Thermodynamics of an Exactly Solvable Model for Superconductivity in a DopedMott Insulator

Jinchao Zhao, Luke Yeo, Edwin W. Huang, and Philip W. PhillipsDepartment of Physics and Institute of Condensed Matter Theory,

University of Illinois at Urbana-Champaign, Urbana, 61801 Illinois, USA

Computing superconducting properties starting from an exactly solvable model for a doped Mottinsulator stands as a grand challenge. We have recently1 shown that this can be done starting fromthe Hatsugai-Kohmoto2 (HK) model which can be understood3 generally as the minimal model thatbreaks the non-local Z2 symmetry of a Fermi liquid, thereby constituting a new quartic fixed pointfor Mott physics. In the current work, we compute the thermodynamics, condensation energy, andelectronic properties such as the NMR relaxation rate 1/T1 and ultrasonic attenuation rate. Keydifferences arise with the standard BCS analysis from a Fermi liquid: 1) the free energy exhibits alocal minimum at Tp where the pairing gap turns on discontinuously above a critical value of therepulsive HK interaction, thereby indicating a first-order transition, 2) a tri-critical point emerges,thereby demarcating the boundary between the standard second-order superconducting transitionand the novel first-order regime, 3) Mottness changes the sign of the quartic coefficient in the Landau-Ginzburg free-energy fuctional relative to that in BCS, 4) as this obtains in the strongly interactingregime, it is Mott physics that underlies the generic first-order transition, 5) the condensation energyexceeds that in BCS theory suggesting that multiple Mott bands might be a way of enhancingsuperconducting, 6) the heat-capacity jump is non-universal and increases with the Mott scale, 7)Mottness destroys the Hebel-Slichter peak in NMR, and 8) Mottness enhances the fall-off of theultrasonic attenuation at the pairing temperature Tp. As several of these properties are observedin the cuprates, our analysis here points a way forward in computing superconducting properties ofstrongly correlated electron matter.

I. INTRODUCTION

A truly remarkable feature of superconductivity in ele-mental metals is that although the superconducting orderparameter, Ψ = ∆eiθ, has two components, a phase, θand an amplitude, ∆, both turn on at the same tempera-ture. This behaviour is captured by the mean-field treat-ment of BCS4 which predicts a second-order phase tran-sition in which superconducting fluctuations of the pairamplitude satisfy the Ginzburg criterion, δ(∆)2/∆ 1.Should this criterion be met, pairing and phase coherenceare synonymous. From a technical standpoint, satisfyingthe Ginzburg criterion is quite surprising as the latter isgoverned by a divergence in the pair susceptibility whilethe former stems from a solution to a mean-field inte-gral equation. Experimentally, no more than a 1K dif-ference between these temperatures in BCS materials isobserved. Hence, the Ginzburg criterion is really a state-ment about the accuracy of mean-field theory.

It is well known that the cuprates violate the Ginzburgcriterion or equivalently the BCS dictum that pairingwithout long-range order is impossible5–8. While thistrend was thought to vanish in the overdoped regime, re-cent spectroscopic measurements9 on the bilayer cuprate(Pb,Bi)2Sr2CaCu2O8+δ (Bi-2212) have verified that evenin such samples remnants of a particle-hole symmetricsuperconducting gap persist up to Tpair = 86K wherethe superconducting transition is Tc = 63K. The focuson a particle-hole symmetric gap as the signature of thepairing gap is designed to disentangle superconductivity-related gaps from the wider range of phenomena as-sociated with the pseudogap10 that need not11–13 have

this symmetry. Within the cuprate family, Bi-2212 andYBa2Cu3O7−δ (YBCO) have the highest ratio of Tpair/Tcof 1.6 while in (La,Sr)2CuO4 (LSCO) the ratio is 1.2. Al-though 2D disordered materials are expected to have awide range where thermal superconducting fluctuationsobtain14,15, the efficient cause of the disconnect betweenphase coherence and the gap turn-on remains unsettledin the cuprates. For example, the discrepancy has beenattributed9 to the existence of a flat band at (π, 0). Fur-ther, it is unclear how to think about a pairing gap with-out a phase transition. The question arises, what is theorder of the transition in which pairing and phase coher-ence are decoupled? It is questions of this type that weaddress in this work.

In trying to understand the source of the discrepancybetween pairing and phase coherence, it is worth cata-loguing other instances of deviations from BCS supercon-ductivity in the cuprates. Two features stand out: 1) thecolor change16 and 2) a violation of the Glover-Ferrell-Tinkham17,18 (GFT) sum rule. Regarding the latter, in astandard BCS superconductor, condensation leads to lossof spectral weight at energy scales no more than ten timesthe pairing energy. Not so in the cuprates. Bontemps andcolleagues19 have directly observed that in underdoped(but not overdoped) Bi-2212, the Glover-Ferrel-Tinkhamsum rule is violated and the optical conductivity must beintegrated to 20,000cm−1 to recover the spectral weightlost upon condensation into the superconducting state.Similarly, Rubhaussen, et. al20 and others21 have shownthat changes in the optical conductivity occur at energies3eV (roughly 100∆ where ∆ is the maximum supercon-ducting gap) away from the Fermi energy at Tc. Finally,

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van der Marel and colleagues16 have seen an accelera-tion in the depletion of the high energy spectral weightaccompanied with a compensating increase in the low-energy spectral weight at and below the superconduct-ing transition. Specifically the integrated weight of theoptical conductivity over the lower Hubbard band scales(< 1eV ) increases below the superconducting temper-ature, whereas the high-energy component ([1eV, 2eV ])decreases. Since the integrated weight determines thenumber of charge carriers, the color change indicates thathigh-energy scales contribute to the superfluid density incontrast to the standard BCS picture. Consequently, thesuperfluid density in the cuprates is derived not just fromlow energy physics. In essence, it involves UV-IR mixing.

What all of this seems to indicate is that the departuresfrom BCS superconductivity in the cuprates are tetheredto the Mott state. What is difficult then is to solve areasonable model for a Mott insulator which capturesthis range of deviations from the BCS paradigm. It isin attempting to answer this array of questions that wehave focused1 on the Hatsugai-Kohmoto (HK)2 model,an exactly solvable model for a Mott insulator. The HKmodel,

HHK =∑kσ

ξknkσ + U∑k

nk↑nk↓, (1)

is essentially the Hubbard model in momentum space.As we pointed out previously3, this model is importantbecause it represents the simplest way of breaking thehidden Z2 symmetry of a Fermi liquid. We illustrate thisfrom the the basic Hamiltonian for a Fermi liquid

HFL =∑p

ψ†p(εp − εF )τ3ψp + · · · . (2)

Here ψ†p = (c†p↑, c−p↓) and τ3 is the standard z-Pauli ma-trix. For electrons at the Fermi surface, εp = εF , thisHamiltonian obeys the symmetry nk↑ → −nk↑ whereonly only one of the spin currents changes sign. Con-sequently, the interaction term of the form in the HKmodel maximally breaks this symmetry. Based on this,we showed previously3 that the HK model represents afixed point for Mott physics that even encompasses Hub-bardology. A straightforward Fourier transform of theHubbard on-site interaction reveals that it contains theHK interaction term. As we have shown previously3,it is this term that is most relevant in the renormaliza-tion sense and the only one that maximally breaks thehidden Z2 symmetry of a Fermi liquid. For these rea-sons, we have focused on revealing its superconductingproperties1. While our previous work revealed T = 0properties of the HK model appended with a pairing termwith couplinng constant, g, none of the thermodynamicswere obtained. Nonetheless, several non-BCS propertieswere apparent: 1) limg→0

2∆Tc→∞ rather than the BCS

ratio of 3.52, 2) Composite quasiparticle excitations con-sisting of doublons and holons rather than the standardparticle-hole excitations of BCS, and 3) a suppression ofthe superfluid density relative to that of BCS.

