Dynamics of Second Order Phase Transitions and Formation of Topological Defects. W. H. Zurek Los Alamos

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1 Dynamics of Second Order Phase Transitions and Formation of Topological Defects W. H. Zurek Los Alamos

2 QUANTUM ISING MODEL Lattice of spin 1/2 particles interacting with an external force (e.g., magnetic field along the x axis) and with each other (ferromagnetic Ising interaction along the z axis): a H = J(t) σ x l W σ z σ z l l +1 l l, = ( + ) / 2 Quantum phase transition occurs as J(t) decreases. Then K ( + )( + )( + )L, analogue of the symmetric vacuum, is no longer favored energetically: K or K (or any superposition thereof) are the ground states. one of two canonical models of for quantum phase transitions S. Sachdev, Quantum Phase Transitions, CUP 1999

3 Symmetry Breaking and Defects H = J(t) σ x l W σ z σ z l l +1 l Broken symmetry states after the phase transition: Quantum case: True ground state is their superposition Also possible kinks : When broken symmetry vacuum is chosen locally l topological defects can form

4 Plan Introduce the quantum Ising model (done) Dynamics of symmetry breaking in thermodynamic phase transitions: general theory (Jim Anglin, Nuno Antunes, Luis Bettencourt, Fernando Cucchietti, Bogdan Damski, Uwe Dorner, Jacek Dziarmaga, Pablo Laguna, Augusto Roncaglia, Augusto Smerzi, Andy Yates, Peter Zoller ) Experimental situation (liquid crystals, superfluids, superconductors, other possibilities) Apply thermodynamic approach to quantum Ising model & compare with numerical simulations Introduce a purely quantum approach and compare with thermodynamic approach and with numerical simulations

5 Spontaneous Symmetry Breaking V Ginzburg Landau (ϕ) = εϕ 2 + ϕ 4 During the transition ε changes sign (for instance, relative temperature increasing from -1 to +1). Choice of the phase of ϕ -- which may be the phase of the wave function of the condensate -- is the choice of the broken symmetry state ( vacuum ).

6 Local choices may not be globally compatible: topological defects can form during quench! ˆ ξ Density of vortices ( strings ): n =1/ ˆ ξ 2 (Kibble, 76)

7 Formation of kinks in a 1-D Landau-Ginzburg system V Ginzburg Landau (ϕ) = εϕ 2 + ϕ 4 with real ϕ driven by white noise. Overdamped Gross-Pitaevskii evolution with ε = t /τ Q and: Γ ϕ = c 2 2 ϕ εϕ + 2ϕ 2 ϕ + noise

8 Local choices may not be globally compatible: topological defects can form during quench! ˆ ξ Density of vortices ( strings ): n =1/ ˆ ξ 2 (Kibble, 76)

9 GENERAL THEORY OF DEFECT FORMATION All second order phase transitions fall into universality classes characterized by the behavior of quantities such as specific heat, magnetic susceptibility, etc. This is also the case for quantum phase transitions. For our purpose behavior of the relaxation time and of the healing length near the critical pointwill be essential; they determine the density of topological defects formed in the rapid phase transition ( the quench ). CRITICAL SLOWING DOWN τ = τ 0 ε CRITICAL OPALESCENCE ξ = ξ 0 ε ε ε

10 Derivation of the freeze out time Relaxation time: determines reflexes of the system. The potential τ = τ 0 ε τ = τ 0 ε ε V Ginzburg Landau (ϕ) = εϕ 2 + ϕ 4 ε Assume: ε = time "quench time" = t τ Q changes at a rate given by: ε = t τ(ε(ˆ Relaxation time is equal to this rate of change when t )) = t ˆ

11 and the corresponding frozen out healing length ˆ ξ.. Hence: Or: τ(ε(ˆ t )) = t ˆ ˆ t = τ 0 (ˆ t /τ Q ) = ˆ t The corresponding length follows: ˆ ξ = ξ 0 / τ 0 τ Q ˆ ε = ξ 0 & ˆ ε = τ 0 τ Q 4 τ Q τ 0 adiabatic adiabatic ˆ ξ = ξ 0 / ˆ ε ˆ t ˆ ε I M P U L S E I M P U L S E τ ˆ t ˆ ε adiabatic ξ = ξ 0 ε ε = t /τ Q adiabatic ε

