Mikhail Kiselev

Ginzburg-Landau glimpse on coexistingmagnetic/superconducting phases of

unconventional superconductors

Kourovka - XXXVII, February 28, 2018

Outline

• unconventional superconductors: phase diagram and coexisting phases

• Landau theory 1: SDW vs SC

• Landau theory 2: effects of disorder

• Landau theory 3: nematic phase

• Landau theory 4: topological superconductivity

• Landau theory 5: stripes vs checkerboard

• from Landau theory to Ginzburg - Landau theory

• open questions and perspectives

CrystalstructureofIron-basedsuperconductors

11 111 122 1111

AfterPaglione,Greene,Nature

CrystalStructure:TetragonalI4/mmm

Fe,Ni

As,PLa,Sm,Ce

O• 2D square lattice of Fe• Fe - magnetic moment • As-similar then O in cuprates

But As not in plane!

Fe

As

Cuprates vs pnictidesafter Chubukov and Basov, Nature Physics(2010)

Mott insulator/Heisenberg AFM

metal

para-magneticmetal

magnetictransition(collinearAForder)

structuraltransition(tetragonal-orthorhomic)

Ba(Fe1-xCox)2As2

Measuredphasediagram

S.Nandietal,PRL(2010)

Ba1-xKxFe2As2

magnetictransition(collinearAForder)

para-magneticmetal

Wen,Li,Annu.Rev.Condens.MatterPhys.(2011)

phasediagramtakenfromμSR

H.Luetkenssetal,Nat.Matt.8,305(2009)

para-magneticmetal

LaO1-xFxFeAs

magnetictransition(collinearAForder)

structuraltransition(tetragonal-orthorhomic)

Phasediagram

NdFeAs(O1-xFx) (x=0.1)A. Kaminski et al.

Hole pockets near (0,0) Electron pockets near (p,p)

dHVaARPES

LaFeOPA. Coldea et al,

Ba06K04Fe2As2H. Ding et al.

A. Kordyuk et alLiFeAs

Itinerantpicture:LDAelectronicstructure(LaFeAsO)

Fe2+ 3d6-states

S.Lebegue,PRB75,035110(2007);D.J.Singhand M.-H.Du,PRL100,237003(2008);I.I.Mazin etal.,PRL101,057003(2008)

Weak CEF splitting: all 5 orbitals (dx2-y2, d3z2-r2, dxy, dxz+dyz) are near the Fermi level

1. Both electron and hole pockets do exist (quantum oscillations, ARPES)

2. There is a Fermi surface even at zero doping (ARPES)

h

eΓ

M

A.A.Kordyuk,LowTemp.Phys.(2012)

Proposedmultibandpairinggapsymmetries

AfterPaglione,Greene(2010)