Because the model we explore is highly amenable tocomputation, we can with certainty catalog how the fi-nite temperature properties derived from Mottness leadto substantial deviations from BCS theory. First, we es-tablish that the turn-on of the gap and the divergenceof the susceptibility occur at distinct temperatures. Theformer appears to obtain at a first-order transition whilethe latter tends to a global second-order transition ofthe superconducting state. We show that this effect van-ishes when U = 0, thereby making this a true conse-quence of Mottness. We make contact with earlier resultson multi-band superconductors that found a first-ordertransition22–25. We trace the first-order nature of thepairing transition to a singly-occupied holon band thatarises purely from Mott physics. Second, we computethe heat capacity jump at Tc as well as the condensa-tion energy. Unlike BCS theory, we find that Econd/∆is stronger than in BCS theory. Finally, we compute theultrasonic attenuation as well as T1 near the supercon-ducting transition. We are able to show that the Hebel-Slichter26 peak in BCS theory vanishes in the stronglycorrelated limit as seen widely in the cuprates27–29. Al-though the absence of this peak in the cuprates30,31 hasbeen attributed to spin fluctuations, we argue here thatit is just a consequence of Mottness, the splitting of thespectral weight over two correlated bands. Subsequentexperiments are discussed.

II. SUPERCONDUCTIVITY IN THE HKMODEL

Superconductivity in the cuprates necessitates a solu-tion to at least the Cooper instability in a doped Mottinsulator. We have shown1 previously that this can bedone exactly by solving Cooper instability equation thatarises from the HK analogue

H = HHK −Hp, Hp =g

V

∑k,k′

b†kbk′ (3)

of the pairing Hamiltonian for a doped Mott insulator.Here bk = c−k↓ck↑ is the s-wave pair creation operatorat zero total momentum. As is well known for Mottsystems, the single-particle Green function for HHK,

Gkσ(iωn) ≡ −∫ β

0

dτ 〈ckσ(τ)c†kσ(0)〉eiωnτ (4)

Gkσ(iωn → z) =1− 〈nkσ〉z − ξk

+〈nkσ〉

z − (ξk + U). (5)

exhibits a bifurcation of the spectral weight betweenlower (l) and upper (u) bands with weights that are deter-mined by the electron filling, 1−〈nkσ〉 and 〈nkσ〉, respec-tively. It is this bifurcation that leads to zeros of the realpart of the Green function. We have also shown1 thatnot only is the Cooper instability exactly solvable but so

3

is the exact pair susceptibility. The exact susceptibility1

χ(iνn) ≡ 1

V

∑k,k′

∫ β

0

dτ eiνnτ 〈Tbk(τ)b†k′〉g (6)

can be expressed, in the normal state, in terms of thebare susceptibility

χ(iνn) =χ0(iνn)

1− gχ0(iνn)(7)

which is given by

χ0(iνn) = χll0 + χuu0 + χlu0 + χul0 (8)

χab0 =1

V

∑k

nak↑nb−k↓

f(ωak) + f(ωb−k)− 1

iνn − ωak − ωb−k(9)

where for ωlk = ξk and ωuk = ξk + U , nukσ = 〈nkσ〉0 andnlkσ = 1 − nukσ, and f(ω) the Fermi function at temper-ature T , the superscripts ab may represent ll, uu, lu, orul.

A consequence of the expression of the pair suscepti-bility is that the divergence at χ0 = 1/g is expected tobe a second-order transition to the bulk superconductingstate. As we will see, this divergence is not coincidentwith the turn-on of the gap. To calculate the suscep-tiblity, we will work with the exact finite temperatureoccupancy,

〈nkσ〉 =1

2〈nk〉 =

e−βξk + e−β(2ξk+U)

1 + 2e−βξk + e−β(2ξk+U), (10)

so as to give the correct temperature dependence of χ(T )explicitly. In Fig. 1, we plot the zero-frequency baresusceptibility χ0(T ) as well as χ(T ). The divergence ofχ(T = Tc) would imply a diverging length scale and thusa possible second-order phase transition temperature Tc.Above Tc there is no extra sigularity in the susceptibility,which is the same case as in the BCS theory. To reiterate,the onset of the gap, a mean-field notion, is distinct fromthe divergence of the susceptibility.

A. Mean-field Theory of HK

Given that the susceptibility calculation is exact, wecan assert that the ground state of the HK model with apairing term is superconducting. To describe this state,we resort to a mean-field description in the spirit of BCS.In our previous work, we implemented the mean-fieldby an appropriately chosen pairing wave function. Togo beyond such a ground-state treatment, we explicitlydiagonalize H, obtaining all eigenstates and measuringobservables in grand canonical ensemble, including theirfull temperature dependence. The procedure we use hasbeen outlined in the Supplementary Materials of Ref. 1and by Zhu, et al.32. In constructing the mean-field,we take advantage that the HK Hamiltonian does not

0 0.0025 0.005 0.0075 0.01

1/g

T/W

χ0

↓ divergence of χ

χ=χ0/(1-gχ

0)

FIG. 1. The temperature dependence of χ0(left) and χ(right)for a 3D-HK model with superconducting pairing. The en-ergy dispersion was approximated by a parabola dispersion,and the chemical potential was set to half filling of the lowerHubbard band, µ = 0.5W where W is the bandwidth of thelower Hubbard band. We take W = 1 in the following calcula-tion. The temperature Tc at which χ = χ0/(1−gχ0) diverges(χ0 = 1/g) represents the true superconducting transitiononset.

mix the various k-states. In terms of the pair amplitude,∆ ≡ (g/V )

∑k bk, we formulate the mean-field,

Hp =g

V

∑kk′

b†kbk′

=V

g∆†∆

=V

g(δ∆† + ∆

∗)(δ∆ + ∆)

≈∑k

(∆b†k + ∆∗bk)− V

g|∆|2,

(11)

entirely on the pairing term where we have introducedthe average of the pairing amplitude to be ∆ = 〈∆〉. Inthe last step, mean-field amounts to ignoring the second-order fluctuation O(δ∆2) term. The mixing between dif-ferent momentum sectors averages out because of themomentum-diagonal structure of HK. Thus the mean-field(MF) HK Hamiltonian can be block diagonalized1,32

H ≈∑

k∈HFBZ

HMFk +

V

g|∆|2

HMFk =

∑s=±

ξk∑σ

c†skσcskσ + Uc†sk↑csk↑c†sk↓csk↓

−(

∆∗c−sk↓csk↑ + ∆c†sk↑c

†−sk↓

).

