Quantum phase transitions of insulators, superconductors and metals in two dimensions

Quantum phase transitions of insulators, superconductors and metals in two dimensions Talk online: sachdev.physics.harvard.edu HARVARD

Outline 1. Phenomenology of the cuprate superconductors (and other compounds) 2. QPT of antiferromagnetic insulators (and bosons at rational filling) 3. QPT of d-wave superconductors: Fermi points of massless Dirac fermions 4. QPT of Fermi surfaces: A. Finite wavevector ordering (SDW/CDW): Hot spots on Fermi surfaces B. Zero wavevector ordering (Nematic): Hot Fermi surfaces

Theory of quantum criticality in the cuprates Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001). Thermally fluctuating SDW Magnetic quantum criticality Spin gap d-wave superconductor E. G. Moon and S. Sachdev, Phy. Rev. B 80, 035117 (2009) Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m.

Theory of quantum criticality in the cuprates Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Thermally fluctuating SDW Magnetic quantum criticality Spin gap Strange Metal d-wave Classical spin waves superconductor Neel order Large Fermi surface Quantum critical E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, 067202 Dilute (2001). triplon gas E. G. Moon and S. Sachdev, Phy. Rev. B 80, 035117 (2009) Criticality of the coupled dimer antiferromagnet at x=xs Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m.

Theory of quantum criticality in the cuprates ncreasing SDW Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Thermally fluctuating SDW Magnetic quantum criticality Criticality of the topological change in Fermi surface at x=xm Spin gap Spin density wave (SDW) Strange Metal d-wave superconductor Large Fermi surface Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m. E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001). E. G. Moon and S. Sachdev, Phy. Rev. B 80, 035117 (2009)

TlCuCl 3

TlCuCl 3 An insulator whose spin susceptibility vanishes exponentially as the temperature T tends to zero.

TlCuCl 3 at ambient pressure N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001).

TlCuCl 3 at ambient pressure Sharp spin 1 particle excitation above an energy gap (spin gap) N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001).

Square lattice antiferromagnet H = ij J ij Si S j Ground state has long-range Néel order Order parameter is a single vector field ϕ = η isi η i = ±1 on two sublattices ϕ = 0 in Néel state.

Square lattice antiferromagnet H = ij J ij Si S j J J/λ Weaken some bonds to induce spin entanglement in a new quantum phase

Square lattice antiferromagnet H = ij J ij Si S j J J/λ Ground state is a quantum paramagnet with spins locked in valence bond singlets = 1 2

λ c λ Quantum critical point with non-local entanglement in spin wavefunction M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev.B 65, 014407 (2002).

λ c λ Pressure in TlCuCl3

Excitation spectrum in the paramagnetic phase λ c λ

Excitation spectrum in the paramagnetic phase λ c λ Sharp spin 1 particle excitation above an energy gap (spin gap)

Excitation spectrum in the Néel phase λ c λ

Excitation spectrum in the Néel phase λ c λ Spin waves

Derivation of field theory of critical point

Description using Landau-Ginzburg field theory λ c CFT3 λ O(3) order parameter ϕ ( S = d 2 rdτ τ ϕ) 2 + c 2 ( r ϕ ) 2 +(λ λ c )ϕ 2 + u ϕ 2 2

Excitation spectrum in the paramagnetic phase λ λ c V (ϕ )=(λ λ c )ϕ 2 + u ϕ 2 2 2.0 V (ϕ ) 1.5 Spin S =1 λ >λ c ϕ triplon

Excitation spectrum in the Néel phase V (ϕ ) 0.1 0.2 λ c V (ϕ )=(λ λ c )ϕ 2 + u ϕ 2 2 λ <λ c λ 0.3 ϕ Spin waves ( Goldstone modes) and a longitudinal Higgs particle

TlCuCl 3 with varying pressure Observation of 3 2 low energy modes, emergence of new longitudinal mode (the Higgs boson ) in Néel phase, and vanishing of Néel temperature at quantum critical point Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans Ulrich Gudel, Severian Gvasaliya, Hannu Mutka, and Martin Boehm, Phys. Rev. Lett. 100, 205701 (2008)

Prediction of quantum field theory Potential for ϕ fluctuations: V (ϕ )=(λ λ c )ϕ 2 + u ϕ 2 2 Paramagnetic phase, λ >λ c Expand about ϕ = 0: 2.0 V (ϕ ) 1.5 V (ϕ ) (λ λ c )ϕ 2 ϕ Yields 3 particles with energy gap (λ λ c ) Néel phase, λ < λ c Expand ϕ = 0, 0, (λ c λ)/(2u) + ϕ 1 : V (ϕ ) 0.1 0.2 0.3 ϕ V (ϕ ) 2(λ c λ)ϕ 2 1z Yields 2 gapless spin waves and one Higgs particle with energy gap 2(λ c λ)

Prediction of quantum field theory Energy of Higgs particle Energy of triplon 1.4 = 2 V (ϕ )=(λ λ c )ϕ 2 + u ϕ 2 2 Energy 2*E(p < p c ), E(p > p c ) [mev] 1.2 1 0.8 0.6 0.4 0.2 TlCuCl 3 p c = 7 kbar T = 1.85 K Q=(0 4 0) L (p < p c ) L (p > p c ) Q=(0 0 1) L,T 1 (p < p c ) L (p > p c ) E(p < p c ) unscaled 0 0 1 1.5 2 Pressure (p p c ) [kbar] S. Sachdev, arxiv:0901.4103

O(3) order parameter ϕ λ c CFT3 ( τ ϕ) 2 + c 2 ( r ϕ ) 2 + sϕ 2 + u ϕ 2 2 λ S = d 2 rdτ

Quantum Monte Carlo - critical exponents S. Wenzel and W. Janke, Phys. Rev. B 79, 014410 (2009) M. Troyer, M. Imada, and K. Ueda, J. Phys. Soc. Japan (1997)

Quantum Monte Carlo - critical exponents Field-theoretic RG of CFT3 E. Vicari et al. S. Wenzel and W. Janke, Phys. Rev. B 79, 014410 (2009) M. Troyer, M. Imada, and K. Ueda, J. Phys. Soc. Japan (1997)