Exotic quantum phases and phase transitions
MIT Theoretical Physics Retreat Jan 2012 T. Senthil (MIT)
Thursday, April 5, 2012 Conventional condensed matter physics: Landau’s 2 great ideas
1. Integrity of the electron as a quasiparticle in phases of matter (Fermi liquid metals, band insulators, BCS superconductors, spin density wave states, ……) 2. Notion of ``order parameter’’ to describe phases of matter - related notion of spontaneously broken symmetry - basis of phase transition theory
Thursday, April 5, 2012 Landau Fermi liquid theory
• Electrons in a metal: quantum fluid of fermions Fermi surface kz • Inter-electron spacing ~ 1 A ⇒ Very strong Coulomb repulsion ~ 1-10 eV comparable to kinetic energy ky
But effects dramatically weakened due to Pauli exclusion. kx Important `quasiparticle’ states near Filled states unavailable for Fermi surface scatter only weakly off each other. scattering Describes conventional metals
Thursday, April 5, 2012 Order parameter
• Example - ferromagnetism
Ferromagnet: Paramagnet: Spins aligned Spins disordered
Increase temperature
• Spontaneous magnetization: `order parameter’.
• Ordered phase spontaneously breaks spin rotation symmetry.
Thursday, April 5, 2012 Continuous phase transitions - theoretical paradigm
• Phenomena: Critical singularities, universality, scaling.
Thursday, April 5, 2012 Continuous phase transitions - theoretical paradigm
• Phenomena: Critical singularities, universality, scaling,
• Landau: critical singularities due to long wavelength fluctuations of order parameter field.
Thursday, April 5, 2012 Continuous phase transitions - theoretical paradigm
• Phenomena: Critical singularities, scaling,universality
• Landau: critical singularities due to long wavelength fluctuations of order parameter field.
• Landau-Ginzburg-Wilson: Landau ideas + renormalization group - sophisticated theoretical framework
Thursday, April 5, 2012 Example: Ising magnets
• Order parameter: Spontaneous magnetization
• Critical singularities: long wavelength fluctuations of coarse-grained continuum magnetization density
Desribe within continuum field theory
Thursday, April 5, 2012 Phase transitions – thermal versus quantum
Loss of magnetism on heating: thermal fluctuations
Can also have a phase transition at zero temperature as a function of some tuning parameter.
Qualitative change in the quantum ground state of the many particle system
Driven by quantum zero point motion rather than thermal motion - ``quantum phase transition”
Thursday, April 5, 2012 Quantum phase transitions Example 1: He-4 under pressure
(First order) T = 0 melting transition
Thursday, April 5, 2012 Quantum phase transitions Example 2: Quantum Ising magnet
Quantum critical point
Thursday, April 5, 2012 Quantum phase transitions Example 3: Superfluid-insulator transition of 87Rb atoms in a magnetic trap and an optical lattice potential
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Thursday, April 5, 2012 Quantum Phase Transitions: Generalities
Thursday, April 5, 2012 Quantum Phase Transitions: Generalities
Thursday, April 5, 2012 Quantum Phase Transitions: Generalities
• Universal critical singularities similar to thermal phase transitions • Continuous quantum phase transitions can control the finite temperature physics in a region.
T Quantum-critical
Insulator Superfluid Jc J
Thursday, April 5, 2012 Quantum Phase Transitions: Generalities
• Universal critical singularities similar to thermal phase transitions • Continuous quantum phase transitions can control the finite temperature physics in a region.
T Quantum-critical
Insulator Superfluid Jc J
Thursday, April 5, 2012 Quantum phase transitions: Landau- Ginzburg-Wilson description
• Universal critical singularities: Long wavelength, long time fluctuations of Landau order parameter field.
• Describe by suitable continuum quantum field theory at zero temperature.
Succesful in describing many quantum critical phenomena.
Thursday, April 5, 2012 Conventional condensed matter physics: Landau’s 2 great ideas
1. Integrity of the electron as a quasiparticle in phases of matter (Fermi liquid metals, band insulators, BCS superconductors, spin density wave states, ……) 2. Notion of ``order parameter’’ to describe phases of matter - related notion of spontaneously broken symmetry - basis of phase transition theory
Thursday, April 5, 2012 Modern quantum condensed matter physics
In the last 30 years both of these ideas have been challenged enormously by discoveries such as the fractional quantum Hall effect, high temperature superconductivity, and other phenomena.
Many fundamental questions have been raised (and some answered).
Thursday, April 5, 2012 A sample of some basic questions
1. Does the electron have to survive as a quasiparticle in a phase of matter?
Thursday, April 5, 2012 A sample of some basic questions
1. Does the electron have to survive as a quasiparticle in a phase of matter?
2. Does every quantum phase have `elementary excitations’? (i.e does the low energy excitation spectrum admit a particle-like description)?
Thursday, April 5, 2012 A sample of some basic questions
1. Does the electron have to survive as a quasiparticle in a phase of matter?
2. Does every quantum phase have `elementary excitations’? (i.e does the low energy excitation spectrum admit a particle-like description)?
3. Does a clean metal need to have a sharp Fermi surface at T = 0 ?
Thursday, April 5, 2012 A sample of some basic questions
1. Does the electron have to survive as a quasiparticle in a phase of matter?
2. Does every quantum phase have `elementary excitations’? (i.e does the low energy excitation spectrum admit a particle-like description)?
3. Does a clean metal need to have a sharp Fermi surface at T = 0 ?
4. Can bosons form a metal?
Thursday, April 5, 2012 A sample of some basic questions
1. Does the electron have to survive as a quasiparticle in a phase of matter?
2. Does every quantum phase have `elementary excitations’? (i.e does the low energy excitation spectrum admit a particle-like description)?
3. Does a clean metal need to have a sharp Fermi surface at T = 0 ?
4. Can bosons form a metal?
5. Is the order in a phase necessarily captured by a Landau order parameter? Is symmetry breaking the only route to ordering?
Thursday, April 5, 2012 A sample of some basic questions
1. Does the electron have to survive as a quasiparticle in a phase of matter?
2. Does every quantum phase have `elementary excitations’? (i.e does the low energy excitation spectrum admit a particle-like description)?
3. Does a clean metal need to have a sharp Fermi surface at T = 0 ?
4. Can bosons form a metal?
5. Is the order in a phase necessarily captured by a Landau order parameter? Is symmetry breaking the only route to ordering?
6. Is it always correct that singularities at phase transitions are due to slow fluctuations of the order parameter?
Thursday, April 5, 2012 In these lectures I will describe recent theoretical advances in understanding phenomena that defy some aspect of conventional condensed matter physics.
Plan: 1. Lightning reviews of Bose and Fermi liquids, conventional magnetism 2. Exotic ``Mott” insulators (popularly known as quantum spin liquids) 3. Exotic quantum criticality and non-fermi liquid metals
Thursday, April 5, 2012 Some simple observations on correlated many body systems
Thursday, April 5, 2012 Two extreme limits
Thursday, April 5, 2012 Many interesting phenomena happen in `intermediate’ correlation regime where neither kinetic or interaction energy is clearly dominant.
