APS March Meeting 2018

Strange Half-Metals and Mott Insulators in SYK Models

Arijit Haldar and Vijay B. Shenoy

Department of Physics, Indian Institute of Science, Bangalore 560012shenoy@iisc.ac.in

Based on arXiv:1703.05111

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Acknowledgements

Generous research funding: DST, India

Key contributor:

Arijit Haldar

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OverviewReview and motivation

I Quick recap of half-metals and Mott insulators

Our model: Two SYK dots coupled via interactionsAnalysis and results

Highlights (arXiv:1703.05111)Low temperature instabilities of coupled SYK dotsStrange metal gives way to new phases – strange half metals, andMott insulators in a rich phase diagram

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OverviewReview and motivation

I Quick recap of half-metals and Mott insulators

Our model: Two SYK dots coupled via interactionsAnalysis and results

Highlights (arXiv:1703.05111)Low temperature instabilities of coupled SYK dotsStrange metal gives way to new phases – strange half metals, andMott insulators in a rich phase diagram

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The QuestionSeveral examples in condensed matter physics where non-Fermiliquid/strange metal phases undergo instabilities to produceinteresting new phases...e. g., High Tc superconductors

(Vishik et al. (2015))

SYK (Sachdev-Ye-Kitaev) model(Sachdev and Ye (1994), Kitaev (2015)) – solvableexample of a non-Fermi liquid

QuestionI Can the SYK model throw light on possible instabilities of non-Fermi

liquids?

I Phase transitions in (?between) non-Fermi liquids4 / 15

Recap of Essential Strong Correlation PhysicsFermionic systems with two spin flavors (paramagnetic metalwhen interactions are absent)Strong local repulsive interactions (e. g., Hubbard model) can leadto interesting phases

Energy

Density

ofstates

µ

Mott insulator

↑↓

EnergyDensity

ofstates

µ

Half metal

↑↓

Mott insulator: Charge gapped; spins as low energy degrees offreedom (at commensurate fillings) (Lee, Wen, Nagaosa (2005))

Half metal: One of the spin species is gapped, and the second oneis gapless with a net spin polarization (Galanakis and Dederichs (2015))

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Our Model: Interaction Coupled SYK DotsSYK dots with arbitrary q-body interactions

....

.... ....

....

H = ∑i1,...,iqc ,j1,...,jqc

Hci1,...,iqc ;j1,...,jqc

c†iqc. . . c†i1

cj1. . . cjqc

+∑

α1,...,αqΨ ,γ1,...,γqΨ

Hcα1,...,αqΨ;γ1,...,γqΨ

Ψ†αqΨ

. . .Ψ†α1

Ψγ1 . . .ΨγqΨ

+∑

i1,...,ir,α1,...,αr

HcΨi1,...,ir;α1,...,αr c†ir . . . c†i1

Ψ†α1

Ψα1 . . .Ψαr + h. c..

Dot c with c-fermions, Nc sites, qc-body interactions described byenergy scale Jc.Dot Ψ with Ψ-fermions, NΨ sites, qΨ-body interactions describedby energy scale JΨ (take qΨ ≤ qc)Inter-dot r-body interaction described by energy scale VFraction of sites f = NΨ

Nc⇐= key parameter

Case qc = 2, qΨ = 1 and r = 1 already considered (Banerjee and Altman (2017))

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AnalyticsLarge-N action at inverse temperature β,

S = NΞ =N

1 + f

[− 1β

ln det[−G−1c ]− f

βln det[−G−1

Ψ ] −∑

s=c,Ψ

(−1)qs f1−s

2J2s

2qs

∫ β

0dτGqs

s (−τ)Gqss (τ)

−∑

s=c,Ψ

f1−s

2

∫ β

0dτ Σs(τ)Gs(−τ) −(−1)r

√f

V2

r

∫ β

0dτGr

c(−τ)GrΨ(τ)

]

τ–imaginary time, Gs – Green function, Σs – self energySelf consistency condition

Σs(τ) =(−1)qs+1J2s Gqs−1

s (−τ)Gqss (τ)

+ (−1)r+1(√

f )sV2Gr−1s (−τ)Gr

s̄(τ), s = c,Ψ

Use a “conformal ansatz” (Sachdev PRX (2015) and references therein)

Gs(τ) = −Cssgn τ|τ |2∆s

, s = c,Ψ

τ – imaginary time, ∆s – fermion dimension, Cs – constant7 / 15

Analytics...contd.Uncoupled dots V = 0

∆s =1

2qs≡ ∆0

s , C2qss =

1J2s

K(∆s), s = c,Ψ

K(x) = 1π (1

2 − x) tan(πx)

Analysis for V � Jc, JΨ: crucial parameter

r? =2qcqΨ

qc + qΨ

If r > r?, coupling V is irrelevant on both dots

∆s = ∆0s , C2qs

s =1J2s

K(∆s), s = c,Ψ

If r = r?, coupling V is marginal on both dots

∆s = ∆0s , C2qs

s depends on Js,V and f

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Analytics...contd.For qΨ < r < r?, V is relevant on one flavor and marginal on theother depending on the value of f

f∆c = ∆0

c, ∆Ψ = 1r − ∆0

c

fh =K( 1

r−∆0

Ψ)

K(∆0Ψ)

fl = K(∆0c)

K( 1r−∆0

c)

Non-Fermi liquid Non-Fermi liquid

No conformalsolution

∆Ψ = ∆0Ψ, ∆c = 1

r − ∆0Ψ

I No conformal solution for fl ≤ f ≤ fhI Can change the nature of the non-Fermi liquid by tuning f , but a

“non conformal phase” intervenes!I Key question: What is the nature of the intervening non-conformal

phase?

