Superconductivity near the Mott transition
T. Senthil (MIT)
Tuesday, July 8, 14 Superconductivity, its friends and its enemies, near the Mott transition
T. Senthil (MIT)
Tuesday, July 8, 14 Useful other Boulder lectures
Cuprate phenomena + some theory
1. M. Randeria, http://boulderschool.yale.edu/sites/ default/files/files/Randeria-Boulder-Lecture1.pdf
2. S. Kivelson, 2014
3. A. Paremakanti, 2014
Quantum spin liquids, quantum criticality.
1. TS, http://boulder.research.yale.edu/Boulder-2008/ Lectures/Senthil/Boulder1.pdf
2. Patrick Lee, http://icam-i2cam.org/index.php/research/ file/lee1
Tuesday, July 8, 14 Plan
Lecture 1
1. Examples of superconductivity and related phenomena near Mott transition
2. Magnetism and Mott insulators
Lecture 2
Metals and superconductors near the Mott transition
-(i) some general questions -(ii) some theoretical answers.
Tuesday, July 8, 14 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.
Tuesday, July 8, 14 Useful theoretical model: the Hubbard model
Electrons on lattice sites i with 1 electron per site on average
ni(ni 1) H = t c†c + h.c + U i j 2
ni = number of electrons at site i.
t U: Hopping wins; Fermi liquid metal. U t: Repulsion wins; Mott insulator
Tuesday, July 8, 14 Complications in many real Mott insulators
1. Orbital degeneracy: More than one atomic orbital may be available for the electron to occupy at each site.
2. Multi-band model may be more appropriate starting point (definitely so if there is orbital degeneracy)
3. Spin-orbit interactions
4. (Obviously) must include long range Coulomb +......
In this lecture I will primarily consider situations in which many of these complications (mainly 1-3) are likely unimportant. Fortunately the cuprates fall in this class!
Tuesday, July 8, 14 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
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Some classic Mott insulating materials: transition metal oxides (eg: NiO, MnO, V2O3, La2CuO4, LaTiO3,...... ) of 3d series, some sulfides (NiS2), ...... 3d orbitals close to nucleus: large on-site repulsion compared to inter-site hopping. Will meet some other interesting examples later.
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.
Tuesday, July 8, 14
Many examples of superconductivity occurring in the vicinity of a Mott insulator.
Tuesday, July 8, 14 Tuesday, July 8, 14 Quasi-2D organics κ-(ET)2X
ET
dimer model X Mott insulator
t t t’ X = Cu(NCS)2, Cu[N(CN)2]Br, Cu2(CN)3….. anisotropic triangular lattice t’ / t = 0.5 ~ 1.1
Tuesday, July 8, 14 Pressure tuned superconductivity in the organics
Pressure decreases U/t.
Mott transition is induced by tuning U/t at fixed density of one electron per site.
κ-Cu[N(CN)2]Cl t’/t = 0.75
Tuesday, July 8, 14
This possibility to select the signals of the two phases allowed us to demonstrate that both are in a paramagnetic insulating state at p=1 bar, and display a Mott transition at distinct applied pressures[13]. We could further study their magnetic behaviour at ambient pressure using data on the 13C nuclear spins, which are directly coupled to the electron spins of the t1u molecular orbitals. The coupling being dipolar [21], it induces a characteristic anisotropic broadening of the 13C NMR which allowed us to 3- monitor the C60 magnetism. As can be seen in figure 3 this anisotropic shift Kax evidences a Curie–Weiss paramagnetism of the 3- C60 , which does not differ for the two isomers above 100K [13]. These data are compatible with a low spin S=1/2 effective moment as expected from figure 3 for a JT stabilized state. Furthermore the structural difference between the two isomers results in distinct magnetic ground states at p=1 bar. The In any case a different school of thoughtbipartite has beenA15 suggesting structure for displays long that electron-electrona Néel order below TN=47 K [12,13,46], while the frustration of the correlations might even drive superconductivityfcc lattice in such results compounds in a[2-3]. spin It freezinghas even been only proposed below 10 K for the standard fcc- Cs3C60 [13,47]. that AnC60 should be Mott insulators and are only metallic due to a non stoichiometry of the alkalis [36]. Although our result for the charge segregation in the cubic quenched phase of CsC60 [37] could be supportive of this possibility, no experimental evidence along this line could be found so far in AnC60. It has rather been seen that Tc peaks at n=3 if one varies the alkali doping [38]. Fig. 3. Paramagnetic behaviour detected by To better qualify the proximity to a Mott insulating state, attempts to produce expanded fulleride SQUID data (full line, right scale) and from the compounds have been done for long. A successful route has been to insert NH3 neutral molecules to 13 expand the A3C60 lattice. It was found that (NH3)K3C60 is insulating at ambient pressure [39], and anisotropic shift Kax of the C NMR in the two becomes SC with Tc up to 28K under pressure [40]. Furthermore at 1bar it was shown to be an AF isomeric Cs3C60 phases [13]. No significant [11]. However this insertion of NH3 induces an orthorhombic distortion of the lattice which is retained, difference can be detected above 100 K and the though modified, under applied pressure [41]. So, although this system undergoes a Mott transition to a metallic state, this occurs in an electronic structure where the t1u level degeneracy has been lifted by effective moment corresponds to the low spin the specific spatial order of the C60 balls. Let us recall then that, for a single orbital case the Mott JTD state sketched in insert. The magnetization transition occurs for U/W=1, while for N degenerate orbitals the critical value Uc is expected to be 1/2 increase below 47 K is that of the A15 phase larger [33], typically Uc=N W. Therefore Uc is reduced in the case of (NH3)K3C60 with respect to that of A3C60, for which the t1u orbital degeneracy is preserved. Though this complicates any direct Néel state. comparison between these two cases, these results establish that A3C60 are not far from a Mott transition, and hence that electronic correlations cannot be neglected.
