Percolation approach to many-body localization M. Ortuo Universidad - - PowerPoint PPT Presentation

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

European Regional Development Fund

FIS2009-13483

• 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

Single-particle Hamiltonian H0 =

i ǫic† i ci + i,j tc† j ci, where t = 1 is our

unit of energy and ǫi ∈ [−W /2, W /2] Diagonalize H0 and obtain a localized basis |α The total Hamiltonian is in this basis H =

• α

φαc†

αcα + 1

2

• αβγη

Vαβγηc†

αcβc† γcη

Short range potential: nearest neighbours for fermions (V = 1) Periodic boundary conditions First compute Vαβγη

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Percolation approach to many-body localization ICTP 4 / 17

Matrix elements

Relevant information contained in the distribution of Vα,β,γ,η/(Eα + Eβ − Eγ − Eη) Distribution

1 2 3 4 0.1 1 10 100

W 4 5 6 7 8 9

|V/| Number of transitions / site

Figure: Distribution of the number of transitions

• f strength |V/∆φ| for several disorders W.

1 5 10 10-1 100 101

20 40 60 80 100

Number of resonances / site Localization Length

3/2

• M. Ortuño (Universidad de Murcia)

Percolation approach to many-body localization ICTP 5 / 17

Matrix elements

Relevant information contained in the distribution of Vα,β,γ,η/(Eα + Eβ − Eγ − Eη) Distribution

1 2 3 4 0.1 1 10 100

W 4 5 6 7 8 9

|V/| Number of transitions / site

Figure: Distribution of the number of transitions

• f strength |V/∆φ| for several disorders W.

We simplify this to the number of resonances, i.e., configurations with |Vα,β,γ,η/(Eα + Eβ − Eγ − Eη)| > 1 Resonances

1 5 10 10-1 100 101

L 20 40 60 80 100

Number of resonances / site Localization Length

3/2 Figure: Number of resonances per site as a 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 (108). 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 (108). Only have to store three active layers Calculate:

Size of the cluster Ω Variance of the number of particles crossing a (virtual) boundary σ2

• M. Ortuño (Universidad de Murcia)

Percolation approach to many-body localization ICTP 6 / 17

Localized regime

We measure the accumulated probability P(Ω) = ∞

Ω P(Ω′)dΩ′ as a function of

the normalized cluster size Ω/ΩL, where ΩL = L

L/2

• ≈ 2L

Disorder W = 9

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 Accumulated probability log(/L)

W = 9 L 20 30 40 50 60 70

Figure: P(Ω) vs. log(Ω/ΩL) for W = 9.

Disorder W = 8

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 Accumulated probability log(/L)

W = 8 L 20 30 40 50 60 80

Figure: P(Ω) vs. log(Ω/ΩL) 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

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 Accumulated probability log(/L)

W = 7 L 20 30 40 50 80 100

Disorder W = 5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 10-2 10-1 100 Accumulated probability log(/L)

W = 5 L 20 30 40 60 80 100

Disorder W = 6

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 10-6 10-5 10-4 10-3 10-2 10-1 100 Accumulated probability log(/L)

W = 6 L 20 30 40 50 60 80

Disorder W = 4

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Accumulated probability log(/L)

W = 4 L 20 30 40 60 80

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Percolation approach to many-body localization ICTP 8 / 17

Nature of the states

States are localized for W 6 There is a transition around Wc ≈ 6 States are metallic for W < 4 Fluctuations are responsible for the complexity of the situation

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Percolation approach to many-body localization ICTP 9 / 17

Percolation in the hypercube

Consider a hypercube of dimension L Occupy each edge with probability p Study the distribution of the size of the clusters Hypercube

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1E-3 0.01 0.1 1 Accumulated probability log /L L=24 Figure: P(Ω) vs. log(Ω/ΩL) 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

Fluctuations

4 5 6 7 8 1E-3 0.01 0.1

L 20 30 40 50 60

2/L W Figure: Variance of the number of particles crossing a surface divided by L as a function of disorder W.

We study the variance σ2 of the number of particles crossing a surface In the insulating regime σ2 → constant In the metallic regime σ2 ∝ L

• M. Ortuño (Universidad de Murcia)

Percolation approach to many-body localization ICTP 11 / 17

Number fluctuations

Fluctuations

4 5 6 7 8 1E-3 0.01 0.1

L 20 30 40 50 60

2/L W Figure: Variance of the number of particles crossing a surface divided by L as a function of disorder W.

We study the variance σ2 of the number of particles crossing a surface In the insulating regime σ2 → constant In the metallic regime σ2 ∝ L There is a transition (6 < Wc < 6.5) Behavior in the extended phase is not clear

• M. Ortuño (Universidad de Murcia)

Percolation approach to many-body localization ICTP 11 / 17

Diagonalization of the Hamiltonian

Local integrals of motion

We want to diagonalize the Hamiltonian H =

• α

φαnα + 1 2

• αβγη

Vαβγηc†

αcβc† γcη

The aim is to design an explicit procedure to find operators of the form ˜ nα = U †nαU = nα +

• αβγη

aχβγηc†

χcβc† γcη + · · ·

such that H can be written as H =

• α

bα˜ nα +

• αβ

bαβ˜ nα˜ nβ + · · ·

• M. Ortuño (Universidad de Murcia)