12 Formation of kinks in a 1-D Landau-Ginzburg system V Ginzburg Landau (ϕ) = εϕ 2 + ϕ 4 with real ϕ driven by white noise. Overdamped Gross-Pitaevskii evolution with ε = t /τ Q and: Γ ϕ = c 2 2 ϕ εϕ + 2ϕ 2 ϕ + noise

13 Kinks from a quench V Ginzburg Landau (ϕ) = εϕ 2 + ϕ 4 with real ϕ Overdamped Gross-Pitaevskii evolution with driven by white noise. ε = t /τ Q Γ ϕ = c 2 2 ϕ εϕ + 2ϕ 2 ϕ + noise and: ˆ τ 0 = Γ ε = Γ /τ Q d ˆ ξ = ξ 4 τ /Γ 0 Q ϕ x Defect separation: Laguna & WHZ, PRL 97

14 Kink density vs. quench rate n =1/( fξ ˆ ) = (1/ fξ ) 4 Γ /τ 0 Q The observed density of kinks scales with the predicted slope, but with a density corresponding to: f~ n Similar values of the factor f multiply ˆ ξ in 2-D and 3-D numerical experiments. τ Q

15 Vortex line formation in 3-D (Antunes, Bettencourt, & Zurek, PRL 1999)

16 Experimental evidence? Liquid crystals (Yurke, Bowick, Srivastava, ) Superfluid 4 He (McClintock et al.) Superfluid 3 He (Krusius, Bunkov, Pickett, ) Josephson Junctions (Monaco, Rivers, Mygind ) Superconducting loops (Carmi, Polturak ) Superconductors in 2D (Maniv, Polturak )..

17 Superfluid Helium 4 He isotope: Hendry et al (1995) Dodd et al (1998) Stirring in 95 experiment No signal in 98 (vortices disappear too fast?) 3 He isotope: Bauerle et al (1996) Ruutu et al (1996) Observed vortices Could not test scaling

18 Carmi & Polturak (1999/2003): 1cm 1cm film of YBCO Measured total flux: vortices antivortices Superconductors

19 Josephson Junction Loops Carmi et al (2000): Loop of 214 segments of YBCO high-tc superconductor cooled with liquid nitrogen Heat with light beam to normal phase Switch off light: Phase transition to superconducting phase Found 7.4±0.7 flux quanta Monaco et al ( ): Two rings of Nb/Al superconductor next to each other cooled with liquid helium Measured the flux through the gap Agreed with Kibble-Zurek scaling

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21 PARTIAL SUMMARY: 1. Topological defects as petrified evidence of the phase transition dynamics. 2. Universality classes: The mechanism is generally applicable. 3. Initial density of defects after a quench using KZ approach. 4. Numerical simulations. 5. Experiments.

22 Formation of 2-D defect clusters in presence of gauge fields (Hindmarsh & Rajantie) Stephens, Bettencourt & Zurek, 2002

23 QUANTUM ISING MODEL Lattice of spin 1/2 particles interacting with an external force (e.g., magnetic field along the x axis) and with each other (ferromagnetic Ising interaction along the z axis): a H = J(t) σ x l W σ z σ z l l +1 l l, = ( + ) / 2 Quantum phase transition occurs as J(t) decreases. Then K ( + )( + )( + )L, analogue of the symmetric vaccum, is no longer favored energetically: K or K (or any superposition thereof) are the ground states. one of two canonical models of for quantum phase transitions S. Sachdev, Quantum Phase Transitions, CUP 1999

24 CRITICAL REGION OF THE QUANTUM ISING MODEL The character of the ground state changes when, in the model Hamiltonian; H = J(t) x σ l W z z σ l σ l +1 the two couplings are equal, that is, when: l J(t) W =1. In quantum phase transition the parameter ( relative coupling ): ε = J(t) W 1 plays the role of the relative temperature (T-Tc)/Tc: To induce phase transition one can lower the field and, hence, J(t). l

25 The gap and the critical behavior The gap (between the ground state and the lowest excited state) plays an essential role. In quantum Ising model it is given by: Δ = 2 W J(t) = 2W ε This is the energetic price of flipping a single spin above Jc or of a pair of kinks in a symmetry broken phase. L L L L Note that the gap is easily related with the relative coupling. Relaxation time and healing length in the critical region can be expressed in terms of the gap.

