Summary of Results

• Intermediate Coupling: • s-wave superconductivity! On site and nearest

neighbor• Enhanced q=(0,0) nematic fluctuations• Stronger with electron doping

• Strong Coupling:• Insulator with antiferro orbital order (AFO)

near 1/2 filling.• Not captured by RPA

3

g

f

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

FIG. 2. Phase diagram of the model (1) as a function ofinteraction g and filling fraction f at inverse temperature� = 8. The red dashed line indicates the boundary of theregion where the superconducting order parameter extrap-olates to a finite value in the thermodynamic limit. Theblue dashed line marks the boundary of phase with antiferro-nematic order. The red / blue coloring is the interpolatedequal time correlation function of the s-wave superconductiv-ity / antiferro-nematic order on a 10⇥ 10 lattice; white spaceis outside of range sampled. Dots indicate simulated pointsalong 10 values of the chemical potentials µ = �1, . . . , 4. Theblack dashed line marks half filling.

Metropolis Monte Carlo algorithm. The fermion de-terminant DetG�1(') decouples into spin sectors sincethe kinetic energy does not mix spins and 'i couplesequally to ", # through �ni. The spin sectors are equal bytime reversal for any field configuration ', DetG�1(') =DetG�1

" (')DetG�1

# (') = |DetG�1

" (')|2 > 0, whichguarantees that the partition function can be sampledin a sign-problem-free manner at any filling.

We perform sweeps through the space-time lattice andupdate the Hubbard-Stratonovich field ' on each site. As' couples di↵erent orbitals, we perform rank-two Wood-bury updates [36] when calculating G" on a given time-slice. We use the one-sided Jacobi Singular-Value Decom-position algorithm [37] for numerical stabilization [38]on every second time-slice. In order to reduce ergodic-ity problems at strong interactions, we run the DQMCsimulation in parallel for various interactions and use aparallel-tempering algorithm [39], which proposes to ex-change ' configurations between simulations at di↵erentinteraction strength g after each sweep [40]. For the datapresented here, we have simulated systems with periodicboundary conditions up to L2 = 10 ⇥ 10 in spatial size(200 orbitals) with an inverse temperature of up to � = 8(�EF ⇠ 40); the imaginary time step is �⌧ = 1/16.

Phase diagram.— We mapped the phase diagram ofmodel (1) as a function of interaction strength g andfilling fraction f (Fig. 2), showing regions of supercon-ducting and antiferro-nematic order. We are consideringa finite temperature system in two spatial dimensions, soonly quasi-long-range order exists. Since our simulationsare on lattice sizes smaller then the scale of these fluctu-

g

C⌧=0

(⇡,⇡

)

g

C

⌧=

0(⇡,⇡)/L

2

FIG. 3. Equal time nematic correlation function averagedaround q = (⇡,⇡) rises rapidly, signaling onset of the long-range order. The left inset shows a cartoon of the antiferro-nematic ordering pattern. The second inset indicates the con-vergence of the correlation function, when normalized by L2,as expected from long-range order. Note the even-odd e↵ectdue to periodic boundary conditions.

ations, our finite size extrapolations indicate long rangeorder of the T = 0 ground state.We first discuss the phase diagram in vicinity of half-

filling f = 0.5, which corresponds to two electrons persite. In the limit of strong coupling g � 3.75 we see de-velopment of long-range antiferro-nematic order. This isfully consistent with the intuition from the strong cou-pling expansion of a fully polarized state in the orbitalbasis with a checkerboard ordering pattern (Fig. 3 inset).The onset of order is confirmed by considering the growthof the equal time nematic correlation function

C⌧=0

(q) =1

L2

X

i,j

eiq·(i�j)h�ni�nji. (2)