Neither `particle’ nor `wave’ picture clearly superior in this regime.
The physics described in these lectures is due to the competition between kinetic and interaction energies in this regime.
Thursday, April 5, 2012 Example: ``Cuprate” High temperature Superconductors
Metal: (Mott) Insulator: Kinetic energy wins; Interaction energy Electrons as waves wins; Electrons as particles
All the interesting things happen en route from insulator to good metal.
Thursday, April 5, 2012 Delocalized limit: Correlated bose and fermi liquids – a brief discussion
Thursday, April 5, 2012 Review: free Bose gas at T = 0
Bose condensation: all N particles in k = 0 state.
n(k)/N k-space occupation: delta function at k =0.
k
Question for you: What is ground state wavefunction ? 29
Thursday, April 5, 2012 Bose liquid, i.e, bosons with repulsion
Thursday, April 5, 2012 Ground state momentum distribution
n(k)/N n(k)/N
Thursday, April 5, 2012 Fermions: Review of free Fermi gas
Ground state: fill up lowest energy k-states. Sharp Fermi surface separates occupied from unoccupied states.
n(k)
1
KF k
Ground state wavefunction: “Slater determinant”
ψSlater(r1σ1,.....rN σN )=det[χi (rjσj)] 32
Thursday, April 5, 2012 Interacting Fermi liquid
Thursday, April 5, 2012 Special case: Gutzwiller wavefunction
Thursday, April 5, 2012 An interesting point of view
Thursday, April 5, 2012 A interesting point of view (cont’d)
Thursday, April 5, 2012 Slave boson mean field theory
Thursday, April 5, 2012 Example: lattice Hubbard model
ni (ni 1) H = t c† c + h.c + U − (1) − ij iα jα 2 ij α i Slave boson representation ciα = bifiα. Factorize electron hopping as
b†b f † f + b†b f † f (2) i j iα jα i j iα jα Boson carries electron charge => Interaction term becomes a boson-boson interaction
38
Thursday, April 5, 2012 Thursday, April 5, 2012 Slave boson/Gutzwiller mean field theories are a useful way to incorporate correlations into a description of metals.
Examples: success in capturing some of the phenomenology of the cuprates.
Thursday, April 5, 2012 Localized limit: Mott insulators and conventional quantum magnetism – a brief discussion
Thursday, April 5, 2012 What is a Mott insulator?
Insulation due to jamming effect of Coulomb repulsion
Coulomb cost of two electrons occupying same atomic orbital dominant
⇒Electrons can’t move if every possible atomic orbital site is already occupied by another electron.
Odd number of electrons per unit cell: band theory predicts metal.
Thursday, April 5, 2012 A useful theoretical model
Thursday, April 5, 2012 When Mott insulator? Periodic Table of Elements
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
1 1 Atomic # 2 2 K 1 H Symbol He Hydrogen Name Solid Metals Nonmetals Helium 1.00794 Atomic Mass C 4.002602 Alkali metals metals earth Alkaline metals Transition Poor metals nonmetals Other Noble gases 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2 K 1 2 Hg Liquid Lanthanoids 3 4 5 6 7 8 L 2 Li Be B C N O F Ne Lithium Beryllium Boron Carbon Nitrogen Oxygen Fluorine Neon 6.941 9.012182 H Gas 10.811 12.0107 14.0067 15.9994 18.9984032 20.1797
2 2 Actinoids 2 2 2 2 2 2 K 11 8 12 8 13 8 14 8 15 8 16 8 17 8 18 8 L
1 2 Unknown 3 4 5 6 7 8 M
Rf 3 Na Mg Al Si P S Cl Ar Sodium Magnesium Aluminium Silicon Phosphorus Sulfur Chlorine Argon 22.98976928 24.3050 26.9815386 28.0855 30.973762 32.065 35.453 39.948
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 K 19 8 20 8 21 8 22 8 23 8 24 8 25 8 26 8 27 8 28 8 29 8 30 8 31 8 32 8 33 8 34 8 35 8 36 8 L 8 8 9 10 11 13 13 14 15 16 18 18 18 18 18 18 18 18 M 4 K 1 Ca 2 Sc 2 Ti 2 V 2 Cr 1 Mn 2 Fe 2 Co 2 Ni 2 Cu 1 Zn 2 Ga 3 Ge 4 As 5 Se 6 Br 7 Kr 8 N Potassium Calcium Scandium Titanium Vanadium Chromium Manganese Iron Cobalt Nickel Copper Zinc Gallium Germanium Arsenic Selenium Bromine Krypton 39.0983 40.078 44.955912 47.867 50.9415 51.9961 54.938045 55.845 58.933195 58.6934 63.546 65.38 69.723 72.64 74.92160 78.96 79.904 83.798
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 K 37 8 38 8 39 8 40 8 41 8 42 8 43 8 44 8 45 8 46 8 47 8 48 8 49 8 50 8 51 8 52 8 53 8 54 8 L 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 M 8 8 9 10 12 13 14 15 16 18 18 18 18 18 18 18 18 18 N 5 Rb 1 Sr 2 Y 2 Zr 2 Nb 1 Mo 1 Tc 1 Ru 1 Rh 1 Pd 0 Ag 1 Cd 2 In 3 Sn 4 Sb 5 Te 6 I 7 Xe 8 O Rubidium Strontium Yttrium Zirconium Niobium Molybdenum Technetium Ruthenium Rhodium Palladium Silver Cadmium Indium Tin Antimony Tellurium Iodine Xenon 85.4678 87.62 88.90585 91.224 92.90638 95.96 (97.9072) 101.07 102.90550 106.42 107.8682 112.411 114.818 118.710 121.760 127.60 126.90447 131.293
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 K 55 8 56 8 72 8 73 8 74 8 75 8 76 8 77 8 78 8 79 8 80 8 81 8 82 8 83 8 84 8 85 8 86 8 L 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 M 18 18 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 N 6 Cs 8 Ba 8 57–71 Hf 10 Ta 11 W 12 Re 13 Os 14 Ir 15 Pt 17 Au 18 Hg 18 Tl 18 Pb 18 Bi 18 Po 18 At 18 Rn 18 O Caesium 1 Barium 2 Hafnium 2 Tantalum 2 Tungsten 2 Rhenium 2 Osmium 2 Iridium 2 Platinum 1 Gold 1 Mercury 2 Thallium 3 Lead 4 Bismuth 5 Polonium 6 Astatine 7 Radon 8 P 132.9054519 137.327 178.49 180.94788 183.84 186.207 190.23 192.217 195.084 196.966569 200.59 204.3833 207.2 208.98040 (208.9824) (209.9871) (222.0176)
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 K 87 8 88 8 104 8 105 8 106 8 107 8 108 8 109 8 110 8 111 8 112 8 113 8 114 8 115 8 116 8 117 118 8 L 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 M 32 32 89–103 32 32 32 32 32 32 32 32 32 32 32 32 32 32 N 7 Fr 18 Ra 18 Rf 32 Db 32 Sg 32 Bh 32 Hs 32 Mt 32 Ds 32 Rg 32 Uub 32 Uut 32 Uuq 32 Uup 32 Uuh 32 Uus Uuo 32 O Francium 8 Radium 8 Ru herfordium 10 Dubnium 11 Seaborgium 12 Bohrium 13 Hassium 14 Meitnerium 15 Darmstadtium 17 Roentgenium 18 Ununbium 18 Ununtrium 18 Ununquadium 18 Ununpentium 18 Ununhexium 18 Ununseptium Ununoctium 18 P (223) 1 (226) 2 (261) 2 (262) 2 (266) 2 (264) 2 (277) 2 (268) 2 (271) 1 (272) 1 (285) 2 (284) 3 (289) 4 (288) 5 (292) 6 (294) 8 Q
For elements with no stable isotopes, the mass number of the isotope with the longest half-life is in parentheses.
Periodic Table Design and Interface Copyright © 1997 Michael Dayah. http://www.ptable.com/ Last updated: May 27, 2008
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 57 8 58 8 59 8 60 8 61 8 62 8 63 8 64 8 65 8 66 8 67 8 68 8 69 8 70 8 71 8 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 19 21 22 23 24 25 25 27 28 29 30 31 32 32 La 9 Ce 9 Pr 8 Nd 8 Pm 8 Sm 8 Eu 8 Gd 9 Tb 8 Dy 8 Ho 8 Er 8 Tm 8 Yb 8 Lu 9 Lanthanum 2 Cerium 2 Praseodymium 2 Neodymium 2 Promethium 2 Samarium 2 Europium 2 Gadolinium 2 Terbium 2 Dysprosium 2 Holmium 2 Erbium 2 Thulium 2 Ytterbium 2 Lutetium 2 138.90547 140.116 140.90765 144.242 (145) 150.36 151.964 157.25 158.92535 162.500 164.93032 167.259 168.93421 173.054 174.9668
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 89 8 90 8 91 8 92 8 93 8 94 8 95 8 96 8 97 8 98 8 99 8 100 8 101 8 102 8 103 8 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 Ac 18 Th 18 Pa 20 U 21 Np 22 Pu 24 Am 25 Cm 25 Bk 27 Cf 28 Es 29 Fm 30 Md 31 No 32 Lr 32 Actinium 9 Thorium 10 Protactinium 9 Uranium 9 Neptunium 9 Plutonium 8 Americium 8 Curium 9 Berkelium 8 Californium 8 Einsteinium 8 Fermium 8 Mendelevium 8 Nobelium 8 Lawrencium 9 (227) 2 232.03806 2 231.03588 2 238.02891 2 (237) 2 (244) 2 (243) 2 (247) 2 (247) 2 (251) 2 (252) 2 (257) 2 (258) 2 (259) 2 (262) 2
Michael Dayah For a fully interactive experience, visit www.ptable.com. [email protected]
Classic Mott insulating materials: transition metal oxides (eg: NiO, MnO, V2O3, La2CuO4, LaTiO3,...... ) of 3d series, some sulfides (NiS2), rare earth insulators,...... 3d (or 4f) orbitals close to nucleus: large on-site repulsion compared to inter-site hopping.
Recent additions: 5d transition metal oxides (eg: Sr2IrO4) Atomic 5d orbitals more extended than 3d, 4d - so why Mott? Mott insulation due to combination of strong spin-orbit + intermediate correlation.
Thursday, April 5, 2012
Electronic Mott insulators
Thursday, April 5, 2012 Fate of electron spins in a Mott insulator
Common: Neel Antiferromagnetism (spontaneously breaks global spin SU(2) symmetry)
Interesting situations when such order is ``geometrically frustrated” by lattice structure and/or low dimensions.
Can get states that preserve spin SU(2) symmetry to T = 0 (``quantum paramagnets”) 46
Thursday, April 5, 2012 Spin ladders: A simple example of a quantum paramagnet
Thursday, April 5, 2012 Other quantum paramagnets: Valence Bond Solid (VBS) states
• Ordered pattern of valence bonds breaks lattice translation symmetry.
• Ground state smoothly connected to band insulator
• Elementary spinful excitations have S = 1 above spin gap.
(CuGeO3, TiOCl,...... )
Thursday, April 5, 2012 Recap: Conventional States of quantum magnetism Ferromagnetism: May be 600 BC ...... | ↑↑↑↑ Antiferromagnetism: 1930s ...... Key concept of broken symmetry. | ↑↓↑↓
Prototypical ground state wavefunction: direct product of local degrees of freedom
Short range quantum entanglement.
1930s- present: elaboration of broken symmetry and other states (eg: Valence Bond Solids) with short range entanglement 49
Thursday, April 5, 2012 Last ≈ 10 years
Experimental discovery of a qualitatively new kind of magnetic matter.
Popular name: ``quantum spin liquid”
Prototypical ground state wavefunction Not a direct product of local degrees of freedom. +
Quantum entanglement is long ranged in space.
+ ......
* In d > 1 50
Thursday, April 5, 2012 What is a quantum spin liquid ?
Rough description: Spins do not freeze but fluctuate in time and space due to quantum zero point motion.
Resonance between many different configurations (like in benzene) In each configuration each spin forms an entangled pair with one other partner spin.
Thursday, April 5, 2012 Some natural questions
Can quantum spin liquids exist in d > 1?
Do quantum spin liquids exist in d > 1?
Thursday, April 5, 2012 Some natural questions
Can quantum spin liquids exist in d > 1? Theoretical question
Do quantum spin liquids exist in d > 1? Experimental question
Thursday, April 5, 2012 Some natural questions
Can quantum spin liquids exist in d > 1? Theoretical question: YES!!
Do quantum spin liquids exist in d > 1? Experimental question: Remarkable new materials possibly in spin liquid phases
Thursday, April 5, 2012 Why are quantum spin liquids interesting?
1. Exotic excitations
Excitations with fractional spin (spinons), non-local emergent interactions described through gauge fields
As rich in possibility if not richer than the fractional quantum Hall systems………..but requires less extreme conditions (eg: no strong B-fields)
Thursday, April 5, 2012 Why are quantum spin liquids interesting?
2. Ordering not captured by concept of broken symmetry
- new concepts of `topological order’ and generalizations
Order is a global property of ground state wavefunction
Possibility of encoding information non-locally. ?? Useful for computing??
Thursday, April 5, 2012 Why are quantum spin liquids interesting?
3. Platform for onset of many unusual phenomena
Eg: (i) Superconductivity in doped Mott insulators ?? Relevant to cuprates ??
(ii) Non-fermi liquid phenomena in correlated d or f-electron metals.
Thursday, April 5, 2012 Why are quantum spin liquids interesting?
