Modeling superconductivity in the background of a spin-vortex checkerboard

Anastasia V. Aristova,∗ † Vivek K. Bhartiya,† and Boris V. Fine‡

Skolkovo Institute of Science and Technology, Nobel Str. 3, 143026 Moscow, Russia(Dated: 6 August, 2019)

We introduce a microscopic model aimed at describing the behavior of fermionic excitations inthe background of a magnetic texture called “spin-vortex checkerboard”. This texture was proposedpreviously as a possible alternative to stripes to interpret the experimental phenomenology of spinand charge modulations in 1/8-doped lanthanum cuprates. The model involves two kinds of inter-acting fermionic excitations residing in spin-rich and spin-poor regions of the modulated structure.It is a generalization of another model developed earlier for the so-called “grid checkerboard”. Theprincipal terms of our model describe the decay of fermionic pairs belonging to spin-poor regions intosingle fermions occupying spin-rich regions and vice versa. These terms induce intricate fermioniccorrelations throughout the system but fall short of inducing superconductivity unless arbitrarilysmall hopping terms are added to the model Hamiltonian. We present the mean-field solution ofthe model, including, in particular, the temperature dependence of the energy gap. The latteris found to be in a good overall agreement with available experimental data for high-Tc cupratesuperconductors.

I. INTRODUCTION

Interplay between superconductivity and the onset ofelectronic spin and charge modulations in cuprate super-conductors remains one of the intriguing and unresolvedissues in the field of high-temperature superconductivity.Manifestations of electronic modulations are reported ina broad doping range for several families of cuprates —most noticeably around the doping level of 1/81–14.

For 1/8-doped lanthanum cuprates, the modulatedstructure is widely believed to exhibit one-dimensionalpattern often referred to as “stripes”1,2. Yet the principalaspects of the same experimental evidence are also con-sistent with the possibility of two-dimensional modula-tions called “checkerboards”15–22. The experiment-basedarguments discriminating between stripes and checker-boards in 1/8-doped lanthanum cuprates are, at present,rather indirect. At the same time, the issue cannot be re-solved on purely theoretical grounds, because it requiresaccuracy of the calculations of the ground-state energynot achievable by first-principles theories. A particu-larly focused effort to investigate the consequences of thecheckerboard scenario was made in Ref.19. That analy-sis was based on a particular kind of checkerboard called“grid”. Later, the grid checkerboard was shown to be in-consistent with the results of spin-polarized neutron scat-tering experiment of Ref.23. This experiment, however,did not rule out another version of a checkerboard repre-senting a two-dimensional arrangement of spin vortices20

shown in Fig.1(a). Somewhat similar noncollinear spintextures were also considered in Refs.24–29. Recentlyan analogous superstructure called ”spin-vortex crystal”was proposed to exist in iron-based superconductors30–33.

Spin-vortex checkerboard in the context of cuprateswas introduced in Ref.20 . Its various properties wereanalyzed in Refs.20–22,34. So far, however, this analy-sis has not touched the superconducting properties. InSection II of the present article, we introduce a micro-scopic model aimed at describing superconductivity in

the background of spin-vortex checkerboard. The modelis a generalization of another one proposed in Ref.19 forthe grid checkerboard. In Sections III and IV, we findthe mean-field solution of the generalized model in thesame way as done in Ref.19. However, our subsequentanalysis in Section V differs from that of Ref.19 in onesignificant respect, namely, we find that both the origi-nal model and its generalization made in Section II donot induce non-zero superfluid stiffness. (In this regard,we correct an error made in Ref.19.) We further showin Section V B that non-zero superfluid stiffness emergesonce arbitrarily small hopping terms are included in themodeling.

Experimentally, the onset of static spin modulationsin 1/8-doped lanthanum cuprates — spin vortices orstripes — largely suppresses three-dimensional supercon-ductivity but appears to coexist with two-dimensionalsuperconductivity35–38. In Section VI, we show that, sim-ilar to the stripe scenario, the suppression of the three di-mensional superconductivity in spin-vortex scenario canbe explained by the displacement of the modulation pat-terns in the adjacent CuO2 planes.

As further argued in Section VI, the generic settingin cuprates beyond 1/8-doped lanthanum family, pos-sibly, involves the fluctuating counterpart of the staticspin-vortex texture used in the present work. These fluc-tuations are likely caused by system’s proximity to thethreshold of electronic phase separation39. They are ex-pected to couple spin, charge and lattice degrees of free-dom — see e.g.40. In this respect, our model illustratesthe potential of the general two-component scenarios41,42

in the limit of initially localized components for describ-ing the superconductivity in cuprates. In such scenarios,the first component represents unpaired fermions, whilethe second component represents preformed fermionicpairs. The spin-vortex checkerboard just suggests us thecoupling connectivity between the low-energy fermionicstates. We assume that in other cuprate families, theinhomogeneous patterns have a two-dimensional charac-

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ter, checkerboard-like or more disordered, where regionsof stronger and weaker antiferromagnetic correlations al-ternate with other, but the size of these regions may bedifferent for different dopings, and the doping dependenceof that size may be different for different cuprate fami-lies. This would be consistent with rather diverse phe-nomenology summarized in Ref.13. The dimensionalityof charge modulations was, in particular, discussed forthe yttrium family of curates in Refs.43–45.

Finally, in Section VII, we compare the temperatureevolution of the energy gap obtained from the mean-field solution of our model with experiments in bismuthcuprates. Several important but technical calculationsbehind the reported results are placed in the Appendices.

II. MICROSCOPIC MODEL: PRINCIPALTERMS

The model to be considered has two different kindsof fermionic states physically located in magnetic andnon-magnetic parts of the underlying spin texture. Thegeneral reasoning for constructing the model is the sameas in Ref.19. Namely, the entire texture is divided intoplaquets having different kinds of spin background, andthen, for each plaquet, only one-particle fermionic statesclosest to the chemical potential are retained. Given thatplaquets are rather small, it can be estimated19 that thespacing of one-particle energies within each plaquet is ofthe order of 40 meV, which implies that, for temperaturesmuch smaller than 400K, it is appropriate to retain onlythe levels closest to the chemical potential.

As shown in Fig.1(b), spin vortex checkerboard can berepresented as a square superlattice of 8d×8d, where d isthe period of the underlying square lattice of Cu atoms.We denote the total number of such unit cells in the sys-tem as N . Each unit cell in Fig.1(b) is further dividedinto four spin-polarized plaquets and four spin unpolar-ized plaquets. We expect that the lowest one-particlestates in spin-polarized plaquets are non-spin-degenerate,and hence we include exactly one state per plaquet. Werefer to two of the resulting states as “a-states” and tothe remaining two as “c-states”. Two different kinds ofa-states are distinguished by index η = ±1, and c-states— by index ζ = ±1. Two a-states or two c-states withdifferent values of η or ζ respectively are expected to haveorthogonal spin wave functions that can be obtained fromeach other by spin inversion. The lowest-energy states ofspin-unpolarized plaquets around the cores of spin vor-tices are assumed to be spin-degenerate. We, therefore,place two fermionic states on each such plaquet with spins“up” or “down” along any chosen direction. We call them“b-states”. Since the spin texture contains four nonequiv-alent kinds of spin-vortex cores, we distinguish the cor-responding b-states by index α = 1, 2, 3, 4 and by spinindex ↑ or ↓ — see Fig. 2.

We now construct a low-energy Hamiltonian similarto that of Ref.19. We expect that charge carriers in

(a)

(b)

FIG. 1. (Color online). (a) Spin-vortex checkerboard. Eacharrow represents average spin polarization on a square latticeformed by Cu atoms within CuO2 planes. (b) Partition of thespin-vortex checkerboard into plaquets associated with a-, b-,and c-states. Thick lines indicate the borders of unit cells ofthe modulated structure. Each unit cell (also shown in Fig.2)includes two a-states, two c-states and four b-states.

3

FIG. 2. (Color online). Unit cell from Fig. 1(b) with labels α,η and ζ as introduced in the text. Colors represent differentquasiparticle states as follows: b-states with α = 1 (pink),α = 2 (yellow), α = 3 (green), and α = 4 (gray); even a-state[η = 1] (orange), odd a-state [η = −1] (brown); even c-state[ζ = 1] (blue), odd c-state [ζ = −1] (cyan).

the background of spin-vortex checkerboard are heavilydressed by antiferromagnetic fluctuations, which shouldlead to relatively large effective masses. Charge carriersare also supposed to experience strong Coulomb repul-sion, which should further prevent them from hopping.Therefore, in the zeroth order, we neglect single-fermionhopping between checkerboard plaquets. (Hopping willbe treated in Section V B as a small correction.) Asfar as the interaction terms are concerned, we use thesame ”heavy-mass/Coulomb-repulsion” argument to in-clude in the model only those terms that do not changethe center-of-mass/center-of-charge positions of the twofermions involved. We also neglect a possible ”on-site”coupling of two b-states on the same plaquet — in ourpreliminary analysis, this term would not change the es-sential features of the solution, but we still plan to inves-tigate it elsewhere. The above assumptions amount toa relatively crude overall approximation, which shouldnevertheless allow us to capture the principal behaviorof the variational solution and, at the same time, avoidintroducing too many adjustable parameters.