The behavior of C⌧=0

(q) at q = (⇡,⇡) is shown in Fig. 3.To reduce finite size e↵ects, we show C⌧=0

(q) averagedover three neighboring points q,q+2⇡/Lex,q+2⇡/Leywhich coincide in the thermodynamic limit. We also con-firmed the onset of order via the Binder ratio [41] for theboson field conjugate to �n at zero frequency (not shown).The AFN order rapidly disappears when the system is

doped away from half-filling, or the interaction strengthis decreased. In contrast to the expectations from weakcoupling RPA, we do not observe any nematic ordering atother wave-vectors. Instead, when the long range AFNdisappears, we observe a large region with non-zero su-perconducting order. In order to probe the supercon-ducting order, we study the equal-time pair correlationfunction ⇠ h�ab(i)�

†cd(j)i, where the specific form of

the �ab(i) depends on the symmetry of pairing. Weconsider all possible irreducible representations of latticepoint group D

4

involving on-site, nearest neighbor and

Summary of Approach

• Idea: Enhanced nematic fluctuations lead to Superconductivity

• Minimal model with orbital / band structure of FeSe + Nematic Fluctuations

• 2 band model dxz, dyz orbitals:

• Sign Problem Free • Solve Exactly with

determinant Quantum Monte Carlo (DQMC)

Expe

rim

ents

Experimental Summary• Bulk FeSe:

• nematic (Tn ~90K) but no magnetic order. • Low temperature superconductor (Tc~8K).

• Monolayer of FeSe on STO: • nematic order suppressed • high Tc superconductor (Meissner effect at

65K [Ref 1])

• K surface doped FeSe• Also high Tc superconductor (Tc ~ 25 – 40 K) [Ref. 1]• Intrinsic electronic effect – not STO phonon

[Refs. 2,3,4,5]

Superconductivity and Nematic Fluctuations in a model of FeSe monolayers: A Determinant Quantum Monte Carlo Study [arxiv:1512.08523]

Ashvin VishwanathUniversity of California at Berkeley

Funding & Collaborators

Philipp Dumitrescu UC Berkeley

Band Structure Normal State

undoped+nematic

Band Structure Superconductor

e doped+symmetric

Our

App

roac

hQ

uant

um M

onte

Car

loSu

mm

ary

Maksym Serbyn Gordon and Betty Moore fellow UC Berkeley

Richard Scalettar UC Davis

1. Z. Zhang, Y.-H. Wang, Q. Song, C. Liu, R. Peng, K. Moler, D. Feng, Y. Wang, Science Bulletin 60, 1301 (2015). 2. Y. Miyata, K. Nakayama, K. Sugawara, T. Sato, and T. Takahashi, Nat Mat 14, 775 (2015). 3. R.Peng,H.C.Xu,S.Y.Tan,H.Y.Cao,M.Xia,X.P. Shen, Z. C. Huang, C. H. P.

Wen, Q. Song, T. Zhang, B. P. Xie, X. G. Gong, and D. L. Feng, Nat Comm 5 (2014). 4. Z. R. Ye, C. F. Zhang, H. L. Ning, W. Li, L. Chen, T. Jia, M. Hashimoto, D. H. Lu, Z.-X. Shen, and Y. Zhang, (2015), arXiv:1512.025265. J. Shiogai, Y. Ito, T. Mitsuhashi, T. Nojima, and A. Tsukazaki, Nat Phys 12, (2016)

6. H. Yamase and R. Zeyher, PRB 88, 180502 (2013). 7. M. A. Metlitski, D. F. Mross, S. Sachdev, T. Senthil, PRB 91, 115111 (2015).8. S. Lederer, Y. Schattner, E. Berg, S. A. Kivelson, PRL 114, 097001 (2015). 9. Y. Schattner, S. Lederer, S. A. Kivelson, E. Berg, (2015), arXiv:1511.03282

Ref

.

Conclusions & Open Issues• A case study:

• Superconductivity established in a multi-orbital model with enhanced nematic fluctuations

• Does Sc arise from nematic fluctuations?

• Anisotropic pairing on hole fermi surface?• Role of onsite Coulomb?