4. Excellent experimental setting for exploration of violation of long cherished notions of condensed matter physics
-quasiparticles with fractional quantum numbers and unusual statistics
- the very existence of a quasiparticle description
- inadequacy of Landau order parameter to describe phases and phase transitions of correlated matter
Thursday, April 5, 2012 Can quantum spin liquids exist?
1. Solution of concrete quantum spin models within 1/N expansions (Read, Sachdev ’91)
2. Effective field theory descriptions ( Wen ’91; Balents,Fisher,Nayak, ’99; TS, Fisher’00,…..)
3. Solution of simple effective models (Kitaev’97; Moessner,Sondhi’01; Balents, Fisher, Girvin’02; Motrunich, TS’02, Wen’03….)
4. Numerical calculations on ``realistic” models (Misguich,Lhuillier’98; Isakov,Paramekanti,Kim,Sen,Damle’07, Yan, Huse, White,’11)
Thursday, April 5, 2012 Where might we find quantum spin liquids?
• Geometrically frustrated quantum magnets
• ``Intermediate’’ correlation regime Eg: Mott insulators that are not too deeply into the insulating regime
Thursday, April 5, 2012 Where might we find quantum spin liquids?
• Geometrically frustrated quantum magnets
Quantum spins on a Kagome lattice (Young Lee lab, MIT)
• ``Intermediate’’ correlation regime Eg: Mott insulators that are not too deeply into the insulating regime I will focus on these.
Thursday, April 5, 2012 Some candidate materials
κ (ET) Cu (CN) − 2 2 3 Quasi-2d, approximately isotropic triangular lattice; best studied candidate spin liquids EtMe3Sb[Pd(dmit)2]2
Na4Ir3O8 Three dimensional frustrated lattice
ZnCu3(OH)6Cl2 2d Kagome lattice (`strong’ Mott insulator) Volborthtite, ......
Thursday, April 5, 2012 Some candidate materials
κ (ET) Cu (CN) − 2 2 3 Quasi-2d, approximately isotropic triangular lattice; best studied candidate spin liquids EtMe3Sb[Pd(dmit)2]2
Na4Ir3O8 Three dimensional frustrated lattice
Close to pressure driven Mott transition: `weak’ Mott insulators
ZnCu3(OH)6Cl2 2d Kagome lattice (`strong’ Mott insulator) Volborthtite, ......
Thursday, April 5, 2012 Some phenomena in experiments
Quantum spin liquid materials are all electrical insulators.
Despite this many properties other than electrical conduction are very similar to that of a metal.
Two examples at low temperature:
1. Entropy very similar to that of a metal at low temperature
2. Conduct heat just like a metal even though they are electrical insulators.
Very strange...... not known to happen in any ordinary insulator.
Thursday, April 5, 2012 REPORTS
exponential decay of the NMR relaxation indicates spin correlation and spin-mediated heat transport. apeakstructureatTg ~1K,whichcharacterizes inhomogeneous distributions of spin excitations Indeed, highly unusual transport properties includ- the excitation spectrum. (22), which may obscure the intrinsic properties ing the ballistic energy propagation have been re- The low-energy excitation spectrum can be of the QSL. A phase transition possibly associated ported in a 1D spin-1/2 Heisenberg system (25). inferred from the thermal conductivity in the low- with the charge degree of freedom at ~6 K further The temperature dependence of the thermal temperature regime. In dmit-131, kxx /T at low 2 complicates the situation (23). Meanwhile, in conductivity kxx divided by Tof a dmit-131 single temperatures is well fitted by kxx /T=k00 /T+bT EtMe3Sb[Pd(dmit)2]2 (dmit-131) such a transi- crystal displays a steep increase followed by a (Fig. 2C), where b is a constant. The presence of a tion is likely to be absent, and a much more homo- rapid decrease after showing a pronounced maxi- residual value in kxx /Tat T→0K,k00/T, is clearly geneous QSL state is attained at low temperatures mum at Tg ~1K(Fig.2A).Theheatiscarried resolved. The distinct presence of a nonzero k00 /T ph (4, 5). As a further merit, dmit-131 (Fig. 1B) has primarily by phonons (kxx )andspin-mediated term is also confirmed by plotting kxx/T versus T spin acousinmaterialEt2Me2Sb[Pd(dmit)2]2 (dmit-221) contributions (kxx ). The phonon contribution (Fig. 2D). In sharp contrast, in dmit-221, a corre- with a similar crystal structure (Fig. 1C), which can be estimated from the data of the nonmagnetic sponding residual k00/T is absent and only a pho- exhibits a nonmagnetic charge-ordered state with state in a dmit-221 crystal with similar dimensions, non contribution is observed (26). The residual spin alargeexcitationgapbelow70K(24). A com- which should have a negligibly small kxx .In thermal conductivity in the zero-temperature limit ph parison between these two related materials will dmit-221, kxx /T exhibits a broad peak at around immediately implies that the excitation from the therefore offer us the opportunity to single out 1K,whichappearswhenthephononconduction ground state is gapless, and the associated correla- genuine features of the QSL state believed to be grows rapidly and is limited by the sample bound- tion function has a long-range algebraic (power-law) realized in dmit-131. aries. On the other hand, kxx/T of dmit-131, which dependence. We note that the temperature depen- ph Measuring thermal transport is highly advan- well exceeds kxx /T of dmit-221, indicates a sub- dence of kxx /Tin dmit-131 is markedly different tageous forSome probing the low-lyingphenomena elementary stantial in contribution experiments of spin-mediated heat con- from that in k-(BEDT-TTF)2Cu2(CN)3,inwhich excitations in QSLs, because it is free from the duction below 10 K. This observation is reinforced the exponential behavior of kxx /Tassociated with nuclear Schottky contribution that plagues the by the large magnetic field dependence of kxx of the formation of excitation gap is observed (18). heat capacity measurements at low temperatures dmit-131, as discussed below (Fig. 3A). Figure Key information on the nature of elementary (21). Moreover, it is sensitive exclusively to itin- 2B shows a peak in the kxx versus T plot for dmit- excitations is further provided by the field depen- ph S. Yamashita et al, Naterant Phys, spin excitations2008 that carry entropy, which 131,M. which Yamashita is absent et in al, dmit-221. Science We 2010 therefore dence of kxx.Becauseitisexpectedthatkxx is provides important information on the nature of the conclude that k spin and k spin/T in dmit-131 have hardly influenced by the magnetic field, particu-