(12)

The summation over k is carried out inside half of the firstBrillouin zone(HFBZ), while the boundary terms withmomenta restricted to the edge of the first Brillouin zoneare neglected as they are suppressed by a factor of 1/Nand hence vanish in the thermodynamic limit.

4

subspace eigenvalue degeneracy basis

PB E4k ≡ 2ξk + U 2 |1100〉 , |0011〉

ST E5k ≡ 2ξk 3

|1010〉 , |0101〉 ,(|1001〉 − |0110〉)/

√2

Mixing

E1k ≡ E1

E2k ≡ E2

E3k ≡ E3

1

1

1

|0000〉 ,(|1001〉+ |0110〉)/

√2,

|1111〉

Odd×4E6

k ≡ E−E7

k ≡ E+

4

4

|1000〉 , |1110〉|0100〉 , |1101〉|0010〉 , |1011〉|0001〉 , |1011〉

TABLE I. Block decomposition and the energy levels of theHKSC mean-field hamiltonian

The decomposed Hamiltonian lives in a Fock space Fk

which contains 4 fermion currents, |nk↑, nk↓, n−k↑, n−k↓〉and spans a 16-dimensional space32. Due to the fermionparity conservation of HMF

k , this Fock space can be de-composed by parity into Fk = F oddk ⊕ F evenk . The evensector is further block diagonalized into 3 subspaces,F evenk = FPBk ⊕ FSTk ⊕ FMix

k , as shown in Table I :

1. The states in the Pauli Blocking (PB) states, pos-sess an energy level at E4

k = 2ξk + U with degen-eracy 2, which have definite total electron num-ber 2(not participating the superconducting pair-ing) and is blocked by the repulsion U .

2. The states in the Spin Triplet (ST) states, possessan energy level at E5

k = 2ξk with degeneracy 3,which also have two electrons(not participating thesuperconducting pairing).

3. The 3-dimensional particle number mixing (Mix-ing) states related by the off-diagonal supercon-ducting pairing.

The Hamiltonian matrix in the 3-dimensional mixingstates subspace is 0 −

√2∆ 0

−√

2∆ 2ξk −√

2∆

0 −√

2∆ 4ξk + 2U

(13)

with corresponding 3 energy levels (i = 1, 2, 3)

Ei = 2ξk +2U

3+

4√3Eeven

k cos

(θk +

2π

3i

)(14)

Eevenk =

√(ξk +

U

2

)2

+ ∆2

+U2

12(15)

θk =1

3arccos

[qk(√

3Eevenk

)3]

(16)

qk = U

(U2 +

9

2Uξk +

9

2ξ2k −

9

4∆

2). (17)

Similarly, the odd sector can be written into a direct-sumof 4 equivalent subspace, e.g. |1000〉 , |1110〉. Here theHamiltonian matrix in each 2-dimensional subspace is(

ξk −∆

−∆ 3ξk + U

)(18)

and the odd sectors share the 2 energy levels,

E± = 2ξk +U

2± Eodd

k (19)

Eoddk =

√(ξk +

U

2

)2

+ ∆2. (20)

The energy levels and bases in each subspace are shown inTable I. Thus we have all energy levels of the system. Theground state is recognized as the eigenvector corespond-ing to E1, which is a linear combination of the 3 occu-pancy states: |0000〉 from Ω0⊗Ω0, (|1001〉+ |0110〉)/

√2

from Ω1 ⊗ Ω1 and |1111〉 from Ω2 ⊗ Ω2. Our previousvariational treatment1 concurs with this result.

The work-horse of the mean-field treatment is the self-consistent equation for the pair amplitude:

∆ =g

V

∑k

〈ck↑c−k↓〉. (21)

The right-hand side has a dependence on ∆ through thethermal average over the energy levels enumerated in Ta-ble I. We can also treat the self-consistent equation for∆ as the solution to the extremum of the free energy,

F = −kBT lnZ, (22)

where

Z =∑

i,k∈HFBZ

e−βEik . (23)

Both procedures yield identical results and are tabulatedin Fig. 2. Displayed first is the Free energy as ∆ isvaried. Of first note is that for T > Tp, there is no non-zero solution for the gap. At T = Tp (p for pair), there

are two degenerate minima, with ∆=0 and ∆ = ∆p 6= 0.The technical definition of Tp then is ∂F/∂∆|∆=∆p

= 0

and F (∆p) = F (0). This degeneracy is lifted by loweringthe temperature such that T2 < T < Tp. The solution to∂2F/∂∆2|∆=0 = 0 defines T2 as the inflection point. Aswe will see, T2 will correspond to the divergence of thesusceptibility.

To corroborate this picture, we solve the self-consistentequations for the the gap. The dashed curve in Fig. 2(b)corresponds to BCS theory which has a unique turn-on temperature for the pair amplitude at the red dot,namely T = T2. In HK, however from our evaluationof the free energy, we see multiple solutions for the gapturn-on distinct from the feature at T2. As shown, the∆ − T curve for the mean-field HK exhibits a nontriv-ial back-folding behavior above T2. This multivaluednessconfirms that there are multiple choices of ∆ that make

5

T>Tp T=Tp

T2<T<Tp T=T2

(a)

0 0.1 0.2 0.3 0.4 0.5

0

Δ/Δ0

F

U=0(BCS)

U=1

T=T2 T2<T<Tp

T=Tp T>Tp(b)

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

1.0

T/T2

Δ/Δ0

FIG. 2. (a) The variation of the Free Energy F as a functionof the the pair amplitude ∆ for an HK superconductor atU/W = 1 and g = 0.3. The chemical potential is set athalf filling of the lower band (µ = 0.5). For T > Tp, theglobal minimum of free energy is ∆ = 0. At T = Tp, thefree energy at ∆ = 0 and ∆ = ∆p coincides, thus indicatinga first-order transition. When T2 < T < Tp, ∆ = 0 is stilla local minimum, while the global minimum obtains at finite∆. For T = T2, the only minimum occurs at finite ∆, and∆ = 0 becomes a local maximum(the inflection point). (b)The dashed purple line and the solid blue line are the solutionto the pair amplitude from the self consistent equation atdifferent temperatures(the ∆ − T plot) for BCS(U/W = 0)and HK(U/W = 1). The Green dot shall represent the first-order phase transition and the red dot represents the second-order phase transition. In both cases, the axes are scaledaccording to respective T2, and respective zero temperaturepair amplitude ∆0.

the first derivative of the free energy with respect to thegap vanish. Only one of these gaps yields the globalminimum of the free energy. The equivalence of the freeenergy at ∆ = 0 and ∆p implies a first order phase tran-sition at this temperature Tp(Green dot), which ensuresthe accuracy of mean-field treatment since the first ordertransition does not have any diverging fluctuations. Thedegenerate minimum of the free energy at T = Tp areshown in Fig. 2(a) with the green solid curve. There aretwo mechanisms to lift the degeneracy: 1) decrease the

temperaure as shown in Fig. (2a) or 2) decrease U fromthe value shown in Fig. (2b). What is not shown is thatthere is a critical value of U that is needed to destroy theback-folding in Fig. (2b). This critical value diminishesas the pairing strength, g, decreases. While it is sugges-tive at this point that it is Mott physics that leads to thefirst-order nature of the superconducting transition, wewill confirm this by a detailed evaluation of the Landauexpansion parameters.

The calculations in this section are all based on a mean-field theory, which is exact if the transition is truly firstorder. Since the exact pair susceptibility diverges at atemperature distinct from Tp, we need to consider twopossibilities:

1. The phase transition is first-order at Tp, and thecalculation of the susceptibility cannot be appliedto the theory below the transition temperature atwhich Tp(∆ jumps to a finite value instead of con-tinuously growing from zero when T = Tp). Themean-field calculation is exact in this scenario.

2. The phase transition is second-order at Tc. Theglobal minimum of the free energy at Tp shall beexcluded due to the divergence of fluctuations nearthe second-order phase transition, which destroysthe mean-field theory. Only the non-MF calcula-tion (e.g. susceptibility) is right.

In the first case, the calculation of Tc is internally in-consistent as it presumed that ∆p = 0 for T > Tc. Inthe second case, we take advantage of the exact diagonal-izability and use the Ginzburg criterion to estimate thetemperature range where the fluctuations diverge (mean-field theory breaks down).

B. Ginzburg Criterion

The unusual first-order nature of the phase transitionin an HK superconductor is manifest at the mean-fieldlevel. The crucial question is then: Is this first-orderTransition just an artifact? In other words, how valid isthe mean-field theory, or equivalently, what is its rangeof validity? To address this, we compute the Ginzburgcriterion, presuming that the phase transition is second-order at T = Tc.

Recall that the Landau-Ginzburg description of atraditional superconductor is equivalent to the mean-field treatment of the BCS theory around the transitionpoint33. We demostrate the same result for HK super-conductors. In an s-wave superconductor which is thecase considered here, the order parameter is given by

Ψ(x) ∝ 〈c↑(x)c↓(x)〉. (24)

In the absence of a magnetic field, the Landau free energyis

F [Ψ] =

∫ddx

(a|Ψ(x)|2 +

1

2b|Ψ(x)|4 +K|∇Ψ(x)|2

),

(25)

6

where a is directly related to the susceptibility and Kis the rigidity. Close to a second-order critical point Tc,that is |Ψ| 1, we may neglect the quartic terms. As aresult, the free energy,

F [Ψ,Ψ∗] =

∫ddx

(a|Ψ(x)|2 +K|∇Ψ(x)|2

)=∑k

(a+Kk2)|Ψ(k)|2

Ψ(x) =1√V

∑k

Ψ(k)eikx,

(26)

by performing the Fourier transform of the order param-eter. The Helmholtz free energy A(T ) is given by

A(T ) = −kBT lnZ

= −kBT ln

∫D[Ψ,Ψ∗]e−F [Ψ,Ψ∗]/kBT

= kBT∑k

ln

(kBT

a+Kk2

).

(27)

We write a(T ) = αt with t = T−TcTc

. The singular contri-

bution to the specific heat CV = −TA′′(T ) comes whendifferentiating with respect to T . The dimensionless heatcapacity per unit cell is

c =CVNskB

=α2ad

K2

∫ Λ ddk

(2π)d1

(ξ−2 + k2)2

=α2ad

K2ξ4−d

∫ Λξ ddq

(2π)d1

(1 + q2)2,

(28)

where Λ ∼ a−1 is an ultravilot cut-off and a is the lat-tice constant(set to unit in the following calculation),ξ = (K/a)1/2 = (K/α)1/2|t|−1/2. The fluctuation con-tribution is small when

α2ad

K2ξ4−d 1, (29)

which entails |t| tG where the latter is defined,

tG =(a2 α

K

) d4−d

, (30)

as the Ginzburg reduced temperature. This calculationthen just hinges on knowing α andK and we can estimatethe temperature region where mean-field theory is notvalid.

To obtain α, we calculate the spatial average of theorder parameter

Ψ ≡ g

V

∫ddx〈c↑(x)c↓(x)〉

=g

V

∑k

〈ck↑c−k↓〉

= ∆.

(31)

Thus, the Landau free energy parameters can be calcu-lated by differentiating the free energy density f with

respect to ∆2

a =∂f

∂∆2

∣∣∣∣∆=0

, (32)

and the parameter α can be calculated by linear fittingthe a(T ) around the transition temperature as shown inFig. 3. The key point is that α does not vary appreciablyfor the system parameters U/W and g.

g=0.1

g=0.2

g=0.3

g=0.4

0.0 0.1 0.2 0.3 0.4 0.5 0.61.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

U/Wα

FIG. 3. The dependence of the Landau expansion parameterα = a/t on the HK interaction strength U at different valuesof the superconducting pairing strength g. We claim that atU = 0, α ≈ 2.1 and at U > µ = 0.5, α ≈ 1.3 which arearound the same scale. The magnitude of α does not varyappreciably for all values of g.

We also have the superfluid stiffness defined by

T [θ] =

∫ddx

1

2ρs(∇θ)2, (33)

where θ is the phase of the superfluid Ψ(x) = Ψeiθ. Thusthe relation between K and ρs is

1

2ρs(∇θ)2 = K|∇Ψ|2 = KΨ

2(∇θ)2 (34)

K =ρs

2Ψ2 =

ρs

2∆2 . (35)

From Fig. 3 and Fig. 4, we can read that at µ = 0.5,U > µ, the value of α and ρs is α ≈ 1.3, K ≈ ρs/2∆2 ≈104. From Eq. (30), we may estimate tG ≈ 10−11. Con-sequently, the Ginzburg temperature is sufficiently smallto guarantee the validity of the mean-field calculation,which predicts a first order transition. Hence, the pre-sumption that the phase transition is second-order shallbe ruled out when Tp > T2. We will discuss the conse-quences of this first order transition in more depth.

C. Phase Diagram

To put this all together, we focus on the phase diagramin the T − U plane presented in Fig. 5 with a supercon-

7

T ≈ Tc

T ≈ 0

0 0.2 0.4 0.6 0.8 1

500

1000

5000

104

U/W

ρs/Δ2

FIG. 4. The rigidity at transition temperature(T ≈ Tc, solidline) and at zero temperature(T ≈ 0, dotted points). Thecalculation(see appendix) was performed with µ = 0.5, g =0.2. The value of K was calculated with ∆ = 10−5 and T =Tc(where the susceptibility diverges).

ducting paring strength g = 0.3. We observe that for asmall value of U/W < 0.13, the superconducting transi-tion is second-order (red line) in which the divergence ofthe pair susceptibility and the turn-on of the gap are co-incident. In general, the critical value of U/W , denotedby Ut/W , for the transition to be second order decreasesas g decreases. Hence, it is only the weakly interactingregime in which we find the traditional result that the su-peronducting transition is second order. As pointed outin our previous paper, in no regime (except U = 0) doesa BCS picture apply. For example, even in the weaklyinteracting regime, the transition temperature comparedwith BCS(U = 0) value increases as U increases imply-ing that the multiband nature of the HK model is atplay in driving the enhancement of superconductivity.The second-order transition line ends at the tricriticalpoint located at Ut/W = 0.13 and Tt/W = 0.0255. Itis at this point that the local and global minima merge.For larger values of U > Ut, the transition becomes afirst-order one(green dots). The phase transition tem-perature Tp saturates when U/W > 0.5 which coincidesthe elimination of the single-occupancy region Ω1 anddouble occupancy region Ω2 boundary in the HK model.

We can also compare T2 with Tc. As is evident, theyboth coincide as illustrated in Fig. 5. This is signifi-cant because Tc is computed from a divergence of thesusceptibility while T2 follows from a solution to a mean-field equation. For an HK superconductor with first ordertransition, the temperature T2 represents the eliminationof the metastable state(local minimum).

In all the computations thus far, either the chemicalpotential or the filling has remained fixed. To make con-tact with the cuprates, a phase diagram of Tc versus fill-ing or chemical potential is needed. This can be doneusing the same machinery. Shown in Fig. 6 is a plotof Tc versus filling for two distinct cases: 1) overlappinglower and upper Hubbard bands and 2) no overlap. This

←Tricritical PointTp

T2

Tc

0.0 0.2 0.4 0.6 0.8 1.00.000

0.005

0.010

0.015

0.020

0.025

0.030

U/W

T/W

FIG. 5. Phase disgram in the T − U plane of the HK super-conductor for coupling g = 0.3.The green line corresponds tothe first-order Transition temperature Tp at which the globalminimum of the free energy switches from ∆ = 0 to a fi-nite value. The red line corresponds to T2 at which ∆ = 0changes from a local minimum into a local maximum. Thegray line is the pair susceptibility diverging temperature Tc.The gray and red line coincide, thereby corroborating that thepair-susceptibility divergence is coincident with the inflectionpoint in the free energy.

can be engineered simply by changing the value of U .In general electron and hole doping yield the mirror re-sults. When the bands overlap, a metallic state alwaysensues, though a non-Fermi liquid one1 and Tc is mini-mized at half-filling though it does not vanish. A van-ishing of Tc at the Mott insulating state obtains only forU > W as is shown in the plots in which U = 1 andU = 2. The familiar dome-shaped phase diagram ob-tains as dictated by the dome-shaped superfluid stiffnessreported previously1. The dome shape here is a directconsequence of Mott physics.

U=0.5

U=1

U=2

0.0 0.2 0.4 0.6 0.8 1.00.000

0.001

0.002

0.003

0.004

<nσ>

Tc

FIG. 6. Phase diagram in the Tc − 〈nσ〉 plane of the HKsuperconductor for coupling g = 0.3. The blue line (U = 0.5)represents the overlapping bands or metallic phase in whichU < W . The pink and purple line(U = 1, 2) represent thenon-overlapping bands (Mott insulator ground state at halffilling, 〈nσ〉 = 0.5).

8

D. Landau Expansion Parameters

To complete a theory of a first order transiton, werewrite the Landau free energy

F [∆] = a∆2

+ b∆4

+ c∆6

+O(∆6). (36)

in terms of the 4th order expansion parameter b. Fora first-order transition, b < 0 and the 6th order termbe positive to keep the free energy bounded from below.When the first-order transition obtains, both the valueof the free energy and the first derivative with respect toδ vanish,

F [∆p] = a∆2

p + b∆4

p + c∆6

p = 0 (37)

F ′[∆p] = 2a∆p + 4b∆3

p + 6c∆5

p = 0. (38)

The solution is ∆2

p = −2a/b and b2 = 4ac, thus proving

that a > 0, b = −√

4ac < 0 is required for this kind offirst order transition to exist. Recall that for an HK su-perconductor, extreme factorizability in momentum sec-tions allows us to write the free energy as

F [∆] =∑

k∈HFBZ

Fk[∆]. (39)

As a result, the free-energy expansion parameters can bedecomposed as a =

∑k ak and b =

∑k bk. To organize

our results, we recall that what makes the HK model anon-Fermi liquid is that at zero-temperature, as a resultof U , singly occupied states, Ω1, exist below the chemicalpotential which never obtains in a Fermi liquid. This isdepicted in Fig. 7(a). The results for the momentum-resolved Landau expansion coefficients are shown in pan-els Fig. 7(b,c).

In the BCS case, the negative contribution of ak comesfrom the states around the fermi-surface, where the oc-cupancy changes from 2 to zero. This arises from thesharpness of the Fermi surface. In the HK superconduc-tor, similar sign changes at the occupancy boundariesfrom Ω2 to Ω1 and from Ω1 to Ω0. ak is always possi-tive except around the boundary of different occupancyregion. When the positive contribution exactly cancels,the negative contribution, a vanishes, implying that thefree energy changes from a concave function into a con-vex one around ∆ = 0. Most crucial here, however, isthe behaviour of bk. For the HK superconductors, whilethe distribution of ak follows the BCS case, the valueof bk differs drastically from BCS. In BCS, bk is alwayspositive; thus b > 0 is true for any temperature. In HK,however, inside the single occupancy region Ω1, whichis not present in any fermi liquid, bk becomes negative!Together with a surpression of the positive bk contribu-tion, it is possible that b < 0 and a first order transi-tion emerges. Consequently, it is Mottness that drivesthe first-order nature of the transition in the HK model.Perhaps this is true in general.

0.

0.2

0.4

0.6

0.8

1.

<n k

σ>

-4.

-3.

-2.

-1.

0.

1.

a k

-0.4 -0.2 0.0 0.2 0.4-3-2-1

01234

ξk

b k

U=0.3(HK)

-- U=0(BCS)

Ω0

Ω1

Ω2

ξk

(a)

(b)

(c)

FIG. 7. (a) The occupancy of the HK(U=0.3) and FermiLiquid(FL) at finite temperature T/W = 0.01. (b) The 2ndexpansion parameter ak of the Landau Free Energy over theenergy levels ξk. (c) The 4th expansion parameter bk of theLandau Free Energy over the energy levels ξk.