26 Relaxation time and healing length Relaxation time is simply the inverse of the gap: τ = h Δ 1/ε critical slowing down Once the characteristic velocity is calculated from the coupling W and the distance a between the spins on the lattice: c = 2Wa /h Healing length is given by: a ξ = c τ = 2Wa /Δ 1/ε critical opalescence This scaling is different than in the mean field case. Still, we have now all of the ingredients of the K-Z mechanism

27 and the corresponding frozen out healing length ˆ ξ.. Hence: Or: τ(ε(ˆ t )) = t ˆ ˆ t = ˆ ξ = ξ 0 / τ 0 (ˆ t /τ Q ) = ˆ t τ 0 τ Q The corresponding length follows: ˆ ε = ξ 0 ˆ ξ = ξ 0 / ˆ ε = ξ 0 & ˆ ε = τ 0 τ Q τ Q 4 τ τ 0 Q τ 0 adiabatic adiabatic ˆ ξ = ξ 0 / ˆ ε ˆ t ˆ ε I M P U L S E I M P U L S E τ ˆ t ˆ ε adiabatic ξ = ξ 0 ε ε = t /τ Q adiabatic ε

28 Density of kinks (# of kinks per spin in the Ising chain) as a function of quench rate (WHZ, Dorner, & Zoller; cond-mat/503511; PRL 05)

29 Dynamics: Landau-Zener Energies (according to Dorner, Fedichev, Jaksch, Lewenstein, & Zoller, PRL 03) Linear change of system parameters with velocity ( = quench time) t 0 gap J(t)/W 1 = t / τ Q Landau-Zener transition probability: (f = probability of staying in ground state = fidelity) L-Z - K-Z connection in avoided level crossings was pointed out by Damski (PRL, 05) and analysed furtehr by Damski & WHZ (PRA 06). Application to q. Ising due to WHZ, Dorner, and Zoller ( 05) and by Dziarmaga (PRL, 05) -- exact solution (agrees w. K-Z).

30 QUANTUM (LANDAU-ZENER) APPROACH In an avoided level crossing, the probability of transition that preserves the character of the state but changes the energy level when the external parameter is used to continuously vary the Hamiltonian is given by: Above: p exp πδ2 2h v Δ = E 1 E 2, v = Δ = d(e 1 E 2 ) /dt v 0 In the adiabatic limit ( ) Landau-Zener formula predicts that the system will remain in the same energy eigenstate. Transitions are induced when the change is sufficiently fast.

31 Dynamics: Landau-Zener (f = probability of staying in ground state) Landau - Zener prediction: Fit result: N = 70 N = 50 N = 30 Fit results: ~ 10-16% deviation in the constant; perfect fit to the form of dpependence

32 Density of Kinks in the Quantum Ising Model Kink -Operator: (counts number of domain walls) Fit results:

33 SUMMARY: 1. Phase transition in the quantum Ising model. 2. Initial density of defects after a quench in a normal second order phase transition. 3. Analogous estimates for the quantum Ising model. 4. Quantum calculation.

34 What actually happens. Landau-Zener fit is still very accurate. But the story is much more complicated! (One can expect order of magnitude estimates to be OK, but the accuracy of predictions is much better than order of magnitude ) See also J. Dziarmaga (PRL 05) for an analytic treatment of a closely related case of quantum Ising in a periodic boundary condition case.

35 .CONSEQUENTLY, THE CONDITION FOR THE RATE OF QUENCH SUFFICIENTLY SLOW FOR THE SPIN CHAIN TO LIKELY REMAIN IN THE GROUND STATE: h 2θ π f ( 3πW ) 2 N 2 CAN BE TRANSLATED INTO A CONDITION FOR N, THE NUMBER OF SPINS IN A CHAIN THAT -- GIVEN FIXED QUENCH RATE -- WILL REMAIN IN THE GROUND STATE: ˆ N 3πW π 2hθ ln p FOR COMPARISON, DOMAIN SIZE OBTAINED BEFORE: ˆ N KZ = W 1 2hθ

36 Dynamics: Landau-Zener Landau - Zener prediction: Fit result for f = 99% : W = 10 MHz τ Q ~ 15 µs (N=40)

37 THE SIZE OF THE MINIMUM GAP (TO THE LOWEST ACCESSIBLE STATE ABOVE THE GROUND STATE) FOR N SPINS DESCRIBED BY ISING MODEL HAMILTONIAN: IS: H = J(t) σ x W σ z σ z l l l +1 l Δ min 3πW N THEREFORE, THE GROUND STATE IS PRESERVED WITH FIDELITY p WHEN THE QUENCH IS NO FASTER THAN: l hδ = πδ 2 min ln p BUT Δ = 2 J (t) = 2θ. CONSEQUENTLY.: