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## Music of the (Fermi) Spheres: Gauge Theories of Quantum Matter Simplicity in addition to Co

Sharpe, Melissa, Morning On-Air Personality has reference to this Academic Journal, PHwiki organized this Journal Music of the (Fermi) Spheres: Gauge Theories of Quantum Matter Simplicity in addition to Complexity in Quantum Matter What is The Problem in High Temperature Superconductors An Inverted View Topology in addition to Duality in Superconductors Emergent QED3 Theory of Underdoped Cuprates Zlatko Tesanovic Johns Hopkins University E-mail: zbt@pha.jhu.edu Web: http://www.pha.jhu.edu/people/faculty/zbt.html Collaborators: O. Vafek (Stan as long as d), A. Melikyan (Florida), M. Franz (UBC) Quantum (~ = 1) Condensed Matter Physics Theory of Everything (TOE): Schrödinger Equation: i~ ¶tY= HY H = åi pi2/2m + åhi,ji V(ri rj) Y (r1, ,rN;t) is wavefunction of N=1023 particles Since basic dynamical laws are known, a superstring theorist might describe condensed matter physics as engineering the ultimate insult in physics. All one needs to do is solve the above partial differential equation in 3£1023 + 1 variables in addition to determine Y. Once Y is known, all physically quantities can be evaluated. In short, condensed matter physics is not a fundamental science. H is Hamiltonian of all 1023 particles in a condensed matter system (electrons plus atomic/molecular ions) in addition to V (ri rj) is their Coulomb (electrostatic) interaction. Complexity in addition to Emergence Behavior of interacting system changes dramatically when number of particles becomes large. Entirely new phenomena in addition to new concepts arise in N 1 limit, which cannot be deconstructed down to the level of few particles in addition to basic dynamical laws. Such phenomena in addition to concepts are called emergent. Examples: temperature in addition to entropy, turbulence, friction, fluctuations in stock markets or noise in cellular phones. All are intrinsically collective notions, brought to life through cooperative behavior of many degrees of freedom. This misguided point of view, once common, has disappeared in the last two-three decades. The reason is our deeper underst in addition to ing of complexity in addition to emergent behavior.

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Emergent Quantum Field Theories of Correlated Systems An important example of an emergent phenomenon is a continuous quantum/thermal phase transition takes place only in N 1 limit. Near continuous phase transition, quantum matter exhibits complexity that rivals indeed equals that of a quantum field theory. EXAMPLES OF EFFECTIVE QUANTUM FIELD THEORIES (QFT) IN QUANTUM CONDENSED MATTER: Quantum Magnetism Non-Abelian QFTs Quantum Hall Effect Chern-Simons QFTs Superfluid 4He Cosmic Strings Superfluid 3He St in addition to ard Model Heavy Fermions Supersymmetric QFT Strontium ruthenates Non-Abelian CS QFT High-Tc Superconductors QED3 Beautiful in addition to fundamental symmetries of such theories con as long as mal, relativistic, gauge, chiral, are emergent in addition to are dynamically generated at low energies in addition to long distances. Four Discoveries in Underdoped Cuprates High Temperature Cuprate Superconductors La1-xSrxCu2O4 is an example of HTS compound. Superconducting state is quasi two-dimensional in nature with CuO2 layers being the key ingredient. La in addition to Sr supply chemical environment which shifts charge of CuO2 layers away from 1 electron/Cu. This doping with x holes generates a high-Tc superconductor from an insulating antiferromagnet at x=0. In Hg- or Tl-based cuprates Tc is almost 200 K ! Compare this to Tc » 20 K as long as best superconductors prior to 1986.

Inverted Approach to High-Tc Superconductivity Normal state is strange in addition to non-Fermi liquid in nature. Strong correlations among its fermionic excitations are apparent. In contrast, superconducting state appears BCS-like. Low energy fermions are protected by a large d-wave (pseudo)gap ( ()»3, »T3) Devise a theory with correlated d-wave SC as a reference state. Key questions: i) How does a correlated superconductor become normal ii) What are the normal ground states in natural proximity to such superconductor What is the nature of pseudogap state iii) What is the low energy effective theory within the pseudogap whose role parallels that of a Fermi liquid in conventional metals How Does Superconductor Become Normal Cooper pair wavefunction: hc+ (r)c+ (r)i » D (r) = Dexp(i). In a superconductor, Cooper pairs Bose condense: hDi ¹ 0 . I) In BCS theory, superconductivity disappears when amplitude D ! 0 at T = Tc traditional paradigm applies. Classical vortex wake vortex of an aircraft water vortex

