Report

UNIVERSALITY AND DYNAMIC LOCALIZATION IN KIBBLE-ZUREK SCALING OF THE QUANTUM ISING CHAIN Michael Kolodrubetz Boston University In collaboration with: B.K. Clark, D. Huse (Princeton) A. Polkovnikov, A. Katz (BU) OUTLINE Part I: Kibble-Zurek scaling of the transverse-field Ising chain Part II: Transverse-field Ising chain with a dynamic field TRANSVERSE-FIELD ISING CHAIN One-dimensional transverse-field Ising chain TRANSVERSE-FIELD ISING CHAIN One-dimensional transverse-field Ising chain TRANSVERSE-FIELD ISING CHAIN One-dimensional transverse-field Ising chain Paramagnet (PM) TRANSVERSE-FIELD ISING CHAIN One-dimensional transverse-field Ising chain Paramagnet (PM) Ferromagnet (FM) TRANSVERSE-FIELD ISING CHAIN One-dimensional transverse-field Ising chain Paramagnet (PM) Ferromagnet (FM) Quantum phase transition QUANTUM PHASE TRANSITION [Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt] QUANTUM PHASE TRANSITION [Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt] QUANTUM PHASE TRANSITION [Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt] QUANTUM PHASE TRANSITION [Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt] QUANTUM PHASE TRANSITION , [Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt] QUANTUM PHASE TRANSITION , [Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt] QUANTUM PHASE TRANSITION , Correlation length critical exponent Dynamic critical exponent [Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt] QUANTUM PHASE TRANSITION , Correlation length critical exponent Dynamic critical exponent Ising: QUANTUM PHASE TRANSITION Can these results be extended to non-equilbrium dynamics? , Correlation length critical exponent Dynamic critical exponent Ising: KIBBLE-ZUREK RAMPS Ramp rate Kibble-Zurek Ramp through the critical point at a constant, finite rate KIBBLE-ZUREK RAMPS Ramp rate KIBBLE-ZUREK RAMPS Ramp rate KIBBLE-ZUREK RAMPS Ramp rate Fall out of equilibrium KIBBLE-ZUREK RAMPS Ramp rate Fall out of equilibrium KIBBLE-ZUREK RAMPS Ramp rate Fall out of equilibrium KIBBLE-ZUREK RAMPS Ramp rate Fall out of equilibrium KIBBLE-ZUREK RAMPS Ramp rate KIBBLE-ZUREK RAMPS Ramp rate KIBBLE-ZUREK RAMPS Ramp rate KIBBLE-ZUREK RAMPS Ramp rate KIBBLE-ZUREK RAMPS Ramp rate KIBBLE-ZUREK RAMPS Kibble-Zurek ramps show non-equilibrium scaling [Chandran et. al., Deng et. al., etc.] KIBBLE-ZUREK RAMPS Kibble-Zurek ramps show non-equilibrium scaling (in the limit of slow ramps) [Chandran et. al., Deng et. al., etc.] KIBBLE-ZUREK RAMPS Kibble-Zurek ramps show non-equilibrium scaling (in the limit of slow ramps) More than a theory of defect production! [Chandran et. al., Deng et. al., etc.] KIBBLE-ZUREK RAMPS Kibble-Zurek ramps show non-equilibrium scaling (in the limit of slow ramps) More than a theory of defect production! [Chandran et. al., Deng et. al., etc.] KIBBLE-ZUREK SCALING Excess heat KIBBLE-ZUREK OBSERVABLES KIBBLE-ZUREK OBSERVABLES KIBBLE-ZUREK OBSERVABLES KIBBLE-ZUREK OBSERVABLES KIBBLE-ZUREK OBSERVABLES KIBBLE-ZUREK SCALING KIBBLE-ZUREK SCALING KIBBLE-ZUREK SCALING 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 Work in subspace where each mode (k,-k) is either occupied or unoccupied TRANSVERSE-FIELD ISING CHAIN Sachdev: “Quantum Phase Transitions” Wigner fermionize phase Quadratic Integrable Work in subspace where each mode (k,-k) is either occupied or unoccupied EQUILIBRIUM SCALING “Spin up” (k,-k) unoccupied “Spin down” (k,-k) occupied EQUILIBRIUM SCALING “Spin up” (k,-k) unoccupied “Spin down” (k,-k) occupied EQUILIBRIUM SCALING “Spin up” (k,-k) unoccupied “Spin down” (k,-k) occupied Low energy, long wavelength theory? EQUILIBRIUM SCALING “Spin up” (k,-k) unoccupied “Spin down” (k,-k) occupied Low energy, long wavelength theory KIBBLE-ZUREK SCALING KIBBLE-ZUREK SCALING Low energy, long wavelength theory? KIBBLE-ZUREK SCALING Low energy, long wavelength theory? KIBBLE-ZUREK SCALING Low energy, long wavelength theory KIBBLE-ZUREK SCALING Schrödinger Equation OR Observable KIBBLE-ZUREK SCALING Schrödinger Equation OR Observable Fixed KIBBLE-ZUREK SCALING Schrödinger Equation OR Observable Fixed KIBBLE-ZUREK SCALING KIBBLE-ZUREK SCALING KIBBLE-ZUREK SCALING KIBBLE-ZUREK SCALING KIBBLE-ZUREK SCALING KIBBLE-ZUREK SCALING OUTLINE Part I: Kibble-Zurek scaling of the transverse-field Ising chain Dynamics near QCP gives non-equilibrium critical scaling theory Part II: Transverse-field Ising chain with a dynamic field OUTLINE Part I: Kibble-Zurek scaling of the transverse-field Ising chain Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? Part II: Transverse-field Ising chain with a dynamic field UNIVERSALITY Theory Sachdev et al. (2002) Experiment Greiner group (Harvard) Nagerl group (Innsbruck) UNIVERSALITY Theory Sachdev et al. (2002) Experiment Greiner group (Harvard) Nagerl group (Innsbruck) UNIVERSALITY or Theory Sachdev et al. (2002) Experiment Greiner group (Harvard) Nagerl group (Innsbruck) UNIVERSALITY Ramp the tilt linearly in time or Theory Sachdev et al. (2002) Experiment Greiner group (Harvard) Nagerl group (Innsbruck) UNIVERSALITY Ramp the tilt linearly in time: Solve numerically with DMRG or Theory Sachdev et al. (2002) Experiment Greiner group (Harvard) Nagerl group (Innsbruck) UNIVERSALITY UNIVERSALITY UNIVERSALITY UNIVERSALITY UNIVERSALITY UNIVERSALITY Matches to analytical solution of the Ising chain! OUTLINE Part I: Kibble-Zurek scaling of the transverse-field Ising chain Dynamics near QCP gives non-equilibrium critical scaling theory Dynamics are universal to Ising-like QPTs Part II: Transverse-field Ising chain with a dynamic field OUTLINE Part I: Kibble-Zurek scaling of the transverse-field Ising chain Dynamics near QCP gives non-equilibrium critical scaling theory Dynamics are universal to Ising-like QPTs What are some properties of the scaling functions? Part II: Transverse-field Ising chain with a dynamic field NON-EQUILIBRIUM PROPERTIES Spin-spin correlation function NON-EQUILIBRIUM PROPERTIES Spin-spin correlation function NON-EQUILIBRIUM PROPERTIES NON-EQUILIBRIUM PROPERTIES Ground state NON-EQUILIBRIUM PROPERTIES Ground state NON-EQUILIBRIUM PROPERTIES Ground state NON-EQUILIBRIUM PROPERTIES NON-EQUILIBRIUM PROPERTIES NON-EQUILIBRIUM PROPERTIES NON-EQUILIBRIUM PROPERTIES NON-EQUILIBRIUM PROPERTIES NON-EQUILIBRIUM PROPERTIES NON-EQUILIBRIUM PROPERTIES NON-EQUILIBRIUM PROPERTIES Inverted NON-EQUILIBRIUM PROPERTIES NON-EQUILIBRIUM PROPERTIES NON-EQUILIBRIUM PROPERTIES Antiferromagnetic NON-EQUILIBRIUM PROPERTIES OUTLINE Part I: Kibble-Zurek scaling of the transverse-field Ising chain Dynamics near QCP gives non-equilibrium critical scaling theory Dynamics are universal to Ising-like QPTs Long-time dynamics are athermal Part II: Transverse-field Ising chain with a dynamic field OUTLINE Part I: Kibble-Zurek scaling of the transverse-field Ising chain Dynamics near QCP gives non-equilibrium critical scaling theory Dynamics are universal to Ising-like QPTs Long-time dynamics are athermal Finite size scaling, dephasing, experiments… Part II: Transverse-field Ising chain with a