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NON-EQUILIBRIUM DYNAMIC CRITICAL SCALING OF THE QUANTUM ISING CHAIN Michael Kolodrubetz Princeton University In collaboration with: Bryan Clark, David Huse David Pekker Krishnendu Sengupta QUANTUM STATE OF TRANSVERSE-FIELD ISING MODEL DURING SLOW RAMP IS… Universal Non-thermal Non-equilibrium Dephasing resistant Experimentally viable CLASSICAL PHASE TRANSITIONS “Magnetization” Landau-Ginzburg functional CLASSICAL PHASE TRANSITIONS “Magnetization” CLASSICAL PHASE TRANSITIONS “Magnetization” Thermal fluctuations QUANTUM PHASE TRANSITIONS One-dimensional transverse-field Ising chain QUANTUM PHASE TRANSITIONS One-dimensional transverse-field Ising chain Paramagnet (PM) Ferromagnet (FM) QUANTUM PHASE TRANSITIONS One-dimensional transverse-field Ising chain Paramagnet (PM) Ferromagnet (FM) Quantum fluctuations CRITICAL SCALING [Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt] CRITICAL SCALING [Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt] CRITICAL SCALING , Correlation length critical exponent Dynamic critical exponent [Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt] CRITICAL SCALING Ising: , Correlation length critical exponent Dynamic critical exponent CRITICAL SCALING Ising: , Order parameter critical exponent Correlation length critical exponent Dynamic critical exponent CRITICAL SCALING Ising: , Order parameter critical exponent Correlation length critical exponent Dynamic critical exponent KIBBLE-ZUREK RAMPS Ramp rate KIBBLE-ZUREK RAMPS Ramp rate KIBBLE-ZUREK RAMPS Ramp rate Adiabatic KIBBLE-ZUREK RAMPS Ramp rate Impulse Adiabatic KIBBLE-ZUREK RAMPS Ramp rate Impulse Adiabatic KIBBLE-ZUREK RAMPS Ramp rate Impulse Adiabatic KIBBLE-ZUREK RAMPS Ramp rate Impulse Adiabatic KIBBLE-ZUREK RAMPS Ramp rate Impulse Adiabatic KIBBLE-ZUREK RAMPS METHOD IDEA WHEN IT WORKS “Old-school” Kibble-Zurek and set the “interesting” time and length scales -Ramp to the QCP -Ramp to deep in the FM phase [Kibble 1976, Zurek 1985] KIBBLE-ZUREK RAMPS METHOD IDEA WHEN IT WORKS “Old-school” Kibble-Zurek and set the “interesting” time and length scales -Ramp to the QCP -Ramp to deep in the FM phase Most quantities show scaling collapse when scaled by and Throughout the ramp [Kibble 1976, Zurek 1985] Kibble-Zurek scaling [Deng et. al. 2008, Erez et. al., in prep., Polkovnikov, …] KIBBLE-ZUREK RAMPS Ramp rate Impulse Adiabatic KIBBLE-ZUREK RAMPS Ramp rate Impulse Adiabatic KIBBLE-ZUREK RAMPS Ramp rate Impulse Adiabatic KIBBLE-ZUREK RAMPS Ramp rate Impulse Adiabatic TRANSVERSE-FIELD ISING CHAIN Sachdev: “Quantum Phase Transitions” TRANSVERSE-FIELD ISING CHAIN Sachdev: “Quantum Phase Transitions” Wigner fermionize phase TRANSVERSE-FIELD ISING CHAIN Sachdev: “Quantum Phase Transitions” Wigner fermionize phase TRANSVERSE-FIELD ISING CHAIN Sachdev: “Quantum Phase Transitions” Wigner fermionize phase Quadratic Integrable TRANSVERSE-FIELD ISING CHAIN Sachdev: “Quantum Phase Transitions” Wigner fermionize phase Quadratic Integrable Hamiltonian conserves parity for each mode k TRANSVERSE-FIELD ISING CHAIN Sachdev: “Quantum Phase Transitions” Wigner fermionize phase Quadratic Integrable Hamiltonian conserves parity for each mode k Work in subspace where parity is even TRANSVERSE-FIELD ISING CHAIN Sachdev: “Quantum Phase Transitions” Wigner fermionize phase Quadratic Integrable Hamiltonian conserves parity for each mode k Work in subspace where parity is even TRANSVERSE-FIELD ISING CHAIN TRANSVERSE-FIELD ISING CHAIN TRANSVERSE-FIELD ISING CHAIN