Percolation approach to many-body localization M. Ortuño Universidad de Murcia 28th August 2015, ICTP Andrés Somoza (Universidad de Murcia) Louk Rademaker (KITP, Santa Barbara) European Union FIS2009-13483 European Regional Development Fund M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 1 / 17
Brief outline The model Percolation approach to MBL Diagonalization of the Hamiltonian through displacement transformations M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 2 / 17
Many-body localization With strong enough disorder, all single-particle states are localized. Conductivity is then by hopping between localized states. Mott’s variable range hopping. The standard driving nechanism for hopping is the phonon bath, but any extended, continuous bath could do the same role. Basko et al. (2006) proposed the electron-electron interaction as the driving mechanism above a certain temperature. The problem can be thought of as many-body delocalization in Fock space. Numerical simulations: mainly exact diagonalization (very small systems) M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 3 / 17
Model i ǫ i c † � i , j � tc † Single-particle Hamiltonian H 0 = � i c i + � j c i , where t = 1 is our unit of energy and ǫ i ∈ [ − W / 2 , W / 2] Diagonalize H 0 and obtain a localized basis | α � The total Hamiltonian is in this basis α c α + 1 � φ α c † � V αβγη c † α c β c † H = γ c η 2 α αβγη Short range potential: nearest neighbours for fermions ( V = 1 ) Periodic boundary conditions First compute V αβγη M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 4 / 17
10 1 Number of resonances / site 3/2 10 0 20 40 60 80 100 10 -1 1 5 10 Localization Length Matrix elements Relevant information contained in the distribution of V α,β,γ,η / ( E α + E β − E γ − E η ) Distribution 100 W 4 5 6 10 7 8 9 | V / | 1 0.1 0 1 2 3 4 Number of transitions / site Figure: Distribution of the number of transitions of strength | V / ∆ φ | for several disorders W . M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 5 / 17
Matrix elements Relevant information contained in the We simplify this to the number of distribution of resonances, i.e., configurations with V α,β,γ,η / ( E α + E β − E γ − E η ) | V α,β,γ,η / ( E α + E β − E γ − E η ) | > 1 Distribution Resonances 100 W 10 1 4 Number of resonances / site 5 6 10 3/2 7 8 9 | V / | 10 0 L 20 1 40 60 80 100 10 -1 0.1 0 1 2 3 4 1 5 10 Localization Length Number of transitions / site Figure: Distribution of the number of transitions Figure: Number of resonances per site as a of strength | V / ∆ φ | for several disorders W . function of the localization length ξ for several L . M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 5 / 17
Spreading in configuration space Initial configuration: we occupied L / 2 single-particle states | α � at random | 0 , 1 , 1 , 0 , 0 , 0 , 1 , 0 , 1 , 1 , 0 , · · · � M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 6 / 17
Spreading in configuration space Initial configuration: we occupied L / 2 single-particle states | α � at random | 0 , 1 , 1 , 0 , 0 , 0 , 1 , 0 , 1 , 1 , 0 , · · · � Obtain and store all configurations resonating with the initial one | 0 , 1 , 0 , 1 , 0 , 1 , 0 , 0 , 1 , 1 , 0 , · · · � M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 6 / 17
Spreading in configuration space Initial configuration: we occupied L / 2 single-particle states | α � at random | 0 , 1 , 1 , 0 , 0 , 0 , 1 , 0 , 1 , 1 , 0 , · · · � Obtain and store all configurations resonating with the initial one | 0 , 1 , 0 , 1 , 0 , 1 , 0 , 0 , 1 , 1 , 0 , · · · � Obtain all configurations resonating with any of the previous set (layer) and not included in it. M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 6 / 17
Spreading in configuration space Initial configuration: we occupied L / 2 single-particle states | α � at random | 0 , 1 , 1 , 0 , 0 , 0 , 1 , 0 , 1 , 1 , 0 , · · · � Obtain and store all configurations resonating with the initial one | 0 , 1 , 0 , 1 , 0 , 1 , 0 , 0 , 1 , 1 , 0 , · · · � Obtain all configurations resonating with any of the previous set (layer) and not included in it. Iterate the procedure expanding layer by layer until there are no more resonating configurations or the number of configurations in a layer exceeds a maximum number ( 10 8 ). Only have to store three active layers M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 6 / 17