Optimaldoping

Landau theory

cT T>cT T<

[ ] 2 3 40 1 2 3 4 ...j j j j jF =F +F +F +F +F +

1 0F = 3 0F =

2 ( )ca T TF = -

4 0constF = >

Landautheoryof2ndorderphasetransitions

0D ¹

M

0M ¹

DMagnets Superconductors

Vorontsov-Vavilov-Chubukov theory

A)PureSCstate

B)PureSDWstate

C)MixedSC+SDWstate

Vorontsov,Vavilov,Chubukov PRB2010

Vorontsov-Vavilov-Chubukov theory

Vorontsov,Vavilov,Chubukov PRB2010

Universalsuppressionofmagneticorder

Ph.Materne etal,PRB2015

Ca1�x

Nax

Fe2As2

Universalsuppressionofmagneticorder

Ph.Materne etal,PRB2015

Universalsuppressionofmagneticorder

Ph.Materne etal,PRB2015

Hoyer-Syzranov-Schmalian theoryRoleofdisorder

nocoexistence,1stordertransition coexistencepossible,2ndordertransition

Hoyer,Syzranov,Schmalian,PRB2014

g =�

pumus

� 1<latexit sha1_base64="9+k4jm7xORsgAaNeHgkdJ2MAsZw=">AAACCXicbVBNS8NAEN34WetX1KOX1SJ4sSQiqAeh6MVjBWMLTQib7aZduruJuxuhhJy9+Fe8eFDx6j/w5r9x2+agrQ8GHu/NMDMvShlV2nG+rbn5hcWl5cpKdXVtfWPT3tq+U0kmMfFwwhLZjpAijAriaaoZaaeSIB4x0ooGVyO/9UCkoom41cOUBBz1BI0pRtpIob3Xu/BjiXDu9xDnqMh9dS91noUcZqEqiiM3tGtO3RkDzhK3JDVQohnaX343wRknQmOGlOq4TqqDHElNMSNF1c8USREeoB7pGCoQJyrIx68U8MAoXRgn0pTQcKz+nsgRV2rII9PJke6raW8k/ud1Mh2fBTkVaaaJwJNFccagTuAoF9ilkmDNhoYgLKm5FeI+Mslok17VhOBOvzxLvOP6ed25Oak1Lss0KmAX7IND4IJT0ADXoAk8gMEjeAav4M16sl6sd+tj0jpnlTM74A+szx9LoJrX</latexit><latexit sha1_base64="9+k4jm7xORsgAaNeHgkdJ2MAsZw=">AAACCXicbVBNS8NAEN34WetX1KOX1SJ4sSQiqAeh6MVjBWMLTQib7aZduruJuxuhhJy9+Fe8eFDx6j/w5r9x2+agrQ8GHu/NMDMvShlV2nG+rbn5hcWl5cpKdXVtfWPT3tq+U0kmMfFwwhLZjpAijAriaaoZaaeSIB4x0ooGVyO/9UCkoom41cOUBBz1BI0pRtpIob3Xu/BjiXDu9xDnqMh9dS91noUcZqEqiiM3tGtO3RkDzhK3JDVQohnaX343wRknQmOGlOq4TqqDHElNMSNF1c8USREeoB7pGCoQJyrIx68U8MAoXRgn0pTQcKz+nsgRV2rII9PJke6raW8k/ud1Mh2fBTkVaaaJwJNFccagTuAoF9ilkmDNhoYgLKm5FeI+Mslok17VhOBOvzxLvOP6ed25Oak1Lss0KmAX7IND4IJT0ADXoAk8gMEjeAav4M16sl6sd+tj0jpnlTM74A+szx9LoJrX</latexit><latexit sha1_base64="9+k4jm7xORsgAaNeHgkdJ2MAsZw=">AAACCXicbVBNS8NAEN34WetX1KOX1SJ4sSQiqAeh6MVjBWMLTQib7aZduruJuxuhhJy9+Fe8eFDx6j/w5r9x2+agrQ8GHu/NMDMvShlV2nG+rbn5hcWl5cpKdXVtfWPT3tq+U0kmMfFwwhLZjpAijAriaaoZaaeSIB4x0ooGVyO/9UCkoom41cOUBBz1BI0pRtpIob3Xu/BjiXDu9xDnqMh9dS91noUcZqEqiiM3tGtO3RkDzhK3JDVQohnaX343wRknQmOGlOq4TqqDHElNMSNF1c8USREeoB7pGCoQJyrIx68U8MAoXRgn0pTQcKz+nsgRV2rII9PJke6raW8k/ud1Mh2fBTkVaaaJwJNFccagTuAoF9ilkmDNhoYgLKm5FeI+Mslok17VhOBOvzxLvOP6ed25Oak1Lss0KmAX7IND4IJT0ADXoAk8gMEjeAav4M16sl6sd+tj0jpnlTM74A+szx9LoJrX</latexit>

Hoyer-Syzranov-Schmalian theoryRoleofdisorder

nocoexistenceofSDWandSC

coexistenceofSDWandSCispossible

Hoyer,Syzranov,Schmalian,PRB2014

Interplaybetweentetragonalmagneticorder,stripemagnetismandsuperconductivity

Kang,Wang,Chubukov,Fernandes,PRB2015

C_2orthorhombicC_4tetragonal ~M2

1 = ~M22

~M1 = 0, or ~M2 = 0

Chubukov,Fernandes,Schmalian PRB2015

Nematic orderinFeSe

Nematic orderinFeSe

Chubukov,Fernandes,Schmalian PRB2015

FepnictidesFeSe

Nematic orderinpnictides

Fernandes,Chubukov,Schmalian NaturePhysics2014

Nematic orderinpnictides

Fernandes,Chubukov,Schmalian NaturePhysics2014

Nematic orderinpnictides

Fernandes,Chubukov,Schmalian NaturePhysics2014

Nematic orderinpnictides

Fernandes,Chubukov,Schmalian NaturePhysics2014

Nematic orderinpnictides

Fernandes,Chubukov,Schmalian NaturePhysics2014

HowmanyQPC?