4. Mott insulating state in Cs3C60 isomeric5. compoundsSC state induced under applied pressure and low T phase diagram A much simpler way to expand the C60 fccUnder lattice has an been applied of course pressure, to attempt the macroscopic insertion of Cs, the diamagneti sm is found to appear abruptly in both phases largest alkali ion, by the synthesis procedure used for the other A3C60. But, the bct Cs4C60 and the polymer phase Cs1C60 being stable at roomthough T, the Cs at3C a60 slightlycomposition lower is found critical metastable. pressure In a mixed pc for the fcc phase than for the A15. From NMR data in the composition sample it was found that a CsA153C60 phase becomes we could SC at high evidence pressure with that T thisc ~ 40 appearance K of SC is concomitant with an abrupt loss of [42]. A distinct chemical route finally recentlymagnetism, permitted which to produce definitely samples points containing toward the fcc- a first order transition. The phase diagrams slightly differ Cs3C60, mixed with an A15 isomeric phase. The latter has been found insulating at ambient pressure, and becomes SC under pressure, exhibitingfor a theSC dome two with Cs 3aC maximum60 isomers, Tc of but38K atas about shown 10 kbar in figure 4, they roughly merge together [13,47] if plotted [12]. versus VC60, the unit volume per C60 ball. There, the Tc data can be scaled as well with that for the In similar mixed phase samples, NMR experiments allowed us to separate the specific 133Cs spectra of the twoPressure isomers [13]. As displayedtunedother in fi knowngureSC 2, the infcc- A15 fccA 3phaseC60 ,Cs3C60 hasand a singleit has Cs been site with evidenced non that a similar maximum of Tc versus VC60 applies for cubic local symmetry, hence its seven linesthe quadrupole two structures split spectrum [13,47,48]. (133Cs has a nuclear spin I=7/2). The fcc-Cs3C60 NMR spectra exhibit the usual features found for all other fcc-A3C60, displaying the signals of the orthorhombic (O) site and of two tetrahedral sites (T, T’) [43]. Those are associated with the merohedral disorder [44] of the orientation of the C60 balls, which differentiates two local orientation orderings of the C60 around the alkali [45].
A15 Cs C 1 bar 3 60
T Fig. 4. Phase diagram representing Tc versus the Fig. 2. The unit cells of the A15 and fcc volume V per C ball, and the Mott transition isomeric phases of Cs C are displayed on C60 60 fcc Cs C 3 60 3 60 133 at pc (hatched bar) for the two isomer phases T' the left [12]. Their Cs NMR spectra [13] O taken at 1bar and T=300 K are reported on [47]. The corresponding pressure scales are
Cs NMR Intensity ( arb. units ) the right. The specific features of the spectra 133 shown on the upper scale. Data for the other are discussed in the text. 47.73 47.79 47.85 known fcc-A3C60 are reported as empty symbols. f ( MHz ) The transition to a magnetic insulating (MI) phase occurs only below 10 K in fcc-Cs3C60.
Ganin et al, Nature Materials, 2008, and Nature, 2010.
Ihara, Alloul, et al, PRL 2010.
Tuesday, July 8, 14 Other related ??
Discussion question: Superfluidity in He-3: `melted solid’ fruitful point of view?
Solid ≈ Mott Insulator Note that spin exchange scale of solid ≈ pairing scale in superfluid
Tuesday, July 8, 14 Comments
Vicinity of the electronic Mott metal-insulator transition: many fascinating phenomena including but not limited to superconductivity.
0. Zeroth order fact: The metal-insulator transition itself !
Tuesday, July 8, 14 Comments
Vicinity of the electronic Mott metal-insulator transition: many fascinating phenomena including but not limited to superconductivity.
0. Zeroth order fact: The metal-insulator transition itself !
1. Emergence of strange metals
Coherent Landau quasiparticles emerge (if at all) at a low energy scale.