Spreading in configuration space Initial configuration: we occupied L / 2 single-particle states | α � at random | 0 , 1 , 1 , 0 , 0 , 0 , 1 , 0 , 1 , 1 , 0 , · · · � Obtain and store all configurations resonating with the initial one | 0 , 1 , 0 , 1 , 0 , 1 , 0 , 0 , 1 , 1 , 0 , · · · � Obtain all configurations resonating with any of the previous set (layer) and not included in it. Iterate the procedure expanding layer by layer until there are no more resonating configurations or the number of configurations in a layer exceeds a maximum number ( 10 8 ). Only have to store three active layers Calculate: Size of the cluster Ω Variance of the number of particles crossing a (virtual) boundary σ 2 M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 6 / 17
Localized regime � ∞ Ω P (Ω ′ ) d Ω ′ as a function of We measure the accumulated probability P (Ω) = � L � ≈ 2 L the normalized cluster size Ω / Ω L , where Ω L = L / 2 Disorder W = 9 Disorder W = 8 10 0 10 0 W = 8 10 -1 10 -1 W = 9 L 10 -2 10 -2 L 20 Accumulated probability Accumulated probability 20 30 30 40 10 -3 10 -3 40 50 50 60 10 -4 10 -4 60 80 70 10 -5 10 -5 10 -6 10 -6 10 -7 10 -7 10 -8 10 -8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 log( / L ) log( / L ) Figure: P (Ω) vs. log(Ω / Ω L ) for W = 9 . Figure: P (Ω) vs. log(Ω / Ω L ) for W = 8 . Size L = 20 is quite anomalous Fluctuations in the number of resonances are important and cause long tails M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 7 / 17
Transition region Disorder W = 7 Disorder W = 6 10 0 10 0 W = 7 10 -1 L 10 -1 20 Accumulated probability 10 -2 Accumulated probability 30 10 -2 40 W = 6 50 10 -3 80 L 10 -3 100 20 10 -4 30 40 10 -4 50 10 -5 60 80 10 -5 10 -6 10 -7 10 -6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 log( / L ) log( / L ) Disorder W = 5 Disorder W = 4 10 0 1 0.9 0.8 0.7 0.6 Accumulated probability Accumulated probability 0.5 W = 5 0.4 W = 4 10 -1 L 0.3 L 20 20 30 30 40 0.2 40 60 60 80 80 100 10 -2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 log( / L ) log( / L ) M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 8 / 17
Nature of the states States are localized for W � 6 There is a transition around W c ≈ 6 States are metallic for W < 4 Fluctuations are responsible for the complexity of the situation M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 9 / 17
Percolation in the hypercube Hypercube 1 Accumulated probability 0.1 0.01 Consider a hypercube of L =24 dimension L Occupy each edge with 1E-3 probability p 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Study the distribution of log / L the size of the clusters Figure: P (Ω) vs. log(Ω / Ω L ) for several values of the percolation probability and L = 24 . M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 10 / 17
Number fluctuations We study the variance σ 2 of Fluctuations the number of particles crossing a surface 0.1 In the insulating regime L σ 2 → constant 20 30 40 In the metallic regime 50 σ 2 ∝ L 60 2 / L 0.01 1E-3 4 5 6 7 8 W Figure: Variance of the number of particles crossing a surface divided by L as a function of disorder W . M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 11 / 17
Number fluctuations We study the variance σ 2 of Fluctuations the number of particles crossing a surface 0.1 In the insulating regime L σ 2 → constant 20 30 40 In the metallic regime 50 σ 2 ∝ L 60 2 / L 0.01 There is a transition 1E-3 ( 6 < W c < 6 . 5 ) 4 5 6 7 8 Behavior in the extended W phase is not clear Figure: Variance of the number of particles crossing a surface divided by L as a function of disorder W . M. Ortuño (Universidad de Murcia) Percolation approach to many-body localization ICTP 11 / 17
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