Fernandes,Maiti,Woelfle,Chubukov PRL2013

TopologicalSCandunconventionalpairing

Scheurer,Schmalian NatureCommunications2014

TopologicalSCandunconventionalpairing:scatteringprocesses

TopologicalSCandunconventionalpairing:RG

Scheurer,Schmalian NatureCommunications2014

TopologicalSCandunconventionalpairing

Scheurer,Schmalian NatureCommunications2014

TopologicalSCandunconventionalpairing

Scheurer,Schmalian NatureCommunications2014

�f =X

V sc↵��0↵0(k,p)hc�p�0cp↵0i

�c =X

V sc↵��0↵0(k,p)hf�p�0fp↵0i

m↵0,�0 =X

V sdw↵��0↵0hf†

p0↵cp�i

How many components does the order parameter have?

S++

⇡T↵0,�0 =

XV ⇡↵��0↵0hf†

�k�q↵c†k�i S+�

Karki,Mandal,Kiselev 2015

Kopaev JETP1971

Aronov,Sonin JETP1973

Karki,Mandal,Kiselev 2015

Landau theory: triplet component includedF (�,m,⇡T ) = ↵�|�|2 + ↵mM2 + ↵⇡(⇡

T )2 +A|�|4 +BM4 + C(⇡T )4+

2D1|�|2M2 + 2D2|�|2(⇡T )2 + 2D3M2(⇡T )2

|�|2 = �↵�(BC �D23) + ↵m(D2D3 � CD1) + ↵⇡(D1D3 �BD2)

2(ABC + 2D1D2D3 �BD22 �AD2

3 � CD21)

M2 = �↵�(D2D3 � CD1) + ↵m(AC �D22) + ↵⇡(D1D2 �AD3)

2(ABC + 2D1D2D3 �BD22 �AD2

3 � CD21)

@F

@�⇤ = 0,@F

@M= 0,

@F

@(⇡T )⇤= 0.

(⇡T )2 = �↵�(D1D3 �BD2) + ↵m(D1D2 �AD3) + ↵⇡(AB �D21)

2(ABC + 2D1D2D3 �BD22 �AD2

3 � CD21)

Universalsuppressionofmagneticorder

Ph.Materne etal,PRB2015

Ca1�x

Nax

Fe2As2

Ginzburg- Landau theory

FromLandautoGinzburg - Landautheory

F = FN +

Zd~r

1

2ms|(r� 2ie

cA) |2 + ↵| |2 + �

2| |4 + |B|2

8⇡

�

@(F � FN )

@ ⇤ = 0 1st GL equation

@(F � FN )

@ ~A= 0 2nd GL equation

Ginzburg - Landautheoryinexternalmagneticfield

FO � FN = FM + FL + Fs + FB

FM =

Zd~r

am2

M2 +bm4M4 + �µBM ·B

�

FL =

Zd~r

al2L2 +

bl4L4 + cl(rL)2

�

Fs =

Zd~r

as2| |2 + bs

4| |4 + cs|(r� 2ie

cA) |2

�

Fmix

=

Zd~r

⇥↵1

L2| |2 + ↵2

L2M2 + ↵3

M2| |2⇤

FB =

Zd~r

B2

8⇡ Karki,Mandal,Kiselev 2015

GLequationsandsupercurrent

@(F � FN )

@ ⇤ = 0 1st GL equation

@(F � FN )

@ ~A= 0 2nd GL equation

@(F � FN )

@~L= 0 3rd GL equation

Karki,Mandal,Kiselev 2015

Openquestionsandperspectives

• spintexturesinthevortexcore

• influenceofmagneticimpuritiesonTc andTn

• influenceofresonanceimpuritiesTc andTn

• supercurrent incoexistentphase

• influenceofpi-tripletphaseonmagneticmomentsuppression

• coexistentphasesinexternalmagneticfield

• influenceofpi-tripletonnematic phase

• magnetizationdynamicsinthepresenceofstrongsuperconductingfluctuations

• etc

Coupledmultiple-modetheoryfors± pairing

mechanisminiron-basedsuperconductors

MIKHAIL KISELEV (ICTP,Trieste)IN COLLABORATION WITH:

DMITRY EFREMOV (IFWDresden)JEROEN VAN DEN BRINK (IFWDRESDEN)

STEFAN-LUDWIG DRECHSLER (IFWDRESDEN)KONSTANTIN KIKOIN

MK,D.V.Efremov,S.L.Drechsler,Jeroen vandenBrinkandK.Kikoin.Sci.Rep.6,37508(2016)

G

Nesting vector responsible for SDW correlations betweentwo pockets of FS

S+ - pairing: two nodeless gaps with opposite sign due to phase difference. These gaps see each other via repulsiveSDW-mediated interaction [χ (ω) with q ~ G]q

SC pairing mechanism

(Aronov, Sonin '72; Mazin et al '08; Kuroku et al '08)

EF

43

G

EF

Competingmodes:twobandmodel

SCchannel

SDWchannel

Competingmodes

IntheregionIandIII

= +

+Гsdw

= +Г = u + uПГ

Bosonicmodes.here we consider the two-band model with 6 vertices..

Bethe – Salpeter equation for the anomalous vertex Г3

=Г3

+ +..

.

u3

u3 Couplingofthemagnetic

Fluctuationsandthesuperconductingmodes

Ginzburg-Landaufunctional

Staticlimit See,Vavilov andChubukovSchmalian etal….

ThemagneticfluctuatingmodeiscoupledtoΔ±superconductingmode

ℒ(() = ∫ 𝑑𝑥 Δ./𝐿±/1Δ/ + 𝑚𝐷5/1𝑚 + 1(𝐴 Δ/ 7 + 𝐵𝑚7 + (𝐶1 − 𝐶() Δ/ (𝑚(

𝐿±/1=1

𝑢= + 𝑢>− Π= 𝑞, 𝑖ΩD, 𝑇 → 𝜈(−

𝑖Ω𝛾IJ

+ 𝜏= + 𝑐=𝑞()

𝐷5/1 =1

𝑢1 + 𝑢>− Π1 𝑞, 𝑖ΩD, 𝑇 → 𝜈(−

𝑖Ω𝛾IMN

+ 𝜏IMN + 𝑐IMN𝑞()

Vavilov andChubukov ,2011

ThemagneticfluctuatingmodeiscoupledtoΔ±superconductingmode

ℒ(() = ∫ 𝑑𝑥 Δ./𝐿±/1Δ/ + 𝑚𝐷5/1𝑚 + 1(𝐴 Δ/ 7 + 𝐵𝑚7 + (𝐶1 − 𝐶() Δ/ (𝑚(

A=Δ/ Δ/

Δ/ Δ/

~m ~m

~m ~m

B=+(𝑒 ↔ ℎ)

C1=mm

Δ/ Δ/+(𝑒 ↔ ℎ) +(𝑒 ↔ ℎ)C1=

mm

Δ/

Δ/

See,Vavilov andChubukovSchmalian etal….

FluctuationcorrectiontoTC.• Tocalculatethecorrectiontothecriticaltemperatureconsider:

where

ℒ(() = ∫ 𝑑𝑥Δ./𝐿±STT/1 Δ/ + 𝐴U Δ/ 7

𝑇V = 𝑇VW 1 − 𝜈/1(|𝐾1| − |𝐾(|)

𝐿Z±STT/1 → 𝐿±/1 + 𝐾1 + 𝐾(

Δ/ Δ/𝐾1 =

Δ/ Δ/𝐾( =

+(𝑒 ↔ ℎ)

+(𝑒 ↔ ℎ)Itleadstothecorrectionsto𝑇V

Here𝐾1 > 0, while𝐾( < 0

FluctuationcorrectiontoTS.• Tocalculatethecorrectiontothecriticaltemperatureconsider:

where

ℒ5STT(() = ∫ 𝑑𝑥𝑚�̂�5/1𝑚 + 𝐵 𝑚 7

𝑇= = 𝑇=W 1 − 𝜈(�̂�1 + �̂�()

𝐿Z5/1 → 𝐿5/1 + �̂�1 + �̂�(

�̂�1 =

�̂�( =

+(𝑒 ↔ ℎ)

+(𝑒 ↔ ℎ)~m ~m

~m ~m

Itleadstothecorrectionsto𝑇=

Here�̂�1 > 0, while�̂�( < 0

ThemagneticfluctuatingmodeiscoupledtotwosuperconductingmodeΔ_ andΔS

• ℒ` = ∫ 𝑑𝑥 Δ.SΔ._𝐿S/1 𝐾(𝐾( 𝐿_/1

ΔaΔb

+𝐴c |Δa|7 + Δ_ 7

• ℒ5 = ∫ 𝑑𝑥 𝑚𝐷5/1𝑚 + 𝐵𝑚7

• ℒdDe = ∫ 𝑑𝑥 𝐶1 |ΔS |( + |Δ_ |( 𝑚( + 𝐶( Δ.SΔb + Δ._ΔS 𝑚(

Magneticfluctuationsfirstbreak𝑈 1 ×𝑈 1 → 𝑈(1)

FluctuationcorrectionstoTC andTS.

• Tocalculatethecorrectiontothecriticaltemperatureconsider:

where

𝑇V = 𝑇VW 1 − 𝜈(𝐾1 + 𝐾()

ℒ` = ∫ 𝑑𝑥 Δ.SΔ._𝐿S/1 𝐾(𝐾( 𝐿_/1

ΔaΔb

+𝐴c |Δa h7+ Δ_ 7

�e �̄e𝐾1 =

�e �̄h𝐾( =Here𝐾1 > 0, while𝐾( < 0

Correctionsto𝑇VaresimilartotheΔ/𝑐𝑎𝑠𝑒

+(𝑒 ↔ ℎ)

+(𝑒 ↔ ℎ)

FluctuationcorrectionstoTS.

• Tocalculatethecorrectiontothecriticaltemperatureconsider:

where

ℒ5STT(() = ∫ 𝑑𝑥𝑚�̂�5/1𝑚 + 𝐵 𝑚 7

𝑇= = 𝑇=W 1 − 𝜈(2�̂�1)

𝐿Z5/1 = 𝐿5/1 + 2�̂�1

�̂�1 = +(𝑒 ↔ ℎ)~m ~m

Itleadstothecorrectionsto𝑇=

Itisquitedifferentfromthecaseofinteractionofthemagneticfluctuatingmodewithsinglesuperconductingmode

Forparticle-holesymmetry

𝑇V − 𝑇= = 𝑇W𝜈/1( 𝐾1 + |𝐾(|)

Estimateoftheintegrals

• InOrnstein–Zernikeapproximation:𝐿S 𝑞, 𝜔, 𝑇 ∼ 1

VnopqD=3

rs/rtru

∼ 𝜈/1𝐾1,( ∼1

vruwtx∼ 𝐺𝑖 >� ∼ ru

{|

(

D=2->logdivergencers/rtru

∼ 𝜈/1𝐾1,( ∼1

vruwolog Vw

�o

qs∼ 𝐺𝑖 ( ln𝐺𝑖(() ∼ ru

{|log {|

ru

Taking𝐺𝑖(() = 0.1 oneobtainsrs/rtru

∼ 0.25

SDW

SCtempe

rature

concentration

Summary

MK,D.V.Efremov,S.L.Drechsler,Jeroen vandenBrinkandK.Kikoin.Sci.Rep.6,37508(2016)

Take home message

Conclusions

• Interactionofthemagneticfluctuatingmodewithasinglesuperconductingmodechangesthetetracritical pointinthesecondorderofperturbationtheorynegligibly

• Interactionofthemagneticfluctuatingmodewithtwosuperconductingmodeschangesthetetracritical pointconsiderably,suppressingthe𝑇=.

Thankyou!