Tuesday, July 8, 14 Example: cuprates
No coherent quasiparticles Coherent quasiparticles re-emerge in SC state.
Tuesday, July 8, 14 Example: cuprates
No coherent quasiparticles Underdoped: Across Tc two things happen.
1. Cooper pairs lose phase coherence
2. Electrons themselves also become incoherent
Tuesday, July 8, 14 Discussion question: What really drives Tc in underdoped cuprates?
Conventional wisdom (Emery, Kivelson, 95): Low superfluid density => phase fluctuations of Cooper pair.
A more refined (alternate?) possibility:
Incoherence of electron causes incoherence of Cooper pair.
Below electron coherence scale, Cooper pairs are able to condense.
Tc (and superfluid density) limited by low scale Tcoh of single particle coherence.
Tuesday, July 8, 14 Remark: effects of magnetic field on underdoped cuprate T TS, Lee, 2009
Pseudogap/ Fermi arcs Tcoh ?
Tc
SC Quantum oscillations H Explains why high T, low H ≠ low T, high H
H has suppressed Tc but not Tcoh => reveals new regime not accessed by destroying SC by heating.
Tuesday, July 8, 14 Comments
Vicinity of the electronic Mott metal-insulator transition: many fascinating phenomena including but not limited to superconductivity.
0. Zeroth order fact: The metal-insulator transition itself !
1. Emergence of strange metals
Coherent Landau quasiparticles emerge (if at all) at a low energy scale.
2. Other broken symmetry (eg, broken translation symmetry, electronic liquid crystals, ....) (see,eg, Randeria and Kivelson lectures)
3.. Emergence of strange insulators (see later).
SC near the Mott transition intertwined with many of these other phenomena.
Tuesday, July 8, 14 Plan
Lecture 1
1. Examples of superconductivity and related phenomena near Mott transition Briefly discuss general 2. Magnetism and Mott insulators nature of magnetism in Mott insulators Lecture 2
Metals and superconductors near the Mott transition
-(i) some general questions -(ii) some theoretical answers.
Tuesday, July 8, 14 Magnetism and Mott insulators
Tuesday, July 8, 14 Fate of electron spins in a Mott insulator
Common: Neel Antiferromagnetism (spontaneously breaks global spin SU(2) symmetry)
Interesting situations with low dimension/quantum fluctuations/``geometrically frustration”
Can get states that preserve spin SU(2) symmetry to T = 0 (``quantum paramagnets”) 24
Tuesday, July 8, 14 Spin ladders: A simple example of a quantum paramagnet
Tuesday, July 8, 14 Other quantum paramagnets: ``Spin-Peierls”/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.
Seen in many model calculations (Eg: Sandvik J-Q model on square lattice)
1 1 H = J S~ S~ Q S~ S~ S~ S~ i · j i · j 4 k · l 4 ij ijkl ✓ ◆✓ ◆ Xh i hXi
Materials: CuGeO3, TiOCl, some organic salts, ......
Tuesday, July 8, 14 Most interesting possibility: quantum spin liquids
What is a quantum spin liquid?
Rough definition: Quantum paramagnet which does not break any symmetries.
Better rough definition: Mott insulator with ground state not smoothly connected to band insulator.
Best definition: Mott insulator with ``long range quantum entanglement” in ground state.
Tuesday, July 8, 14 (Important) Digression
Non-local quantum entanglement in macroscopic matter
Tuesday, July 8, 14 Entanglement in quantum mechanics
Two parts A and B of a quantum mechanical system may be ``entangled” with each other.
Example: Spin orientations of two electrons in a simple molecule
unentangled; each spin by itself in a definite quantum state A B entangled: each spin by itself not in a definite B A + quantum state though full system is. A - B
Tuesday, July 8, 14
I would not call entanglement one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.
E. Schrodinger, 1935
Tuesday, July 8, 14 The relation of a part to the whole
Unentangled parts: wavefunction of whole system factorizes into a product of wavefunctions of parts. A B
Other examples: B or even + A A B
Entangled parts: Wavefunction of full system does not B + A factorize as products of wavefunction of parts. A - B Cannot describe one part fully without the other.
Tuesday, July 8, 14 Entanglement and macroscopic matter
How are the different parts of a piece of macroscopic matter entangled quantum mechanically with each other?
B A deep and fundamental question...... A
Importance only became clear in last few years.
Very fruitful in our ongoing attempt to characterize distinct phases of quantum Entanglement between A and B? matter.
Tuesday, July 8, 14 Phases of matter
Macroscopic matter in equilibrium organizes itself into phases.
Solids, liquids, gases......
Magnets......
Superconductors......
Tuesday, July 8, 14 Organizing principles: long range order and broken symmetry
Example: crystalline solid.
Atoms arrange themselves into an ordered array.
Pattern of atomic positions in one region determines atomic positions far away.
Broken symmetry: Microscopic interactions invariant under translating all atoms but equilibrium state is not.