Superconductor Normal Giant Nernst Effect in addition to Quantum Diamagnetism in HTS Vortex fluctuations in addition to enhanced quantum diamagnetism in the pseudogap state !! Z. A. Xu et al., Nature 406, 486 (2000) QED3 Phase Diagram Growing Mott correlations suppress commensurate density fluctuations Mott correlations are reflected in strong quantum phase fluctuations

What is the microscopic theory of HTS Focus on the saddle point of HLdSC associated with d-wave superconductivity effective hopping d-wave pairing (JSi¢Sj) on-site repulsion extended range (Up À t, , t2 /, ) interactions (Coulomb, phonons, ) Formation of Local Spin Singlets on SITES (Strong Attraction, U < 0) CDW SC Formation of Local Singlets on BONDS (Strong Repulsion, U > 0) !! If center-of-mass moves around freely while relative motion is suppressed

Dual Theory of Superconductivity I Introduce vortex/antivortex creation operators Yyv(a)(r,t) . Vortices/antivortices have mass M in addition to quantum tunnel through time t . Frequency of vortex-antivortex pair creation/annihilation processes is set by Dv . Quantum action: in dSC ZT, PRL 93, 217004 (2004) A. Melikyan in addition to ZT, PRB 71, 214511 (2005) Dual Opposite of a Correlated Superconductor I Dual Opposite of a Correlated Superconductor II Phase is not frustrated in addition to has unhindered fluctuations if Strong Hubbard repulsion keeps spin up in addition to down electrons apart:

Dual Theory of Superconductivity I Introduce vortex/antivortex creation operators Yyv(a)(r,t) . Vortices/antivortices have mass M in addition to quantum tunnel through time t . Frequency of vortex-antivortex pair creation/annihilation processes is set by Dv . Quantum action: in dSC F is a dual of a superconducting order parameter in addition to describes vortex loops in 2+1D spacetime. Why is dualization useful In dual language normal state hFi = 0 is a physical superconductor. Dual superfluid, hFi ¹ 0 , is the pseudogap state (intermediate phase is called supersolid : hFi = 0 in addition to hF (r)2i ¹ 0 ). Bosonic Bogoliubov-deGennes spinors Y= (Yv,Ya) carry dual charge ed=2p in addition to couple to Ad ( conservation of vorticity). In field theory limit Y ! F in addition to one obtains a dual Ginzburg-L in addition to au theory (SHE) Strong SC correlations Dual mean-field theory dual superconductor dual normal L. Balents, L. Bartosch, A. Burkov, S. Sachdev & K. Sengupta, PRB 71, 144508 (2005) ZT, PRL 93, 217004 (2004); A. Melikyan in addition to ZT, PRB 71, 214511 (2005) Dual Theory of Superconductivity II Dual Abrikosov-Hofstadter Problem I ZT, PRL 93, 217004 (2004) A. Melikyan in addition to ZT, PRB 71, 214511 (2005)

Quantum particle hopping on a tight-binding lattice in a uni as long as m magnetic field (Peierls, Hofstadter). Translation operators do not commute: TxTy TyTx = exp(ixy) xy = 2(p/q) – flux enclosed by translations Dual Abrikosov-Hofstadter Problem II ZT, PRL 93, 217004 (2004) A. Melikyan in addition to ZT, PRB 71, 214511 (2005) In L in addition to au gauge: HHof has a Bloch periodicity with a size q unit cell Spectrum has q-fold degenerate minima which span a U(q) Hofstadter manifold What is the ground state within this U(q) manifold Quartic in addition to other higher order terms decide ! Magic fractions: f = ½ , ¼ , 7/16 Magic dopings: x = 0, ½ , 1/8 Dual Abrikosov-Hofstadter Problem (x=1/8 $ f=7/16) Modulation in F in addition to Bd $ Modulation in gap function Dij in addition to charge density ni Result is modulated local tunneling density of states (STM) Cooper pair (nodal) density-wave : F is a dual of a superconducting order parameter in addition to describes vortex loops in 2+1D spacetime. Why is dualization useful In dual language normal state hFi = 0 is a physical superconductor. Dual superfluid, hFi ¹ 0 , is the pseudogap state (intermediate phase is called supersolid : hFi = 0 in addition to hF (r)2i ¹ 0 ). Bosonic Bogoliubov-deGennes spinors Y= (Yv,Ya) carry dual charge ed=2p in addition to couple to Ad ( conservation of vorticity). In field theory limit Y ! F in addition to one obtains a dual Ginzburg-L in addition to au theory (SHE) Strong SC correlations Dual mean-field theory dual superconductor dual normal L. Balents, L. Bartosch, A. Burkov, S. Sachdev & K. Sengupta, PRB 71, 144508 (2005) ZT, PRL 93, 217004 (2004); A. Melikyan in addition to ZT, PRB 71, 214511 (2005) Dual Theory of Superconductivity II