We are, finally, left with terms representing on-site en-ergies εa, εb and εc (with εa = εc) and with the followingtwo-kinds of effective interaction terms, namely: two a-states or two c-states adjacent to a given spin vortex coremaking transitions to the two b-states inside the core or

vice versa. The resulting Hamiltonian is:

H =∑i,η

εaa†iηaiη +

∑i,ζ

εcc†iζciζ +

∑i,α,η

εbb†iαηbiαη

+ g∑i,α

[(b†iα↑b

†iα↓aje[i,α]

ako[i,α]+ h.c

)+(b†iα↑b

†iα↓cme[i,α]

cno[i,α]+ h.c

)], (1)

where g is the interaction constant, εa, εb and εc areon-site energies defined with respect to the chemical po-tential µ, which we set equal to zero, index i labels unitcells depicted in Fig 2, and indices η, α and ζ label theplaquets within the unit cell as illustrated in Fig 2. Fol-lowing Ref.19, whenever the specific value of subscriptsη or ζ is fixed, as is the case in the interaction termof Hamiltonian (1), we use subscript “e” for η = 1 orζ = 1 referring to the corresponding plaquets as “even”and subscript “o” for η = −1 or ζ = −1 referring to thecorresponding plaquets as “odd”. Double-subscripts no-tations such as aje[i,α]

imply that a-states labeled as {je}must be adjacent to the b-states labeled as {i, α}.

If all terms containing c-states are removed fromHamiltonian (1), the result would be exactly equivalentto the Hamiltonian considered in Ref19. Since c-statesdo not directly couple to a-states, and since c-stateshave the same connectivity with the b-states as a-states(but shifted), the mean-field solutions of the two modelsare very similar with the only difference being that b-states now experience mean field from both a-states andc-states, which, in turn, makes that mean field two timeslarger, and, as a result, the value of the superconductingtransition temperature becomes modified.

Since the entire mean-field solution has nearly the samestructure and logic as that of Ref.19, below we only in-clude the formal structure of the derivation and the re-sults, leaving the justification mostly to Ref.19.

As explained later in Section V B, the system describedby Hamiltonian (1) will not be superconducting, becauseit will have zero superfluid stiffness. In order to makeit superconducting, arbitrarily small hopping terms willneed to be added. However, the model based on Hamil-tonian (1) is easier to handle, and its solution alreadycaptures the important aspects of the resulting super-conducting phase, such as the energy gap.

III. BOGOLIUBOV TRANSFORMATIONS

In the model considered, each of the fermionic statescouples to relatively few other states, which makes amean-field solution rather approximate. We, neverthe-less, assume that it gives at least the right qualitativepicture of model’s behavior. The first step of this so-lution is to introduce the Bogoliubov transformation for

4

b-states within the same plaquet:

biα↑ = sBiα↑ + weiϕαB†iα↓,

biα↓ = sBiα↓ − weiϕαB†iα↑,(2)

where s and w are positive real numbers satisfying a con-straint arising from canonical fermionic anticommutationrelations

s2 + w2 = 1, (3)

and ϕα are the transformation phases, which are to bedetermined later by minimizing system’s energy.

Substituting Bogoliubov transformation for b-states in(1) and keeping only the thermal averages of terms thatdo not change the occupations of Bogoliubov B-states,we obtain partially averaged Hamiltonian,

Ha = 8εbN[s2nB + w2(1− nB)

]+ εa

∑i,η

a†iηaiη + εc∑i,ζ

c†iζciζ

+ gsw(1− 2nB)

∑α

[(e−iϕα

∑ij

aje[i,α]ako[i,α]

+ h.c)

+ (e−iϕα∑ij

cme[i,α]cno[i,α]

+ h.c)

], (4)

where εB is the energy of B-quasiparticles and

nB =1

exp εBT + 1

, (5)

is their occupation number. We set kB = 1.As explained in Ref.19, in order to assure proper

fermionic anticommutation relations for the Bogoliubovcounterparts of a- and c-states, the Bogoliubov transfor-mation for these states should be made in the quasimo-mentum space. Therefore, we need to rewrite the Hamil-tonian in terms of the real-space Fourier transforms fora- and c-operators. To do the Fourier transforms, we firstchange the notations from aiη, ciη to the notations a(r),c(r), where r indicates the position of the center of therespective plaquet. We also need to define the followingvectors — all in units of underlying crystal lattice periodd:

L = (0, 4), (6)

which connects an even a-state with an adjacent evenc-state, and

R1 = (4, 4),

R2 = (−4, 4),

R3 = (−4,−4),

R4 = (4,−4),

(7)

which connect an even a-state with four adjacent odd a-states. The subscript α in Rα is chosen such that αth

b-states are located between the pairs a-states connectedby vector Rα originated from an even a-state. Now, wedefine the position of each unit cell by the position re ofan even a-state within this cell. Therefore, even a-statesare located at a set of positions {re}, odd a-states at{re +R1}, even c-states at {re +L}, and odd c-states at{re + L + R1}. Finally, we rewrite the Hamiltonian (4)as follows,

Ha = 8εbN[s2nB + w2(1− nB)

]+∑re

{εaa†(re)a(re) + εaa

†(re + R1)a(re + R1)

+εcc†(re + L)c(re + L)

+εcc†(re + L + R1)c(re + L + R1)

}+gsw(1−2nB)

∑α

[(e−iϕα

∑re

a(re)a(re + Rα) + h.c

)

+

(e−iϕα

∑re

c(re + L)c(re + L + R5−α) + h.c

)], (8)

We now explicitly write separate Fourier transforms foreven and odd a- and c-states as follows:

ae(k) =

√1

N

∑re

a(re)e−ikre , (9)

ao(k) =

√1

N

∑re

a(re + R1)e−ik(re+R1), (10)

ce(k) =

√1

N

∑re

c(re + L)e−ik(re+L), (11)

co(k) =

√1

N

∑re

c(re + L + R1)e−ik(re+L+R1). (12)

Since the superlattice periods for each of the above fourkinds of states are the same, the sets of wave vectors{k} are also the same, even though the correspondingstates are shifted with respect to each other in real space.Substituting these transformations to (8), we obtain

H = 8εbN[s2nB + w2(1− nB)

]+ εa

∑k

a†e(k)ae(k)+

εa∑k

a†o(k)ao(k) + εc∑k

c†e(k)ce(k) + εc∑k

c†o(k)co(k)

+ gsw(1− 2nB)∑k

[(ae(k)ao(−k)V (k) + h.c)

+ (ce(k)co(−k)V (k) + h.c)

], (13)

5

where

V (k) =∑α

exp−iϕα−ikRα , (14)

= 2 exp

[−iϕ1 + ϕ3

2

]cos

[kR1 +

ϕ1 − ϕ3

2

]+2 exp

[−iϕ2 + ϕ4

2

]cos

[kR2 +

ϕ2 − ϕ4

2

],

and

V (k) =∑α

exp−iϕα−ikRα−1 (15)

= 2 exp

[−iϕ1 + ϕ3

2

]cos

[kR2 +

ϕ3 − ϕ1

2

]+2 exp

[−iϕ2 + ϕ4

2

]cos

[kR1 +

ϕ4 − ϕ2

2

].

Bogoliubov transformations for a- and c-states can nowbe defined as,

ae(k) = u(k)Ae(k) + v(k)eiφa(k)A†o(−k), (16)

ao(−k) = u(k)Ao(−k)− v(k)eiφa(k)A†e(k), (17)

ce(k) = p(k)Ce(k) + q(k)eiφc(k)C†o(−k), (18)

co(−k) = p(k)Co(−k)− q(k)eiφc(k)C†e (k), (19)

where u(k), v(k) and p(k), q(k) are the real-valued co-efficients for a- and c-states respectively, subjected to aconstraint arising from the fermionic canonical commu-tation relations for A- and C-operators:

u(k)2 + v(k)2 = 1, (20)

p(k)2 + q(k)2 = 1, (21)

and φa and φc are complex phases — all to be found bythe energy minimization.

We now complete the following steps: (i) substitutingthe above canonical transformation for a- and c-states(16-19) into Hamiltonian (13), then (ii) obtaining the en-ergy of the system by summing over the thermal averagesof the diagonal terms — the result is given in Appendix Aby Eq.(A1)— and then (iii) minimizing the resulting en-ergy with respect to the choice of phases φa(k) and φc(k).As explained in Appendix A, these steps lead to condi-tions:

cos[φV (k) + φa(k)] = 1, (22)

cos[φV (k) + φc(k)] = 1, (23)

were φV (k) and φV (k) are the complex phases of V (k)

and V (k) respectively, which, in turn, depend on phases{ϕα}. [Phases φa(k) and φc(k) do not need to be ob-tained explicitly, because they will not enter any quan-tity computed in this paper.] With the above conditions,

the expression for the energy of the system becomes:

E = 8εbN[s2nB + w2(1− nB)

]+ 2εa

∑k

{u2(k)nA(k) + v2(k)[1− nA(k)]

}+ 2εc

∑k

{p2(k)nC(k) + q2(k)[1− nC(k)]

}+ 2gsw (1− 2nB)

∑k

[u(k)v(k)(1− 2nA(k))|V (k)|

+ p(k)q(k)(1− 2nC(k))]|V (k)|], (24)

where,

nA(k) =1

exp εA(k)T + 1

, (25)

nC(k) =1

exp εC(k)T + 1

, (26)

are the Bogoliubov quasiparticle occupation number andεA(k), εC(k) are their energies obtained in the next sec-tion.