Sc

AFO

Microscopic Model Determinant Monte Carlo

• Spin Symmetry avoids Fermion Sign Problem

• Technical improvement - parallel tempering to improve convergence.

• Lattice sizes < 2 x 12 x 12

• Temperature > 1/8 hopping

t1 = -1.0 t3 = -1.2 µ = 0.6t2 = +1.5 t4 = -0.95

Superconductivity and Nematic Fluctuations Strong Coupling:

AntiFerro Orbital Order

NFL

SC

Nematic

Superconductivity from nematic fluctuations?

• Arguments from field theory [7,8] but not all nematic models show Sc [9]

Other explanations for Sc in the model?

• Negative `U’ intra-orbital attraction?

• However - mean field decouplings do no give superconductivity in observed range.

• Eliashberg with nematic modes see enhanced Sc in same model [6]

• Nematic fluctuations enhanced near Sc.• Electron doping enhances Sc, also stronger

nematic couplings.RPA/Eliashberg, same model: Yasmase and Zeyher [6]

3

g

f

FIG. 2. Phase diagram of the model (1) as a function ofinteraction g and filling fraction f at inverse temperature� = 8. The red dashed line indicates the boundary of theregion where the superconducting order parameter extrap-olates to a finite value in the thermodynamic limit. Theblue dashed line marks the boundary of phase with antiferro-nematic order. The red / blue coloring is the interpolatedequal time correlation function of the s-wave superconductiv-ity / antiferro-nematic order on a 10⇥ 10 lattice; white spaceis outside of range sampled. Dots indicate simulated pointsalong 10 values of the chemical potentials µ = �1, . . . , 4. Theblack dashed line marks half filling.

Metropolis Monte Carlo algorithm. The fermion de-terminant DetG�1(') decouples into spin sectors sincethe kinetic energy does not mix spins and 'i couplesequally to ", # through �ni. The spin sectors are equal bytime reversal for any field configuration ', DetG�1(') =DetG�1

" (')DetG�1

# (') = |DetG�1

" (')|2 > 0, whichguarantees that the partition function can be sampledin a sign-problem-free manner at any filling.

We perform sweeps through the space-time lattice andupdate the Hubbard-Stratonovich field ' on each site. As' couples di↵erent orbitals, we perform rank-two Wood-bury updates [36] when calculating G" on a given time-slice. We use the one-sided Jacobi Singular-Value Decom-position algorithm [37] for numerical stabilization [38]on every second time-slice. In order to reduce ergodic-ity problems at strong interactions, we run the DQMCsimulation in parallel for various interactions and use aparallel-tempering algorithm [39], which proposes to ex-change ' configurations between simulations at di↵erentinteraction strength g after each sweep [40]. For the datapresented here, we have simulated systems with periodicboundary conditions up to L2 = 10 ⇥ 10 in spatial size(200 orbitals) with an inverse temperature of up to � = 8(�EF ⇠ 40); the imaginary time step is �⌧ = 1/16.

Phase diagram.— We mapped the phase diagram ofmodel (1) as a function of interaction strength g andfilling fraction f (Fig. 2), showing regions of supercon-ducting and antiferro-nematic order. We are consideringa finite temperature system in two spatial dimensions, soonly quasi-long-range order exists. Since our simulationsare on lattice sizes smaller then the scale of these fluctu-

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00

5

10

15

20

25

306x67x78x89x910x10

2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.20.0

0.1

0.2

0.3

0.4

0.5

g

C⌧=0

(⇡,⇡

)

g

C

⌧=

0(⇡,⇡)/L

2

FIG. 3. Equal time nematic correlation function averagedaround q = (⇡,⇡) rises rapidly, signaling onset of the long-range order. The left inset shows a cartoon of the antiferro-nematic ordering pattern. The second inset indicates the con-vergence of the correlation function, when normalized by L2,as expected from long-range order. Note the even-odd e↵ectdue to periodic boundary conditions.