Tuesday, July 8, 14 General consequences of broken symmetry
Pattern of broken symmetry determines many macroscopic properties of ordered matter.
Examples: rigidity of solids, persistence of currents in a superconductor, etc.
Broken symmetry point of view: unifying theoretical framework for many seemingly distinct properties of matter.
Tuesday, July 8, 14 Magnetism: an illustrative example
Most familiar form of magnetism: ferromagnetism.
Discovered may be around 600 BC.
Microscopic picture: Electron spins inside magnet are all pointed in same direction.
Example of broken symmetry: Microscopic interactions do not pick direction for spin but macroscopic magnetized state has specific spin orientation. Tuesday, July 8, 14 Antiferromagnetism: The more common magnetism
Actually the more common form of magnetism is not ferromagnetism but antiferromagnetism.
Also a broken symmetry state - spin orientation frozen in time but oscillates in space Microscopic interactions allow any orientation.
Despite being more common antiferromagnetism was discovered only in the 1930s!
Ferromagnetism: easily detected.
Antiferromagnetism: need microscopic probes that sense spin orientation with atomic spatial resolution.
Tuesday, July 8, 14 Quantum description of magnetism
The essential properties of these magnetic states of matter is contained in their ground state wavefunction.
Example: Prototypical wavefunctions
Ferromagnet ...... | ⇥ Antiferromagnet ...... | ⇥ ⇥ ⇤
Prototypical wavefunctions capture the pattern of broken symmetry which holds the key to many macroscopic properties of these phases.
38
Tuesday, July 8, 14 Short range entanglement
For familiar magnetic states, prototypical ground state wavefunction factorizes as ...... direct product of local degrees of | ⇥ freedom ...... | ⇥ ⇥ ⇤ Quantum entanglement short ranged in space.
1930s- present: elaboration of broken symmetry and other states with short range entanglement
39
Tuesday, July 8, 14 Emergence of classical physics
Broken symmetry states of magnetism:
Macroscopic description in terms of classical physics of the ``thing” that orders.
Example: spontaneous magnetization of a ferromagnet.
Microscopic quantum spins Macroscopic classical magnet
Tuesday, July 8, 14 Modern times
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.
+ ......
41 * In d > 1 Tuesday, July 8, 14 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.
Envisaged by P. W. Anderson (1973, 1987);
Tuesday, July 8, 14 Long Range Entangled (LRE) phases
Universal information about state not visible by looking only at small local part of system.
Passage from microscopic to macroscopic scales - classical physics does not emerge. (contrast with broken symmetry phases, eg, ferromagnet)
Older very famous example: Fractional quantum Hall states.
Other fascinating examples: Conventional metals: ``Fermi Liquid” state (oldest familiar Long Range Entangled state)
Many new metals: ``Non-fermi liquids”
Tuesday, July 8, 14 Can quantum spin liquid phases exist? Question for theory
Yes!!! (work of many people over last 25 years)
Many dramatic phenomena seen to be theoretically possible.
Examples:
1. Electron can break apart into fractions
2. Emergence of long range quantum mechanical interactions between fractional pieces of electron.
Similar phenomena established in FQHE in two dimensions in strong magnetic fields but now are known to be possible in much less restrictive situations.
Tuesday, July 8, 14 REPORTS
interaction is known to be insufficient to destroy the long-range ordered ground state (6). This has led to the proposals of numerous scenarios which Highly Mobile Gapless Excitations might stabilize a QSL state: spinon Fermi surface (7, 8), algebraic spin liquid (9), spin Bose metal (10), ring-exchange model (11), Z2 spin liquid in a Two-Dimensional Candidate state (12), chiral spin liquid (13), Hubbard model with a moderate onsite repulsion (14, 15), and Quantum Spin Liquid one-dimensionalization (16, 17). Nevertheless, the origin of the QSL in the organic compounds Minoru Yamashita,1* Norihito Nakata,1 Yoshinori Senshu,1 Masaki Nagata,1 remains an open question. 2,3 2 1 1 Hiroshi M. Yamamoto, Reizo Kato, Takasada Shibauchi, Yuji Matsuda * To understand the nature of QSLs, knowledge of the detailed structure of the low-lying elemen- The nature of quantum spin liquids, a novel state of matter where strong quantum fluctuations tary excitations in the zero-temperature limit, par- destroy the long-range magnetic order even at zero temperature, is a long-standing issue in ticularly the presence or absence of an excitation physics. We measured the low-temperature thermal conductivity of the recently discovered gap, is of primary importance (18). Such infor- quantum spin liquid candidate, the organic insulator EtMe3Sb[Pd(dmit)2]2.Asizablelinear mation bears immediate implications on the spin temperature dependence term is clearly resolved in the zero-temperature limit, indicating the correlations of the ground state, as well as the presence of gapless excitations with an extremely long mean free path, analogous to excitations correlation length scale of the QSL. For example, near the Fermi surface in pure metals. Its magnetic field dependence suggests a concomitant in 1D spin-1/2 Heisenberg chains, the elementary appearance of spin-gap–like excitations at low temperatures. These findings expose a highly Doexcitations quantum are gapless spinons (chargeless spin spin- liquid phases exist? unusual dichotomy that characterizes the low-energy physics of this quantum system. 