Percolation approach to many-body localization ICTP 12 / 17

Diagonalization of the Hamiltonian

Local integrals of motion

We want to diagonalize the Hamiltonian H =

• α

φαnα + 1 2

• αβγη

Vαβγηc†

αcβc† γcη

The aim is to design an explicit procedure to find operators of the form ˜ nα = U †nαU = nα +

• αβγη

aχβγηc†

χcβc† γcη + · · ·

such that H can be written as H =

• α

bα˜ nα +

• αβ

bαβ˜ nα˜ nβ + · · · The idea was to perform a basis change of the form cα → cα + V ∆φnαcβc†

γcη

• M. Ortuño (Universidad de Murcia)

Percolation approach to many-body localization ICTP 12 / 17

Diagonalization of the Hamiltonian

Local integrals of motion

We want to diagonalize the Hamiltonian H =

• α

φαnα + 1 2

• αβγη

Vαβγηc†

αcβc† γcη

The aim is to design an explicit procedure to find operators of the form ˜ nα = U †nαU = nα +

• αβγη

aχβγηc†

χcβc† γcη + · · ·

such that H can be written as H =

• α

bα˜ nα +

• αβ

bαβ˜ nα˜ nβ + · · · The idea was to perform a basis change of the form cα → cα + V ∆φnαcβc†

γcη

Louk Rademaker (KITP) arXiv:1507.07276

• M. Ortuño (Universidad de Murcia)

Percolation approach to many-body localization ICTP 12 / 17

Displacement transformation

Displacement transformation: DX(λ) = exp{λ(X † − X)} X = nα1 · · · nαkcβ

† 1cβ2cβ † 3 · · · cβl

(here we concentrate in the particular case X = c†

αcβc† γcη)

DX(λ) = 1 + sin λ(X † − X) + (cos λ − 1)(X †X + XX †)

• M. Ortuño (Universidad de Murcia)

Percolation approach to many-body localization ICTP 13 / 17

Displacement transformation

Displacement transformation: DX(λ) = exp{λ(X † − X)} X = nα1 · · · nαkcβ

† 1cβ2cβ † 3 · · · cβl

(here we concentrate in the particular case X = c†

αcβc† γcη)

DX(λ) = 1 + sin λ(X † − X) + (cos λ − 1)(X †X + XX †) ˜ nδ = D†

X(λ)nδDX(λ) = nδ ± 1

2 sin 2λ(X † + X) ∓ sin2 λ(X †X − XX †) upper sign if δ = β or η and lower sign if δ = α or γ D†

X(λ)(X † + X)DX(λ) = cos 2λ(X † + X) − sin 2λ(X †X − XX †)

D†

X(λ)HDX(λ) =

• α

φαnα +

• (−φα − φγ + φβ + φη)1

2 sin 2λ + 1 2Vαβγη cos 2λ

• ×(X † + X) +
• (φα + φγ − φβ − φη) sin2 λ − 1

2Vαβγη sin 2λ

• (X †X − XX †)

tan 2λ = Vαβγη φα + φγ − φβ − φη

• M. Ortuño (Universidad de Murcia)

Percolation approach to many-body localization ICTP 13 / 17

Trivial example

One particle in two sites

H = φ1n1 + φ2n2 − t(c†

1c2 + c† 2c1)

φ1 −t −t φ2

• Define X = c†

1c2

tan 2λ = −2t φ1 − φ2 ˜ n1 = n1 − sin 2λ(X † + X) + sin2 λ[(1 − n1)n2 − n1(1 − n2)] ˜ n2 = n2 + sin 2λ(X † + X) − sin2 λ[(1 − n1)n2 − n1(1 − n2)] 10|˜ n1|10 = 1 − sin2 λ 10|˜ n2|10 = sin2 λ sin2 λ = 4t2 (φ1 − φ2)2 + 4t2

• M. Ortuño (Universidad de Murcia)

Percolation approach to many-body localization ICTP 14 / 17

Consecutive transformations

Each transformation modifies the remaining quantum terms of H. For example, for X = c†

αcβc† γcη, Y = c† αcic† γcj, Z = c† ηcic† βcj

D†

X(λ)(Y † + Y )DX(λ) = cos λ(Y † + Y ) − sin λ(Z † + Z) + · · ·

One start with the transformation corresponding to the higher |λ|, i.e. higher |Vαβγη/(φα + φγ − φβ − φη)|, and continues performing consecutive transformations with decreasing values of λ until all four operators terms in H have been cancelled (to a certain accuracy). Elliminating terms with a given number of operators de not generate terms with fewer operators. The final unitary transformation is U =

• i

DXi(λi)

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Percolation approach to many-body localization ICTP 15 / 17

Occupation number

As a proof of principle, we calculate Φ0|˜ nα|Φ0 where |Φ0 = c†

β · · · c† γ|0

(we assume that β, · · · γ = α) There seems to be a localized regime for W 6 and probably a transition at Wc ≈ 6. Occupation number

5 6 7 8 9 0.0 0.1 0.2 0.3 0.4 0.5

L 20 30 40 60 80 100

n W Figure: Average occupation number of a local integral of motion as a function of disorder.

• M. Ortuño (Universidad de Murcia)

Percolation approach to many-body localization ICTP 16 / 17

Future possibilities

Calculation of Φ0|˜ nα˜ nβ|Φ0 Quantum Coulomb gap Level statistics Higher dimensions Study convergence of the method

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Percolation approach to many-body localization ICTP 17 / 17