HTS are Nodal d-wave Superconductors (Phase sensitive exps., ARPES, FT-STS, etc.) FT-STS measured D ARPES D(q), Mesot -PRL 83,840 (1999) Courtesy of J.C. Davis Tsuei & Kirtley, 1997 FT-STS from Davis group, Nature 422, 520 (2003). Nodal Fermions Must Be Included !! HTS are unique not only because they have high Tc in addition to strong quantum vortex-antivortex fluctuations (caused by Mott correlations). They also are d-wave superconductors !! Low energy effective theory must contain FERMIONS !! This is in contrast to s-wave/He-type (BOSON ONLY) models of SC fluctuations s-wave ¹ D(k) does not change sign on Fermi surface quasiparticles are gapped: E(k) = § [(e (k)-m)2 + D2]1/2 » § D d-wave dSc-normal d-wave Superconductor Normal nodal quasiparticles vortex-antivortex pairs

Some Notorious Square Roots Singular Gauge Trans as long as mation Eliminate phase from pairing term in favor of : fA(B)(r,t) is the singular part of the phase due to A(B) vortices/antivortices. Average over all A/B divisions gauge choice having the same symmetry as fermion-vortex dynamics. M. Franz in addition to ZT, PRL 84, 554 (2000) Effective gauge theory with two gauge fields: Doppler gauge field vm = (1/2)(¶mfA + ¶mfB) couples to BdG fermion charge Berry gauge field am = (1/2)(¶mfA – ¶mfB) couples to BdG fermion spin Physics Behind vm in addition to am M. Franz in addition to ZT, PRL 87, 257003 (2001) hc/2e vortex/antivortex is seen by fermions as two confined half-Dirac strings corresponding to Doppler in addition to Berry gauge fields v in addition to a . Doppler field vm = (1/2)(¶mfA + ¶mfB) couples to BdG fermion charge. Doppler shift in fermion energies caused by quantum current (vortex/antivortex) fluctuations. Berry gauge field am = (1/2)(¶mfA – ¶mfB) couples to BdG fermion spin. Topological frustration inflicted by hc/2e vortices/antivortices on charge e quasiparticles Berry phase exp(§ ip) = § 1 . + hc/2e vortex/antivortex vm am e,s

Dual Abrikosov-Hofstadter Problem (x=1/8 $ f=7/16) Modulation in F in addition to Bd $ Modulation in gap function Dij in addition to charge density ni Result is modulated local tunneling density of states (STM) Cooper pair (nodal) density-wave : Pseudogap State: Charge Insulator, Spin Conductor Charge current cancels: Jc = 0 . Charge center-of-mass is pinned to the lattice: Abrikosov-Hofstadter dual vortex solid. Spin current Js is finite. Pseudogap is a spin (semi) metal: Spinful BdG nodal fermions with long range gauge field correlations $ chiral QED3. e e charge current spin current O. Vafek in addition to ZT, PRL 91, 237001 (2003) Phase Diagram from QED3 Theory of HTS Antiferromagnet/SDW (Mott ins.) Pairing pseudogap (chiral QED3) dSC Nodal pair DW (chiral QED3) Supersolid Nodal Pair CDW is a charge insulator (Abrikosov-Hofstadter dual solid) but a spin conductor (chiral QED3) thermal metal Using SC order parameter in addition to dual order parameter one obtains the phase diagram :

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