IV. SINGLE-PARTICLE EXCITATIONS,ENERGY GAP AND THE CRITICAL

TEMPERATURE

As argued in Ref19, the chemical potential of the sys-tem is, likely, to coincide with either εb or εa (same asεc), which, given our convention µ = 0, means that eitherεb = 0 or εa = εc = 0. Below, we treat these two casesseparately, referring to them as “Case I” and “Case II”respectively, and also refer to the case of εa = εb = εc = 0as “critical”.

The coefficients of the Bogoliubov transformations forboth Cases I and II are obtained in Appendices B andC, respectively, by minimizing the total energy (24) atfixed quasiparticle occupation numbers46. We then sub-stitute those coefficients back to Eq (24) and obtain theenergy of a Bogoliubov quasiparticle by taking derivativeof the total energy (24) with respect to the quasiparticleoccupation numbers nA(k), nC(k) or nB .

A. Case I: εb = 0

In Case I, the above procedure gives the followingquasiparticle energies:

εA(k) =

√ε2a +

1

4g2(1− 2nB)2|V (k)|2, (27)

εC(k) =

√ε2c +

1

4g2(1− 2nB)2|V (k)|2, (28)

6

εB =g2

16N(1− 2nB)

∑k

[(1− 2nA(k))

εA(k)|V (k)|2

+(1− 2nC(k))

εC(k)|V (k)|2

]. (29)

The mean-field approach now requires finding a non-trivial solution for εA(k), εC(k), and εB from Eqs.(5, 25,26,27, 28, 29). In general, it can only be done numeri-cally, but one can also obtain a closed analytical equationfor the critical temperature Tc using the fact that, nearthe transition, the ordered state is close to the normalstates, which allows one to use the limits

εA(k)→ εa,

εC(k)→ εc,

(1− 2nB)→ εB2Tc

.

This gives

Tc =g2

8

[(exp(|εa|/Tc)− 1

exp(|εa|/Tc) + 1

)1

|εa|

+

(exp(|εc|/Tc)− 1

exp(|εc|/Tc) + 1

)1

|εc|

],

(30)

from which the mean-field Tc can be obtained numeri-cally.

As explained in Ref.19, the energy densities of A- andC-states described by Eqs. (27) and (28) have Van Hovesingularities located in the both cases at the value

∆ =√ε2a + g2(1− 2nB)2, (31)

corresponding to |V (k)| = |V (k)| = 2. We refer to ∆ asthe “energy gap”. As T → Tc, ∆ approaches not zerobut |εa|, which we associate with the pseudogap.

B. Case II: εa = εc = 0

Following the same procedure as for Case I, we obtain:

εB =

√ε2b +

g2

64C2, (32)

εA(k) =g2(1− 2nB)C|V (k)|

16εB, (33)

εC(k) =g2(1− 2nB)C|V (k)|

16εB, (34)

where

C = Ca + Cc, (35)

Ca =1

N

∑k

(1− 2nA(k))|V (k)|, (36)

Cc =1

N

∑k

(1− 2nC(k))|V (k)|. (37)

0 0.2 0.4 0.6 0.8 10

2

2 2

4

6

8

T /Tc

Δ/Tc

FIG. 3. (Color online). Family of theoretical curves for tem-perature dependence of the superconductong gap for differ-ent ratios ∆(0)/Tc. Thick red line corresponds to the criticalratio ∆(0)/Tc = 2

√2. Solid lines above the thick line repre-

sent Case I and below the thick line Case II. The dashed lineshows the standard result of the Bardeen-Cooper-Schrieffertheory47.

Detailed calculations can be found in Appendix C. In thegrid model19, the c−states are absent, hence, Cc = 0, andthe value of Ca is reported to be 0.958.

The same approach as in Case I now gives the criticaltemperature

Tc =g2

4|εb|

(exp(|εb|/Tc)− 1

exp(|εb|/Tc) + 1

), (38)

and the gap parameter

∆ =g2(1− 2nB)C

8εB. (39)

associated with the Van Hove singularity for A- and B-states located at |V (k)| = |V (k)| = 2.

C. Temperature evolution of the energy gap

In Fig. 3, we present temperature dependences of en-ergy gaps for Cases I and II given by Eqs.(31) and (39)respectively. These dependencies were obtained by thenumerical solution of the system of equations Eqs.(5, 25,26,27, 28, 29) for Case I, or Eqs.(5, 25, 26, 32, 33, 34) forCase II.

The families of plots for Cases I and II are connectedthrough the critical case εa = εb = εc = 0, which is repre-sented by the thick red line. This case corresponds to theratio ∆(0)/Tc = 2

√2 ≈ 2.82. Plots above the critical-

case line correspond to Case I: at T = Tc, they all endat nonzero values ∆(Tc) = εa. Plots below the critical-case line correspond to Case II: they all have ∆(Tc) = 0,

7

and, moreover approach closely the canonical BCS de-pendence for εb/g →∞.

Thus, if the assumptions of the present model are valid,the critical-case ratio ∆(0)/Tc = 2

√2 signifies the tran-

sition from the conventional behavior ∆(Tc) = 0 for

∆(0)/Tc < 2√

2 to unconventional behavior ∆(Tc) 6= 0

for ∆(0)/Tc > 2√

2. The value of ∆(0)/Tc = 2√

2 forthe critical case makes important quantitative differencefrom the critical case result ∆(0)/Tc = 4 for the grid-based model of Refs.19,48, which involved only a- and b-states. Such a difference was to be expected, because thecoupling between b- and c-states in the present modelleads to additional energy advantage for the supercon-ducting state and hence higher superconducting transi-tion temperature for the same value of the coupling con-stant g.

V. SUPERCONDUCTING PROPERTIES

A. Anomalous correlation functions

Bogoliubov transformations (2,16-19) can be used toobtain the following anomalous correlation functions forT < Tc:

Ψa(k) = 〈ae(k)ao(−k)〉= u(k)v(k)eφa(k)[2nA(k)− 1], (40)

Ψc(k) = 〈ce(k)co(−k)〉= p(k)q(k)eφc(k)[2nC(k)− 1], (41)

Ψb(riα, rjβ) ≡ 〈biα,−bjα′,+〉= sweiϕiα(1− 2nB)δ(riα − rjβ), (42)

where riα is the position of αth b-state in the ith unitcell,and δ(riα− rjβ) is defined as Kronecker delta on the dis-crete superlattice. Two different components of the SCorder parameter corresponding to a, c, and b-states arethe correlation functions (40), (41), and (42) respectively.

The anomalous averages for a and c-components, inreal space, can be written as following,

Ψa(re[i,α], ro[j,β]) ≡ 〈a(re[i,α]

)a(ro[j,β])〉, (43)

Ψc(r′e[i,α]

, r′o[j,β]) ≡ 〈c(r′e[i,α]

)c(r′o[j,β])〉, (44)

where re[i,α]and ro[j,β] are, respectively, the positions of

an even (η = 1) and odd (η = −1) a-states adjacentto the b-states labelled by indices [i, α] and [j, β], and,likewise, r′e[i,α]

and r′o[j,β] are, respectively, the positions

of an even (ζ = 1) and odd (ζ = −1) c-states adjacentto the same b-states. The anomalous averages given byEqs.(43) and (44) have non-zero values only when theirtwo arguments correspond to states of different kind (i.e.even and odd). They can be expressed as

Ψa(re, ro) =2

N

∑k

Ψa(k)eik(re−ro), (45)

Ψc(r′e, r′o) =

2

N

∑k

Ψc(k)eik(r′e−r′o), (46)

where Ψa(k) and Ψc(k) are given by Eq.(40) and (41).These anomalous averages also obey the following rela-tions:

Ψa(re, ro) = −Ψa(ro, re), (47)

Ψc(r′e, r′o) = −Ψc(r

′o, r′e), (48)

Ψa(re, re) = Ψa(ro, ro) = 0, (49)

Ψc(r′e, r′e) = Ψc(r

′o, r′o) = 0. (50)

Relations (47) and (48) follow from the fermionic anti-commutation rule. Variables re, ro, r′e and r′o in Eqs. (49)and (46)represent different positions of the same kind asre, ro, r′e and r′o respectively.

The coherence length of the order parameters Ψa(re,ro) and Ψc(re, ro) should be inversely proportional to

the characteristic k−space scale of V (k) and V (k), re-spectively. The examination of Eqs.(14) and (15) revealsthat this characteristic scale is π/l, where l = 8d . There-fore, the coherence length associated with Ψa(re, ro) andΨc(r

′e, r′o) can be estimated as the modulation period l.

The coherence length associated with Ψb is equal tozero, which means that only b-states located on the sameplaquet form coherent pairs.