xx xx on July 20, 2010 larly at very low temperatures, the field depen- spin 1.6 1.0 dence is governed by kxx (H)(26). The obtained T 1.6 0.8 ABg D CH-dependence, kxx(H), at low temperatures is 1.4 0.6 1.2 quite unusual (Fig. 3A). At the lowest temperature, 0.8 dmit-131 kxx(H)atlowfieldsisinsensitivetoH but displays 1.2 0.4 0.8 asteepincreaseaboveacharacteristicmagnetic (W/K m) 0.2 1.0 xx field Hg ~2T.AthighertemperaturesclosetoTg, m) 0.4
2 0.6 m) 0.0 2 0.0 0.3 this behavior is less pronounced, and at 1 K kxx(H) 0.8 0.0 T (K) increases with H nearly linearly. The observed www.sciencemag.org (W/K 0 2 4 6 8 10 (W/K /T field dependence implies that some spin-gap–like T (K) /T 0.4 xx 0.6 xx dmit-131 (spin liquid) κ-(BEDT-TTF)2Cu2(CN)3 excitations are also present at low temperatures, 0.4 dmit-221 (non-magnetic) (×2) along with the gapless excitations inferred from 0.2 dmit-221 the residual k00/T.Theenergyscaleofthegapis 0.2 characterized by mBHg,whichiscomparableto kBTg.Thus,itisnaturaltoassociatetheobserved 0.0 0.0 0 2 4 6 8 10 0.00 0.02 0.04 0.06 0.08 0.10 zero-field peak in kxx(T)/T at Tg with the excitation 2 2 T (K) T (K ) gap formation. Downloaded from Next we examined a dynamical aspect of the Fig. 2. The temperature dependence of k (T)/T (A)andk (T)(B)ofdmit-131(pink)anddmit-221 xx xx spin-mediated heat transport. An important ques- (green) below 10 K in zero field [k (T)isthethermalconductivity].Aclearpeakink /T is observed in xx xx tion is whether the observed energy transfer via dmit-131 at Tg ~1K,whichisalsoseenasahumpinkxx.Lowertemperatureplotofkxx(T)/T as a function of T2 (C)andT (D) of dmit-131, dmit-221,Thermal and k-(BEDT-TTF) conductivityCu (CN) (black) (18). A clear residual of elementary excitations is diffusive or ballistic. In Heat capacity 2 2 3 the 1D spin-1/2 Heisenberg system, the ballistic kxx(T)/T is resolved in dmit-131 in the zero-temperature limit. energy propagation occurs as a result of the con- servation of energy current (25). Assuming the Fig. 3. (A)Fielddependenceof 0.23 K thermal conductivity normalized A kinetic approximation, the thermal conductivity 0.70 K spin 0.3 is written as k =Cv ‘ /3,whereC is the spe- These are bothby the exactly zero field value,like [kinxx( Ha) –metal but were 1.0 K xx s s s s kxx(0)]/kxx(0) of dmit-131 at low cific heat, vs is the velocity, and ‘s is the mean free (0)
measured in temperatures.an insulator. (Inset) The heat cur- xx path of the quasiparticles responsible for the ele- 0.2 rent Q was applied within the 2D mentary excitations. We tried to estimate ‘s sim- plane, and the magnetic field H was ply by assuming that the linear term in the thermal (0) } /
perpendicular to the plane. kxx and xx 0.1 conductivity arises from the fermionic excitations, 0.1 B kxy were determined by diagonal ) - in analogy with excitations near the Fermi surface ) H H ( and off-diagonal temperature gra- (
xx 0.0 in metals. The residual term is written as k00/T ~ 2 tan
D D { dients, Tx and Ty,respectively. (kB /daħ)‘s,whered (~3 nm) and a (~1 nm) are 0.0 (B)Thermal-Hallangletanq(H)= interlayer and nearest-neighbor spin distance. We ph -0.1 kxy/(kxx – kxx )as a function of H at assumed the linear energy dispersion e(k) = ħv k, Thursday, April 5, 2012 H 0 2 4 6 8 10 12 s 0.23 K (blue), 0.70 K (green), and g 0H (T) a2DdensityofstatesandaFermienergycom- 1.0 K (red). -0.1 0 2 4 6 8 10 12 parable to J (26). From the observed k00/T,we H (T) 0 find that ‘s reaches as long as ~1 mm, indicating
www.sciencemag.org SCIENCE VOL 328 4 JUNE 2010 1247 Exotic Mott insulators at intermediate correlation
Thursday, April 5, 2012 Approach from insulator
Motrunich, 2005
Thursday, April 5, 2012 PHYSICAL REVIEW B 81, 245121 ͑2010͒
Weak Mott insulators on the triangular lattice: Possibility of a gapless nematic quantum spin liquid
Tarun Grover,1 N. Trivedi,2 T. Senthil,1 and Patrick A. Lee1 1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Department of Physics, Ohio State University, Columbus, Ohio 43210, USA ͑Received 7 March 2010; revised manuscript received 18 May 2010; published 18 June 2010͒ We study the energetics of Gutzwiller projected BCS states of various symmetries for the triangular lattice antiferromagnet with a four-particle ring exchange using variational Monte Carlo methods. In a range of parameters the energetically favored state is found to be a projected dx2−y2 paired state which breaks lattice rotational symmetry. We show that the properties of this nematic or orientationally ordered paired spin-liquid state as a function of temperature and pressure can account for many of the experiments on organic materials. We also study the ring-exchange model with ferromagnetic Heisenberg exchange and find that among the studied ansätze, a projected f-wave state is the most favorable.
DOI: 10.1103/PhysRevB.81.245121 PACS number͑s͒: 75.10.Jm, 71.27.ϩa, 74.20.Mn, 74.70.Kn
I. INTRODUCTION At this point, several questions arise: what is a good de- In the last few years the quasi-two-dimensional ͑2D͒ or- scription of the putative spin-liquid Mott state seen in experi- ments mentioned above? What is the connection between the ganic salts -͑ET͒2Cu2͑CN͒3 and EtMe3Sb͓Pd͑dmit͒2͔2 ͑ab- breviated, respectively, as CN and DMIT in the paper͒ have superconducting ͑SC͒ state and the underlying spin-liquid emerged as possible realizations of Mott insulators in the state that becomes unstable upon applying pressure? What is long sought “quantum spin-liquid” state.1–8 These layered the nature of the finite-temperature transitions/crossovers? materials are believed to be well described by the single We find, using a variational Monte Carlo analysis of the band Hubbard model on a nearly isotropic triangular lattice. energetics of several possible wave functions for a spin At ambient pressure they are Mott insulators which do not Hamiltonian with Heisenberg and ring-exchange interactions order magnetically down to temperatures ϳ30 mK ͑much that the nodal d-wave projected BCS state is the best candi- lower than the exchange Jϳ250 K inferred from high- date for the spin liquid. This state has gapless spin excita- temperature susceptibility͒.1,8 The low-temperature phase is tions and can naturally explain many of the experiments in characterized by a linear T-dependent heat capacity and a CN though a number of open questions remain. We also finite spin susceptibility just like in a metal ͑even though the study the antiferromagnetic J4, ferromagnetic J2 model, and material is insulating͒͑Refs. 1, 4, and 6͒ indicating the pres- find that among the studied ansätze, a projected f-wave state ence of low-lying spin excitations. There is a sharp crossover is most favorable. This may have bearing on the explanation or possibly a phase transition at a low temperature Ϸ5K of the observed gapless spin-liquid behavior in He-3 films.12 signaled by a peak in the heat capacity and the onset of a Ring exchange promotes spin liquids drop in the susceptibility.1,4 Further an external magnetic field induces inhomogeneity that is evidenced by a broaden- A. Summary of results 3 10 ing of the NMR line. Application of moderate pressure Example:Our results Ring are exchange based on the model model on Hamiltonian,2d triangular lattice ͑Ϸ0.5 GPa͒ induces a transition to a superconductor ͑CN͒ 2 ជ ជ or metal ͑DMIT͒. H =2J2 ͚ Sr . SrЈ + J4͚ ͑P1234 + H.c.͒ = J2H2 + J4H4. Broadly speaking a spin-liquid ground state of a Mott ͗rrЈ͘ ᮀ ! ! insulator cannot be smoothly deformed to the ground state of ͑1͒ any electronic band insulator. The theoretical possibility of 9 Many different numerical theoretical methods: quantum spin liquids has been appreciated for a long time. Here Sជr are spin-1/2 operators at the sites of a triangular Many sharply distinct spin-liquid phases are possible. Fur- lattice. The second term sums over all elementary parallelo- Exact diagonalization (LiMing et al, 2000, H.-Y. Yang et al, 2010) ther, any quantum spin-liquid state possesses exotic excita- grams and P1234 performs a cyclic exchange of the four spins tions with fractional quantum number and various associated at the sites of the parallelogram. The multiple ring exchange Variational wavefunctions (Motrunich 2005,...... ) topological structures. The distinction between different is expected to be significant due to the proximity to the Mott quantum spin-liquid phases is reflected in distinctions of the transitionDensity in Matrix the organics. Renormalization It is known Group that the (Sheng three sublatticeet al, 2009). structure of the low-energy effective theory of these excita- Neel order vanishes beyond a critical J4 /J2 Ϸ0.1 ͑Ref. 11͒ tions. that can lead to novel spin-liquid phases with no long-range Currently the most promising candidate materials all seem spinEvidence order. for spin liquid behavior with increasing J4/J2. to share a few key properties. First, they are weak Mott in- We study various paired spin-liquid states for the J2 –J4 sulator that are easily driven metallic by application of pres- model using variational Monte Carlo calculations. In terms sure. Second, they appear to have gapless spin-carrying ex- of wave functions, paired states may be described by 68 citations. We are thus lead to study possible gapless spin- Gutzwiller projected BCS states. Two natural states ͑which liquid behavior in weak Mott insulators to understand these Thursday,retain April the 5, full 2012 symmetry of the triangular lattice͒ are pro- materials. jected singlet dx2−y2 +idxy and nodal triplet fx3−3xy2 wave
1098-0121/2010/81͑24͒/245121͑7͒ 245121-1 ©2010 The American Physical Society Alternate view from the metallic side
Thursday, April 5, 2012 Comments
Thursday, April 5, 2012 Thursday, April 5, 2012 Extreme limit: Gutzwiller projection
Thursday, April 5, 2012 Picture of Mott transition
Electrons swimming in sea of +vely charged Metal ions
Electron charge gets Mott spin liquid pinned to ionic lattice near metal while spins continue to swim freely.
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Thursday, April 5, 2012 Formal theory
Slave boson mean field theory:
Hmf = Hb + Hf (1)
ni(ni 1) H = t b†b + U − (2) b − c i j 2
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Thursday, April 5, 2012 Fluctuations: gauge theory
Thursday, April 5, 2012 Properties of this spin liquid (in d = 2)
(Spinon) Fermi surface + U(1) gauge field: Hard strongly coupled quantum field theory problem.
No a priori small parameter to make approximations.
Large theoretical literature -
Holstein et al, 1973, many papers in the 90s - large-N treatments
Some recent controversy (S.S. Lee 2009, Metlitski, Sachdev 2010): problems with the large-N expansion somewhat similar to non-abelian gauge theory.
Resolution (Mross, McGreevy, Liu, TS, 2010): Combine large-N with another small parameter introduced by Nayak, Wilczek (94).
Controlled double expansion allows calculation of physics. 76
Thursday, April 5, 2012 A further complication
The U(1) gauge field is actually ``compact”, i.e, vector potential is only defined mod 2π.
=> Instanton events where magnetic flux changes by 2π are allowed (i.e magnetic monopoles in space-time).
Famous old result (Polyakov 1976): In d= 2 ``pure” compact U(1) gauge theory is confining due to proliferation of these instanton events.
Bad sign for spin liquid?
Rescue: Gapless spinons which are charged under U(1) gauge group.
Enough gapless matter fields defeat Polyakov confinement mechanism (Hermele, TS, Fisher, Lee, Nagaosa, Wen 2004, S.S. Lee 2009).
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Thursday, April 5, 2012 Properties of this spin liquid (cont’d) (in d = 2)
2 Specific heat C T 3 v ∼ Spin susceptibility χ const ∼ 1 Thermal conductivity κ T 3 . ∼
Sharp 2Kf singularities in both spin density f †σf and spinon density f †f.
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Thursday, April 5, 2012 Properties of BCS paired spin liquids
Spinon pair condensate expels U(1) gauge field.
Spin physics similar to that of corresponding paired superconductor (eg: d-wave paired spinons => spin physics of d-wave BCS SC).
Vortices of spinon pair condensate: topological defects of the paired spin liquid
However for subtle reasons (instantons) the vortices have a Z2 character (a vortex is its own antivortex): ``visons” (Ising-like vortex)
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Thursday, April 5, 2012 Utility of gauge theory
Quantum spin liquid phases have emergent fractional quantum number excitations with emergent non-local interactions.
Gauge theory is (as always!) a convenient mathematical way to encapsulate this non-locality.
More fundamental: Fractional quantum numbers + gauge interactions a property of the non-local quantum entanglement of the ground state wavefunction.
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Thursday, April 5, 2012 Stability of spin liquids
Thursday, April 5, 2012 Stability of spin liquids
Thursday, April 5, 2012 Confinement versus deconfinement
Thursday, April 5, 2012 Confinement versus deconfinement (cont’d)
Thursday, April 5, 2012 Varieties of spin liquids
Thursday, April 5, 2012 A useful distinction: Gapped versus gapless spin spectrum
Thursday, April 5, 2012 In general different quantum spin liquid phases will have very different universal low temperature properties
Clear cut predictions for experiments which can distinguish between different spin liquid states.
Thursday, April 5, 2012 A common sense guide to the phase diagram: when what kind of spin liquid?
Spinon FS Metal t/U
Paired SL SC t/U
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Thursday, April 5, 2012 Experiments: Spinon FS as a universal intermediate temperature `mother’ state
Spinon FS? Low T instability in kappa-ET (Motrunich, 05) at ambient pressure at same temperature scale as SC instability under pressure
Paired SL? (Lee, Lee, TS, 06)
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Thursday, April 5, 2012 Current status
1. What experiments can best reveal if this is what is really going on? Detect the spinon Fermi surface! Ideas: 1. Quantum oscillations in applied magnetic field (Motrunich, 06) 2. Oscillatory magnetic coupling between two ferromagnets separated by spin liquid buffer (Micklitz, Norman, 09) 3. Signatures in phonon spectrum (Mross, TS, 10) 4. Scanning Tunneling Microscopy (Mross, TS, 10)
All are hard!
2. Paired spin liquid state at low T not fully understood. Probably d-wave paired (Grover, Trivedi, TS, Lee, 2010).
Thursday, April 5, 2012 Summary/outlook
• Maturing theoretical understanding of quantum spin liquid phases in d > 1
• Theoretically demonstrable violations of long cherished notions of condensed matter physics
• Interesting candidate materials exist – exciting times ahead!
• Important general lessons for correlated metallic systems
Thursday, April 5, 2012 Unconventional quantum criticality
Thursday, April 5, 2012 The Mott transition
Thursday, April 5, 2012 The simple case: Bosons at integer filling
Thursday, April 5, 2012 Review: Simple Mott transition of bosons
Thursday, April 5, 2012 Approaching the transition
Thursday, April 5, 2012 Thursday, April 5, 2012 Finite temperature phase diagram
T
g
Thursday, April 5, 2012 Landau-Ginzburg-Wilson theory
Thursday, April 5, 2012 More difficult Mott and related quantum phase transitions
Thursday, April 5, 2012 The electronic Mott transition
Difficult old problem in quantum many body physics
How does a metal evolve into a Mott insulator?
Prototype: One band Hubbard model at half-filling on non-bipartite lattice
????? t/U AF insulator; Fermi liquid; No Fermi surface Full fermi surface
Thursday, April 5, 2012 Why hard?
1. No order parameter for the metal-insulator transition
2. Need to deal with gapless Fermi surface on metallic side
3. Complicated interplay between metal-insulator transition and magnetic phase transition
Typically in most materials the Mott transition is first order.
But (at least on frustrated lattices) transition is sometimes only weakly first order - fluctuation effects visible in approach to Mott insulator from metal.
Thursday, April 5, 2012 Quantum spin liquids and the Mott transition
Modern condensed matter physics: possibility of quantum spin liquid Mott insulators with no broken symmetries/conventional long range order.
Theory: Quantum spin liquids can exist; maturing understanding.
Experiment: Several candidate materials; all of them have some gapless excitations.
Opportunity for progress on the Mott transition: study metal-insulator transition without complications of magnetism.
Thursday, April 5, 2012 Possible experimental realization of a second order Mott transition
Thursday, April 5, 2012 Quantum spin liquids and the Mott transition
Some questions:
1. Can the Mott transition be continuous?
2. Fate of the electronic Fermi surface?
????? t/U Spin liquid insulator; Fermi liquid; No Fermi surface Full fermi surface
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Thursday, April 5, 2012 Killing the Fermi surface
????? t/U Spin liquid insulator; Fermi liquid; No Fermi surface Full fermi surface
At half-filling, through out metallic phase, ``Luttinger theorem” => size of Fermi surface is fixed.
Approach to Mott insulator: entire Fermi surface must die while maintaining size (cannot shrink to zero).
If Mott transition is second order, critical point necessarily very unusual.
``Fermi surface on brink of disappearing” - expect non-Fermi liquid physics.
Similar ``killing of Fermi surface” also at some quantum phase transitions in rare earth alloys, and may be also around optimal doping in cuprates.
Thursday, April 5, 2012 How can a Fermi surface die continuously?
Metal
Continuous disappearance of Fermi surface if quasiparticle weight Z Mott insulator vanishes continuously everywhere on the Fermi surface (Brinkman, Rice, 1970). Mott critical point
Concrete examples: Infinite d (Vollhardt, Metzner, Kotliar, Georges 1990s), slave particle theories in d = 2, d = 3 (TS, Vojta, Sachdev 2003, TS 2008)
Thursday, April 5, 2012 Electronic structure at a continuous Mott transition
Crucial question: Electronic excitation structure right at Mott critical point when Z has just gone to zero?
Claim: At critical point, Fermi surface remains sharply defined even though there is no Landau quasiparticle (TS, 2008)
``Critical Fermi surface”
Thursday, April 5, 2012 Why a critical Fermi surface?
Thursday, April 5, 2012 Evolution of single particle gap
Thursday, April 5, 2012 Killing the Fermi surface
Disappearance of Fermi surface through a continuous phase transition requires at critical fixed point
1. Z = 0
2. Fermi surface sharp
Many questions:
Scaling phenomenology?
Calculational theoretical framework?
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Thursday, April 5, 2012 Killing a Fermi surface: other examples
At other T = 0 phase transitions in metals, an entire Fermi surface may again disappear.
Eg: (i) Onset of magnetism in rare earth alloys
(ii) High-Tc cuprates as function of doping?
IF second order, non-fermi liquid very natural.
Thursday, April 5, 2012 Example: Evolution of Fermi surface across the magneticGeometry phase transitionof Fermi in surfaceCeRhIn5 of CeRhIn5
H. Shishido, R. Settai, H. Harima, & Y. Onuki, JPSJ 74, 1103 (2005)
Thursday, April 5, 2012 Example: Evolution of Fermi surface across the magneticGeometry phase transitionof Fermi in surfaceCeRhIn5 of CeRhIn5
H. Shishido, R. Settai, H. Harima, & Y. Onuki, JPSJ 74, 1103 (2005)
Thursday, April 5, 2012 High Tc cuprates
Thursday, April 5, 2012 Scaling phenomenology at a quantum critical point with a critical Fermi surface (TS, 08)
Thursday, April 5, 2012 Critical Fermi surface: scaling for single particle physics
Thursday, April 5, 2012 New possibility: angle dependent exponents
Thursday, April 5, 2012 Leaving the critical point
Thursday, April 5, 2012 Approach from the Fermi liquid
Thursday, April 5, 2012 Specific heat singularity
Thursday, April 5, 2012 Critical 2Kf surface
Thursday, April 5, 2012 Implications of angle dependent exponents
Thursday, April 5, 2012 Finite T crossovers
Thursday, April 5, 2012 Calculational framework
Only currently available framework: Slave particle methods
View electron as composite of `slave’ particles with fractional quantum numbers
Reformulate electron model in terms of slave particles interacting through gauge forces.
Provides concrete examples of phase transitions where an entire Fermi surface disappears continuously.
Successes: Demonstrate critical Fermi surface, emergence of non-fermi liquids (TS, 2008)
Exponents angle independent.
Important as proof of principle, application to experiment with caution.
Thursday, April 5, 2012 An example
Continuous Mott transition between a metal and a spin liquid Mott insulator in 2d (TS, 2008)
Cannot be described as Fermi surface + X
Thursday, April 5, 2012 Other results
Similar models for destruction of Fermi surfaces can be studied in variety of contexts.
1. Chemical potential tuned version (TS, 08; TS and P.A. Lee, 09)
2. Continuous Mott transitions in 3d to a spin liquid (Podolsky, Paremakanti, Kim, TS, 09)
3. Kondo breakdown transitions between heavy Fermi liquids and ``fractionalized” Fermi liquids where f-band becomes spin liquid (TS, Vojta, Sachdev, 03; Paul, Pepin, Norman, 07; Pepin, 08)
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Thursday, April 5, 2012 Incorporating magnetism (important for phase transitions in rare earth alloys)
Can the disappearance of Fermi surface happen at the same critical point as the appearance of magnetic order?
Difficulty: Two different things seem to happen at the same time.
Study possibility of such phenomena in simpler systems. Can ordered phases with two distinct broken symmetries have a direct second order transition?
Thursday, April 5, 2012 General theoretical questions
• Fate of Landau-Ginzburg-Wilson ideas at quantum phase transitions?
• (More precise) Could Landau order parameters for the phases distract from the true critical behavior?
Study phase transitions in insulating quantum magnets - Good theoretical laboratory for physics of phase transitions/ competing orders. (Senthil, Vishwanath, Balents, Sachdev, Fisher, Science 2004)
Thursday, April 5, 2012 Highlights
• Failure of Landau paradigm at (certain) quantum transitions
• Rough description: Emergence of `fractional’ charge and gauge fields near quantum critical points between two CONVENTIONAL phases. - ``Deconfined quantum criticality’’
• Many lessons for competing order physics in correlated electron systems.
Thursday, April 5, 2012 Phase transitions in quantum magnetism
• Spin-1/2 quantum antiferromagnets on a square lattice.
• ``……’’ represent frustrating interactions that can be tuned to drive phase transitions.
Thursday, April 5, 2012 VBS Order Parameter
• Associate a Complex Number
Thursday, April 5, 2012 VBS Order Parameter
• Associate a Complex Number
Thursday, April 5, 2012 VBS Order Parameter
• Associate a Complex Number
Thursday, April 5, 2012 VBS Order Parameter
• Associate a Complex Number
Thursday, April 5, 2012 Neel-valence bond solid(VBS) transition
• Neel: Broken spin symmetry Naïve Landau expectation • VBS: Broken lattice symmetry.
• Landau – Two independent order First order parameters. - no generic direct second order transition. - either first order or phase coexistence. VBS This talk: Direct second order Neel transition but with description not in terms of natural order parameter fields. Neel +VBS
Thursday, April 5, 2012 Neel-Valence Bond Solid transition
• Naïve approaches fail Attack from Neel ≠Usual O(3) transition in D = 3
Attack from VBS ≠ Usual Z4 transition in D = 3 (= XY universality class).
Why do these fail? Topological defects carry non-trivial quantum numbers!
Thursday, April 5, 2012 Attack from VBS (Levin, TS, ‘04 )
Thursday, April 5, 2012 Topological defects in Z4 order parameter
• Domain walls – elementary wall has π/2 shift of clock angle
Thursday, April 5, 2012 Z4 domain walls and vortices
• Walls can be oriented; four such walls can end at point.
• End-points are Z4 vortices.
Thursday, April 5, 2012 Z4 vortices in VBS phase
Vortex core has an unpaired spin-1/2 moment!!
Z4 vortices are spin-1/2 ``spinons’’.
Domain wall energy ⇒ linear confinement in VBS phase.
Thursday, April 5, 2012 Z4 disordering transition to Neel state
• As for usual (quantum) Z4 transition, expect clock anisotropy is irrelevant. (confirm in various limits).
Critical theory: (Quantum) XY but with vortices that carry physical spin-1/2 (= spinons).
Thursday, April 5, 2012 Alternate (dual) view
• Duality for usual XY model (Dasgupta-Halperin) Phase mode - ``photon’’
Vortices – gauge charges coupled to photon.
Neel-VBS transition: Vortices are spinons => Critical spinons minimally coupled to fluctuating U(1) gauge field*.
*non-compact
Thursday, April 5, 2012 Critical theory ``Non-compact CP1 model’’
z = two-component spin-1/2 spinon field
aµ = non-compact U(1) gauge field. Distinct from usual O(3) or Z4 critical theories*. Theory not in terms of usual order parameter fields but involve fractional spin objects and gauge fields.
*Distinction with usual O(3) fixed point due to non-compact gauge field (Motrunich,Vishwanath, ’03)
Thursday, April 5, 2012 Renormalization group flows
Clock anisotropy = quadrupled Instanton fugacity
Deconfined critical fixed point
Clock anisotropy is ``dangerously irrelevant’’.
Thursday, April 5, 2012 Precise meaning of deconfinement
• Z4 symmetry gets enlarged to XY
⇒ Domain walls get very thick and very cheap near the transition.
=> Domain wall energy not effective in confining Z4 vortices (= spinons)
.
Formal: Extra global U(1) symmetry not present in microscopic model :
Thursday, April 5, 2012 Two diverging length scales in paramagnet
``Critical” ξ `` spin ξ VBS L liquid’’ VBS
ξ: spin correlation length
ξVBS : Domain wall thickness.
κ ξVBS ~ ξ diverges faster than ξ
Spinons confined in either phase but `confinement scale’ diverges at transition – hence `deconfined criticality’.
Thursday, April 5, 2012 Other examples of deconfined critical points
1. VBS- spin liquid (Senthil, Balents, Sachdev, Vishwanath, Fisher, ’04) 2. Neel –spin liquid (Ghaemi, Senthil, ‘06) 3. Certain VBS-VBS (Fradkin, Huse, Moessner, Oganesyan, Sondhi, ’04; Vishwanath, Balents,Senthil, ‘04)
4. Superfluid- Mott transitions of bosons at fractional filling on various lattices (Senthil et al, ’04, Balents et al, ’05,…….) 5. Spin quadrupole order –VBS on rectangular lattice (Numerics: Harada et al, ’07;Theory: Grover, Senthil, 07)
……..and many more!
Apparently fairly common
Thursday, April 5, 2012 Numerical evidence: Neel-VBS on square lattice
Thursday, April 5, 2012 A sample scaling plot
Thursday, April 5, 2012 Some lessons
• Striking ``non-fermi liquid’’ (morally) physics at critical point between two competing orders. Eg: At Neel-VBS, spin spectrum is anamolously broad - roughly due to decay into spinons- as compared to usual critical points.
Most important lesson: Failure of Landau paradigm – order parameter fluctuations do not capture true critical physics even if natural order parameters exist.
Strong impetus to radical approaches to non fermi liquid physics at magnetic critical points in rare earth metals (and to optimally doped cuprates).
Thursday, April 5, 2012 Outlook
• Theoretically important answer to 0th order question posed by experiments: Can Landau paradigms be violated at phases and phase transitions of strongly interacting electrons?
• But there still is far to go to seriously confront non-Fermi liquid metals in existing materials……….! Can we go beyond the 0th order answer?
Thursday, April 5, 2012