E. Condensation Energy

The deviation of an HK superconductor from that ofthe BCS type is also manifest from the condensation en-ergy, defined by the energy difference between the super-conducting and normal states,

Econd = ESC − EN . (40)

In a traditional s-wave BCS superconductor, the con-densation energy is well-known to be propotional to thesquare of the pair amplitude EBCScond = −N(0)∆2/2, whereN(0) is the density of states around the chemical poten-tial. In the HK superconductor, however, this relationholds up to a modification to the coefficient

EHKcond = CN(0)∆2/2, (41)

where C is a pure number. The dependence of C on pair-ing strength g and U is plotted in Fig. (8). Generally, the

9

FIG. 8. The dependence of the condensation energy Econd

devided by N(0)∆2 at zero temperature in the g − U plane.The chemical potential is set µ/W = 0.5. The bottom redbold line represents the BCS result C = −1. As is evident, HKsuperconductors have condensation energies that genericallydeviate from the BCS result.

coefficient C have a smaller value in HK superconductorsthan in BCS, demonstrates a stronger tendency to form asuperconductor. This result is surprising but consistentwith what we know about the cuprates in which the pair-ing temperatures are greatly enhanced despite the localrepulsions. In this case, enhancement arises because ofthe singly-occupied region. The minimum value C ≈ −2can be achieved at a specific point U smaller than thechemical potential and quickly goes back to the BCS ra-tio C = −1 as U > µ, where the double occupancy regionΩ2 goes away, and the boundary between Ω1 and Ω2 isremoved. This observation again emphasizes the signifi-cance of the single occupancy region boundary in the HKsuperconductor. It may imply a mechanism to enhancethe superconductivity by adding the occupancy bound-ary, where the Copper pair formation is enhanced.

F. Heat capacity

Even when the superconducting gap develops contin-uously at the BCS transition, the heat capacity C(T )undergoes a jump discontinuity ∆C with the univer-sal ratio ∆C/Cn(Tc) = 12/7ζ(3) ≈ 1.43, where Cn(Tc)is the heat capacity in the normal phase at T = Tc.The discontinuity obtains because the BCS Hamilto-nian is an effective one that varies with the temper-ature through the gap parameter ∆(T ). The same istrue of HK even with a first-order transition as can beseen from the simple argument. With the free energygiven by −T ln tr e−H/T = 〈H〉 − TS, any system with atemperature-independent Hamiltonian has a heat capac-

ity coefficient C/T ≡ ∂TS = β3〈(H − 〈H〉)2〉 that variescontinuously with the spectrum of H. However, becausethe gap is temperature dependent, the heat capacity

C/T = β3

[〈(H − 〈H〉)2〉 −∆

∂∆

∂T〈H〉

], (42)

has an explicit temperature derivative. This term in-troduces a jump discontinuity at T = Tc since ∆(T ) ∼√Tc − T (so ∆∂T∆ ∼ −1) on the superconducting side

of the transition whereas ∆(T ) = 0 (so ∆∂T∆ = 0) onthe normal side.

In the HKSC where the gap changes discontinuously,the heat capacity still undergoes such a jump discontinu-ity, although the size of the discontinuity ∆C no longerscales universally with Cn(Tc). Rather as we will see,it depends on the Mott scale U as well. When the gapchanges discontinuously with temperature, the entropyalso changes discontinuously so that the heat capacityC = T∂TS is singular at the transition. We show its be-havior in Fig. 9, omitting the singularity. As is evident,Motness, as tracked by increasing U , enhances the heatcapacity jump at Tp, the temperature of the first-ordertransition. Similar enhancements will be seen as well forthe ultrasonic attenuation and the spin-lattice relaxationrate.

U/W=0

U/W=0.2

U/W=0.4

U/W=0.6

U/W=0.8

U/W=1

0.97 0.98 0.99 1.00 1.01 1.020

2

4

6

8

10

TTp

CC

n(Tp)

FIG. 9. Heat capacity C near the phase transition at T = Tp,at pair coupling g/W = 0.53 and for representative values ofthe Mott coupling U . C is normalized by its value Cn(Tp) onthe normal side of the phase transition.

G. NMR

The key feature of BCS is computability with themean-field formalism. The same can be done here aswe know all the excited states and hence can calculateall of the experimental quantities delineated by BCS. Asan example, we compute the spin-lattice relaxation ratewhich in a BCS superconductor exhibits a peak30 belowTc. The spins of atomic nuclei relax by exchanging energywith their environment. As the spins in a superconductor

10

are in phase, there is an enhancement below Tc. The re-laxation rate 1/T1 of nuclei in an electronic environmentis related to the transverse dynamic spin susceptibility ofthe quasiparticles, by33

1

TT1= limω→0

2kB

γ2e h

4

∑q

|AH(q)|2 Imχs(q, ω)

ω(43)

where γe is the electron gyromagnetic ratio, AH(q) isthe hyperfine coupling of the contact interaction withelectron spins, and Imχs is the imaginary part of thespin susceptibility.

In a normal metal state, Imχs(ω)ω ∼ N(0)2. This leads

to a linear dependence of the nuclear relaxation rate ontemperature, referred to as Korringa relaxation law34

1

TT1∼ 2kB

γ2e h

4N(0)2∑q

|AH(q)|2 = constant. (44)

In a BCS superconductor, we need to take account of the

FIG. 10. The NMR relaxation rate 1/TT1(normalized by the1/TT1 value at the transition temperature (T = Tt) in normalstate) as a function of temperature at superconducting pairingstrength g = 0.53, with varying value of U . The temperatureis scaled by the gap-oppening temperature, Tt = T2(second-order transition) for U = 0, U = 0.2, and Tt = Tp(first ordertransition) for others. To regulate the divergence of the deltafunction when doing momentum summation, we introduceda small imaginary damping rate iε to the frequency whereε/W = 0.01.

strongly energy-dependent quasiparticle density of states

N(E) → N(0) |E|√E2−∆2

. If we further assume a point

contact interaction , A(q) = A is a constant with respectto q, the relaxation rate becomes

1

TT1∝∫ ∞

∆

dE(df

dE)

E2

E2 −∆2, (45)

which generates the Hebel-Slichter peak right below thetransition temperature Tc, which is shown in Fig. 10 asU = 0(blue curve).

By contrast, in the HKSC model, there is no fermionicquasipartical excitation of the ground state. As a result,we shall calculate the susceptibility χs(q, ω) from its ba-sic definition. The dynamical susceptibility in imaginarytime is

χsab(q, iνn) = 〈Ma(q)Mb(−q)〉

=

∫ β

0

〈Ma(q, τ)Mb(−q, 0)〉eiνnτ .(46)

For a spin-isotropic system, we have χsab(q) = δabχs(q).

Thus, we can calculate the z-axis response to a field ap-pllied along z: χs(q) = 〈Mz(q)Mz(−q)〉. Since

Mz(q) =∑k

(c†k−q,↑ck,↑ − c†k−q,↓ck,↓), (47)

we find that

χs(q, τ) = 〈Mz(q, τ)Mz(−q, 0)〉 = χr(q, τ)− χa(q, τ),(48)

where

χr(q, τ) =∑k,σ

〈c†k−q,σ(τ)ck,σ(τ)c†k,σ(0)ck−q,σ(0)〉

χa(q, τ) =∑k,σ

〈c†k−q,σ(τ)ck,σ(τ)c†−k+q,σ(0)c−k,σ(0)〉

(49)

are the regular and anomalous parts of the correlationfunction. The anomalous term is non-zero only if theground state no longer conserves the particle number asin the superconducting phase. The imaginary part ofspin susceptibility takes the form