TRANSVERSE-FIELD ISING CHAIN TRANSVERSE-FIELD ISING CHAIN TRANSVERSE-FIELD ISING CHAIN TRANSVERSE-FIELD ISING CHAIN Low energy, long wavelength theory KIBBLE-ZUREK RAMPS Low energy, long wavelength theory? Ramp rate Impulse Adiabatic KIBBLE-ZUREK RAMPS Low energy, long wavelength theory Ramp rate Impulse Adiabatic KIBBLE-ZUREK RAMPS Low energy, long wavelength theory Ramp rate Impulse Adiabatic KIBBLE-ZUREK SCALING LIMIT Schrödinger Equation OR Observable Fixed KIBBLE-ZUREK SCALING LIMIT KIBBLE-ZUREK SCALING LIMIT KIBBLE-ZUREK SCALING LIMIT KIBBLE-ZUREK SCALING LIMIT KIBBLE-ZUREK OBSERVABLES Excess heat Spin-spin correlation function KIBBLE-ZUREK OBSERVABLES Excess heat Spin-spin correlation function KIBBLE-ZUREK OBSERVABLES KIBBLE-ZUREK OBSERVABLES KIBBLE-ZUREK OBSERVABLES KIBBLE-ZUREK OBSERVABLES KIBBLE-ZUREK OBSERVABLES KIBBLE-ZUREK OBSERVABLES KIBBLE-ZUREK OBSERVABLES FINITE-SIZE SCALING FINITE-SIZE SCALING FINITE-SIZE SCALING Finite size effects can be ignored FINITE-SIZE SCALING FINITE-SIZE SCALING EQUILIBRIUM VIA DYNAMICS KZ scaling function If dynamic scaling functions exist, they must have the equilibrium critical exponents Equilibrium scaling function FINITE-SIZE SCALING FINITE-SIZE SCALING LANDAU-ZENER DYNAMICS LANDAU-ZENER DYNAMICS LANDAU-ZENER DYNAMICS FINITE-SIZE SCALING LANDAU-ZENER DYNAMICS ATHERMAL PROPERTIES ATHERMAL PROPERTIES ATHERMAL PROPERTIES ATHERMAL PROPERTIES ATHERMAL PROPERTIES Inverted ATHERMAL PROPERTIES ATHERMAL PROPERTIES DEPHASING Protocol Ramp to create excitations Freeze the Hamiltonian Wait DEPHASING … … Protocol Ramp to create excitations Freeze the Hamiltonian Wait DEPHASING … … Protocol Ramp to create excitations Freeze the Hamiltonian Wait DEPHASING … … Protocol Ramp to create excitations Freeze the Hamiltonian Wait DEPHASING … … Protocol Ramp to create excitations Freeze the Hamiltonian Wait DEPHASING … … Protocol Ramp to create excitations Freeze the Hamiltonian Wait DEPHASING … … Protocol Ramp to create excitations Freeze the Hamiltonian Wait Dephasing in integrable model: Generalized Gibb’s ensemble (GGE) DEPHASING … … Dephasing in integrable model: Generalized Gibb’s ensemble (GGE) Does dephasing occur during the KibbleZurek ramp? DEPHASING … … Dephasing in integrable model: Generalized Gibb’s ensemble (GGE) DEPHASING … … as Dephasing in integrable model: Generalized Gibb’s ensemble (GGE) DEPHASING … … as Dephasing in integrable model: Generalized Gibb’s ensemble (GGE) DEPHASING Cubic ramp: … … DEPHASING Cubic ramp: … … as DEPHASING Cubic ramp: … … as UNIVERSALITY UNIVERSALITY UNIVERSALITY Additional terms change (renormalize) the nonuniversal aspects of the critical point They do not change critical scaling Critical exponents Scaling functions Debated for non-integrable system dynamics UNIVERSALITY UNIVERSALITY UNIVERSALITY Paramagnet Antiferromagnet UNIVERSALITY Paramagnet Antiferromagnet Ramp the tilt ( ) linearly in time UNIVERSALITY UNIVERSALITY UNIVERSALITY UNIVERSALITY CONCLUSIONS Solved dynamic critical scaling behavior of the TFI chain Athermal negative correlations Phase-locked high order ramps Strong numerical evidence for universality Tilted boson model has same scaling functions Experimentally accessible Athermal features robust against open boundary conditions Open b.c. simplifies measurement Time scales already available [Simon et. al., 2007] DEPHASING VIA QUASIPARTICLES DEPHASING VIA QUASIPARTICLES OPEN BOUNDARY CONDITIONS OPEN BOUNDARY CONDITIONS OPEN BOUNDARY CONDITIONS UNIVERSALITY Remove spin ups on neighboring sites