ations, our finite size extrapolations indicate long rangeorder of the T = 0 ground state.We first discuss the phase diagram in vicinity of half-

filling f = 0.5, which corresponds to two electrons persite. In the limit of strong coupling g � 3.75 we see de-velopment of long-range antiferro-nematic order. This isfully consistent with the intuition from the strong cou-pling expansion of a fully polarized state in the orbitalbasis with a checkerboard ordering pattern (Fig. 3 inset).The onset of order is confirmed by considering the growthof the equal time nematic correlation function

C⌧=0

(q) =1

L2

X

i,j

eiq·(i�j)h�ni�nji. (2)

The behavior of C⌧=0

(q) at q = (⇡,⇡) is shown in Fig. 3.To reduce finite size e↵ects, we show C⌧=0

(q) averagedover three neighboring points q,q+2⇡/Lex,q+2⇡/Leywhich coincide in the thermodynamic limit. We also con-firmed the onset of order via the Binder ratio [41] for theboson field conjugate to �n at zero frequency (not shown).The AFN order rapidly disappears when the system is

doped away from half-filling, or the interaction strengthis decreased. In contrast to the expectations from weakcoupling RPA, we do not observe any nematic ordering atother wave-vectors. Instead, when the long range AFNdisappears, we observe a large region with non-zero su-perconducting order. In order to probe the supercon-ducting order, we study the equal-time pair correlationfunction ⇠ h�ab(i)�

†cd(j)i, where the specific form of

the �ab(i) depends on the symmetry of pairing. Weconsider all possible irreducible representations of latticepoint group D

4

involving on-site, nearest neighbor and

4

(a) (b)

Ps q,r

Ps q,r

FIG. 4. Finite size scaling of the on-site s-wave equal timepair correlation at maximal-distance P s

r=(L/2,L/2)

and zero-momentum P s

q=0

. (a) In the region which we identify as asuperconductor (µ = 0.6, g = 3.59) Pr, Pq extrapolate to afinite value in the thermodynamic limit. (b) In the regionof the AFN phase (µ = 0.6, g = 3.91), the pair correlationfunctions scale to zero in the thermodynamic limit.

next nearest neighbor sites and found non-vanishing paircorrelation function for the order parameter with s-wave(A

1

) symmetry. The dominant response is the on-sitepairing, where the only non-vanishing pairing is withinthe same orbitals with equal sign (A

1

⇥ A1

irreduciblerepresentation),

�s(i) =1

2cia↵(i�

y↵�)(⌧

0

ab)cib� . (3)

Here � and ⌧ are the Pauli matrices acting in the spinand orbital basis, and ⌧0 is an identity matrix. The orderparameter �s(i) coexists with the extended s-wave pair-ing between nearest neighbors, �s-ex(i), where the gapchanges sign between orbitals (B

1

⇥B1

representation),

�s-ex(i) =1

2

X

e

d(e) ci+e,a↵(�y↵�)(⌧

zab)cib� , (4)

as is reflected by ⌧z matrix. In these notations vector eruns over nearest neighbors and d(e) denotes the dx2�y2 -wave symmetry form-factor, d(±x) = 1, and d(±y) =�1. For µ � 2, the extended s-wave pairing also extendsto the next nearest neighbor sites (along diagonals) forand has a dxy-wave form factor along with the ⌧x pairingin the orbital basis (B

2

⇥B2

representation).The equal time (⌧ = 0) pair correlation function for

the on-site s-wave is defined as

P s

r =1

L2

X

i

h�s(i+ r)�s(i)i, P s

q =1

L2

X

r

eiq·rP s

r (5)

in the coordinate and Fourier space respectively, whereboth sums are performed over all lattice points. In thethermodynamic limit the value P s

q at q = 0 must con-verge to the pair correlation function at the maximum

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.5

1.0

1.5

2.0

2.5 6x67x78x89x910x10

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.00

0.02

0.04

0.06

0.08

0.10

0.12 6x67x78x89x910x10

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.00

0.02

0.04

0.06

0.08

0.10

0.12 6x67x78x89x910x10

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

g

C

0(0,0)/L

2

(a)

