Temperature Dependence Of Cuprate Superconductors' Order Parameter
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Temperature Dependence Of Cuprate Superconductors’ Order Parameter
Alexander Mihlin and Assa Auerbach
Physics Department, Technion, Haifa 32000, Israel (Dated: October 17, 2018)A model of charged hole-pair bosons, with long range Coulomb interactions and very weak inter-layer coupling, is used to calculate the order parameter Φ of underdoped cuprates. Model parametersare extracted from experimental superfluid densities and plasma frequencies. The temperature de-pendence Φ( T ) is characterized by a ’trapezoidal’ shape. At low temperatures, it declines slowlydue to harmonic phase fluctuations which are suppressed by anisotropic plasma gaps. Above thesingle layer Berezinski-Kosterlitz-Thouless (BKT) temperature, Φ( T ) falls rapidly toward the threedimensional transition temperature. The theoretical curves are compared to c -axis superfluid den-sity data by H. Kitano et al., (J. Low Temp. Phys. , 1241 (1999)) and to the transverse nodalvelocity measured by angular resolved photoemmission spectra on BSCCO samples by W.S. Lee etal. , (Nature , 81 (2007)), and by A. Kanigel, et al. , (Phys. Rev. Lett. , 157001 (2007)). PACS numbers: 74.72.-h,74.20.-z,74.78.Fk
I. INTRODUCTION
Unconventional superconductivity in cuprates is oftenmeasured by deviations from Bardeen, Cooper and Schri-effer’s (BCS) phenomenology [1]. A case in point is theorder parameterΦ( T ) = X η d ( η ) h c † r ↑ c † r + η ↓ i , (1)where d ( η ) is the intra-layer pairing function with d -wavesymmetry, and uniformity is assumed in suppressing the r dependence of Φ. In BCS theory, the order parame-ter is inextricably related to a gap in the quasiparticleexcitations , whose maximal value is given by∆ BCS ( T ) = ¯ V Φ( T ) . (2)where ¯ V is an interaction parameter. ∆ BCS is the pairbreaking energy which sets the scale of the transitiontemperature T c . However, BCS theory is a mean fieldapproximation which neglects all phase fluctuations.In underdoped cuprates, there is compelling evidencethat T c is driven by phase fluctuations [2]. Uemura’sempirical scaling law T c ∝ ρ abs ( T = 0) [3] and the ob-servation of a superfluid density jump in ultrathin un-derdoped cuprate films [4, 5, 6, 7] are consistent with thebehavior of a bosonic superfluid, captured by an effective xy model.In this paper we calculate the temperature dependent or-der parameter of an effective Hamiltonian of charged lat-tice bosons (CLB). The CLB model incorporates essentialingredients of underdoped cuprates including extremelyweak interlayer coupling, and long range Coulomb inter-actions.Our main result is that Φ( T ) exhibits a trapezoidal shape in the weak interlayer coupling limit, as depicted in Fig.1. At low temperatures Φ( T ) decreases slowly due toeffects of anisotropic plasma frequency gaps. The effectsof long range charge interactions, however, do not drive FIG. 1: Temperature dependences of normalized supercon-ducting order parameters. A trapezoidal shape is obtained forCharged Lattice Bosons (black color online) model (Eq.5) foranisotropy ratio α = 10 − and κ = 150. We see that coulombinteractions suppress thermal phase fluctuations relative tothe classical xy model, Eq.(3), (red color online), depicted for α = 10 − . The rapid fall toward T c is calculated within in-terlayer mean field theory (see text). BCS theory for d and s -wave order parameters is depicted for comparison, (greenand blue colors online, respectively). the transition. The transition is driven by proliferationof vortex loops above the two dimensional Berezinskii-Kosterlitz-Thouless (BKT) [8] temperature T BKT , wherethe order paramater falls rapidly toward T c .Phase fluctuation theories have been previously appliedto cuprates, with special attention to the intra-layer su-perfluid density ρ abs ( T ) [9, 10, 11]. In order to explain thelinearly decreasing temperature dependence, additionalgapless (nodal) fermionic excitations were argued to beessential [12].