We now obtain three quantities that characterize theshort-range pair correlations:

Ψa(i,α) ≡ Ψa(re[i,α], ro[i,α]

) ≡ 〈ae[i,α]ao[i,α]

〉, (51)

Ψc(i,α) ≡ Ψc(r′e[i,α]

, r′o[i,α]) ≡ 〈ce[i,α]

co[i,α]〉, (52)

Ψb(iα) ≡ Ψb(riα, riα) ≡ 〈biα,↓biα,↑〉. (53)

In Case I, the explicit expression for Ψb(iα) can be ob-tained by substituting the values of s and w given byEqs.(C3) into Eq.(42) for riα = rjβ , which gives

Ψb(iα) =1

2eiϕiα(2nB − 1). (54)

One can then obtain both Ψa(i,α) and Ψc(i,α) by makinguse of the fact that

Ψ∗b(iα)Ψa(i,α) =Eaint4gN

(55)

Ψ∗b(iα)Ψc(i,α) =Ecint4gN

(56)

where Eaint and Ecint are the interaction parts of the en-ergy (24):

Eaint = 2gsw(1− 2nB)

×∑k

u(k)v(k)(1− 2nA(k))|V (k)|, (57)

Ecint = 2gsw(1− 2nB)

×∑k

p(k)q(k)(1− 2nC(k))]|V (k)|. (58)

8

After Eaint and Ecint are evaluated with the help ofEqs.(B1-B4), one can use Eqs.(54,55,56) to obtain:

Ψa(i,α) =g(1− 2nB)eiϕiα

8N

×∑k

[1− 2nA(k)]|V (k)|2

εA(k), (59)

Ψc(i,α) =g(1− 2nB)eiϕiα

8N

×∑k

[1− 2nC(k)]|V (k)|2

εC(k). (60)

In Case II, the expressions analogous to (59,60,54) are

Ψa(i,α) =1

4eiϕiα , (61)

Ψc(i,α) =1

4eiϕiα , (62)

Ψb(iα) = −gCeiϕiα(1− 2nB)

8εB. (63)

B. Superfluid phase stiffness and emergence ofsuperconductivity

The model introduced in Section II is defined in termsof localized fermionic states with a Hamiltonian that doesnot contain hopping terms or interaction terms chang-ing the center of mass of the particles. As a result, themean-field solution obtained in the preceding sectionshas zero superfluid stiffness and hence cannot sustainsuperconductivity49. In order to illustrate this, let usconsider the following phase transformation of fermionicoperators:

a(r) = a(r)e−iθ(r) a†(r) = a†(r)eiθ(r) (64)

bσ(r) = bσ(r)e−iθ(r) b†σ(r) = b†σ(r)eiθ(r) (65)

c(r) = c(r)e−iθ(r) c†(r) = c†(r)eiθ(r), (66)

where θ(r) is the position-dependent quantum phase,subscript σ represents the spins of the b-states. All otherformer subscripts “i”,“e”,“o”, and “α” are unambigu-ously determined by the position r. Technically, the ab-sence of the superfluid stiffness originates from the factthat the Hamiltonian of our model is invariant undertransformation (64-66) with

θ(r) = G · r, (67)

where G is an arbitrary vector. Therefore, there existsa continuous set of mean-field solutions of equal energy,which can be obtained from each other by applying theabove set of transformations.

Despite having zero superfluid stiffness, the mean-fieldsolution of the model based on Hamiltonian (1) massively

suppresses phase fluctuations of the pair correlations thateventually lead to superconductivity. This happens fortwo reasons. First, the solution fixes the relative phase ofthe fermionic pairs occupying a- and c-states with respectto the b-states. Second, each a- or c-state participates ina fermionic pair with four adjacent states of the samekind but opposite spins, which implies that the phasefluctuations other than the linear ones having form (67)across the entire system do cost energy. The above con-straints leave the simultaneous phase fluctuations of a-,b- and c-states with θ(r) given by Eq.(67) as a rathernarrow channel, through which superconductivity is sup-pressed. As we illustrate below, this channel is removedand the superconductivity is recovered once arbitrarilysmall hopping terms are added to the model.

In general, the hopping terms can connect two nearestregions (a ↔ b and c ↔ b), two next-to-nearest regions(a ↔ c or b ↔ b), two next-to-next-nearest regions, etc.Because our goal in this section is to describe the mainqualitative effects of hopping, we would like to considerthe terms whose impact is simplest to calculate. Thuswe include into consideration the hopping terms betweentwo nearest a-states with the same spins and betweentwo nearest c-states with the same spins:

H ′ = H +Ht (68)

where

Ht = t∑ra,α

a†(ra + Rα + Rα+1)a(ra)

+ t∑rc,α

c†(rc + Rα + Rα+1)c(rc) (69)

Here t is the hopping parameter, and ra and rc run overall a- and c-plaquets respectively.

For the model with Hamiltonian (68), the variationalmean-field solution can be constructed in the same wayas in the preceding sections — with only one difference,namely, the energies of unpaired fermions occupying a-and c-states now form a band with the k-dependence:

E(k) ≡ εa + 2t [cos(2kxL) + cos(2kyL)] (70)

Here we again assume εa = εc. The Bogoliubov trans-formations are still defined by Eqs.(2,16,17,18,19). Asshown in Appendix D, for small t, the k-dependence ofbare energies (70) changes the values of the Bogoliubovcoefficients u(k), v(k), p(k) and q(k) only very little.The variational wave function of the system can still beexpressed as

|Ψ〉 =∏k

(u(k)− v(k)eiφa(k)a†o(−k)a†e(k)

)·

·(p(k)− q(k)eiφc(k)c†o(−k)c†e(k)

)·

·∏iα

(s− weiϕiαb†iα↓b†iα↑)|vac〉, (71)

9

and the resulting equation for the critical temperaturechanges also only slightly. We note, however, that thehopping part of Hamiltonian Ht introduces additionalconstraints on phases ϕi, which are important for com-puting the superfluid stiffness. These constraints are de-rived in Appendix E.

In order to calculate the phase stiffness of the abovemean-field solution, we introduce an auxiliary wave func-tion |Ψ〉 obtained from |Ψ〉 with the help of trans-formation (64-66). Namely, in the right-hand-side ofEq.(71), we use the inverse Fourier transform for a- andc-operators defined by Eqs.(9-12), then replace all latticesubscripts with the position argument r, and, finally, sub-stitute the creation operators a†(r), b†σ(r), and c†(r) with

their respective counterparts a†(r), b†σ(r), and c†(r) de-fined by Eqs.(64-66). The state represented by the wave

function |Ψ〉 carries supercurrent. It is the lowest-energy

state of the Hamiltonian H ′ which has the same form asH ′ but in terms of a(r), bσ(r) and c(r).

We define the zero-temperature phase stiffness J in thefollowing way:∑

r

J

2(∇θ)24L2 = 〈Ψ|H ′|Ψ〉 − 〈Ψ|H ′|Ψ〉+ o((∇θ)2),

(72)where θ is the phase appearing in transformations (64-66), and the sum in the left-hand side is taken over all2L×2L unit cells of the spin vortex checkerboard. (In the

continuum limit, this sum is replaced by

∫J

2(∇θ)2d2r.)

In Appendix F, we obtain the following expression forthe value of J corresponding to the Hamiltonian (68) inthe case where εb = µ ≡ 0:

J =2t

N

∑k

cos(2kxL)× sign(E(k)

)×

×

(√E2(k)

E2(k) + 14g

2|V (k)|2+

+

√E2(k)

E2(k) + 14g

2|V (k)|2

)(73)

We explicitly evaluated this expression in Case I underadditional assumption |t| � |g| � |εa|. In the leadingorder, the result is:

For4|tεa|g2

6 1,

J =4t2

|εa|, (74)

while, for4|tεa|g2

> 1,

J =|t|g2

ε2a. (75)

In this work, we do not attempt to calculate the su-perconducting properties at finite temperatures. Belowwe only discuss qualitative considerations about the re-sulting thermodynamic behavior:

Without hopping, the phase transition that we ob-tained in the mean-field approximation for Hamilto-nian (1) turns into a crossover, whose sharpness is limitedby the phase fluctuations of the order parameter. Theaddition of a small hopping part (69) to Hamiltonian (1)introduces into the system finite superfluid stiffness andthe ability to carry current at sufficiently low temper-atures. In such a setting, we expect the temperatureTph, at which in-plane superfluid stiffness appears, to besignificantly smaller than the the mean-field transitiontemperature Tc. Strong pair fluctuations should thus bepresent in the temperature range between Tph and Tc.However, in a three-dimensional setting of stacked lay-ers and in the presence of Coulomb interaction betweencharges, the above fluctuations should be somewhat sup-pressed.

Once the overall phase of the superconducting solu-tion is stabilized, the strong spatial dependence of theanomalous correlation functions described in Section V Acan lead to an unorthodox interpretation of the experi-ments sensitive to the sign of the superconducting phasein cuprates — see the discussion in Ref.19. It also im-plies that the standard factorization of the Cooper-pairwave functions in terms of the center-of-mass dependenceand the relative-coordinate dependence is not applicable,which also means that the classification of Cooper pairsin terms of spin singlets and spin triplets does not apply.This in turn, may lead to non-trivial spin susceptibilityof the superconducting state, which requires a separateinvestigation extending beyond the scope of the presentpaper. Here, we would only like to remark that the cur-rent experimental knowledge of the spin susceptibility ofsuperconducting cuprates is largely based on the Knightshift nuclear magnetic resonance (NMR) experiments50,which are subject to a number of assumptions about thechemical shifts and about certain accidental cancellationof hyperfine coefficients. According to a recent review51,the overall Knight-shift phenomenology in cuprates isstill evolving, and this may lead to a re-examination ofthe standard NMR interpretations adopted in the past.

