Introduction to Quantum Materials Leon Balents, KITP QS3 School, June 11, 2018 Quantum Materials • What are they? Materials where electrons are doing interesting quantum things • The plan: • Lecture 1: Concepts in Quantum Materials • Lecture 2: Survey of actual materials Themes of modern QMs • Order • Topology • Entanglement • Correlations • Dynamics Order: symmetry • Symmetry: a way to organize matter • A symmetry is some operation that leaves a system (i.e. a material) invariant (unchanged • In physics, we usually mean it leaves the Hamiltonian invariant U<latexit sha1_base64="z44s+OqrR28LnNVtYut8w3Yl5bw=">AAAB+nicbVBNS8NAEN34WetXqkcvi0XwVBIR9CIUvfRYwbSFNpbNZpIu3WzC7kYptT/FiwdFvPpLvPlv3LY5aOuDgcd7M8zMCzLOlHacb2tldW19Y7O0Vd7e2d3btysHLZXmkoJHU57KTkAUcCbA00xz6GQSSBJwaAfDm6nffgCpWCru9CgDPyGxYBGjRBupb1e8+15I4hgkbmAPX+FG3646NWcGvEzcglRRgWbf/uqFKc0TEJpyolTXdTLtj4nUjHKYlHu5gozQIYmha6ggCSh/PDt9gk+MEuIolaaExjP198SYJEqNksB0JkQP1KI3Ff/zurmOLv0xE1muQdD5oijnWKd4mgMOmQSq+cgQQiUzt2I6IJJQbdIqmxDcxZeXSeus5jo19/a8Wr8u4iihI3SMTpGLLlAdNVATeYiiR/SMXtGb9WS9WO/Wx7x1xSpmDtEfWJ8/44aScA==</latexit> †HU = H Order and symmetry • Why symmetry? • It is persistent: it only changes through a phase transition • It has numerous implications: • Quantum numbers and degeneracies • Conservation laws • Brings powerful mathematics of group theory • The set of all symmetries of a system form its symmetry group. Materials with different symmetry groups are in different phases Ising model • A canonical example z z x z z H = J σ σ hx σ σ<latexit sha1_base64="qkGFhco5brjSZMvHUYBkOShCXm8=">AAACF3icbVDLSsNAFJ34rPUVdSnIYBHcWBIRdFl047KCfUATy2Q6SYfOI8xMlBq68yf8Bbe6dyduXbr1S5w+BG09cOFwzr3ce0+UMqqN5306c/MLi0vLhZXi6tr6xqa7tV3XMlOY1LBkUjUjpAmjgtQMNYw0U0UQjxhpRL2Lod+4JUpTKa5NPyUhR4mgMcXIWKnt7gWaJhzd3MNA0aRrkFLyDh79qG235JW9EeAs8SekBCaott2voCNxxokwmCGtW76XmjBHylDMyKAYZJqkCPdQQlqWCsSJDvPRHwN4YJUOjKWyJQwcqb8ncsS17vPIdnJkunraG4r/ehGf2mziszCnIs0MEXi8OM4YNBIOQ4Idqgg2rG8Jwora2yHuIoWwsVEWbSj+dASzpH5c9r2yf3VSqpxP4imAXbAPDoEPTkEFXIIqqAEMHsATeAYvzqPz6rw57+PWOWcyswP+wPn4BguHoBk=</latexit> σ − i j − i ! ij i Z2 symmetry <latexit sha1_base64="fYBqiH2eniqeMADXi+eIXUyZPnE=">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</latexit> Xh i X kT/J PM σz =0 <latexit sha1_base64="qzim7GBagBk1qh+ULk66r2UEJfU=">AAACFnicbVC7SgNBFL3rM8ZX1FKEwSBYhV0RtBGCNpYRzAOy6zI7mWyGzMwuM7NCXFL5E/6CrfZ2Ymtr65c4eRSaeODC4Zx7ufeeKOVMG9f9chYWl5ZXVgtrxfWNza3t0s5uQyeZIrROEp6oVoQ15UzSumGG01aqKBYRp82ofzXym/dUaZbIWzNIaSBwLFmXEWysFJYOfI5lzCnyNYsFDtndg68mygVyw1LZrbhjoHniTUkZpqiFpW+/k5BMUGkIx1q3PTc1QY6VYYTTYdHPNE0x6eOYti2VWFAd5OM3hujIKh3UTZQtadBY/T2RY6H1QES2U2DT07PeSPzXi8TMZtM9D3Im08xQSSaLuxlHJkGjjFCHKUoMH1iCiWL2dkR6WGFibJJFG4o3G8E8aZxUPLfi3ZyWq5fTeAqwD4dwDB6cQRWuoQZ1IPAIz/ACr86T8+a8Ox+T1gVnOrMHf+B8/gAWKp7x</latexit> i z = σ h i <latexit