Introduction to Contemporary Quantum Matter Physics Lecture 3: Heavy

Lectures by: Profs. Marc Janoschek & Johan Chang Course Outline: Feb. 15-19, 2021 Monday Tuesday Wednesday Thursday Friday Room ZOOM ZOOM ZOOM ZOOM ZOOM 10-10h45 Lecture 1 Lecture 4 Lecture 7 Lecture 10 Lecture 13 Johan Johan Johan Marc Johan Fermi-liquids Strongly Supercond. Magnetism Anomalous Correlated Hall effect Insulators 11-11h45 Lecture 2 Lecture 5 Lecture 8 Lecture 11 Lecture 14 Marc Marc Marc Marc Johan Kondo-physics Quantum Supercond. Skyrmions Thermal Hall Phase Effect and Transitions Conductivity Lunch Break Lunch Break Lunch Break Lunch Break Lunch Break

13h30- Lecture 3 Lecture 6 Lecture 9 Lecture 12 Lecture 15 14h15 Marc Johan Johan Marc Johan Heavy Fermions Non-Fermi Nematicity Spin Liquids Charge Order liquids Exercise Class Exercise Class 14h30-16 14h30-16 Further Reading

• P. Coleman, “Heavy Fermions: at the edge of magnetism, ” in Handbook of Magnetism and Advanced Magnetic Materials (Vol I), J. Wiley and Sons (2007). • D. I. Khomskii, “Basic Aspets of the Quantum Theory of Solids,” Cambridge University Press, (2010). • P. Coleman, “Introduction to Many-Body Physics,” Cambridge: Cambridge University Press (2015). doi:10.1017/CBO9781139020916. • Z. Fisk, H. R. Ott, T. M. Rice & J. L. Smith, “Heavy- metals,“ Nature 320, 124–129 (1986). • J. D. Thompson and Z. Fisk, J. Phys. Soc. Jpn. 81, 011002 (2012). • C. Pfleiderer, Rev. Mod. Phys. 81, 1551 (2009). • Silke Paschen & Qimiao Si, “Quantum phases driven by strong correlations,” Nature Reviews 3, 9–26 (2021). “More is Different” or the Importance of Emergence

e-

-31 Mass = 9.109 × 10 kg Understand the behavior of > 1023 -19 Charge = -1.602 × 10 C electrons in a solid Spin = ½ Interactions: gravity electromagnetic weak From Free Electrons to Complex Electronic Systems

e-

-31 Mass = 9.109 × 10 kg Understand the behavior of > 1023 -19 Charge = -1.602 × 10 C electrons in a solid Spin = ½ Interactions: gravity electromagnetic weak

P. Coleman, “Introduction to Many-Body Physics,” Cambridge: Cambridge University Press (2015). Landau’s to the rescue

A.J. Schofield, Contemporary Phys.,40, 95-115 (1990) q Consider non-interacting of electrons and “turn on interactions slowly”. q Electrons interacts with surrounding in such way that the net effect of the interactions transform them into electron-like with increased effective mass.

Well defined properties of a Fermi liquid:

ρ ~ T² C/T = γ χ = χ0 The First Heavy Material: CeAl3

) Cu 2 K - /mole mJ ( �� C/T � = �. �� ���� ��

T2 (K2)

E. S. R. Gopal, ‘Specific Heats at Low Temperature’ Plenum Press, NY (1966)

K. Andres et al. Phys. Rev. Lett. 35, 1779 (1975)

� Reminder: � = ��� + ��� = �� + �� with �~����

���� � � ��� �.�� ⇒ �� = ⁄�.����� = ���� !!!!!! ���� Heavy Fermion Materials Host a Plethora of Exotic Physics!!! CeRhIn : Unconventional SC & 5 UTe2: Putative Chiral Superconductor Nematic Order

Nem atic S. Ran et al., Science 365, 684-687 (2019) ) D. Aoki et al., J. Phys. Soc. Jpn. 88, 043702 (2019) kbar H // [001] (T) S. Ran et al., arXiv:1905.04343 P ( M. Janoschek, Nature Physics 15, 1211–1212 (2019) F. RonningWhatet al., Nature do548, all313–317 listed(2017) materials have in common?

SmB : Topological Kondo URu2Si2: “Hidden” Order 6

K. H. Kim et al., PRL. 91, 256401 (2003). W. Knafo et al., Nat. Comm. 7 13075 (2016) and Lathanide Intermetallics f-electron Systems as Prototypes For Strong Electronic Correlations

➔ 4f and 5f valence electrons of rare-earth and actinide elements are typically well- localized around their nucleus (i. e. electrons do not contribute to conduction).

➔ The localized unpaired electrons form a magnetic moment. Phil Anderson’s Quantum Elephant

Electrical Resistance of Prototypical HF Metal such as CeAl3 ”Elephantine Version by P. W. Anderson”

~T2

P. W. Anderson, “Epilogue” in “Valence Instabilities and Narrow Band Phenomenon,” p 389-396, Plenum, NY, editor Ron Parks, (1977) → Materials in which the increased ‘effective’ mass can be 10-1000 x free electron mass!

→ This causes complex electronic behavior apparent in the electrical resistivity.

→ High effective mass implies small electronic energies EF ~ 1/me.

→ Only materials for which Fermi liquid behavior is observed due to increased effective electron mass.

Further reading: D. I. Khomskii. Basic Aspets of the Quantum Theory of Solids . Cambridge University Press, 2010. P. Coleman, “Heavy Fermions: electrons at the edge of magnetism” in Handbook of Magnetism and Advanced Magnetic Materials (Vol I), J. Wiley and Sons, 2007 Phil Anderson’s Quantum Elephant How do we get the large effective mass?

CeAl3

~T2

P. W. Anderson, “Epilogue” in “Valence Instabilities and Narrow Band Phenomenon,” p 389-396, Plenum, NY, editor Ron Parks, (1977)

→ By now we know the resistivity increase for low temperatures may be explained by the …

→ But how to explain the decrease for � ⟶ �?

→ And the high effective mass of the electrons?

→ Note: Naively CeAl3 is Al metal with 25 % Ce impurities! One impurity per unit cell!!!! Enter the Kondo Lattice

P. Coleman, “Heavy Fermions: electrons at the edge of magnetism, ” in Handbook of Magnetism and Advanced Magnetic Materials (Vol I), J. Wiley and Sons (2007). Understanding Strongly Correlated Quantum Elephants

Onset of coherence Below Tcoh (this implies no back-scattering) Onset of electronic scattering off f-electron moments

~T2 Formation of heavy electronic quasiparticles

Image courtesy of P. Coleman Nature Materials 11, 185–187 (2012) Observation of Kondo Lattice Coherence (Charge Channel)

URu2Si2

A. R. Schmidt, et al. Nature, 465, 570 (2012)

→ Creation of “heavy” electron band & delocalization of f-electrons in conduction band. Observation of Kondo Lattice Coherence (Charge Channel)

ARPES: CeCoGe1.2Si0.8

Image courtesy of Shin-ichi KIMURA (Osaka Univ.) & Takahiro Ito (Nagoyo Univ.) Probing the Kondo Lattice with Neutron Scattering (Spin Channel)?

CePd3 with Kondo temperature TK ≈ 600 K (~55 meV)

T < T K TK

T > TK

TK

Fanelli, Lawrence et al, J Phys: Condens. Matter 26 (2014) 225602 Revealing Electronic Coherence in CePd3 by means of Neutron Spectroscopy

T = 100 K T = 400 K

E. A. Goremychkin eet al., Science 359, 186 (2018) Electron pockets Hole pockets @ � (0 0 0) @ X (½ 0 0) R ( ½ ½ ½ ) M(½ ½ 0 )

→ DFT+DMFT calculation for CePd3 shows hybridization between f-state in conduction band at low temperature resulting in coherent bands

→ At high temperature f-weight becomes broad continuum in energy (no coherence)

Original Idea to measure interband excitations: S. K. Sinha, J. F. Cooke, A. P. Murani, Proceedings of the 1984 Workshop on “High-Energy Excitations in Condensed Matter,” Tech. rep., Los Alamos National Laboratory (1984). J. F. Cooke, J. A. Blackman, Physical Review B 26, 4410 (1982). Revealing Electronic Coherence in CePd3 by means of Neutron Spectroscopy E = 35 meV L=1 E= 35 meV L=3/2 E= 55 meV L=3/2 T = 100 K T = 400 K

1 3

2 2 1 3

E. A. Goremychkin eet al., Science 359, 186 (2018)

→ Neutron scattering cross-section for this inter-band scattering is proportional to the density of states for the occupied and unoccupied electronic bands:

A few examples of possibly scattering vectors:

1: Q = (0 0 0): ℏω = 35 meV 2: Q = � (0 0 0) - M (½ ½ 0) = (-½ -½ 0): ℏω = 35 meV 3: Q = R (½ ½ ½) - X (½ 0 0) = (0 ½ ½): : ℏω = 55 meV and higher Original Idea to measure interband excitations: S. K. Sinha, J. F. Cooke, A. P. Murani, Proceedings of the 1984 Workshop on “High-Energy Excitations in Condensed Matter,” Tech. rep., Los Alamos National Laboratory (1984). J. F. Cooke, J. A. Blackman, Physical Review B 26, 4410 (1982). Temperature dependence of Coherent Band Scattering

T = 6 K

T = 300 K

→ As expected when measured above the coherence temperature the structure of the scattering in momentum is lost, and the scattering can be described by single impurity Kondo scattering.

→ Note: A Kondo lattice exhibits coherence in charge and spin channels! A few comments on Energy Scales

→ Note 40 T corresponds to a small energy scale of 3.5 meV (typically 1 T ~ 1 K ~ 10 μeV)!!!

