v  INTRODUCTION – DISORDER AND THE QUANTUM HALL EFFECT; THE USE OF TOPOLOGICAL QUANTUM NUMBERS FOR INTEGER QUANTUM HALL TRANSITIONS OF NON- INTERACTING ELECTRONS

v  DISORDER-DRIVEN FRACTIONAL QUANTUM HALL – INSULATOR TRANSITION: USING TOPOLOGICAL QUANTUM NUMBERS

v  A NEW APPROACH: USING QUANTUM ENTANGLEMENT － ENTANGLEMENT ENTROPY AND ENTANGLEMENT LEVEL STATISTICS

v  CONCLUDING REMARKS

**SUPPORTED BY US-DOE

DISORDER DRIVEN FRACTIONAL QUANTUM HALL TRANSITIONS**

Ravin Bhatt, Princeton University Workshop on Anderson Localization in Topological Insulators

PCS, Institute for Basic Science, Daejeon, September 7, 2016

Prehistory: Disorder and the Quantum Hall Effect

•  Disorder is essential for quantization of Hall conductivity over a finite range of parameters (filling, magnetic field etc.).

•  Non-interacting electrons in the high-field limit (ħωc, gµBH >> ED, the disorder energy scale) exhibit only IQH phases, with transitions between them characterized by a divergence of the localization length ξ ~ ε−ν, where ε = (E-En)/ED is the distance to the nth Landau level at En, in units of the disorder energy, and ν is a universal exponent.

•  Various numerical methods have been used in the past to determine ν. These include transfer matrix methods which study strips and use the crossover from 2D to 1D behavior and extract ν, as well as purely 2D methods which study the scaling behavior of the Hall conductance, the diagonal conductance or topological quantum numbers.

Huo and Bhatt, PRL 68, 1375 (1992)

Estimates of Localization Length Exponent

From Obuse, Gruzberg and Evers, PRL 109, 206804 (2012) supplement

Effect of Disorder on the Fractional Quantum Hall Effect

•  Though disorder is essential for the QH effect, reducing disorder in experimental systems exposes finer and finer details of the quantum Hall phase diagram, from IQH plateaus emerge primary FQH plateaus and then more exotic states with smaller gaps are revealed.

•  e-e interactions are essential for understanding FQH phases; thus even in the high field limit, disorder does not uniquely set the energy scale, and one can induce phase transitions with disorder strength.

•  Unfortunately, unlike the non-interacting problem, sizes that can be numerically studied for the many-body system are very small, and they are even more limited than the uniform case because of the lack of translational symmetry in the presence of disorder.

•  The first such study, motivated by the promising results in the non-interacting case, involved the use of topological quantum numbers (Chern numbers).

RH

(h/e

2 )

R (a

.u.)

B (T)

classical

Experimental Data IQHE (left)

FQHE (below)

Better Samples: More FQH phases

ν = 1/3 FQH state to Insulator: Collapse of Gap with Disorder

•  Exact Diagonalization of 2DES (polarized) with finite thickness and Coulomb interactions.

•  Calculate Chern numbers C for many-body eigenstates [σxy = C (e2/h)].

•  Look at gap between ground and excited state manifolds.

•  Two types of gap: Spectral (Es) and Mobility (Em)

Sheng et al. PRL 90, 256802 (2003); Wan et al. PRB 72, 075325 (2005)

Sheng, Wan, Rezayi, Yang, Bhatt and Haldane (2003, 2005)

•  For zero disorder, 3-fold degeneracy of eigenstates with total Chern number = 1 => σxy = 1/3 (e2/h)

•  Eigenstates for small disorder in groups of three, which maintain the total Chern character of the manifold, which gets scrambled at high disorder, leading to an insulating ground state with C = 0 for large disorder.

•  Two types of gap: Spectral (Es) and Mobility (Em)

ABOVE: Spectral (Es) and Mobility (Em) Gaps

versus disorder W

RIGHT: Total Chern number Distribution for

disorder W = 0.06

Sheng, Wan, Rezayi, Yang,

Bhatt & Haldane (2003, 2005)

Quantum Entanglement Studies(Zhao Liu & RNB, arXiv:1607.04762)

•  N electrons on (square) torus, with flux I.N (I=3 => ν =1/3) •  Haldane Pseudo-potential Interactions (i.e. V1 for ν =1/3;

V1, V3, V5 for ν =1/7); also Coulomb Interaction. •  Gaussian white noise, strength W. •  Ground state 3-fold degenerate at W = 0; focus on the GS

manifold (3 lowest states, |Ψi>) for W ≠ 0, and describe system by mixed state ρ = Σi |Ψi><Ψi|.

•  Orbital cut divides system into two parts A & B; reduced density matrix of part A, ρA = (TrB ρ) has eigenvalues {λi}. This yields the von Neumann entropy SA = − Σi λi ln λi and entanglement spectrum ξi = − lnλi. Look at disorder-averaged <SA(W)> as well as entanglement level-statistics.

Entanglement Entropy •  Entanglement entropy expected to exhibit

“Area Law”, i.e. scale with perimeter of cut, L ~ N½ in both FQH and insulating phases.

•  FQH state has a sub-leading N-independent topological term in addition.

•  Nature of GS changes at transition => Sharp variation in S near the transition?

Size-scaling of

Entanglement Entropy

Entanglement Entropy versus W (V1)

<dS/dW> versus W (V1)

WHY LOOK AT dS/dW? •  In thermal phase transitions, where temperature T is

a control parameter, specific heat C = T(dS/dT) often exhibits singularity at the transition.

•  For a quantum phase transition driven by disorder (strength W), the corresponding quantity is dS/dW, or for finite size, the ensemble average <dS/dW>.

•  Finite Size Scaling: <dS/dW> ~ L.F [L1/ν(W – Wc)] ~ N1/2 f [N1/2ν(W – Wc)]

Peak height versus size

h ~ N1.35

Scaling of entire dS/dW curve

Entanglement Entropy versus W (Coulomb)

<dS/dW> versus W (Coulomb)

Peak scales as N1.2

<dS/dW> versus W (1/7 state)

Entanglement Level Statistics

•  Entanglement Spectrum ξi = - ln λi

•  Level Spacing sn = ξn+1 – ξn

•  Look at distribution P(s) in different parts of the entanglement spectrum ξ as function of disorder W.

•  Is there level repulsion?

P(s) at W = Wc

P(s) for W < Wc

P(s) for W > Wc

P(s) at W >> Wc – 1 (W =10)

P(s) at W >> Wc – 2 (W =100)

P(s) at W >> Wc – 3 (W = ∞)

CONCLUDING REMARKS •  Disorder driven transitions in systems with e-e interactions is a

challenging problem for numerical studies – N-P complete, and no symmetries present in the uniform case.

•  Using topological (Chern) numbers to characterize eigenstates, for the 1/3 FQH state to insulator transition, two gaps were found – the spectral gap and the mobility gap, which close at somewhat different values of disorder. These efforts focused on locating the transition, and characterization of phases on either side, not on critical exponents.

•  Studies of quantum entanglement suggest this is a useful method to locate and quantitatively study disorder driven transitions in the FQH regime.

•  The derivative of the entanglement entropy with disorder scales with a large power of the system size, and appears to diverge in the thermodynamic limit. Scaling of the entire curve is found to be good, but gives an exponent that violates the CCFS bound (like numerical studies of Many-body Localization in 1D systems) – large corrections due to subleading terms?

•  Distribution of entanglement level spacings appears not to be as sensitive to the topological phase-insulator transition .

•  Update on non-interacting electrons localization length exponent?