(b)

(c)

C⌧=0

(0,0)

Ps r=(L/2,L/2)

Ps-ex

r=(L/2,L/2)

g

g

g

FIG. 5. The (a) uniform nematic correlation function, (b)on-site s-wave pair correlation function, and (c) nearest neigh-bor extended s-wave have very similar dependence on the in-teraction strength for fixed value of µ = 0.6. The onset andtermination of on-site superconducting order coincides withthe similar trends in nematic susceptibility. In (a), the insertshows the correlation function normalized by 1/L2 showingthe decrease with system size and lack of long-range order.

separation P s

r , r = (L/2, L/2), provided there is long-range superconducting order. At small L, P s

q=0

includesmostly short range contributions and overestimates theorder parameter. Figure 4(a) demonstrates a case whereboth quantities extrapolate to finite value as 1/L ! 0,moreover these quantities become closer to each other forlarger system sizes as expected. In contrast, in the AFNphase, the pair correlation function is non-zero only dueto finite size e↵ects and extrapolate to zero in the ther-modynamic limit, see Fig. 4(b).Origin of superconductivity.— We observed an ex-

Hint = �g

2

X

i

(ni,x

� ni,y

)2

Figure 4 | Phase diagram of potassium-coated FeSe single crystal. a, the Lifshitz

transitions of Fermi surface in two-iron (upper panel) and one-iron (lower panel) BZ. The red

arrows denote the vectors of inter-pocket scattering. b, Phase diagram of potassium-coated

FeSe single crystal. The black dashed lines represent the two Lifshitz transitions. The Tc is

around 8.4 K for bulk FeSe25 and the SCh-e dome is illustrated by red solid line according to

ref. 26.

Figure 4 | Phase diagram of potassium-coated FeSe single crystal. a, the Lifshitz

transitions of Fermi surface in two-iron (upper panel) and one-iron (lower panel) BZ. The red

arrows denote the vectors of inter-pocket scattering. b, Phase diagram of potassium-coated

FeSe single crystal. The black dashed lines represent the two Lifshitz transitions. The Tc is

around 8.4 K for bulk FeSe25 and the SCh-e dome is illustrated by red solid line according to

ref. 26.

Figure 4 | Phase diagram of potassium-coated FeSe single crystal. a, the Lifshitz

transitions of Fermi surface in two-iron (upper panel) and one-iron (lower panel) BZ. The red

arrows denote the vectors of inter-pocket scattering. b, Phase diagram of potassium-coated

FeSe single crystal. The black dashed lines represent the two Lifshitz transitions. The Tc is

around 8.4 K for bulk FeSe25 and the SCh-e dome is illustrated by red solid line according to

ref. 26.

RAPID COMMUNICATIONS

HIROYUKI YAMASE AND ROLAND ZEYHER PHYSICAL REVIEW B 88, 180502(R) (2013)

-g

T

1

2

3

4

X

Y M

Γ

0

0.05

0.1

0.15

0.2

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

N ON

ON+SCSC

FIG. 1. (Color online) Phase diagram in the T -g plane for g0 = 0.The thick solid line separates the normal (N) from the orbital nematic(ON) phase, which at low temperatures is modulated, as indicated bythe crosses. The dashed line is the instability line of the modulatedto the homogeneous nematic state. The circles, smoothly joined bya thin solid line, separate the superconducting (SC) from the normaland nematic states. The inset shows the Fermi lines of the four pocketsin the normal state.

