1. Dephasing catastrophe in 4 – ε: A possible instability of the ergodic phase?

2. Dephasing in quasi-1D wires (redux)2: Non-Markovian noise and itinerant spin interactions

Matthew S. Foster Rice University

1. Liao and Foster, PRL 120, 236601 (2018)

2. Davis and Foster, in preparation

Yunxiang Liao U. Maryland, College Park

Seth Davis Rice University

- Basko, Aleiner, Altshuler (2006) - Gornyi, Mirlin, Polyakov (2005)

• Can MBL occur in higher dimensions? MBL is stable in certain 1D systems:

- J. Z. Imbrie (2014)

• Nature of the MBL-ergodic transition in higher dimensions? 1D MBL transition:

- Vosk, Huse, Altman (2015) - Potter, Vasseur, Parameswaran (2015) - Serbyn and Moore (2015) - Dumitrescu, Goremykina, Parameswaran, Serbyn, Vasseur (2019)

• Can MBL be destabilized by rare thermal fluctuations?

- de Roeck, Huveneers, Mueller, Schiulaz (2015) - de Roeck and Huveneers (2017)

Many-body localization (MBL)

• Weak localization correction (orthogonal metal class)

• Even a good metal ( ) would localize at all temperatures in 2D

without dephasing ( ).

MBL in 2D? Try to approach from the ergodic (diffusive metal) side

• Weak localization correction (orthogonal metal class)

• Even a good metal ( ) would localize at all temperatures in 2D

without dephasing ( ).

• Isolated interacting, disordered system: ergodic phase must serve as its own heat bath (dephasing mechanism)

• Differential equation for the Cooperon:

• Dephasing is due to thermal fluctuations of the charge density

MBL in 2D? Try to approach from the ergodic (diffusive metal) side

• Differential equation for the Cooperon:

• Path integral representation

MBL in 2D? Try to approach from the ergodic (diffusive metal) side

• Differential equation for the Cooperon:

• Path integral representation

• Dynamically screened long-range Coulomb interactions:

Noise kernel is (approximately) Markovian (bath is Ohmic):

Note: effectively classical Markovian bath—neglects ultraviolet quantum corrections that, in fact, have to be canceled by other Pauli-blocking self-energy terms. See e.g. Aleiner, Altshuler, Vavilov (2002); J. von Delft (2007). UV requires care in d > 1…

MBL in 2D? Try to approach from the ergodic (diffusive metal) side

• Differential equation for the Cooperon:

• Path integral representation

• Dynamically screened long-range Coulomb interactions:

Noise kernel is (approximately) Markovian (bath is Ohmic):

• Always dephases. Solve path integral exactly

MBL in 2D? Try to approach from the ergodic (diffusive metal) side

Altshuler, Aronov, Khmelnitsky (1982)

• Short-range interacting ( ), isolated system.

Noise kernel is non-Markovian (diffusive!):

• Does it always dephase? Exact solution? • SCBA (2D):

Is it true? Or does dephasing fail at sufficiently low-T and/or for sufficiently small diffusion constant D?

MBL in 2D? Try to approach from the ergodic (diffusive metal) side

Narozhny, Zala, Aleiner (2002)

• Non-Markovian action is similar to a self-avoiding random walk:

“Self-dephasing random walk”

IDEA: View as geometric stat mech problem, look for scaling

• Finite dephasing length at large T due to “self-interactions”

• Use RG to look for a new critical point at low but nonzero T

“Self-dephasing random walk”

• Technique: replicated path integral

Cooperon:

Thermal density fluctuation: Vertices:

Coupling strength (in d = 2):

Dimensional analysis (z = 2, d dimensions):

“Self-dephasing random walk”

• Technique: replicated path integral

Cooperon:

Thermal density fluctuation: Vertices:

Upper critical dimension is d = 4! Lower critical dimension? (d = 1 for SAW)

Dimensional analysis (z = 2, d dimensions):

“Self-dephasing random walk”

“Self-dephasing random walk”

Liao and MSF 2018

“Self-dephasing random walk”

Vertex corrections (neglected in SCBA; disallowed in

AAK/Markovian case)

Liao and MSF 2018

• Non-trivial fixed point:

“Self-dephasing random walk”

Liao and MSF 2018

• Non-trivial fixed point:

“Self-dephasing random walk”

• “Order parameter/mass”: Dephasing rate

• Correlation length exponent

• Chayes bound Liao and MSF 2018

Cooperon at the non-trivial fixed point:

• Scaling dimension:

• Scaling form:

• Weak localization correction:

Failure of dephasing at finite temperature T*: “toy” MBL transition!