sha1_base64="3OR9pNEstHxh2r/d7Mf5mcR+AwE=">AAACGHicbVC7SgNBFJ31GeMramnhYBCswq4I2ghBG8sI5gHZdZmd3GyGzMwuM7NCXFL6E/6CrfZ2Ymtn65c4eRSaeODC4Zx7ufeeKOVMG9f9chYWl5ZXVgtrxfWNza3t0s5uQyeZolCnCU9UKyIaOJNQN8xwaKUKiIg4NKP+1chv3oPSLJG3ZpBCIEgsWZdRYqwUlg78VLMLnxMZc8C+ZrEgIbt78NVECUtlt+KOgeeJNyVlNEUtLH37nYRmAqShnGjd9tzUBDlRhlEOw6KfaUgJ7ZMY2pZKIkAH+fiRIT6ySgd3E2VLGjxWf0/kRGg9EJHtFMT09Kw3Ev/1IjGz2XTPg5zJNDMg6WRxN+PYJHiUEu4wBdTwgSWEKmZvx7RHFKHGZlm0oXizEcyTxknFcyvezWm5ejmNp4D20SE6Rh46Q1V0jWqojih6RM/oBb06T86b8+58TFoXnOnMHvoD5/MHlqSgXQ==</latexit> i symmetric h i thermal FM “order phase symmetry broken transition parameter” σz = m =0 h<latexit sha1_base64="S5uWgTYSXVLTBuxJzCl6jrxQXNs=">AAACIXicbVDLSgNBEJyNrxhfqx69DAYhp7Argl6EoBePEcwDsjHMTnqTITOz68ysEEN+wZ/wF7zq3Zt4E29+iZNkD5pY0FBUddPdFSacaeN5n05uaXlldS2/XtjY3NrecXf36jpOFYUajXmsmiHRwJmEmmGGQzNRQETIoREOLid+4x6UZrG8McME2oL0JIsYJcZKHbcUcCJ7HHCgWU+QDrt9CNRMOcdBIrDAgYQ77HXcolf2psCLxM9IEWWodtzvoBvTVIA0lBOtW76XmPaIKMMoh3EhSDUkhA5ID1qWSiJAt0fTj8b4yCpdHMXKljR4qv6eGBGh9VCEtlMQ09fz3kT81wvF3GYTnbVHTCapAUlni6OUYxPjSVy4yxRQw4eWEKqYvR3TPlGEGhtqwYbiz0ewSOrHZd8r+9cnxcpFFk8eHaBDVEI+OkUVdIWqqIYoekTP6AW9Ok/Om/PufMxac042s4/+wPn6AfslowU=</latexit> i i ± 6 QCP hx/J Symmetries in QMs • Basic symmetries of our world: • space-time (Lorentz/Poincare) symmetry • spatial isotropy and translations • time reversal • Charge/particle number conservation • Approximate symmetries (sometimes) • spin-rotation • various internal quantum numbers • These things are broken down to varying degrees in different QMs RuCl3 LiNaSO4 Tourmaline Proustite Li2ZrF6 Coquimbite Portlandite Fluocerite-(La) (#172) P63 (#173) P6 (#174) P6/m (#175) P63/m (#176) P622 (#177) P6122 (#178) P6522 (#179) Cystallography tetracosakis(µ2-Methoxo)- catena-[2,2'-(biphenyl-4,4'- dodecakis(µ2-proline)-dodeca- iron(iii) dodecaperchlorate AgF )(H2O)4 •NephCrystaleline structure:LiNaCO3 diyldiimino)di 230benzene crystallographic-1,3,5-triol] Fluorapatite space groups3 LaBTB m2 (#187) P6c2 (#188) P62m (#189) P62c (#190) P6/mmm (#191)P6/mcc (#192) P63/mcm (#193) P63/mmc (#194) • Classifies the arrangements of atoms (which break the symmetries of free space) BaTi(Si O ) Na O KCaF(CO3) • Wallpaper3 9 groups2 2 inSrBe 2d3O4 (c.f. 2dAlB2 materials)Beryl ZrI3 Graphite 3 (#203) Im3 (#204) Pa3 (#205) Ia3 (#206) P432 (#207) P4232 (#208) F432 (#209) F4132 (#210) I432 (#211) • Basic input to many things • Phonons, elasticity, band structure... Dodecasil Na1-x•WO3 LOADSPyrite of extremelyYttria BIF useful-9-Cu stuffBe3 Pon2 BilbaoPCN-20 Te(OH)6 NiHg c (#219) I43d (#220)crystallographicPm3m (#221) Pn3n (#22 server...2) Pm3n (#223) Pn3m (#224) Fm3m (#225) Fm3c (#226) Fd3m (#227) • Structural phase transition = change of space group. Katoite Mn3B7O13I hydrogarnet ZIF-71-RHO Co-Squarate V6SnSi (NH4)[(Mo12O36)(AsO4)Mo(MoO)] NaCl LTA Spinel J B Goodenough tilting of the CuO, octahedra-which aretetragonal 3. Electron correlatlon energies (c/a =- 1)"along a [ 1lo] axis to transform the structure fromthe tetragonal symmetry of figure l(a)to the Substitution of a larger Ba2+ or Sr2+ ion for La3+ in orthorhombic symmetry of figure l(b) below IT; = 530 K La,CuO, not only expands the mean A-0 bond length, [4]. Bending of the Cu-0-Cu bond angle from 180" butalso oxidises the CuO,planes; removal of anti- Chemical and