In the following we will study superconductivity fromnematic fluctuations. We consider the usual Fock diagramfor the electronic self-energy where the bosonic propagatordescribes nematic fluctuations. Since superconductivity is aFermi surface effect, we may restrict the momenta to theregion near the Fermi line of each pocket. In our case thereare two hole pockets near the ! and M points and twoelectron pockets near the X and Y points. We denote themby i = 1 . . . 4, respectively (see the inset in Fig. 1). Assumingthat the superconducting order parameter is constant on eachindividual pocket, the Eliashberg equations for the gap "i (iωn)and the renormalization function Zi(iωn) read as

"i(iωn)Zi(iωn) = −πT!

j,n′

Nj

gij (iωn − iωn′ )|ωn′ |

"j (iωn′ ),

(3)

Zi(iωn) = 1 − πT!

j,n′

Nj

ωn′

ωn

gij (iωn − iωn′ )|ωn′ |

. (4)

ωn is a fermionic Matsubara frequency and Nj is the densityof states of pocket j at the Fermi energy. gij (iωn − iωn′ ) isobtained from the microscopic pairing potential Wαβ(k,k′,iνn)by averaging k and k′ independently over the Fermi lines ofpockets i and j , respectively, and by summing over α and β.W is given by

Wαβ(k,k′,iνn) = V 2αβ(k,k′)g(k − k′,iνn), (5)

where νn denotes a bosonic Matsubara frequency. V representsthe form factor

Vαβ (k,k′) =!

γ ,δ

U †αγ (k)(τ3)γ δUδβ(k′). (6)

Uαβ(k) is a unitary matrix which diagonalizes H0, τ3 is a Paulimatrix, and g(k − k′,iνn) is given by

g(k − k′,iνn) = g2+(k − k′,iνn)1 − g+(k − k′,iνn)

+ g0. (7)

The first term in Eq. (7) is the retarded interaction mediated bynematic fluctuations, and the second one is an instantaneousterm accounting for repulsive Coulomb interactions. + standsfor a single bubble of noninteracting nematic particle-holeexcitations.

The transition temperature Tc to superconductivity isobtained from the condition that the largest eigenvalue of thematrix

M(i,n; j,n′) = −πT Nj gij (iωn − iωn′ )/|ωn′ |/Zi(iωn) (8)

is equal to one.The momentum-averaged pairing potentials gij (iνn) are

important ingredients in the calculation of Tc. Figure 2 showsthe dependence of gij on the Matsubara frequency νn, treatedas a continuous variable, for various temperatures. The curvesfor g44(iωn) of Fig. 2(a) are always in the normal state.With decreasing temperature the correlation length of nematicfluctuations increases, which implies that g44(iωn) increasesmonotonically with decreasing temperature. At low tempera-tures g44(iνn) assumes huge values at νn = 0 and then decaysvery fast with increasing frequency on an energy scale muchsmaller than t . In this frequency range g44(iνn) varies onlyslowly with temperature, which indicates that the attractivepairing interaction is rather insensitive to temperature and tocritical fluctuations. For g = −1.8 [Fig. 2(b)], the nematicinstability occurs at the temperature Tn ≈ 0.125. Here g44(iνn)first increases monotonically with decreasing temperatureuntil Tn is reached. Entering the nematic state, g44(iνn) is

-25

-20

-15

-10

-5

0

0 0.2 0.4 0.6 0.8 1

(b) g=-1.8,

νn

g44~

iνn

()

-3

-2

-1

0

0 2 4 6 8 10

g14~

iνn

()

νn

T=0.30T=0.20T=0.10T=0.05T=0.001

-60

-50

-40

-30

-20

-10

0

0 0.2 0.4 0.6 0.8 1

(a) g=-1.7,

νn

g44~

iνn

()

-2

-1

0

0 2 4 6 8 10

g14~

iνn

()

νn

T=0.30T=0.20T=0.10T=0.05T=0.001

g =00

g =00

FIG. 2. (Color online) Retarded pairing interactions g44 (main di-agrams) and g14 (insets) as a function of νn for different temperaturesfor (a) g = −1.7 and (b) g = −1.8. In (b) the nematic state occursbelow T = Tn ≈ 0.125.

180502-2