A possible instability of the ergodic phase? 2. Dephasing in quasi1D wires (redux...
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1. Dephasing catastrophe in 4 – ε: A possible instability of the ergodic phase?
2. Dephasing in quasi1D wires (redux)2: NonMarkovian 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 MBLergodic 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)
Manybody 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 longrange 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 Pauliblocking selfenergy 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 longrange 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)

• Shortrange interacting ( ), isolated system.
Noise kernel is nonMarkovian (diffusive!):
• Does it always dephase? Exact solution? • SCBA (2D):
Is it true? Or does dephasing fail at sufficiently lowT and/or for sufficiently small diffusion constant D?
MBL in 2D? Try to approach from the ergodic (diffusive metal) side
Narozhny, Zala, Aleiner (2002)

• NonMarkovian action is similar to a selfavoiding random walk:
“Selfdephasing random walk”

IDEA: View as geometric stat mech problem, look for scaling
• Finite dephasing length at large T due to “selfinteractions”
• Use RG to look for a new critical point at low but nonzero T
“Selfdephasing random walk”

• Technique: replicated path integral
Cooperon:
Thermal density fluctuation: Vertices:
Coupling strength (in d = 2):
Dimensional analysis (z = 2, d dimensions):
“Selfdephasing 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):
“Selfdephasing random walk”

“Selfdephasing random walk”
Liao and MSF 2018

“Selfdephasing random walk”
Vertex corrections (neglected in SCBA; disallowed in
AAK/Markovian case)
Liao and MSF 2018

• Nontrivial fixed point:
“Selfdephasing random walk”
Liao and MSF 2018

• Nontrivial fixed point:
“Selfdephasing random walk”
• “Order parameter/mass”: Dephasing rate
• Correlation length exponent
• Chayes bound Liao and MSF 2018

Cooperon at the nontrivial 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
• Selfconsistent (“RGimproved PT”) solution: running D?
“Selfdephasing random walk”
Liao and MSF 2018

Part 2 Dephasing in quasi1D wires (redux)2: NonMarkovian noise and itinerant spin interactions
Seth Davis and Matthew S. Foster (in preparation)
Seth Davis Rice University

• Bathaveraged 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 infrareddivergent 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 “superdephasing”:
2). Second order “superduperrephasing”:
∴ 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 “superdephasing”:
2). Second order “superduperrephasing”:
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: spinexchange (triplet channel) Fermi liquid interactions [e.g., in a quasi1D wire with spin SU(2) symmetry—silver, not gold]
• 2D version with selfconsistency (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:
• Centerofmass 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 quasi1D 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. BetheSalpeter calculation of the
cooperon, Phys. Rev. B 76, 195332 (2007). Geometry dependence • D. Natelson, R. L. Willett, K. W. West, and L. N. Pfeiffer, Geometrydependent 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 Kondotemp 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 lowtemperature 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 rephasing”)
2. Isolated, interacting, disordered fermion system: weak localization is selfdephased by thermal fluctuations of the density, spin, etc.
3. Screened longranged Coulomb interactions: Effective Markovian bath. Dephases at any nonzero temperature (AAK 1982).
4. Shortranged interactions for a conserved hydrodynamic mode: thermal selfdephasing bath is diffusive, strongly nonMarkovian.
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 rephasing”)
2. Isolated, interacting, disordered fermion system: weak localization is selfdephased by thermal fluctuations of the density, spin, etc.
3. Screened longranged Coulomb interactions: Effective Markovian bath. Dephases at any nonzero temperature (AAK 1982)
4. Shortranged interactions for a conserved hydrodynamic mode: thermal selfdephasing bath is diffusive, strongly nonMarkovian.
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 quasi1D gives alternating dephasing, rephasing contributions
Liao and MSF
PRL (2018)
Davis and MSF In preparation
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