Φ( T ), however, behaves differently than ρ abs ( T ). In thetwo dimensional limit, for example, Φ must vanish at all T > ρ abs jumps to a finite value below T c . Also, nodal fermionshave a small effect on Φ( T ). This is demonstrated inFig.1, which shows s and d wave order parameters be-having very similarly within BCS theory.We propose experimental probes for the order parame-ter, without relying on BCS theory and Eq. (2). Atweak interlayer coupling, we argue that Φ( T ) should beproportional to the square root of the c -axis superfluiddensity.Angular Resolved Photoemmission Sprectroscopy(ARPES) finds a ”pseudogap” ∆ pg in the electronicspectrum, which persists well above T c [14, 15, 16].Apparently, ∆ pg ( T ) is not proportional to Φ( T ), (thelatter of course vanishes at T c ), which violates Eq. (2).Pseudogap phenomena are often interpreted as shortrange pairing correlations well above T c .To address ARPES data, we employ a Boson-Fermion(BF) model which was derived from the Hubbard model[17] by contractor renormalization. The model describesthe CLB system, Andreev-coupled to fermion quasipar-ticles which occupy small hole pockets (or ’arcs’). Sim-ilar BF models were arrived at by other approaches[18, 19, 20]. Within our model, Φ( T ) is proportional tothe transverse nodal velocity v ⊥ ( T ). In Fig. 5, we findreasonable agreement between the theoretical curves andBSCCO ARPES data for v ⊥ ( T ) of Refs. [21, 22]. Fur-ther tests of the trapezoidal shape closer to T c would bedesirable.The paper is organized as follows: The CLB model isintroduced in Section II. The order parameter is calcu-lated within the harmonic phase fluctuations approxima-tion to obtain the low temperature regime. In SectionIII, the interlayer mean field theory is applied to com-pute the suppression of the order parameter near T c . InSection IV we relate the model parameters to experimen-tal data for several commonly studied cuprates. SectionV compares the theory to experiments, using an effectiveBoson-Fermion model to interpret the ARPES data. Weconclude with a brief summary and discussion.In Appendix A, we provide details of the analytical fit tothe harmonic phase fluctuations result. In Appendix Bwe estimate the temperature region near T c where three-dimensional critical fluctuations are important (Ginzburgcriterion). II. CHARGED LATTICE BOSONS
The xy Hamiltonian is a lattice model of boson phasefluctuations, H xy = − J X r , η cos( ϕ r − ϕ r + η ) + α X r , c cos( ϕ r − ϕ r + c ) ! (3)where r resides on a layered tetragonal lattice. η and c are in-plane and interlayer nearest neighbor vectors oflengths a and c respectively. The lattice constant a is in effect a coarse grained parameter chosen to be largerthan the coherence length ξ . J is the bare intra-layersuperfluid density, and α ≪ T ) = Φ h cos( ϕ r ) i (4)where Φ is the zero temperature value. The quantumCLB model is given by H clb = H xy [ ϕ ] + 12 X r , r ′ V ( r − r ′ ) n r n r ′ (5)where n r is the occupation number of a charge 2 e bosonon site r , obeying the commutation relation,[ n r , ϕ r ′ ] = iδ r , r ′ (6)Long range Coulomb interactions V ( r ) are given by theFourier components V q = X r e − i qr V ( r ) = 16 πe vǫ b q . (7)where v ≡ a c is a unit cell volume and ǫ b ( q , ω q ) is the ef-fective dielectric function in the appropriate wave vectorand frequency scale.At low temperatures, we can expand the CLB action toquadratic order and obtain the harmonic phase fluctua-tions (HPF) action, S hpf [ ϕ ] = 12 ~ T X q n ω n + ω p ( q ) V q ϕ q ω n ϕ − q − ω n (8)where ω n = 2 πnT / ~ are bosonic Matsubara frequencies.The plasmon dispersion, as derived by Kwon et. al. [10],is ω p ( q ) ≡ ω ab q ab + ω c q c q ω ab = 16 πe Jǫ b ~ cω c = 16 πe cαJǫ b ~ a , (9)where q ab and q c are the planar and c -axis wave-vectorsrespectively.The HPF order parameter is given byΦ hpf ( T ) = Φ e − h ϕ r i (10)where the local phase fluctuations are given by, h ϕ r i = 1 Z Z D ϕ ϕ r e −S hpf [ ϕ ] = v Z d q (2 π ) V q ~ ω p ( q ) (cid:18) sinh( ~ ω p ( q ) /T )cosh( ~ ω p ( q ) /T ) − (cid:19) (11)At extremely low temperatures, T ≪ ~ ω c , all thermalphase fluctuations are frozen out. However, as we shallshow in Section V, the experimentally interesting regimeof large anisotropy, has a wide separation of plasma en-ergy scales, such that ~ ω c ≪ T BKT ∼ T c ≪ ~ ω ab (12)For our regime, we fit Eq. (11) by the analytical approx-imation (see Appendix A), h ϕ i T,α = (cid:18) TJ (cid:19) ( a − a | ln( α ) | ) e − a ~ √ ω ab ω c /T (13)For the simplified case of a = c , the coefficients are givenby: a ≈ . , a = − . a ≈ .
35 (14)Thus, expression (10) reduces to the classical result ofHikami and Tsuneto (HT) [23], (shown later in Eq. (24))in the limit T ≫ a ~ √ ω ab ω c . In the experimentally rele-vant regime, Φ hpf decreases significantly slower than theclassical model, as demonstrated in Fig. 1. III. INTERLAYER MEAN FIELD THEORY
The HPF action (8) cannot describe the order parameternear T c since it does not include vortex excitations. Inthe narrow regime of T BKT ≤ T ≤ T c proliferation ofwidely separated two dimensional vortex pairs dramati-cally reduces the order parameter.For anisotropies of order α ∼ − − − , a straighfor-ward numerical calculation of Eq. (3) is encumbered byfinite size limitations. Instead, we employ the interlayermean field theory (IMFT) [24], described by a single layerhamiltonian in an effective field h : H imft ( h ) = H d ( h ) + h αJ H d ( h ) = − J D X r η cos( ϕ r − ϕ r + η ) − h D X r cos( ϕ r )(15) Variational detrminition of h yields the IMFT equation h = 2 αJ h cos ϕ r i = 2 αJ Φ d ( T, h ) (16)where the magnetization of a single two dimensionallayer, Φ d ( T, h ), is, in principle, the exact field dependentorder parameter of the single layer CLB model. SolvingEq. (16) for h ( T ), yields the three dimensional tempera-ture dependent order parameterΦ imft ( T ) = Φ d ( T, h ( T )) . (17)The transition temperature T c is given by T c = min T { T ; Φ imft ( T ) = 0 } (18)Solution of eq. (16) for samll anisotropies requires precisedetermination of Φ d ( h, T ) for very weak fields h near T c . This is obtained by using the asymptotic criticalproperties of the order parameter near T BKT , which isnot far from T c in the small α limit. A. BKT critical properties
The two dimensional classical xy model undergoes a BKTtransition [8] at T BKT ≈ . J [52, 53]. Vortex pairproliferation changes the phase correlation temperaturedependence from power law to exponential decay, h cos( ϕ r − ϕ ) i ∼ r − η ( T ) T < T
BKT e − r/ξ ( T ) T > T
BKT (19)where at low temperatures, η ≃ T πJ , T ≪ T BKT (20)Above T BKT the correlation length diverges as ξ d ∝ exp (cid:16) β/ √ t (cid:17) χ d ( t ) = B χ J exp (cid:16) νβ/ √ t (cid:17) t ≡ ( T − T BKT ) /T BKT β = 3 / , ν = 7 / β and ν were derived by Kosterlitz[27].In order to match the transition region to the low tem-perature HPF order parameter, we need to determine thenon-universal amplitude B χ of χ d ( T ). B χ was deter-mined numerically. We evaluated Φ d ( h, t ) by a Monte-Carlo simulation with Hamiltonian (15). Good conver-gence was achieved with 10 spin tilts per ( T, h ) point,
FIG. 2: Determination of B χ from Monte-Carlo data. Thedifferent fitting functions F ( B χ , T ) (solid lines, red color on-line), are defined in Eq.(22). The curve with parameter value B χ = 0 .
