus T . ) diﬀraction in single crystal studies prior to the work re- partial order above pc: -5 −5 4 ported here were at best ∼ 10 , being ultimately lim- large ‘droplets’ ited by the beam divergence and monochromaticity. Syn- value a0 = 4.58 A˚ in terms of three contributions

(*10 chrotron experiments are moreover limited to tempera- 2 buta(T ,nop) expansion!!! a 0 tures above a few K due to heating effects and they pro- = κp/3 + a1(T ) + a2(T, p) (1) vide information on the surface of the samples only. a0 T→0 -4 Here we show that the resolution limit of conventional The first term, κp/3, describes the pressure dependence scattering experiments may be readily overcome by neu- of the lattice ignoring temperature dependent contribu- 0 tron5Larmor di10ffraction, for15which only20the basic p25rinci- tions, where κ is the volume compressibility. The second ple of operation was demonstrated so far [23]. In Larmor term, a (T ) = αT 2, describes the conventional thermal p (kbar) CP, Böni,1 Keller, Rößler, Rosch, Science 316, 1871 (2007) diffraction the precession of the spin is attached to the expansion, where α is insensitive to pressure and higher neutron path as an ’internal clock’. In comparison to order contributions are not required to describe our data. synchrotron experiments it is unique by providing infor- The third term, a2, accounts for all other contributions, mation on the distribution of lattice constants over the in particular those related to magnetic and electronic entire sample volume. For a detailed introduction we properties. refer to the supplementary information, where we also The temperature dependence of the lattice constant at present measurements on single crystal Cu as a proof of ambient pressure is shown in the inset of Fig. 1. Above principle [24]. 35 K a quadratic temperature dependence is observed, Our study was carried out on the thermal neutron res- where the coefficient α ≈ 3.2 · 10−8K−2 as measured onance spin-echo triple-axis spectrometer, TRISP [25], for three different samples is in excellent agreement with at the neutron source Heinz Maier-Leibnitz (FRM II) the bulk data reported in Ref. [26, 27]. The magneto- at the Technical University of Munich. TRISP offers a expansion below Tc is a few % smaller than that reported world-wide unique combination of high intensity, ultra- in Ref. [26]. < < low background and spin resonance capability necessary In the range 35 K ∼ T ∼200 K, i.e., up to the for Larmor diffraction. highest temperature measured systematically, we find a In the following we report a comprehensive study of the quadratic temperature dependence at all pressures. For lattice constant of several single crystals of MnSi at pres- the four combinations of pressure cells, pressure trans- sures up to 21 kbar and temperatures down to 0.5 K. A mitter and samples studied the coefficient α is always total of four different combinations of pressure cells, pres- constant as function of pressure for each set-up, but dif- sure transmitters and samples were studied. For further fers up to 25% between set-ups. This may be traced to information on the samples, pressure cells and technical pressure changes as function of temperature due to dif- set-up as well as issues such as pressure changes during ferences of thermal expansion of the pressure cell and cool-down and small pressure inhomogeneities we refer to the sample (see supplement). Taking into account the the ’Methods and Supplementary Information’ section at set-up dependence of α we find that isothermal changes the end of this paper. of a(T, p)/a0 are well described by κp/3, where the pres- −7 −1 We discuss our data of the lattice constant, a(T, p), sure as inferred from Tc(p) [17] yields κ ≈ 5.3·10 bar as normalized to its ambient pressure and temperature consistent with ultrasound measurements at Tc [28]. Partial Order in the Non-Fermi Liquid Regime

40 MnSi T (ρ, χ) T (Larmor) c c,L T (ENS) T (Larmor) 30 0 TE

20 T (K)

10 FL NFL

0 0 5 10 15 20 25 p (kbar)

CP, Reznik, Pintschovius, v. Löhneysen, Garst, Rosch Nature 427, 227 (2004) CP, Reznik, Pintschovius, Haug PRL 99, 156406 (2007) Neutron Scattering vs mu-SR

40 MnSi T (ρ, χ) T (Larmor) c c,L T (ENS) T (Larmor) 30 0 TE

20 T (K)

10 FL NFL

0 0 5 10 15 20 25 p (kbar)

with Uemura, Nature Physics 3, 34 (2007) Formation of a Topological Non-Fermi Liquid (?) Hall Effect under Pressure Hall-Effect under Pressure (ﬁrst results)

MnSi

Neubauer et al., PRL 102 186602 (2010) Lee et al., PRL 102 186601 (2010) Hall-Effect under Pressure Revisited

p(kbar)     (a) 100 pc5 7.6 RRR 45 6.7  ) ≈   

m 50 T≈0.83 T 5.9

c C 5.1   Ω

n 0 ⇥ ( 4.6

y x 3.4 

ρ -50 2.9  -100 2.6   2.2 (b) 100 pc8 12.3 RRR≈150     ) 11.5 50

m T 0.96 T ≈ C 10.3 c   Ω 0 9.3 n ( ⇥

y 7.8 x  ρ -50 6.1  3.7 -100  

0.5 (c) 0.2 pcM     ) RRR≈70

. 0 u  . T≈0.95 T f C 4.0

/  B ⇥

7.5 µ 0 ( 10.2 M

  11.8 

-0.2 

(d) B B pcM c2 c2  0.4 RRR 70     ≈ 0

T 0.95 T  ≈ C H 4.0   d ⇥ / 7.5

M 0.2

  d 10.2

11.8 

 0 -1.0 -0.5 0 0.5 1.0  B (T) int 

Ritz, Halder, Franz, Bauer, Wagner, Bamler, Rosch, CP PRB 87, 134424 (2013) Hall-Effect under Pressure Revisited

MnSi, B=0.25 T, p c8 MnSi, B=0.25 T, p c5 75

) (a) fc (d) fc/fh m c 50 Ω n (

C F ,

y 25 x ρ 0

) (b) zfc/fh (e) zfc/fh m c 50 Ω n (

C F

Z 25 , y x ρ 0

75 (c) p(kbar) (f)

) p(kbar) m 0.5 9.3 2.2 5.9 c 50

Ω 3.7 10.3 3.4 6.7 n

( 6.1 11.5 y 5.1 7.6 x 25 ρ 7.8 12.3 Δ

0 0 10 20 30 0 10 20 30 T(K) T(K)

Ritz, Halder, Franz, Bauer, Wagner, Bamler, Rosch, CP PRB 87, 134424 (2013) Hall-Effect under Pressure Revisited

Ritz, Halder, Franz, Bauer, Wagner, Bamler, Rosch, CP PRB 87, 134424 (2013) Overall-Behavior into the NFL Regime

Ritz, Halder, Franz, Bauer, Wagner, CP, Nature 497, 231 (2013) Hall-Effect in the NFL Regime

Ritz, Halder, Franz, Bauer, Wagner, CP, Nature 497, 231 (2013) Formation of a Topological Non-Fermi Liquid

40 T (ρ, χ) T (Larmor) c c,L T (ENS) T (Larmor) 30 0 TE

20 T (K)

10 FL NFL

0 0 5 10 15 20 25 p (kbar)

topological Hall signal

Ritz, Halder, Franz, Bauer, Wagner, CP, Nature 497, 231 (2013) Untold Stories Instead of Conclusions

● emergent electrodynamics ● skyrmion ﬂow ● ab-initio Hall effect ● inelastic neutron scattering ● polarised neutron scattering ● magnetic resonance ● Raman scattering ● routes towards skyrmions ● role of magnetic anisotropy ● uniaxial pressure tuning ● composition tuning ● elasticity moduli ● new systems ● thin ﬁlms ● manipulating skyrmions ● ....