Imχ(q, ν − i0+) =∑k,σ

∑i,j,i′,j′

(pjkp

i′

k−q − pikpj′

k−q

)[∣∣∣M ijk,σ

∣∣∣2 ∣∣∣M i′j′

k−q,σ

∣∣∣2 +M ijk,σM

ji−k,σM

i′j′

k−q,σMj′i′

−k+q,σ

]δ(ν − (ωjik − ω

j′i′

k−q)),

(50)

where (Eik,∣∣ψik⟩)i=1,...,16 is the eigensystem of Hk, pik =

e−βEik/Zk is a Boltzmann weight, ωijk = Eik − E

jk is an

excitation energy, and M ijkσ = 〈ψik|ckσ|ψ

jk〉. Thus we can

11

write the NMR relaxation rate as

1

TT1∝∑q

Im[χ(q, ν − iδ)]ν

∣∣∣∣ν→0

=∑q

∑k,σ

∑i,j,i′,j′

(pik + pjk)(pi′

k−q + pj′

k−q)

f ′(ωijk )M i,j,i′,j′

k,q,σ δ(ωjik − ωj′i′

k−q),

(51)

where M i,j,i′,j′

k,q,σ is the term inside the square braket from

Eq.(50) and f(x) = 1/(eβx + 1) is the Fermi-distributionfunction. The NMR relaxation rate of the HKSC modelcan be calculated numerically. Thus we use the ex-pression given above together with the self-consistentpair amplitude to obtain the temperature dependence of1/TT1. Fig. 10 showes the temperature dependence of1/TT1 at g = 0.53. For U < 0.4, we have a second-order superconducting transtion and as U increases, theHS peak shrinks and is completely absent when The firstorder transtion takes place at U ≥ 0.4. At sufficientlylow temperature, 1/TT1 has an exponential dependenceon temperature. Once again, we see that Mottnesss isthe culprit in leading to significant deviations from thestandard BCS theory. Previous work31 has attributedthe absence27–29 of the HS peak in the cuprates to spinfluctuations. In the current work, such fluctuations areabsent. Hence, we advocate that Mott physics alone, thetendency to have single occupancy below the chemicalpotential, is sufficient to kill the Hebel-Slichter peak.

III. ULTRASONIC ATTENUATION

Alongside the nuclear spin relaxation rate, anotherstandard observable that probes the superconducting gapis the attenuation rate α of ultrasonic phonons transmit-ted through the sample. The attenuation rate,

α(q) = Γ(q, ν = ωq), (52)

of phonons with momentum q is given by the on-shelldecay rate Γ(q, ν = ωq) of the phonon propagator, whereωq ∼ vphq is the phonon dispersion at small momenta q.Typical experiments are conducted with phonon frequen-cies on the order of f ∼ 107Hz35. Supposing a speed ofsound vph ∼ 104m/s in the sample, these phonons haveenergy Eph/kB = h2πf/kB ∼ 5 × 10−4K and momen-

tum q = 2πf/s ∼ 6× 10−7A−1

. Since Eph/kB Tc and

q kF ∼ A−1

, we focus on the limit

α ≡ α(q → 0). (53)

At weak electron-phonon coupling, the decay rate

Γ(q, ν) = Imχ(c)(q, ν − i0+) (54)

is given in turn by the electronic charge susceptibility

χ(c)(q, iνn) ≡∫ β

0

dτ eiνnτχ(c)(q, τ) (55)

evaluated at the bosonic Matsubara frequency νn =2πn/β. In imaginary time,

χ(c)(q, τ)

≡∑

k,k′,σ,σ′

〈c†k−q,σ(τ)ck,σ(τ)c†k′+q,σ′(0)ck′,σ′(0)〉0,c (56)

= χr(q, τ) + χa(q, τ), (57)

where χr and χa are the regular and anomalous corre-lation functions introduced in the previous section. Thesign between the two terms is reversed relative to the spinsusceptibility. This prevents any coherence peak in theultrasonic attenuation rate here, just as it does for BCSsuperconductors. At small momenta q, the attenuationrate takes the form

α(q) ∼ ωq∑k,σ

∑i,j,i′,j′

(pik + pjk)(pi′

k + pj′

k ) (58)

× f ′(ωjik )M i,j,i′,j′

k,0,σ δ(vphq − (ωjik − ω

j′i′

k−q)),

analogous to Eq. (51) for the spin relaxation rate. Itsconvergence in the limit q → 0 can be seen as followsfor the standard context36, i.e. in d = 3 dimensions andwith an isotropic quadratic dispersion ξk = k2/2m − µ.The sum is dominated by the excitation between theground state and the lowest-lying odd-parity level, forwhich (i, j) = (i′, j′) and the delta function resolves to

δ(vphq − q · ∇kωjik ). Pulling this back to a form that

can be formally integrated over k, i.e. δ(k − k0)/|· · ·|,then extracts a factor of 1/q, resulting in an overall q-dependence given by the product ωq/q ∼ vph.

For general parameters the attenuation rate α mustbe evaluated numerically, as shown in Fig. 11. In linewith the NMR relaxation rate calculation, we take anisotropic dispersion that is cut off at some magnitudeof the crystal momentum. Unlike the NMR calculation,however, the probe momentum q is taken asymptoticallyto zero instead of being summed over, so it is necessary inthis case to perform the integral directly in momentumspace. For the figure, we have used vph/(aW ) = 10−2

and q = 10−7π/a. The resulting attenuation rate αdecreases monotonically in the superconductor from thenormal-phase value αn(Tp), changing discontinuously atthe phase transition when the gap ∆ opens discontinu-ously. At low temperatures it decays exponentially ase−∆/T . This allows the gap to be extracted from the ul-trasonic attenuation rate in the HKSC. Once again, wesee that as the overall precipitous fall-off of the ultra-sonic attenuation rate is accented as the Mott parameterU increases. In principle, this trend is experimentallytestable.

IV. FINAL REMARKS

The primary difficulty in unlocking how superconduc-tivity arises in a doped Mott insulator is computation

12

TABLE II. Summary of the superconducting properties in the HK model and a Fermi liquid(FL). χ represents the pairsusceptibility, ∆ is the pairing gap, H. S. stands for the Hebel-Slichter26 peak, a key feature of BCS superconductors.

property FL Mottness (HK)

χ divergent at Tc Tc(= T2)

∆ opens at Tc Tp(> T2)

limg→0 2∆0/kBTc 3.52 ∞Econd/N(0)∆2 -1 [−2,−1]

Quasiparticles Bogoliubons PHYonsa

Ginzburg reduced tG ∼ 10−9 ∼ 10−11

NMR 1/TT1 H-S peak no H-S peak

Landau Expansion a = αt b, b > 0 a = αt, b < 0, c > 0

Ultrasonic attenuation, α ∼ e−∆/T below Tc ∼ e−∆/T below Tp < Tca linear combinations of Holon and Doublon1

b t = T−TcTc

and α > 0 for both cases

U/W=0

U/W=0.2

U/W=0.4

U/W=0.6

U/W=0.8

U/W=1

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

TTp

αα

n(Tp)

FIG. 11. Ultrasonic attenuation rate α in the superconduct-ing phase, at pair coupling g/W = 0.53 and for representativevalues of the Mott coupling U . α is normalized by its valueαn(Tp) on the normal side of the phase transition at T = Tp.The curves for U/W = 0.6 and 0.8 are difficult to see herebecause they are covered by the U/W = 1 curve.

with a controlled theory. The HK model enables suchan analysis as it represents a fixed point for quarticFermionic theories that break the Z2 symmetry of Fermiliquids. Since this includes the Hubbard model, it suf-fices to analyse the tractable HK model. An analogy withFermi liquids is relevant here. The relevant physics of aFermi liquid follows from the free quadratic theory of Eq.(2) as all short-range repulsions are irrelevant. That suchrepulsions are irrelevant follows from the simple fact thatFermi liquids are local in momentum space. Destructionof this state, except for pairing, requires an equally localinteraction in momentum space. The HK interaction isjust the most relevant interaction in momentum spacethat suffices as it maximally breaks the Z2 symmetry ofa Fermi liquid. Carrying out a pairing analysis from thisstarting point should reveal the key differences with howsuperconductivity obtains from doped Mott insulator asopposed to a Fermi liquid.

Table (II) catalogs the differences between supercon-ductivity from a FL with the Mott counterpart. As de-termined here, the first key dfference is the appearanceof two energy scales, the pairing temperature, Tp andthe temperature at which the susceptiblity diverges. Inthe absence of Mottness, only a single scale character-izes superconductivity. Nonetheless, in the HK modelwe still find that the mean-field theory is essentially ex-act as the Ginzburg reduced temperature is vanishinglysmall. A key prediction here is that Mottness makesthe underlying transition first order. This can be con-firmed by careful mesurements of the laten heat in thecuprates. Another key prediction here is that the Hebel-Slichter peak is killed by the strong correlations of theMott state. While it had been speculated that antifer-romagnetic correlations diminish the relaxation rate31,what we find here is that in a model that has Mottphysics but no antiferromagnetism, the HS peak doesnot survive. Experimentally, the best NMR data27–29 in-dicate that on the Cu or O sites, no Hebel-Slichter peakexists. Our work indicates that the suppression of theHS peak is due entirely the bifurcation of the spectruminto upper and lower Hubbard bands. Such a bifurca-tion prevents the coherence that is typically thought tobe the mechanism behind the HS peak. It is these strongcorrelations of the Mott state that lead to a deviationas well from the standard Bogoliubov quasiparticles andthe onset of the composite excitations, PYHons1, as thenew quasi-excitations above the superconducting groundstate. Such correlations also enhance the condensationenergy and lead to a divergence of limT→0 2∆0/Tc in theHK superconductor. As all of these trends are traceableto the strong correlations of the Mott state, we concludethat Table (II) should provide the blueprint for superd-conductivity in doped Mott insulators.

Acknowledgements PWP thanks DMR-1919143and DMR-2111379 for partial funding of this project.E.W.H. was supported by the Gordon and Betty MooreFoundation EPiQS Initiative through the grants GBMF4305 and GBMF 8691.

13

APPENDIX

A. stiffness

The superfluid stiffness is defined by

F [θ] = F [0] +

∫ddx

1

2ρs(∇θ)2, (59)

where F = − 1β logZ is the free energy and θ is the phase

of the superfluid. Consider applying the following twist

c†i → c†ieiφri,x . (60)

For charge-ne superconductivity, ∇θ = nφx and we cancalculate the stiffness by

ρs =1

n2

1

N

∂2F [φ]

∂φ2

∣∣∣∣φ=0

= − 1

n2

1

N

1

β

[∂2φZ

Z−(∂φZ

Z

)2]φ=0

.

(61)

Here we have set the lattice constant to 1, and N is thetotal number of unit cells. In order to derive the correctstiffness for an HK superconductor, we work with theHamiltonian

H =∑kσ

εkc†kσckσ +

∑k

Unk↑nk↓ +Hp, (62)

where Hp is the general superconducting pairing term

Hp = g∑q

∆†q∆q (63)

and ∆q =∑

k ck+q↑c−k↓ is the copper pair creation op-erator with momentum q. Under the twist we introducedin Eq.(60), the fourier transform of the fermion operatorbecomes

c†k =1√N

∑i

c†ieik·ri

→ 1√N

∑i

c†ieik·ri+iφri,x

= c†k+φex.

(64)

Thus the Hamiltonian under the twist

H[φ] =∑kσ

εkc†k+φexσ

ck+φexσ

+∑k

Unk↑nk↓ +Hp[φ](65)

Hp[φ] = g∑q

∆†q[φ]∆q[φ] (66)

∆q[φ] =∑k

ck+q+φex↑c−k+φex↓ = ∆q+2φex , (67)

where the HK term and the superconducting pairingterms do not change under the twist. Thus,

H[φ]−H[0] =∑kσ

(εk − εk−φex)c†kσckσ. (68)

We expand the Hamiltonian to second-order in φ to ob-tain

H[φ]−H[0] = −φJx + 12φ

2Tx (69)

Jx =∑

kσ(∂kxεk)c†kσckσ (70)

Tx =∑

kσ(∂2kxεk)c†kσckσ, (71)

with Jx the total current in the x direction and Tx is thekinetic energy arising from hopping in the x direction.The partition function and its derivatives are

Z[φ] = tr e−βH[φ]

= tr e−βH[0]Te∫ β0dτφJx(τ)− 1

2φ2Tx(τ)]

(72)

∂φZ[φ]

Z

∣∣∣∣φ=0

=1

Ztr e−βH[0]T

[∫ β

0

dτJx

]= β〈Jx〉 = 0

(73)

∂2φZ[φ]

Z

∣∣∣∣∣φ=0

=1

Ztr e−βH[0]T

(∫ β

0

dτJx

)2

−∫ β

0

dτTx

= β

(∫ β

0

dτ〈Jx(τ)Jx〉

)− β〈Tx〉.

(74)

Finally, the superfluid stiffness is

ρs =1

n2N

(〈Tx〉 −

∫ β

0

dτ〈Jx(τ)Jx〉

). (75)

The decrease of the superfluid stiffness in the HK model1

was calculated at T = 0. For finite temperature, espe-cially near Tc, we expect that the ratio ρs/∆

2 remainsfinite. Fig. 12 shows the temperature dependence of thesuperfluid stiffness for multiple value of U/W . Near zerotemperature the HK pairing reduce the stiffness by ap-proximately 2 times. The stiffness close to Tc, however,was increased drastically up to 10 times. The increasedstiffness guarantees that the Ginzburg reduced tempera-ture is small in the HK model, and thus proved the appli-cability of the mean-field theory of HK superconductingmodel.

14

U=0

U=0.2

U=0.5

U=0.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2

500

1000

5000

104

T/Tc

ρs/Δ2

FIG. 12. Finite temperature superfluid stiffness. For the sakeof illustrating the ginzburg criterion, we keep the multiplesolutions to the self-consistant gap equation even though onlyone of them represents the true global minimum of the freeenergy. The first order transition happens at Tp where thesolid lines change into dotted points for U = 0.2, 0.5, 0.8.

15

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