• Does it survive to ?

• WL is first correction. Must analyze “AAK” problem at each order

• Self-consistent (“RG-improved PT”) solution: running D?

“Self-dephasing random walk”

Liao and MSF 2018

Part 2 Dephasing in quasi-1D wires (redux)2: Non-Markovian noise and itinerant spin interactions

Seth Davis and Matthew S. Foster (in preparation)

Seth Davis Rice University

• Bath-averaged Cooperon:

• Switch to COM, relative coordinates:

Path integral formulation

• Markovian bath:

• Cooperon return probability:

Path integral formulation: Markovian (AAK)

• Markovian bath:

• Cooperon return probability: • AAK in 1D: Airy eigenfunctions

Path integral formulation: Markovian (AAK)

Altshuler, Aronov, Khmelnitsky (1982)

• Diffusive bath:

• Bath action: • Cumulant expansion:

We are way below the upper critical dimension d = 4.

Expect infrared-divergent integrals (as in d = 2). (Lower critical dimension?)

…in fact, no divergences…

Diffusive bath: 1. Bare cumulant expansion

• Diffusive bath:

• Bath action: • Cumulant expansion:

1). First order “super-dephasing”:

2). Second order “super-duper-rephasing”:

∴ Bare Cumulant expansion doesn’t converge (in time η)! (no surprise)

Diffusive bath: 1. Bare cumulant expansion

• Assume bare Cooperon already dephased (massive—e.g. SCBA) • Beyond Born: calculate diffusive bath corrections,

with massive bare Cooperon

Diffusive bath: 2. SCBA and beyond

• Assume bare Cooperon already dephased (massive—e.g. SCBA) • Beyond Born: calculate diffusive bath corrections,

with massive bare Cooperon Result: mass is a spectator, does not otherwise change expansion!

1). First order “super-dephasing”:

2). Second order “super-duper-rephasing”:

Diffusive bath: 2. SCBA and beyond

Diffusive bath: 3. Physical IR regularization by Markovian bath

• Diffusive bath:

• Diffusive bath action: • Cumulant expansion:

Diffusive bath: 3. Physical IR regularization by Markovian bath

• Diffusive bath:

• Diffusive bath action: • Cumulant expansion: Consider combined Markovian, Diffusive baths:

• Markovian bath: screened Coulomb interactions ala 1D AAK

• Diffusive bath: spin-exchange (triplet channel) Fermi liquid interactions [e.g., in a quasi-1D wire with spin SU(2) symmetry—silver, not gold]

• 2D version with self-consistency (mean field):

Narozhny, Zala, Aleiner (2002)

Diffusive bath: 3. Physical IR regularization by Markovian bath

• Diffusive bath:

• Diffusive bath action: • Cumulant expansion: Consider combined Markovian, Diffusive baths:

• Center-of-mass R(τ) vertex operator correlators unaffected

• Relative ρ(τ) correlators: Use Airy eigenfunction expansion

Seth Davis Rice University

• Diffusive bath: • Cumulant expansion:

0). AAK (Markovian)-dephased: 1). First order dephasing:

Markovian and Diffusive baths: 3. Cumulant expansion

• Diffusive bath: • Cumulant expansion:

0). AAK (Markovian)-dephased: 1). First order dephasing:

Markovian and Diffusive baths: 3. Cumulant expansion

• Diffusive bath: • Cumulant expansion:

0). AAK (Markovian)-dephased: 1). First order dephasing:

2). Second order “rephasing”:

Markovian and Diffusive baths: 3. Cumulant expansion

• Diffusive bath: • Cumulant expansion:

0). AAK (Markovian)-dephased: 1). First order dephasing:

2). Second order “rephasing”:

Temperature dependence same as Markovian AAK (neglecting renormalization of spin triplet coupling )

Markovian and Diffusive baths: 3. Cumulant expansion

1. Rephasing at second order in the cumulant expansion due to crossed, nested diagrams: Relationship to vertex corrections in field theory language?