dynamic field OUTLINE Part I: Kibble-Zurek scaling of the transverse-field Ising chain Dynamics near QCP gives non-equilibrium critical scaling theory Dynamics are universal to Ising-like QPTs Long-time dynamics are athermal Finite size scaling, dephasing, experiments… Part II: Transverse-field Ising chain with a dynamic field DYNAMIC-FIELD ISING CHAIN Basic idea: Add (classical) dynamics to the transverse field DYNAMIC-FIELD ISING CHAIN Basic idea: Add (classical) dynamics to the transverse field DYNAMIC-FIELD ISING CHAIN Basic idea: Add (classical) dynamics to the transverse field “Friction” = back-action of spins on field DYNAMIC-FIELD ISING CHAIN Basic idea: Add (classical) dynamics to the transverse field “Friction” = back-action of spins on field Mass is extensive ( ) Mean-field coupling between field and spins DYNAMIC-FIELD ISING CHAIN Basic idea: Add (classical) dynamics to the transverse field “Friction” = back-action of spins on field Mass is extensive ( ) Mean-field coupling between field and spins What happens when field tries to pass through the critical point? DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN as DYNAMIC-FIELD ISING CHAIN as Field motion arrested by QCP! DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN Dynamics dominated by critical behavior DYNAMIC-FIELD ISING CHAIN Dynamics dominated by critical behavior Linearize the Hamiltonian DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN What happens for other models? DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN Crossover tunable via… …dimensionality DYNAMIC-FIELD ISING CHAIN Crossover tunable via… …dimensionality …critical exponents DYNAMIC-FIELD ISING CHAIN Crossover tunable via… …dimensionality …critical exponents Possibility of as OUTLINE Part I: Kibble-Zurek scaling of the transverse-field Ising chain Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Part II: Transverse-field Ising chain with a dynamic field Field is trapped at QCP by critical absorption OUTLINE Part I: Kibble-Zurek scaling of the transverse-field Ising chain Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Part II: Transverse-field Ising chain with a dynamic field Field is trapped at QCP by critical absorption Dynamics of field during trapping? DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN Overdamped/underdamped? DYNAMIC-FIELD ISING CHAIN Measure velocity at QCP Overdamped/underdamped? DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN Hypothesis Initial momentum is the relevant scale for dynamics DYNAMIC-FIELD ISING CHAIN Hypothesis Initial momentum is the relevant scale for dynamics DYNAMIC-FIELD ISING CHAIN Hypothesis Initial momentum is the relevant scale for dynamics DYNAMIC-FIELD ISING CHAIN Hypothesis Initial momentum is the relevant scale for dynamics OUTLINE Part I: Kibble-Zurek scaling of the transverse-field Ising chain Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Part II: Transverse-field Ising chain with a dynamic field System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse OUTLINE Part I: Kibble-Zurek scaling of the transverse-field Ising chain Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Part II: Transverse-field Ising chain with a dynamic field System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse Analytical understanding of late-time dynamics? DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN Dephasing DYNAMIC-FIELD ISING CHAIN Dephasing Are long-time dynamics welldescribed by the dephased ensemble? (generalized Gibbs ensemble / GGE) DYNAMIC-FIELD ISING CHAIN Manually dephase Are long-time dynamics welldescribed by the dephased ensemble? (generalized Gibbs ensemble / GGE) DYNAMIC-FIELD ISING CHAIN Manually dephase Are long-time dynamics welldescribed by the dephased ensemble? (generalized Gibbs ensemble / GGE) DYNAMIC-FIELD ISING CHAIN Manually dephase Are long-time dynamics welldescribed by the dephased ensemble? YES! DYNAMIC-FIELD ISING CHAIN Approximate dynamics by adiabatic PT Based on unpublished work by Anatoli Polkovnikov and Luca D’Alessio DYNAMIC-FIELD ISING CHAIN Approximate dynamics by adiabatic PT Based on unpublished work by Anatoli Polkovnikov and Luca D’Alessio Start from stationary state of DYNAMIC-FIELD ISING CHAIN Approximate dynamics by adiabatic PT Based on unpublished work by Anatoli Polkovnikov and Luca D’Alessio Start from stationary state of Go to the “moving frame” of Frame that locally diagonalizes DYNAMIC-FIELD ISING CHAIN Approximate dynamics by adiabatic PT Based on unpublished work by Anatoli Polkovnikov and Luca D’Alessio Start from stationary state of Go to the “moving frame” of Frame that locally diagonalizes DYNAMIC-FIELD ISING CHAIN Approximate dynamics by adiabatic PT Based on unpublished work by Anatoli Polkovnikov and Luca D’Alessio Start from stationary state of Go to the “moving frame” of Frame that locally diagonalizes Treat term via 2nd order time-dependent PT DYNAMIC-FIELD ISING CHAIN Approximate dynamics by adiabatic PT DYNAMIC-FIELD ISING CHAIN Approximate dynamics by adiabatic PT Need to know… Initial condition on , Mode occupation numbers DYNAMIC-FIELD ISING CHAIN DYNAMIC-FIELD ISING CHAIN OUTLINE Part I: Kibble-Zurek scaling of the transverse-field Ising chain Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Part II: Transverse-field Ising chain with a dynamic field System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse Late-time dynamics are given by dephasing OUTLINE Part I: Kibble-Zurek scaling of the transverse-field Ising chain Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Part II: Transverse-field Ising chain with a dynamic field System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse Late-time dynamics are given by dephasing FUTURE DIRECTIONS Analytically understand the dynamics via APT FUTURE DIRECTIONS Analytically understand the dynamics via APT Remove the offset potential Is it RG relevant? FUTURE DIRECTIONS Analytically understand the dynamics via APT Remove the offset potential Is it RG relevant? Tune the scaling of excess heat FUTURE DIRECTIONS Analytically understand the dynamics via APT Remove the offset potential Is it RG relevant? Tune the scaling of excess heat Generalize Ising model to higher dimensions Use models with other critical exponents FUTURE DIRECTIONS Analytically understand the dynamics via APT Remove the offset potential Is it RG relevant? Tune the scaling of excess heat Generalize Ising model to higher dimensions Use models with other critical exponents What happens if as FUTURE DIRECTIONS Analytically understand the dynamics via APT Remove the offset potential Is it RG relevant? Tune the scaling of excess heat Generalize Ising model to higher dimensions Use models with other critical exponents What happens if as Relationship to the Higgs boson? SUMMARY Part I: Kibble-Zurek scaling of the transverse-field Ising chain Part II: Transverse-field Ising chain with a dynamic field DYNAMIC-FIELD ISING CHAIN DYNAMIC-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 TRANSVERSE-FIELD ISING CHAIN