Cluster mean-field approach to the steady-state phase diagram of dissipative spin systems. Davide Rossini. Scuola Normale Superiore, Pisa (Italy)

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1 Cluster mean-field approach to the steady-state phase diagram of dissipative spin systems Davide Rossini Scuola Normale Superiore, Pisa (Italy) Quantum simulations and many-body physics with light Orthodox Academy of Crete June 5 th, 2016

2 In collaboration with Jiasen Jin Alberto Biella Leonardo Mazza Oscar Viyuela Jonathan Keeling Rosario Fazio Univ. Dalian, China (former SNS) SNS ENS, Paris (former SNS) UC Madrid St. Andrews, UK ICTP Trieste arxiv: See also Alberto's poster

3 Dissipative quantum many-body systems Quantum many-body systems: strong interactions + coupling with the environment competition between coherent dynamics & incoherent processes

4 Dissipative quantum many-body systems Quantum many-body systems: strong interactions + coupling with the environment competition between coherent dynamics & incoherent processes Probing this kind of physics in a controllable platform? For example using ultracold atoms and/or molecules e.g. excited Rydberg atoms driven ultra-cold atoms in optical superlattices [S. Diehl, A. Micheli, A. Kantian, B. Kraus, H.P. Büchler, P. Zoller, Nat. Phys. (2008)] or by manipulating non-linear photon dynamics optomechanical arrays of cavities [M. Ludwig, F. Marquardt, Phys. Rev. Lett. (2013)] coupled QED cavities [A. A. Houck, H. E. Tureci, J. Koch, Nat. Phys. (2012)]

5 at equilibrium minimize the ground-state energy free-energy analysis (Ginzburg-Landau) quantum/thermal fluctuations discontinuity in thermodynamic properties e.g. different ordering / symmetry long-range correlations are crucial Critical phenomena in a dissipative context the system may want to reach a steady state (t ) ordering (if exists) has a dynamical origin short-range correlations could be important

6 Our framework Z 2 symmetry breaking dissipative phase transitions a lattice model of interacting dissipative spin-1/2 particles mean-field treatment cluster mean field

7 Model We consider an XYZ anisotropic Heisenberg model in the presence of incoherent dissipative spin-flips (z axis) Hamiltonian anisotropy generates a non-trivial competition between coherent many-body couplings & dissipation experimentally implementable with trapped ions mean-field study performed in T.E. Lee, S. Gopalakrishnan, M.D. Lukin, Phys. Rev. Lett. 110, (2013)

8 The master equation has a symmetry which may spontaneously break in ordered phases (e.g.: paramagnetic / ferromagnetic transition in the Ising model) We study the paramagnet (PM) / ferromagnet (FM) steady-state phase transition Single-site mean-field phase diagram T.E. Lee, S. Gopalakrishnan, M.D. Lukin, PRL (2013)

9 Physical mechanism importance of short-range correlations

10 The 1 x 1 mean-field single-site mean field the ferromagnet extends over a semi-infinite range of J y for the magnetization vanishes (PM) progressive deterioration of the purity : pure state (fully polarized along z) fully mixed state (unpolarized)

11 The 2 x 1 cluster mean-field two-site mean field the ferromagnet extends over a finite range of J y the PM at stabilizes over an extended region different nature of the two PM regions (purity): nearly pure state (fully polarized along z) nearly fully mixed state (unpolarized)

12 The 2 x 1 cluster large J y coherent part dissipative part The steady state for is almost fully mixed: therefore correlations dynamically-induced purity reduction drastically modify the steady-state structure for dynamical suppression of ferromagnetic ordering

13 The 2 x 1 cluster large J y ferromagnet Initial condition: paramagnet dynamics does not strongly affect the magnetization dynamics strongly affects long time instability in the initial condition

14 Results for different dimensionalities

15 One-dimensional geometry Due to extremely reduced dimensionality, mean field should fail Reminiscence of features predicted by mean field Presumably no phase transition

16 One-dimensional geometry Spin ordering signaled by the structure factor (k = 0 FM k = π AFM) MF Mean field prediction: PM to FM transition

17 One-dimensional geometry RK / QT MPO + MF Finite-size scaling of numerical data power-law decay RK / QT MPO + MF

18 One-dimensional geometry PM reminiscent xx correlators FM reminiscent clearly exponential large r

19 Two-dimensional geometry What is the fate of mean field? Presumably there is a phase transition (existence of a symmetry-broken phase)

20 Two-dimensional geometry Mean-field phase diagram Cluster mean-field on a 2D square lattice Extension of the symmetry-broken phase drastically reduced Boundaries & topology of the phase diagram change a lot

21 Two-dimensional geometry no need to evaluate correlators with mean field: the system spontaneously breaks the symmetry 2x2: FM phase does not close CMF artifact? 3x3: FM phase closes and then re-opens

22 Two-dimensional geometry FM phase seems to survive in the thermodynamic limit no large J observed for larger clusters

23 Two-dimensional geometry FM phase corroborated by correlation functions: PM phase: exponential decay FM phase: saturation crossover region: unclear at small size j+r j

24 Stability of cluster mean field A strange feature appears when FM region closes! First-order transition within the ordered phase? (metamagnetism) Bistability effects We perform a linear stability analysis of 2x2 cluster mean field

25 Linear stability Linearize the master equation for small fluctuations around the steady-state value of the cluster mean-field solution: in momentum space [expand fluctuations as plane waves] We analyze the eigenvalues of CMF solution is stable if the real part of all the eigenvalues is negative Single-site mean field stability already employed in A. Le Boité, G. Orso, C. Ciuti, PRL 110, (2013) M. Schiró, C. Joshi, M. Bordyuh, R. Fazio, J. Keeling, H. Türeci, PRL 116, (2016)

26 Linear stability real part of the most unstable eigenvalue (dark colors = stable)

27 Higher dimensionalities L = 2x2 cluster mean-field L = 2x2x2 MF results approached if dimensionality is increased BUT a finite extension of the FM phase is predicted even in 4D L = 2x2x2x2

28 Summary Quantum symmetry breaking (Ising) dissipative transition 1D: no symmetry-broken phase 2D: ordered phase drastically shrinks 3D 4D: toward mean-field? XYZ Heisenberg + incoherent spin-flips mean-field results are fragile to a more accurate analysis importance of short-range correlations in dynamical ordering