1/2 quasiparticles) characterized by a linear en- ergy dispersion and a power-law decay of the spin pin systems confined to low dimensions long-range magnetic order down to a temperature correlationQuestion (19), whereas in the integer spin for case experiment exhibit a rich variety of quantum phenome- corresponding to J/12,000, where J (~250 K for such excitations are gapped (20). In the organic on July 20, 2010 Sna. Particularly intriguing are quantum both compounds) is the nearest-neighbor spin compound k-(BEDT-TTF)2Cu2(CN)3,wherethe spin liquids (QSLs), antiferromagnets with quan- interaction energy (exchangeYes coupling) - many (3, 5). In interesting a first putative QSL statecandidate was reported (3), thematerials pres- in last few years!! tum fluctuation–driven disordered ground states, triangular lattice antiferromagnet, the frustration ence of the spin excitation gap is controversial which have been attracting tremendous attention brought on by the nearest-neighbor Heisenberg (18, 21). In this compound, the stretched, non- for decades (1). The notion of QSLs is now firmly established in one-dimensional (1D) spin sys- tems. In dimensions greater than one, it is widely A Pd(dmit)2 molecule believed that QSL ground states emerge when in- teractions among the magnetic degrees of freedom www.sciencemag.org are incompatible with the underlying crystal ge- ometry, leading to a strong enhancement of quan- Non-magnetic layer (EtMe Sb, Et Me Sb) tum fluctuations. In 2D, typical examples of systems 3 2 2 where such geometrical frustrations are present are the triangular and kagomé lattices. Largely trig- gered by the proposal of the resonating-valence- bond theory on a 2D triangular lattice and its possible application to high-transition temperature cuprates Downloaded from Some layered inorganic minerals (2), realizing QSLs in 2D systems has been a B C long-sought goal. However, QSL states are hard to achieve experimentally because the presenceLayered organic (Y. Lee, Nocera et al, 2007) of small but finite 3D magnetic interactions ZnCu3(OH)6Cl2 usually results in some ordered (or frozen) state.crystals (ET)2Cu2(CN)3 Kanoda et al, 2003-now Two recently discovered organic insulators, Herbertsmithite k-[bis(ethylenedithio)-tetrathiafulvalene] Cu (CN) 2 2 3 EtMe3Sb[Pd(dmit)2]2 Kato et al, 2008 [k-(BEDT-TTF)2Cu2(CN)3](3)andEtMe3Sb[Pd t (dmit)2]2 (4, 5), both featuring 2D spin-1/2 Cu3V2O7(OH)2·2H2O (Z. Hiroi et al, 2010) Heisenberg triangular lattices, are believed to be promising candidate materials that are likely to host QSLs. In both compounds, nuclear magnetic Volborthite resonance (NMR) measurements have shown no Fig. 1. The crystal structure of EtMe3Sb[Pd(dmit)2]2 and Et2Me2Sb[Pd(dmit)2]2.(A)Aviewparallel to the 2D magnetic Pd(dmit)2 layer, separated by layers of a nonmagnetic cation. (B)Thespin structure of the 2D planes of EtMe3Sb[Pd(dmit)2]2 (dmit-131), where Et = C2H5,Me=CH3,and 1Department of Physics, Graduate School of Science, Kyoto 2 dmit = 1,3-dithiole-2-thione-4,5-dithiolate. Pd(dmit)2 are strongly dimerized (table S1), forming University, Kyoto 606-8502, Japan. RIKEN, Wako-shi, Saitama spin-1/2 units [Pd(dmit) ] – (blue arrows). The antiferromagnetic frustration gives rise to a state in 351-0198, Japan. 3Japan Science and Technology Agency, 2 2 Precursory Research for Embryonic Science and Technology which none of the spinsThree are frozen dimensional down to 19.4 mK (4). (C)Thespinstructureofthe2Dplanesof (JST-PRESTO), Kawaguchi, Saitama 332-0012, Japan. Et2Me2Sb[Pd(dmit)2]2 (dmit-221). A charge order transition occurs at 70 K, and the units are 0 2– *To whom correspondence should be addressed. E-mail: separated as neutraltransition [Pd(dmit)2]2 and metal divalent dimers oxide [Pd(dmit)2]2 .Thedivalentdimersform [email protected] (M.Y.); matsuda@ intradimer valence bonds, showing a nonmagnetic spin singlet (blue arrows) ground state with a scphys.kyoto-u.ac.jp (Y.M.) very large excitation gap (24). Na4Ir3O8 1246 4JUNE2010 VOL328 SCIENCE www.sciencemag.org (H. Takagi et al, 2008)
Tuesday, July 8, 14 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.
Tuesday, July 8, 14 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 phenomena low-lying 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 excitations 2008 that carry entropy, which 131,M. whichYamashita is absent et inal, 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 [k xxin(H) a– metal but 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)
were measuredtemperatures. in an (Inset) insulator. 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 Tuesday, July 8, 14 kxy/(kxx – kxx )as a function of H at H 0 2 4 6 8 10 12 assumed the linear energy dispersion e(k) = ħvsk, 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 Towards understanding experiments
Low-T properties of metals are determined by mobile electrons obeying Pauli exclusion principle.
In an insulator there cannot be mobile electrons.
A promising idea: perhaps there are emergent particles obeying Pauli exclusion that carry the electron spin but not its charge inside these materials.
Such phenomena are known to be theoretically possible in LRE phases (but are prohibited if there is only SRE)
Tuesday, July 8, 14 Picture of a particular quantum spin liquid
Electrons swimming in sea of +vely charged ions Metal
Electron charge gets pinned A quantum to ionic lattice while spins spin liquid continue to swim freely.
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Tuesday, July 8, 14 Future prospects: short term
1. Combined theory/experiment effort to characterize currently existing quantum spin liquids.
??Directly demonstrate non-local entanglement in experiment??
2. Theory predicts possibility of wide variety of such exotic phases of magnetic matter.
Tuesday, July 8, 14 Future prospects: long term
In the last 3 decades, growing number of experimental discoveries* have dethroned all the ``textbook” paradigms of the old field of solid state physics.
Some of these we understand; most of these we do not.
Our eyes have been opened to a new truly quantum world of 1023 electrons.
Characterizing ``pattern of entanglement” in macroscopic quantum matter promises to be as rich and profound as the previous century’s efforts at characterizing broken symmetry.
*FQHE (1982), high temperature superconductivity (1987), strange metals where electron-like charge carriers do not exist, quantum spin liquid magnets
Tuesday, July 8, 14 Entanglement and Phases of quantum matter
``Short range entangled” ``Long range entangled”
Conventional Gapped Gapless* phases/critical phases `topologically ordered’ points Eg: Band insulators, phases Eg: Fermi/ non-fermi liquids superfluids/ Eg: FQHE superconductors, antiferromagnets, ......
*Not just Goldstone
Tuesday, July 8, 14 Entanglement and Phases of quantum matter
``Short range entangled” ``Long range entangled”
Conventional Almost Gapped Gapless phases/critical phases conventional `topologically ordered’ points Eg: Band insulators, phases: phases Eg: Fermi/ non-fermi liquids superfluids/ Topological Eg: FQHE superconductors, insulators antiferromagnets, ......
Tuesday, July 8, 14 End of digression
Tuesday, July 8, 14 Remarks
A wide variety of distinct kinds of quantum spin liquid phases can exist - distinct physical properties and low energy effective field theories.
A gross distinction: Gapped versus gapless
Gapped spin liquids - many interesting properties (topological order, etc); not the focus here.
Gapless spin liquids: relevant to experiments and as a platform for understanding emergence of metal/superconductors near the Mott transition.
Tuesday, July 8, 14 A useful theoretical framework
1 ~ ↵ Slave particle construction S~ = f † f r 2 r↵ 2 r
fr↵: fermionic ‘spinon’ with spin ↵.
Constraint fr†fr = 1 ensures physical Hilbert space.
Redundant description, e.g., can let f ei✓r f . r↵ ! r↵ Full redundancy: SU(2) gauge transformation
Tuesday, July 8, 14 A useful theoretical framework (cont’d)
Strategy: Put f in some mean field state with a quadratic Hamiltonian.
Examples: 1. ‘spinon metal’ H = t f †f + h.c Spin physics similar to metal MF f r r0 Xrr0 2. ‘Paired’
HMF = tf fr†fr0 + rr0 (fr fr0 fr fr0 )+h.c " # # " rr X0 Spin physics similar to superconductor
Mean field theory for highly non-trivial quantum spin liquid insulators
Tuesday, July 8, 14 Fluctuations
Mean field Hamiltonian breaks gauge redundancy down to a subgroup.
Fluctuations beyond mean field: must couple f to gauge fields in that subgroup.
Effective field theory: spinon + fluctuating gauge fields.
Use to address stability (`lower critical dimension’, etc) and predict testable physical properties.
Example:
1. `Spinon metal’:
Spinon Fermi surface + fluctuating U(1) gauge field.
2. Paired spin liquid
Spinons + fluctuating Z2 gauge field.
Tuesday, July 8, 14 A physical description: Quantum spin liquids near the Mott transition
Start with the metal.
Interacting Ferm ifluid: Incorporate correlations w ith Jastrow factor
⇤ ( r ⇥ ,.....r ⇥ )= f( r r ) ⇤ ( r ⇥ ,.....r ⇥ )(1) F 1 1 N N i j Slater 1 1 N N ij Y Specialcase:G utzwillerapproximation to lattice H ubbard m odel;choose
fij = g ij (2 )
with g<1toweighdowndoubleoccupancyofanysite.
Can think of f(ri rj) as wave function of a boson fluid. Boson coordinates are slaved to electrons.
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Tuesday, July 8, 14 Quantum spin liquids near the Mott transition
Wavefunction of insulator: Replace f(r r ) by wave function of boson insulator i j BI
F = BI Slater
BI suppresses charge fluctuations. Extreme case: Remove all charge fluctuation - Gutzwiller projection PG to no double occupancy to get spin wave function.
SL = PG Slater
Can repeat with BCS wavefunction instead of Slater for the `paired’ spin liquid.
Wavefunctions closely connected to those from slave particle approach.
Tuesday, July 8, 14 Though quantum paramagnets may exist the cuprate (and many other) Mott insulators are actually antiferromagnetically ordered.
Nevertheless it will be useful to consider the emergence of metals and superconductors from various kinds of Mott insulators, not just antiferromagnets.
Tuesday, July 8, 14 Doping a quantum spin liquid
Spinon metal → Fermi Liquid
Paired spin liquid → Superconductor.
Wavefunctions:
Doped spinon metal = PG Slater (now not at half-filling ) Wavefunction of (correlated) Fermi liquid.
Doped paired spin liquid = PG BCS Wavefunction of (correlated) superconductor.
Tuesday, July 8, 14 High Tc cuprates: how does a Fermi surface emerge from a doped Mott insulator?
Evolution from Mott insulator to overdoped metal : emergence of large Fermi surface with area set by usual Luttinger count.
Mott insulator: No Fermi surface Overdoped metal: ADMR, quantum Large Fermi surface oscillations (Hussey), ARPES (Damascelli,….)
Tuesday, July 8, 14 High Tc cuprates: how does a Fermi surface emerge from a doped Mott insulator?
Large gapless Fermi surface present even in optimal doped strange metal albeit without Landau quasiparticles .
Mott insulator: No Fermi surface
Tuesday, July 8, 14 High Tc cuprates: how does a Fermi surface emerge from a doped Mott insulator?
Large gapless Fermi surface present also in optimal doped strange metal albeit without Landau quasiparticles .
Even in the pseudogap regime the minimum gap features (nodal Fermi arcs, antinodal gaps) in ARPES are apparently located at large Fermi surface!
Mott insulator: No Fermi surface In SC state, the d-wave gap is centered on the large Fermi surface down to low doping.
Tuesday, July 8, 14 A basic question
Quite generally, large Fermi surface visible (at least at short time scales) already in underdoped.
How should we understand the emergence of the large Fermi surface in a doped Mott insulator?
Tuesday, July 8, 14 Motivates general study of how metal emerges from a Mott insulator.
Tuesday, July 8, 14 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
Tuesday, July 8, 14 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.
Tuesday, July 8, 14 Quantum spin liquid Mott insulators:
Opportunity for progress on the Mott transition - study metal-insulator transition without complications of magnetism.
Tuesday, July 8, 14 Some candidate spin liquid materials
(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 `hyperkagome’ lattice
ZnCu3(OH)6Cl2 2d Kagome lattice (`strong’ Mott insulator) Volborthtite, ......
Tuesday, July 8, 14 Some candidate materials
(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 `hyperkagome’ lattice
Close to pressure driven Mott transition: `weak’ Mott insulators
ZnCu3(OH)6Cl2 2d Kagome lattice (`strong’ Mott insulator) Volborthtite, ......
Tuesday, July 8, 14 Possible experimental realization of a second order(?) Mott transition
Tuesday, July 8, 14 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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Tuesday, July 8, 14 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 Kondo breakdown transition in heavy fermion metals, and may be also around optimal doping in cuprates.
Tuesday, July 8, 14 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: DMFT in infinite d (Vollhardt, Metzner, Kotliar, Georges 1990s), slave particle theories in d = 2, d = 3 (TS, Vojta, Sachdev 2003, TS 2008)
Tuesday, July 8, 14 Quantum spin liquids and the Mott transition
Some questions:
1. Can the Mott transition be continuous at T = 0?
2. Fate of the electronic Fermi surface?
????? t/U Spin liquid insulator; Fermi liquid; No Fermi surface Full fermi surface
Only currently available theoretical framework to answer these questions is slave particle gauge theory.
(Mean field: Florens, Georges 2005; 77 Spin liquid phase: Motrunich, 07, S.S. Lee, P.A. Lee, 07)
Tuesday, July 8, 14 Slave particle framework
Split electron operator cr = brfr↵
Fermi liquid: b =0 h i6
Mott insulator: br gapped
Mott transition: br critical
In all three cases fr↵ form a Fermi surface. Low energy e↵ective theory: Couple b, f to fluctuating U(1) gauge field.
Tuesday, July 8, 14 Example: lattice Hubbard model
ni (ni 1) H = t c† c + h.c + U (1) ij i j 2 ij i X X ⇣ ⌘ X Slave boson representation ci = bifi . Factorize electron hopping as
b†b f † f + b†b f † f (2) h i ji i j i jh i j i Boson carries electron charge => Interaction term becomes a boson-boson interaction
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Tuesday, July 8, 14 `Mean field’ description
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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Tuesday, July 8, 14 Description of correlated metal
b condensed, =0 6 =>cr =fr Electron Green’s function
Tuesday, July 8, 14 Quantum spin liquids and the Mott transition
1. Can the Mott transition be continuous?
2. Fate of the electronic Fermi surface?
Analyse fluctuations: Concrete tractable theory of a continuous Mott transition; demonstrate critical Fermi surface at Mott transition;
Definite predictions for many quantities (TS, 2008, Witczak-Krempa, Ghaemi, Kim, TS, 2012).
- Universal jump of residual resistivity on approaching from metal - Log divergent effective mass - Two diverging time/length scales near transition - Emergence of marginal fermi liquids
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Tuesday, July 8, 14 Superconductivity near a Mott transition
Some basic questions
1.How does a metal emerge from a Mott insulator?
2. Why superconductivity?
Simple physical picture (Anderson1987): Superexchange favors formation of singlet valence bonds between localized spins.
Doped Mott insulator: Hole motion in background of valence bonds.
Tuesday, July 8, 14 Cartoon pictures
Large doping: Hubbard-U not very effective in t blocking charge motion Expect `large Fermi surface’ with area set by 1-x.
What happens as doping is reduced to approach Mott insulator?
Tuesday, July 8, 14 Cartoon pictures
Low doping: Most of the time most electrons unable to hop to neighboring sites t due to Mott-blocking. If electrons stay localized next to each other long enough, will develop superexchange which will lock their spins into singlets.
Electron configuration changes at long times – conveniently view as motion of holes in sea of singlets.
Resulting state: metallic but with a spin gap due to valence bond formation => `pseudogap metal”.
Tuesday, July 8, 14 Why superconductivity?
Crucial Anderson insight:
Singlet valence bond between localized spins: A localized Cooper pair.
`Pairing’ comes from superexchange due to a repulsive Hubbard interaction.
If spins were truly localized, Cooper pairs do not move => no superconductivity.
Nonzero doping: allow room for motion of valence bonds => superconductivity!
Hole picture: Coherent hole motion in valence bond sea
Tuesday, July 8, 14 Fate of collection of valence bonds Two general possibilities: Valence bonds can crystallize to form a solid VBS state (``Valence Bond Solid”) (with doping OR a `bond Stay liquid to form a `Resonating Valence centered’ Bond’ stripe)
Ongoing debates on which one is more relevant but very formation of valence bond crucial ingredient in much thinking about cuprates.
RVB state = quantum spin liquid
Tuesday, July 8, 14 Does valence bond formation provide a legitimate theoretical route for superconductivity in a repulsive doped Mott insulator? See Kivelson Many different kinds of studies (work of large number of lectures people):
1.1d doped spin ladder: Zero doping – spin gapped insulator due to valence bond formation. Dope – (power law) superconductor .
2. Quasi-1d: Weakly coupled ladders
3. Inhomogenous 2d: Checkerboard Hubbard model
4. Superconductivity in doped VBS Mott insulators (`large-N’ methods): spontaneously generate weakly coupled ladders.
5. Superconductivity in doped spin liquid Mott insulators (i.e insulators with one electron per site)
Tuesday, July 8, 14 Superconductivity in doped spin liquids: mean field
Tuesday, July 8, 14 Superconductivity in doped spin liquids: variational wavefunctions
Tuesday, July 8, 14 Common features of superconductivity in doped (paramagnetic) Mott insulators
Tuesday, July 8, 14 Refined basic theory questions
Is superconductivity with gapless nodal excitations possible in a doped Mott insulator?
Only currently known route is by doping a gapless spin liquid Mott insulator.
Eg: Z2 spin liquid with d-wave pairing of fermionic spinons → d-wave SC with nodal quasiparticles.
Many similarities to physics of cuprates (more detail: Paremakanti lectures).
Tuesday, July 8, 14 Comments
Quantum spin liquids a useful platform to understand the emergence of metals and superconductors from a Mott insulator.
``Recognizable caricature” (to borrow from Sidney Coleman) of cuprate (and organics) physics
Direct relevance to cuprates?
Tuesday, July 8, 14 Transition to SC: onset of coherence
ARPES results
Tuesday, July 8, 14 Onset of coherence in transport
Tuesday, July 8, 14