Questions: Markovian and Diffusive Baths

Dephasing in quasi-1D wires: Saturation controversy

Experimental observation (Webb et al.) • P. Mohanty, E.M.Q. Jariwala and R. A. Webb, Intrinsic decoherence in mesoscopic systems, Phys. Rev. Lett. 78 3366 (1997). • P. Mohanty and R. A. Webb, Decoherence and quantum fluctuations, Phys. Rev. B 55 13452 (1997). • P. Mohanty and R. A. Webb, Low temperature anomaly in mesoscopic Kondo wires, Phys. Rev. Lett. 84 4481 (2000). • D. S. Golubev and A. D. Zaikin, Quantum decoherence in disordered mesoscopic systems, Phys. Rev. Lett. 81 1074 (1998).

Canonical theory • I. L. Aleiner, B. L. Altshuler, and M. E. Gershenson, Interaction effects and phase relaxation in disordered systems,

Waves Random Media 9, 201 (1999). • F. Marquardt, J. von Delft, R. A. Smith, and V. Ambegaokar, Decoherence in weak localization. I. Pauli principle in influence

functional, Phys. Rev. B 76, 195331 (2007). • J. von Delft, F. Marquardt, R. A. Smith, and V. Ambegaokar, Decoherence in weak localization. II. Bethe-Salpeter calculation of the

cooperon, Phys. Rev. B 76, 195332 (2007). Geometry dependence • D. Natelson, R. L. Willett, K. W. West, and L. N. Pfeiffer, Geometry-dependent dephasing in small metallic wires,

Phys. Rev. Lett. 86 1821 (2001). Theory of Kondo impurities on dephasing • G. Zarand, L. Borda, J. von Delft, and N. Andrei, Theory of inelastic scattering from magnetic impurities,

Phys. Rev. Lett. 93 107204 (2004). • T. Micklitz, A. Altland, T.A. Costi, and A. Rosch, Universal dephasing rate due to diluted Kondo impurities,

Phys. Rev. Lett. 96 226601 (2006). • T. Micklitz, T.A. Costi, and A. Rosch, Magnetic field dependence of dephasing rate due to diluted Kondo impurities,

Phys. Rev. B 75 054406 (2007). Measurements of Kondo-temp dephasing regime • G. M. Alzoubi and N. O. Birge, Phase coherence of conduction electrons below the Kondo temperature,

Phys. Rev. Lett. 97, 226803 (2006). • F. Mallet et al., Scaling of the low-temperature dephasing rate in Kondo systems, Phys. Rev. Lett. 97 226804 (2006).

Questions: Markovian and Diffusive Baths

1. Rephasing at second order in the cumulant expansion due to crossed, nested diagrams: Relationship to vertex corrections in field theory language?

2. Enhancement of additional first order dephasing due to spin coupling renormalization (multifractal enhancement of matrix elements)?

Finkel’stein (1983) Castellani, DiCastro, Lee, and Ma (1984)

Summary

1. Idea: try to approach MBL from the ergodic side; look for a precursor in the slowing of dephasing (“weak re-phasing”)

2. Isolated, interacting, disordered fermion system: weak localization is self-dephased by thermal fluctuations of the density, spin, etc.

3. Screened long-ranged Coulomb interactions: Effective Markovian bath. Dephases at any nonzero temperature (AAK 1982).

4. Short-ranged interactions for a conserved hydrodynamic mode: thermal self-dephasing bath is diffusive, strongly non-Markovian.

5. Localization occurs at all temperatures in 1D and 2D, without dephasing.

Summary

1. Idea: try to approach MBL from the ergodic side; look for a precursor in the slowing of dephasing (“weak re-phasing”)

2. Isolated, interacting, disordered fermion system: weak localization is self-dephased by thermal fluctuations of the density, spin, etc.

3. Screened long-ranged Coulomb interactions: Effective Markovian bath. Dephases at any nonzero temperature (AAK 1982)

4. Short-ranged interactions for a conserved hydrodynamic mode: thermal self-dephasing bath is diffusive, strongly non-Markovian.

5. Localization occurs at all temperatures in 1D and 2D, without dephasing.

6. Dephasing with a diffusive bath: strongly coupled auxiliary quantum field theory for d = 1,2,3!

7. Role of vertex corrections, possible nontrivial fixed point

8. Combined Markovian, diffusive baths in quasi-1D gives alternating dephasing, rephasing contributions

Liao and MSF

PRL (2018)

Davis and MSF In preparation

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