VI. THEORETICAL SCENARIOS FORCUPRATES

A. 1/8-doped lanthanum cuprates

The model presented in this article is primarily rele-vant to the properties of 1/8-doped lanthanum cuprateswhere the static spin and charge order is stabilized, suchas La1.875Ba0.125CuO4 (1/8-doped LBCO). On cooling,the latter system exhibits static charge order at temper-atures below TCO = 54K. On further cooling at zeromagnetic field, 1/8-doped LBCO sample investigated in

10

Ref.35 exhibits the onset of static spin order and a simul-taneous sharp drop of in-plane resistivity by a factor of10 at temperature TSO ≈ 40K. At yet lower temperature,the in-plane resistivity decays to unmeasurably small val-ues following the Berezinskii-Kosterlitz-Thouless-like52,53

scaling with characteristic temperature TBKT = 16K.Finally, at temperature Tc = 4K, the sample exhibitsbulk superconductivity. Another sample reported by thesame group in Ref.38 exhibits similar behavior as far asthe initial resistivity drop and the bulk superconduct-ing transitions are concerned, but, in the temperaturerange between 4K and 40K, the temperature dependenceof resistivity appears to be sample-dependent. The latternon-universality, probably, indicates a “fragile” quantummechanism of resistivity in the interval between the tem-perature of the initial resistivity drop and the tempera-ture of the onset of bulk superconductivity.

A theoretical interpretation of the above phenomenol-ogy in the framework of the stripe scenario was presentedin Refs.36,54. It was based on the idea of strongly sup-pressed Josephson coupling between the adjacent CuO2

layers caused by the in-plane change of the sign of the SCorder parameter between the adjacent stripes and thenby the mismatch of that sign between the adjacent layers.

A similar suppression of the Josephson coupling be-tween adjacent layers can exist in the checkerboard sce-nario. As shown in Ref.19 for the grid checkerboard andin the present work for the spin-vortex checkerboard,the sign of the SC order parameters of the a- (and c-)fermionic components is supposed to change in real space,remaining zero on average - thus falling under the defi-nition of the pair-density wave proposed in Ref.54. Thesign of SC order for the b-component may also change inreal space but it may or may not lead the zero real-spaceaverage. If layers described by our model are stacked onthe top of each other, then, due to the Coulomb repul-sion, the b-plaquets are supposed to be located over the a-or c-plaquets - see Fig.4. This mismatch suppresses theJosephson tunneling between the adjacent planes, whichcan lead to different transition temperatures for the on-sets of the two-dimensional and three-dimensional super-conductivity. (We note here that the above-mentionedmismatch cannot be perfect, because the adjacent CuO2

planes in lanthanum cuprates are shifted by half a crystalperiod.)

Apart from the difference between stripes and checker-boards, another crucial difference between our proposaland that of Ref.54 is that the model of Ref.54 is basedon one fermionic component forming Cooper pairs in-side non-magnetic regions, while our model involves twofermionic components — one residing in the magneticregions (a-/c-states) and the other one in non-magneticregions (b-states), and, moreover, the Hamiltonian of ourmodel is dominated by the center-of-mass-conserving in-teraction between the two components, which leads toanomalously low in-plane superfluid stiffness. As dis-cussed at the end of the preceding section, our model isthen supposed to exhibit a higher-temperature crossover

FIG. 4. (Color online) Two checkerboard patterns in the ad-jacent planes. Violet squares schematically denote plaquets,corresponding to b-states, and green squares denote antifer-romagnetic a or c plaquets.

associated with the appearance of anomalous fermionicaverages (42)-(44) with fluctuating phases and the lower-temperature transition that leads to establishing thelong-range phase coherence. We associate the abovehigher-temperature crossover with the initial resistivitydrop in 1/8-doped LBCO at T ∼ 40K, while the lower-temperature transition is assigned to the observed BKT-like crossover.

Now we address several issues that naturally arise inthe context of the above scenario:

(i) Two-dimensional modeling versus three-dimensional character of real materials. In thepresent work, we only obtain a mean-field descriptionof a two-dimensional model. On the one hand, it isknown that long-range fluctuations and the proliferationof topological defects turn the transitions obtained fromsuch a solution into a crossover. On the other hand, theCuO2 planes described by our model are stacked in athree-dimensional setting and, as a result, coupled toeach other via Coulomb interaction, lattice strain, spinexchange and the hopping of charge carriers. This, inturn, suppresses both the long-range fluctuations andthe formation of topological defects, thereby, making themean-field solution adequate at least semi-quantitatively.

(ii) Reason for the resistivity drop to coincide with themean-field transition temperature in our model. Let usconsider Case IA of our model in the limit g � εa = εcwith small hopping between a-states and between c-states, as also done in Section V B. (In terms of the con-centration of charge carriers, Case IB with g � |εa| = |εc|is, likely, closer to the situation in 1/8-doped LBCO, butthe mathematics of this case is the same as that of CaseIA, while the latter is more intuitive to discuss.) In CaseIA, Tc � εa and, therefore, just above Tc, the occu-pations of the a- and c-states are exponentially small,which means large resistivity. Below Tc, the anomalousaverages of a- and c- states (43,44) appear, which meansthat Cooper pairs can now occupy these states. Due toanomalously low superfluid stiffness the phase of theseCooper pairs fluctuates. Hence, instead of carrying a su-percurrent, they carry normal current but the resultingstate still has resistivity much smaller than that above

11

Tc.(iii) Coincidence between the mean-field transition tem-

perature in our model and the onset of static spin modu-lations. In Ref.35, the drop of resistivity in zero magneticfield was observed to coincide with the onset static spinmodulations. Since we attribute the same resistivity dropto the mean-field transition in our model, the questionarises: what can our model have to do with the onset ofstatic spin modulations? In the model, we do not obtainthe spin background but rather treat it as imposed ex-ternally. We can imagine, however, that, in a more com-plete theory, spin-background degrees of freedom can beincluded in the modeling together with the terms con-sidered by us. In such a case, the energy gain due tothe onset of superconducting correlations can simultane-ously become a factor stabilizing the spin background,thereby making it static. We would then expect that themagnetic field that suppresses the onset of superconduct-ing fluctuations should simultaneously suppress the onsetof static spin order. Such a behavior can be contrastedwith the weak sensitivity to magnetic fields of spin ordersdriven by the short-range exchange interactions.

B. Other cuprates

Let us now turn to cuprates beyond doping 1/8and beyond the lanthanum family. As shown inRef.39, cuprates are generically close to the thresholdof Coulomb-frustrated phase separation into nanoregionsof stronger antiferromagnetic correlations and lower den-sity of charge carriers and nearly non-magnetic nanore-gions attracting charge carriers. It is, therefore, reason-able to assume that nanoscale charge inhomogeneities aregenerically present in cuprates, but they are not neces-sarily completely static or periodic. (Yet the evidencefor charge modulations that are both static and periodicwas reported for a variety of cuprates3,5,6,10,13,14.) It isfurther plausible that, in the general case, clusters withstronger antiferromagnetic correlations are not magnet-ically ordered with respect to each other. In fact, eachof them might be in a singlet state as far as the totalspin is concerned (see Ref.70). As mentioned earlier, ourmodel in the present form is lacking variables describingthe degrees of freedom of the spin background. We ex-pect that, generic situation in cuprates is such that theeffective hopping between different nanoregions of inho-mogeneous electronic background is larger than that for1/8-doped lanthanum cuprates. This larger hopping canbe sufficient to make Tph, Tc to approach each other,but still not large enough to dominate over the principalcenter-of-mass conserving interaction term appearing inHamiltonian (1). Therefore, apart from suppressing thesuperconducting phase fluctuations, the model with thedynamic spin background can, quite plausibly, have thevariational solution close in its principal features to theone obtained in the present work, in particular, as far asthe gap of the fermionic excitations is concerned.

VII. COMPARISON OF MODEL PREDICTIONSWITH EXPERIMENTALLY MEASURED

ENERGY GAPS

We now demonstrate that the energy gap obtained inSection IV C from the variational solution of the modelbased on Hamiltonian (1) exhibits temperature depen-dence quite similar to that of the superconducting energygap experimentally observed in a variety of cuprates. InFigs. 5 and 6, we make comparisons with the availableexperimental results for break junctions (BJ)55,61,62 andthe interlayer tunneling (ILT)56–60,63–69 for the bismuthfamily of cuprates. Figure 5 includes the data sets re-viewed in Ref.48, while Fig. 6 covers the experiments donelater. The predictions of the grid-based model are alsoplotted in Figs. 5 and 6.

The model predictions, when limited to Cases I or IIonly, require two input parameters ∆(0) and Tc, whichhelp us to determine g and |εa| for Case I, or |εb| for CaseII. The choice between Cases I and II is made on thebasis of the ratio ∆(0)/Tc being larger or smaller than

2√

2.All plots in Figs. 5 and 6 demonstrate either very

good or satisfactory agreement between the predictionsof the present model and the experiment. In compar-ison with the predictions of the grid-based model, theagreement with experiment has improved overall. Itshould be remarked here that the experimental datathemselves are subject to a number of uncertainties, in-cluding, in particular, the overheating effect for the ILTmeasurements56–60,63,64,68,71–77.

We further remark that, despite the significant experi-mental difficulty of measuring ∆(T ) close to T = Tc, thevery notion of the existence fo the critical ratio ∆(0)/Tcwhich separates the dependencies ending with ∆(Tc) = 0from those ending with ∆(Tc) 6= 0 appears to be reason-ably supported by experiments, and, moreover, the value2√

2 for such a critical ratio obtained in this work leadsto more consistent predictions than the critical value 4obtained in Ref.48 for the grid-based model.

Finally, in Fig. 7, we present the empirical dependenceof the model parameters g and εa on the doping level.We extract this dependence from the ILT experiments re-ported in Ref.68, where the values of doping were explic-itly indicated. The parameters g and εa were obtainednumerically by solving Eqs.(30) and (31) with εb = 0,nB = 0 and with experimentally determined input pa-rameters ∆(0) and Tc. The resulting plot shows that, inthe doping range between 0.09 and 0.2, g depends on thedoping level rather weakly, while εa depends strongly —it decreases nearly linearly with increasing doping and, ifextrapolated, reaches zero around the doping level 0.25,i.e. close to the level, above which superconductivity dis-appears. Such a behavior of εa is consistent with thebehavior of the pseudogap expected from other experi-ments. The eventual disappearance of superconductivityat higher dopings implies, that, if the present model isrelevant, then the interaction constant g should steeply

12

(a) UD 83K (b) OP 95K (c) OD 87.1K (d) UD 70K (e) UD 79K (f) OP 85K (g) OD 77K

0 1 1 1 1 1 1 10

2

4

6

8

T /Tc

Δ/Tc

(h) OD 93K (i) OD 89K (j) 77K

HgB2-INTERCALATED

(k) UD 84.4K (l) UD 85K (m) UD 92.8K (n) OD 93K

0 1 1 1 1 1 1 10

2

4

6

8

T /Tc

Δ/Tc

(o) 80K (p) OD 86K (q) UD 96K (r) OP 107K (s) OD 107K (t) OP 32K

0 1 1 1 1 10 10

2

4

6

8

T /Tc

Δ/Tc

FIG. 5. (Color online). Temperature evolution of superconducting gap ∆(T ) - part 1 (reviewed in Ref.48). Circles representexperimental data sets for break junction (BJ) and interlayer tunnelling (ILT). [Open circles imply that the data pointscorrespond to very broad and small SC peaks.] Solid red line represents theoretical results of the current work, green dashedline previous theoretical work48. The experimental data sets are taken from the following references– (a,b) BJ - Miyakawa etal.,55; (c) ILT - Suzuki et al.,56; (d-g) ILT - Suzuki and Watanabe57; (h,i) ILT- Krasnov et al.,58; (j) ILT- Krasnov59; (k-n)ILT- Krasnov,60; (o) BJ- Vedeneev et al.,61; (p) BJ- Akimenko et al.,62 [plots (a-p) are for Bi-2212]; (q-s) ILT for Bi-2223from Yamada et al.,63; and (t) ILT for Bi-2201-La0.4 from Yurgens et al.64. UD refers to underdoped samples, OP- optimallydoped, OD - overdoped. The superconducting critical temperature also indicated in each frame. Horizontal marks in eachframe indicate the critical ratio ∆/Tc = 2

√2.

decrease to zero for doping levels beyond the plottingrange of Fig. 7 (i.e. above 0.2).

VIII. CONCLUSIONS

In the present paper, we generalized the microscopicmodel proposed in Ref.19 for the grid background to thebackground formed by the checkerboard of spin vortices.The technical difference is that the former involves twokinds of fermionic states, while the later involves three,even though two of the three are similar. We have shown

that the predictions of the grid-based model for the tem-perature evolution of the energy gap largely remain in-tact. The most important difference between the spin-vortices-based model and the grid-based model turns outto be the critical ratio ∆(0)/Tc above which the temper-ature dependence of the energy gap ends at the value∆(Tc) 6= 0, which, in turn, is, probably, related to the

pseudogap. For spin vortices, this critical value is 2√

2,while, for grid, it is 4. We have further demonstratedthat the predictions of the spin-vortices-based model forthe temperature evolution of the energy gap exhibit good

13

(a) OP 93K (b) UD 80K (c) OD 88.3K (d) UD 63.5K (e) OP 77K (f) OD 86.5K

0 1 1 1 1 1 10

2

4

6

8

T /Tc

Δ/Tc

(g) UD 50K (h) UD 60.3K (i) OP 74.9K (j) OD 86K (k) OD 78.2K (l) OD 78K

0 1 1 1 1 1 10

2

4

6

8

10

12

T /Tc

Δ/Tc

(m) UD 71K (n) UD 80K (o) UD 89K (p) OD 79K

0 1 1 1 1 100

2

4

6

8

T /Tc

Δ/Tc

FIG. 6. (Color online) Temperature evolution of superconducting gap ∆(T ) - part 2 (experiments after 2005). Circles representexperimental data sets for interlayer tunnelling (ILT). [Open circles imply that the data points correspond to very broadand small SC peaks.] Solid red line represents theoretical results of the current work, green dashed line previous theoreticalwork48. The experimental data sets are taken from the following references– (a,b) Krasnov65; (c) Bae et al.,66; (d,e,f) Kambaraet al.,67; (g-l) Suzuki et al.,68; and (m-p) Ren et al.,69. All plots except (d,e,f) are for Bi2Sr2CaCu2O8+δ; (d,e,f) are forBi1.9Pb0.1Sr2CaCu2O8+δ.

agreement with experiments, and, moreover, this agree-ment is somewhat better than that for the grid-basedmodel.

An important difference of the present work from thatof Ref.19 is the treatment of the superfluid stiffness of theresulting solution. We have shown that nonlocal charac-ter of the interaction term of the kind used in Ref.19 isnot sufficient to induce non-zero superfluid stiffness. Thelatter appears only once arbitrarily small hopping termsare added to the model. The resulting behavior then im-

plies large superconducting fluctuations of the kind that,possibly, exist in 1/8-doped LBCO.

In the broader context of cuprate superconductivity,the model considered in this work is still rather over-simplified. However, one can use it to develop intuitionabout more realistic settings that must involve the fluc-tuations of the spin background, as well as other interac-tions between fermions.

14

◦ ◦ ◦ ◦◦ ◦

△

△ △

△△

△

◦ g

△ ϵa

0.10 0.12 0.14 0.16 0.18 0.200

10

20

30

40

Doping

g,ϵ a

[meV

]

FIG. 7. (Color online). Doping dependence of the model pa-rameters g and εa for the calculations presented in Fig. 6(g-l) to describe ILT experiments of Suzuki et al.68 forBi2Sr2CaCu2O8+δ. The model parameters were obtained bysolving numerically Eqs.(30) and (31) with εb = 0, nB = 0and with experimentally determined input parameters ∆(0)and Tc. The doping level is as indicated in Ref.68

ACKNOWLEDGMENTS

The authors are grateful to J. Haase, M. Jurku-tat, J. Kohlrautz and A. V. Rozhkov for discussions.This project was funded by the Skoltech NGP Program(Skoltech-MIT joint project).

Appendix A: Total energy of the system

In this Appendix, we elaborate on the derivation steps(ii) and (iii) mentioned after Eq.(20).

Substituting the canonical transformation for a- andc-states (16-19) in (13), we obtain

E = 8εbN[s2nB + w2(1− nB)

]+ 2εa

∑k

{u2(k)nA(k) + v2(k)[1− nA(k)]

}+ 2εc

∑k

{p2(k)nC(k) + q2(k)[1− nC(k)]

}+ 2gsw (1− 2nB)

∑k[

u(k)v(k)(1− 2nA(k))|V (k)| cos[φV (k) + φa(k)]

+ p(k)q(k)(1− 2nC(k))|V (k)| cos[φV (k) + φc(k)]

],

(A1)

where all variables are defined in Section (III).Two interaction terms in the above expression

have phase-dependent factors cos[φV (k) + φa(k)] andcos[φV (k) + φc(k)]. Eventually, the variational ground-state energy obtained by finding u(k), v(k), p(k) and

q(k) will monotonically decrease with the increasing ab-solute value of these terms. This implies that the varia-tional energy will be minimized for | cos[φV (k)+φa(k)]| =1 and | cos[φV (k) + φc(k)]| = 1. Choosing the sign ofcosines in these relations is just a matter of sign conven-tion for the Bogoliubov transformation coefficients laterconverting into the sign of the products swu(k)v(k) andswp(k)q(k). We adopt the convention that the signs ofthe above products are negative in the state minimizingthe total energy.

Appendix B: Case I

For εb = 0, the Bogoliubov transformation parameterss and w enter the energy (24) only as a term proportionalto sw. For such a case, given the constraint s2 +w2 = 1,

the minimization of energy (24) gives, s =√

12 , w =

−√

12 . The relative negative sign of s and w implies later

the positive relative sign for the pairs of transformationparameters {u(k), v(k)} and {p(k), q(k)}. For εa ≥ 0,the minimization of energy (24) with respect to u(k),v(k), p(k), and q(k), finally, gives

u(k) =

√√√√1

2+

1

2

√1

1 +T 2a (k)Q2a(k)

,

v(k) =

√√√√1

2− 1

2

√1

1 +T 2a (k)Q2a(k)

,

(B1)

p(k) =

√√√√1

2+

1

2

√1

1 +T 2c (k)Q2c(k)

,

q(k) =

√√√√1

2− 1

2

√1

1 +T 2c (k)Q2c(k)

,

(B2)

where,

Ta(k) = g(1− 2nB)(1− 2nA(k))|V (k)|,Qa(k) = 2εa(1− 2nA(k)),

(B3)

Tc(k) = g(1− 2nB)(1− 2nC(k))|V (k)|,Qc(k) = 2εc(1− 2nC(k)).

(B4)

The total energy E of the system in this case can beexpressed as

E = −∑k

[(1− 2nA(k)) εA(k)

+ (1− 2nC(k)) εC(k)− εa − εc], (B5)

which is an implicit function of |V (k)| and |V (k)|. Both

|V (k)| and |V (k)| are a function of four phases ϕ1, ϕ2,

15

ϕ3 and ϕ4 entering Eqs. (14, 15). Therefore the energyequation should be further minimized with respect to thevalues of these phases. Such minimization imposes onlyone constraint19

ϕ2 + ϕ4 − ϕ1 − ϕ3

2=π

2+ πn. (B6)

Appendix C: Case II

In this case, εa = εc = 0 in Eq. (24) and, as a result,the minimization of energy gives u(k) = v(k) = p(k) =

q(k) = 1/√

2. The minimization with respect to s subjectto condition (3), now gives

s4 − s2 +T 2

4(Q2 + T 2)= 0 (C1)

where we introduced the following parameters:

Q ≡ 8εbN(2nB − 1),

T = Ta + Tc,Ta = g(1− 2nB)CaN,

Tc = g(1− 2nB)CcN.

(C2)

The parameters Ca and Cc are defined by Eqs.(36,37).The solution of the biquadratic equation (C1) that min-imizes the total energy under condition εb ≥ 0 is

s =

√√√√1

2+

1

2

√1

1 + T 2

Q2

,

w = −

√√√√1

2− 1

2

√1

1 + T 2

Q2

,

(C3)

We obtain εB by varying energy (24) with respect to nB :

εB ≡1

8N

dE

dnB=

εb[s2−w2] + 2gsw(−2)

∑k

{u(k)v(k)(1− 2nA(k))|V (k)|

+p(k)q(k)(1− 2nC(k))(k)]|V (k)|}.

The substitution of the parameters defined byEqs.(C3,C2) now gives

εB = εb

√Q2

Q2 + T 2− 2

T2√Q2 + T 2

1

(1− 2nB)T ,

which, after some manipulations, leads to Eq.(32).The quasiparticle excitation energies for A- and C-

states are calculated in a similar way — as εA(k) ≡12

dEdnA(k) and εC(k) ≡ 1

2dE

dnC(k) , which gives Eqs.(33) and

(34) respectively. The total energy of the system thusbecomes

E = −2N [(1− 2nB) εB − εb]. (C4)

In this case, phases ϕα also obey the constraint (B6).

Appendix D: Effects of non-zero hopping on thevariational solution

In this Appendix, we discuss the properties of the su-perconducting variational solution for Hamiltonian givenby Eq. (68). We introduce Bogoliubov transformationsin exactly the same way as was done in Section III. Thenthe energy of the system has the following form:

E = 8εbN[s2nB + w2(1− nB)

]+ 2

∑k

E(k){u2(k)nA(k) + v2(k)[1− nA(k)]

}+ 2

∑k

E(k){p2(k)nC(k) + q2(k)[1− nC(k)]

}+ 2gsw (1− 2nB)

∑k

[u(k)v(k)(1− 2nA(k))|V (k)|+

+ p(k)q(k)(1− 2nC(k))]|V (k)|], (D1)

where E(k) is the band energy given by equation (70).The only aspect that is different here in comparison withEq.(24) is the k-dependence of the band energies for thea- and c-states.

We are unable to find an analytic expression for the Bo-goliubov transformation coefficients in the general case,but we still can do it in Case I (εb=0), where the mini-

mization of energy (D1) gives s =√

12 , w = −

√12 . The

relative negative sign of s and w implies later the positiverelative sign for the pairs of transformation parameters{u(k), v(k)} and {p(k), q(k)}. The minimization of en-ergy (D1) with respect to u(k), v(k), p(k), and q(k),leads to the result having the same form as the one givenby equations (B1)-(B2), with the only difference beingthat the parameters Ta(k), Qa(k), Tc(k) and Qc(k) arenow defined as:

Ta(k) = g(1− 2nB)(1− 2nA(k))|V (k)|,Qa(k) = 2E(k)(1− 2nA(k)),

(D2)

Tc(k) = g(1− 2nB)(1− 2nC(k))|V (k)|,Qc(k) = 2E(k)(1− 2nC(k)).

(D3)

The distinction between Eqs.(D2,D3) and Eqs.(B3, B4)is that the former contain the band energy E(k) given byEq.(70), while the latter contain the on-site energies εa

16

and εc. Equation (30) for the critical temperature is nowreplaced by the following one:

Tc =g2

32N

∑k

[(exp (|E(k)|/Tc)− 1

)(exp (|E(k)|/Tc) + 1

)× 1

|E(k)|(|V (k)|2 + |V (k)|2

)](D4)

Appendix E: Additional constraints on phases ϕicaused by hopping

Energy minimization for Hamiltonian (68) leads to ad-ditional constraints on phases ϕα appearing in Bogoli-ubov transformations (2). In this Appendix, we derivethese constraints for the case εb = 0.

For convenience, let us introduce new phases:

α ≡ ϕ1 − ϕ2 − ϕ3 + ϕ4

2, (E1)

β ≡ ϕ1 + ϕ2 − ϕ3 − ϕ4

2, (E2)

γ ≡ −ϕ1 + ϕ2 − ϕ3 + ϕ4

2, (E3)

and then express |V (k)|2 and |V (k)|2 from Eqs. (14),(15) as

|V (k)|2 = 4(

1 + cos (2kxL+ α) · cos (2kyL+ β)+

+ cos γ ·[

cos (2kxL+ α) + cos (2kyL+ β)])

(E4)

and

|V (k)|2 = 4(

1 + cos (2kxL+ α) · cos (2kyL− β)+

+ cos γ ·[

cos (2kxL+ α) + cos (2kyL− β)]). (E5)

In the presence of non-zero hopping t, the expression(B5) for the total energy of the system is modified asfollows:

E = −∑k

[(√E2(k) +

1

4g2|V (k)|2 − E(k)

)+

+

(√E2(k) +

1

4g2|V (k)|2 − E(k)

)], (E6)

where E(k) is the band energy given by Eq.(70).Necessary conditions for phases α, β and γ to minimize

the total energy (E6) are:

∂E

∂α= 0,

∂E

∂β= 0,

∂E

∂γ= 0.

(E7)

(E8)

(E9)

In general, the system of equations (E7)-(E7) has morethan one solution for a given set of parameters εa, g andt. The solution representing the global minimum shouldthen be selected by computing the corresponding valuesof E.

We were able to find analytically three types of solu-tions of the system (E7)-(E9).Solutions of type A:

α = πn, β = πm, γ = πk. (E10)

Solutions of type B:

α = πn, β = π(n− 1) + 2πm, γ = πk +π

2. (E11)

Solutions of type C (exist, only if4|tεa|g2

6 1):

α = 2πn, β = 2πm, cos γ = −4tεag2

;

α = (2n+ 1)π, β = (2m+ 1)π, cos γ =4tεag2

.

(E12)

(E13)

Here k, m and n are arbitrary integer numbers. We can-not prove analytically that the system (E7)-(E7) has noother solutions, but we checked numerically that the min-imum of E corresponds to the solutions of either type A,or B, or C.

Let us now prove that the sets of α, β and γ given byEqs.(E10-E13) are, indeed, the solutions of the system(E7)-(E9).

The substitution of Eq.(E6) into Eqs.(E7) and (E8)gives, respectively,∑

k

sin(2kxL+ α)× cos(2kyL+ β)√E2(k) + 1

4g2|V (k)|2

+

+ cos γ∑k

sin(2kxL+ α)√E2(k) + 1

4g2|V (k)|2

+

+∑k

sin(2kxL+ α)× cos(2kyL− β)√E2(k) + 1

4g2|V (k)|2

+

+ cos γ∑k

sin(2kxL+ α)√E2(k) + 1

4g2|V (k)|2

= 0, (E14)

and

−∑k

cos(2kxL+ α)× sin(2kyL+ β)√E2(k) + 1

4g2|V (k)|2

−

− cos γ∑k

sin(2kyL+ β)√E2(k) + 1

4g2|V (k)|2

+

+∑k

cos(2kxL+ α)× sin(2kyL− β)√E2(k) + 1

4g2|V (k)|2

+

+ cos γ∑k

sin(2kyL− β)√E2(k) + 1

4g2|V (k)|2

= 0. (E15)

17

FIG. 8. (Color online) Schematic representation of transfor-mation (E21) in the Brillouin zone of spin vortex checker-board. Each of eight triangle regions is rotated around thenearest purple dot (± π

4L,± π

4L) by 180 degrees — thereby or-

ange regions map into green ones and vice versa.

We now observe that α = πn, β = πm and arbitrary γsatisfy both Eq.(E14) and Eq.(E15), because these equa-tions would then contain only summations of functionsthat are odd with respect to either kx or ky, while thesummation regions are symmetric. Thus all sets of α, βand γ entering (E10) -(E13) satisfy Eqs.(E14) and (E15),and, hence, Eqs.(E7) and (E8)

The substitutions of Eq.(E6) into Eq.(E9) yields:

sin γ(∑

k

cos(2kxL+ α) + cos(2kyL+ β)√E2(k) + 1

4g2|V (k)|2

+

+∑k

cos(2kxL+ α) + cos(2kyL− β)√E2(k) + 1

4g2|V (k)|2

)= 0. (E16)

The solutions of type A given by Eq.(E10) imply thatsin γ = 0. Hence they, obviously, satisfy Eq.(E16).

Now let us show that the solutions of type B also satisfyEq.(E16). Once we substitute (E11) to (E16), we get anexpression, proportional to

∑k

cos(2kxL)− cos(2kyL)√E2(k) + g2 cos(2kxL) cos(2kyL)

. (E17)

This expression is zero, because it changes sign under thetransformation (kx, ky) −→ (ky, kx).

Finally, let us consider solutions of type C given by(E12) and (E13) and prove that they satisfy Eq.(E16).In order to do it, let us examine the expressions under

the square roots in the denominators in Eq.(E16):

E2(k) +1

4g2|V (k)|2 = ε2a+

+ 4tεa[

cos (2kxL) + cos (2kyL)]+

+ 4t2[

cos (2kxL) + cos (2kyL)]2

+

+ g2(

1 + cos (2kxL+ α) · cos (2kyL+ β)+

+ cos γ[

cos (2kxL+ α) + cos (2kyL+ β)])

(E18)

and

E2(k) +1

4g2|V (k)|2 = ε2a+

+ 4tεa[

cos (2kxL) + cos (2kyL)]+

+ 4t2[

cos (2kxL) + cos (2kyL)]2

+

+ g2(

1 + cos (2kxL+ α) · cos (2kyL− β)+

+ cos γ[

cos (2kxL+ α) + cos (2kyL− β)]). (E19)

When we substitute (E12) or (E13) into (E18)-(E19), the terms 4tεa

[cos (2kxL) + cos (2kyL)

]and

g2 cos γ[

cos (2kxL+ α) + cos (2kyL± β)]

cancel eachother. For this reason, the sum over k in Eq.(E16) for α,β and γ given by (E22) becomes proportional to∑

k

[cos (2kxL) + cos (2kyL)

]×

×{ε2a + 4t2 [cos (2kxL) + cos (2kyL)]

2+

+ g2[1 + cos (2kxL) cos (2kyL)

]}− 12

. (E20)

Now we note that, after the simultaneous change of thesigns of cos (2kxL) and cos (2kyL), the function underthe sum also changes sign. A possible transformationachieving this is :

(kx, ky) −→( π

2Lsign(kx)− kx,

π

2Lsign(ky)− ky

)(E21)

It is illustrated in Fig. 8. The fact that transformation(E21) maps the Brillouin zone onto itself, while the func-tion summed in Eq.(E20) changes sign, means that thesum itself is equal to zero and, hence, Eq.(E16) is satis-fied. This finishes the proof that Eqs.(E10-E13) representthe solutions of the system (E7-E9) and thus correspondto the extrema of E.

We have conducted extensive numerical tests of theminima of E in the space of parameters εa, g and t. In allthese tests, the minimuma corresponded to the solutionsof type A, B or C given by Eqs.(E10-E13). The numeri-cally found “phase diagram” assigning the regions of theparameter space (εa, t) to different types of solutions forg = 1 is shown in Fig. 9.

In Section V B and in Appendix F, we focus on thelimit |t| � |g| � |εa|. In this case, the minima of E arethe following:

18

FIG. 9. (Color online) Numerically obtained “phase diagram”of the type of extrema (A,B, or C) given by Eqs.(E10-E13)that realize the global minimum of energy (E6) for g = 1 andfor the values of parameters (εa, t) indicated on the axes.

If4|tεa|g2

6 1, then

cos γ = − 4tεag2 , α = 2πn, β = 2πm;

orcos γ = 4tεa

g2 , α = (2n+ 1)π, β = (2m+ 1)π.

(E22)

If4|tεa|g2

> 1, then cos γ = − sign (tεa), α = 2πn, β = 2πmorcos γ = sign (tεa), α = (2n+ 1)π, β = (2m+ 1)π.

(E23)The transition between the two kinds of minima is illus-trated in Fig. 10 through the evolution of | cos γ| as afunction of t for εa = 10 and g = 1.

Appendix F: Calculation of superfluid phase stiffnessfor Hamiltonian (68)

In this section, we obtain zero-temperature superfluidphase stiffness for Hamiltonian (68) with εb = 0.

Let us assume for simplicity that linearly changing inspace phase θ has gradient only along the x-direction.We also introduce variable θ0 to denote the change ofphase θ across the 2L × 2L unit cell of the spin-vortexcheckerboard. The gradient of θ can now be expressed as

|∇θ| = ∂θ

∂x=

θ02L

(F1)

FIG. 10. (Color online) Numerically obtained | cos γ | realizingthe global minimum of energy (E6) as a function of t for g = 1and εa = 10. It illustrates solutions (E22)-(E23) in the limit

|t| � |g| � |εa|: at t = ± g2

4εa= ±0.025 the minimum switches

from type C given by Eq.(E22) to type A given by Eq.(E23).

and, as a result, the energy (72) of the system associatedwith the phase gradient can be written as

∑r

J

2(∇θ)24L2 =

J

2Nθ20. (F2)

Using Eq.(72), we then obtain

J =1

N

∂2

∂θ20

(〈Ψ|H ′|Ψ〉 − 〈Ψ|H ′|Ψ〉

)∣∣∣∣θ0=0

. (F3)

We now observe that 〈Ψ|H ′|Ψ〉 = 〈Ψ|H ′|Ψ〉, where H ′ isthe Hamiltonian obtained by replacing operators a, b andc in Hamiltonian (68) with operators a, b and c definedby Eqs.(64-66). This allows us to rewrite Eq.(F3) as

J =1

N

∂2

∂θ20

(〈Ψ|H ′ −H ′|Ψ〉

)∣∣∣∣θ0=0

, (F4)

where

H ′ −H ′ = Ht −Ht =

=∑k

2t[cos(2kxL+ θ0)− cos(2kxL)](a+e (k)ae(k)+

+ a+o (k)ao(k) + +c+e (k)ce(k) + c+o (k)co(k)) =

=∑k

2t[cos(2kxL)(cos(θ0)− 1)− sin(2kxL) sin(θ0)]×

×(a+e (k)ae(k)+a+o (k)ao(k)+c+e (k)ce(k)+c+o (k)co(k)).(F5)

For the mean-field ground state in the presence of hop-ping elements, one can use Eqs.(B1) and (B2) with in-

19

gredients from Eqs.(D2) and (D3) to obtain

〈Ψ|a+e (k)ae(k)|Ψ〉 = 〈Ψ|a+o (k)ao(k)|Ψ〉 = v2(k) =

=1

2

(1− sign

(E(k)

)√ E2(k)

E2(k) + 14g

2|V (k)|2

)(F6)

and

〈Ψ|c+e (k)ce(k)|Ψ〉 = 〈Ψ|c+o (k)co(k)|Ψ〉 = q2(k) =

=1

2

(1− sign

(E(k)

)√ E2(k)

E2(k) + 14g

2|V (k)|2

). (F7)

After substituting Eqs.(F6) and (F7) into Eq.(F4), we,finally, get

J =2t

N

∑k

cos(2kxL)× sign(E(k)

)×

×

(√E2(k)

E2(k) + 14g

2|V (k)|2+

+

√E2(k)

E2(k) + 14g

2|V (k)|2

). (F8)

(Here and below, we do not include terms that give zeroafter the summation.)

Now let us study expression (F8) in the limit |t| �|g| � |εa| and find its dependence on t. Expanding it in

the powers of g/εa and t/g, we get, in the leading order,

J = − tg2

4Nε2asign (εa)

∑k

(|V (k)|2 + |V (k)|2) cos(2kxL).

(F9)For α, β, γ corresponding to the energy minima (E22)

and (E23), we obtain from Eqs.(E4) and (E5) that,

if4|tεa|g2

6 1:

|V (k)|2 = |V (k)|2 = 4(

1 + cos(2kxL) · cos(2kyL)−

− 4tεag2[

cos(2kxL) + cos(2kyL)])

; (F10)

and,

if4|tεa|g2

> 1:

|V (k)|2 = |V (k)|2 =

= 4(

1 + cos(2kxL) · cos(2kyL)−

− sign(tεa)[

cos(2kxL) + cos(2kyL)]). (F11)

This leads to the following result:

if4|tεa|g2

6 1:

J =8t2

N |εa|∑k

cos2 (2kxL) =4t2

|εa|; (F12)

if4|tεa|g2

> 1:

J = sign(εa) sign(tεa)2tg2

Nε2a·∑k

cos2 (2kxL) =|t|g2

ε2a.

(F13)

∗ anastasia.aristova@skolkovotech.ru† A. V. Aristova and V. K. Bhartiya contributed equally to

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