→ This allows to tune through five phases with distinct properties!!! Why

� → This is because for heavy fermions �� = ���� ~ is a small energy scale!!! ����

→ What are typical energy scales in solids (binding energies, crystal fields?) Typical Energy Scales in Solids

10-4 10-3 10-2 10-1 100 101 102 Energy per Bond (eV)

Interaction 3d (meV) 4f (meV) 5f (meV) Coulomb 1,000-10,000 1,000-10,000 1,000-10,000 Spin-Orbit 10-100 100 300 Crystal Field 1,000 10 100 Exchange 100 1 10

→ In heavy fermions the Kondo energy and the coherence energy are typically ≲ 1 meV!

→ Where does this small energy scale come from? How come it “makes all the music”? What is the Origin of the Small Emergent Energy Scale in HF?

→ The competition of two large energy scales, notably the Coulomb interaction U and the hybridization V results in local moment formation.

→ The resulting moment leads to the emergence of new small but relevant energy scales! Renormalization Theory

Kenneth G. Wilson Phil W. Anderson

→ Renormalization group theory teaches us to describe complex condensed matter systems using simple models that reproduce the relevant low energy physics! Kondo and RKKY Interaction Kondo Interaction Ruderman–Kittel–Kasuya–Yosida (RKKY) Interaction

~ ρJ2 ~ 1/ρ exp(-1/J ρ) Simeth Figure credit: Wolfgang

…suppresses (screens) …correlates magnetic magnetic moments moments over long-distance!

→ Once local moments are formed in heavy fermion metals, a second magnetic interaction may arise! → The interplay of magnetic moments and the spins of the conduction electrons can result in long-range order through the Ruderman-Kittel Kasuya-Yosida (RKKY) interaction Quantum Criticality & Prototypical Doniach Phase Diagram Ruderman–Kittel–Kasuya–Yosida (RKKY) Kondo Interaction Interaction ~ ρJ2 ~ 1/ρ exp(-1/J ρ)

S Doniach Physica B 91 231 (1977)

→ RKKY and Kondo interactions drive f-electron material phase diagram. → When they mutually cancel we frequently observe emergent behavior. → Note: While this is the commonly used model to understand the behavior in QCPs little quantitative characterization of RKKY and Kondo interactions so far.

→ How do we experimentally tune the two interactions? Quantum Phase Transitions in Heeavy Fermions

CeRhIn5 CePd3 δ-Pu

Further reading: D. I. Khomskii. Basic Aspets of the Quantum Theory of Solids . Cambridge University Press, 2010. C. Pfleiderer, Rev. Mod. Phys. 81, 1551 (2009). H. v. Löhneysen et al., Rev. Mod. Phys. 79, 1015 (2007) P. Coleman and A. J. Schofield, Nature 433, 226-229 (2005) M. B. Maple, MJ et al., J. Low. Temp. Phys. 161, 4-54 (2010) Heavy Fermion Ground State is Prototypical Metallic Quantum Matter Topological Magnetism Unconventional, Exotic and Topological SC Orbital and Hidden Order Topological Matter

T. Kurumaji, et al.,, Science 365, 914 (2019). T. Asaba, MJ, et al., Physical Review B 102 (2020). J. A. Mydosh, Rev. Mod. Phys. 83, 1301 (2011) P. Das, MJ et al., New J Phys 15, 053031 (2013).

C. Pfleiderer, Rev. Mod. Phys. 81, 1551 (2009). M. Kenzelmann, et al. Science 321, 1652 (2008). S. Gerber et al., Nat. Phys. 10 126 (2014) S. Ran et al., Science 365, 684-687 (2019) F. Haslbeck, et al., Phys. Rev. B 99, 014429 (2019). M. Janoschek, Nat →Phys 15,Due 1211 to (2019). the narrow electronic bandwidth HF materials are easy to tune by external parameters (T, P, H, x, …). M. Dzero et al., Complex Structural Behavior of Pu Annu. Rev. Condens. Matter Phys. 7, 249 (2016). P. Rosa, MJ et al., Npj Quantum Materials 5 (2020) Candidates Quantum Criticality Electronic Nematic Order

S. Nakatsuji, et al. Phys. Rev. Lett. 96, 087204 (2006) H. v. Löhneysen et al., Y. Tokiwa et al., Nature Materials, 356–359 (2014) Rev. Mod. Phys. 79, 1015 (2007) F. Ronning, et al. Nature 548, 313 (2017) MJ et al., Science Advances e1500188 (2015) M. B. Maple et al., D. M. Fobes et al., Nature Physics 14, 456–460 (2018) J. Low. Temp. Phys. 161, 4 (2010) MJ et al., PNAS 114, E268 (2017) S. Seo, et al., Phys. Rev. X 10, 011035 (2020)