structural relationships in high-T, materials permits a matching of the sizes of the basal-plane lattice bondingelectrons from theplane shortens the Cu-0 parameters of the two intergrowth layers without com- bondlength. Consequently the tolerance factor t pression of the Cu-0 bond length. increasesLa,CuO, with and y Nd,CuO,;in the system the former La,-,Sr,CuO,, is a parent com-so I; plane Cu-0 bonds whereas a reduction would add anti- decreasespound forwith p-type increasing superconductivity, y in thephase the latterdiagram for n-typeof bonding c~,*~-,,~electrons and thus expand the in-plane figuresuperconductivity. 3. Moreover, as both t and the oxidation state of Cu-0 bonds against the internal compressive force. As 2.4. The T'-tetragonal structure the CuO,Stoichiometric planes increases, La,CuO, the drivinghasthe tetragonal force(T) for insert- a result, La,CuO, is readily oxidised, but attempts to structure of figure l(a) at temperatures T > IT;; aco- reduce it have failed. A t < 1 at high temperatures allows theinsertion of ingoperative excess oxygen tilting decreases,of the CuO, and octahedra stoichiometric at lower oxygen tem- excess interstitial oxygen into the La202+dlayers. concentrationsperatures (T are< IT;) readilyyields the achieved orthorhombic in air (0) in structurethe com- However, a Coulomb repulsion between the interstitial positionalof figure range l(b) [3,0.07 41. < The y < phase 0.15 changewhereas, at for T =x IT;< 0.07,is a 2.2. Oxidation and c-axis oxide ions forces a strong relaxation of the it istypical necessary second-order, to annealsoft-mode in transition.N, to avoidIn insertion both of Even with electronordering, a t < 1 occurs at tem- c-axis oxygen neighbouring an interstitial, so onlya excessstructures, oxygen. (Cu0,)'- Annealing planes in (or 0, sheets allows Textension < IT;) alternate of the peratures high enough (T > 400 "C) for the insertion of small concentrationcan be accommodated before a rangewith of twooxygen (Lao)' stoichiometry layershaving tothe yrocksalt = 0.27;structure. for y 0.27, > interstitial oxygen between adjacent La0 planes;the Thisconfiguration has two immediate consequences phase transition occurs within the La20,+d layers. The a high oxygen pressureappears to be necessary to interstitial oxygen atoms occupy the sites coordinated that are common to all the p-type copper oxide super- layer transforms from a rocksalt to a fluorite configu- prevent oxygen loss [14,151. Oxygen-deficient by four La andfour c-axis oxygen A similar insertion conductors: (i) the layers are alternately charged posi- [S]. ration by adisplacement of the c-axis oxygen to the La, -,S~,CUO,-~has oxygen vacancies at the c-axis of interstitial oxygen occurs in La,NiO, The inter- tively and negatively, which creates an interlayer [7]. interstitial sites; in the absence of any c-axis oxygen, the sites. nal electric field parallel to the c axis stabilises an inter- internal electric field that shifts the energies within one oxygen is coordinated by only four Ln3+ ions (figure