For r < qΨ, V is relevant on both dots

K(∆c)1r −∆c

= f , ∆Ψ =1r−∆c

fermion dimension is f dependent! (For further details see 1703.05111)

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Numerics

Numerical solution of the self consistency equations

Σs(τ) =(−1)qs+1J2s Gqs−1

s (−τ)Gqss (τ)

+ (−1)r+1(√

f )sV2Gr−1s (−τ)Gr

s̄(τ), s = c,Ψ

Work at fixed chemical potential µ = 0Quantities of interest

I Spectral function ρs(ω)I Entropy SI Number density nI Polarization P = nc − nΨ

I Specific heat CVI Compressibility κI “Magnetic susceptibility” χ

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Numerical Resultsqc = qΨ = 3, r = 2, Jc = JΨ = V = 1

0.01

0.05

0.1

0.15

0.2

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

continuous transistion line

ρcρΨ

-2 0 2

ρcρΨ

-2 0 2

ρcρΨ

-2 0 2

ρcρΨ

-2 0 2ω

T

f

0.05

0.1

0.15

0.2

0.45

0.6

0.75

0.9(b) n

0.05

0.1

0.15

0.2

0

0.3

0.6

0.9(c) P

0.05

0.1

0.15

0.2

0

0.3

0.6

0.9(d) CV

0.05

0.1

0.15

0.2

0

1.5

3

4.5(e) κ

0.01

0.05

0.1

0.15

0.2

0 0.25 0.5 0.75 1

0

1.5

3

4.5(f) χ

High temperature phase isstrange metal with a largeentropy akin to usual SYK; heren = 1,P = 0 independent of f

At a critical temperature Tc(f ), asecond order (Landau like)transition occurs – instability ofthe strange metal

Phase below Tc depends on f

For f � 1, a c-strange half metal(c− SHM) emerges – c isgapless, Ψ is gapped

For f ≈ 1, a Mott insulator (MI)phase where both c and Ψ aregapped occurs

A very low temperatures, a firstorder transition separates thec− SHM and MI; the first orderline end in a critical point

Other quantities calculated areall consistent with this picture

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Physics of Instability

Insights by considering Jc = JΨ = 0,V 6= 0, whose free energy

Ξ =

√f

1 + fV2

r

∫ β

0dτ Gr

c(β − τ)GrΨ(τ)

For f = 1, a particle hole symmetric solution implies

Gs(β − τ) = Gs(τ) =⇒ ρs(−ω) = ρs(ω) s = c,Ψ

Interaction problem viewed as two “classical strings” with longranged interactionsParticle hole symmetric solution – good for entropyInteraction energy is reduced by breaking particle hole symmetry

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A Variational AnsatzClassical string analogy suggests a variational ansatz

Gvars (iωn) =

1iωn + dsξ

; ds=c = −1, ds=Ψ = 1

Key results of variational studyI For r = 1 there is no instability; dots must be coupled by interactions

for the low temperature instabilityI For Jc = JΨ = 0, f = 1, instability via second order phase transition for

r = 2, 3, but first order for r ≥ 4! (confirmed by full numerics)

T

(a) P [Jc=JΨ=0, f=1.0]

r=2 num.r=2 var.r=5 num.r=5 var.

0

0.25

0.5

0.75

1

0 0.04 0.08 0.12 0.16 0.2

Tf

0.02

0.04

0.06

0.08

0.1

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

continuous transition

-order line

I Promises a very rich phase diagram forqc = 7, qΨ = 3, r = 4, Jc = JΨ = V = 1 with a plethora of phases,critical lines and multicritical points

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Numerical Resultsqc = 7, qΨ = 3, r = 4, Jc = JΨ = V = 1

0.01

0.02

0.04

0.06

0.08

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

-order line

-order line

continuous transition

ρcρΨ

-2 0 2

ρcρΨ

-2 0 2

ρcρΨ

-2 0 2

ρcρΨ

-2 0 2ω

T

f

0.02

0.04

0.06

0.08

0.5

0.6

0.7

0.8(b) n

0.02

0.04

0.06

0.08

0

0.2

0.4

0.6

0.8(c) P

0.02

0.04

0.06

0.08

0

0.3

0.6

0.9(d) CV

0.02

0.04

0.06

0.08

0

1.5

3

4.5(e) κ

0.010.02

0.04

0.06

0.08

0 1 2 3

0

4

8

12(f) χ

Rich phase diagrampredicted byvariational study isindeed found!SHM and MI phasesappear at lowtemperatureThere are many phasetransitions, bothcontinuous and firstorder, with multicriticalpoints

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SummaryWhat we study

Two SYK dots each with arbitrary q body interactions, coupledwith r body interactions; key parameter, ratio f of number of sites

What we learn (1703.05111)Analytical result – possible to go from one type of strange metal toanother by tuning fNumerics: The coupled strange metals are unstable at lowtemperatures – giving way to new phase such as strange halfmetals and Mott insulators

Physical insights from a variational approachFuture work: Realize such physics in lattice systems (see 1710.00842)

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