072 is chosen as the best fit to ( T − T BKT ) /T BKT .The dashed (gren color online) line is the two dimensionalorder parameter in the presence of an ordering field ¯ h . sampling every 10 tilts and averaging over the last 5000configurations. We define a fitting function F ( B χ , T ) = (cid:18) νβ ln ( J Φ d ( h, T ) / ( hB χ )) (cid:19) (22)The fitting procedure which is depicted in Fig. 2, yields B χ ≃ . . (23)For finite interlayer coupling, the classical xy model or-ders at T c ( α ) > T BKT . Hikami and Tsuneto [23] evalu-ated the order parameter for small α ≪
1, and obtainedΦ cl ( T ) = Φ α η/ (4 − η ) ≃ Φ e − T πJ | ln α | . (24)In Fig. 1, Φ cl ( T ) of Eq. (24) is plotted in comparisonto the CLB model. The classical model decreases muchfaster since it does not contain the plasma gaps in thethermal phase fluctuations.The IMFT equation for T c is2 αJχ d ( T c ) = 1 . (25)Using Eq. (21) for χ d ( T ) and the value (23) for B χ , theshift of T c is T c − T BKT ∼ (cid:18) βν ln(2 B χ α ) (cid:19) T BKT , (26)The IMFT is consistent with the renormalization groupanalysis of Hikami and Tsuneto [23]. We note, however, that a large vortex core energy can increase the shift of T c above the value given by Eq. (26) [25, 26].The critical field-exponent was derived by Kosterlitz [27]Φ d ( T BKT ) ∝ h /δ , δ = 15 . (27)Combining this result with the IMFT equation (16) yieldsΦ( T BKT ) ∝ α / ( δ − = α / . (28)Thus, by Eqs. (26) and (28), the order parameter dropsrapidly between T BKT and T c , with an average slope of d Φ( T ) /dT ∼ −| ln( α ) | . B. Matching at the crossover
In the crossover region, Φ d is given by the harmonicmean of the temperature and field dependent singularitiesat T BKT .Φ d ( T, h ) = Φ hpf ( T ) hχ d ( T ) + (cid:18) h h (cid:19) δ ! − (29)Eq. (29) correctly captures the singularities of the vari-ables ( t, h ) at the BKT transition. h is chosen to matchthe order parameter smoothly at T BKT , h = 2 αJ Φ hpf ( T BKT ) , (30)IMFT, as a mean field theory cannot properly capture three dimensional critical exponents of the xy model.Nevertheless, as shown in Appendix B, the criticalregime, by Ginzburg’s criterion is limited to T c − T < T
BKT / | ln α | , (31)which is difficult to resolve experimentally, in the systemsof interest. C. Fermionic excitations
The CLB model ignores effects of fermionic particle-holeexcitations, which are clearly observed in ARPES andtunneling. In underdoped cuprates, most of their spec-tral weight is associated with wave vectors around theantinodes, (( π, , (0 , π )), with energies at the pseudogapscale ∆ pg ≫ T c . Contribution of these excitations to de-pletion of the order parameter temperature is of order T / ∆ pg ≪ low energy (nodal)excitations might play an important role. This has beenshown to be the case for the temperature dependence ofthe superfluid density ρ abs ( T ) [9, 11, 28]).However, nodal excitations are weakly coupled to theorder parameter. Consider, for example, the BCS gapequation,1 λ = X k | d ( k ) | E k (∆( T )) tanh ( E k (∆( T )) /T ) , (32)where E k = p ( ǫ k − µ ) + | d ( k )∆( T ) | , (33)and λ is the BCS coupling constant. The pair wavefunction factor | d ( k ) | vanishes on the nodal lines k =( ± k, k ). This suppresses contributions from the nodalregions to the thermal depletion of the gap. As a re-sult, s -wave and d -wave order parameters have very sim-ilar temperature dependence as shown by Won and Maki[29] and depicted in Fig. 1. Although here we do notappeal to BCS theory, this observation depends only onthe weak coupling between nodal fermions and the orderparameter, imposed by the pair wave function symmetry. IV. EXPERIMENTAL PARAMETERS
The cuprates exhibit very large anisotropy between in-plane and interlayer Josephson couplings J c and J ab ,which can be experimentally determined by the in-plane and interlayer zero temperature London penetra-tion depths λ ab and λ c , λ ab = (cid:18) πe ~ c d J ab (cid:19) − λ c = (cid:18) πe d ~ c a J c (cid:19) − , (34)where d and a are effective lattice constants, e is theelectron charge and c the speed of light. The anisotropyratio for cuprates is in the range, α ≡ J c J ab = (cid:18) λ ab aλ c d (cid:19) ∼ − − − . (35)Our phenomenological assignment of J ab and J c , ne-glects quantum corrections which become sizable nearthe critical doping toward the insulating phase. Analternative measure of J ab and J c is given by rela-tions (9) and the experimental measurements of ω ab and ω c by optical and microwave conductivities (cf. Refs.[30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]). Thus, theanisotropy parameter, α , of Eq. (35) can be determined.Table I contains typical experimental values of relevantquantities at zero temperature (except for Ω and α whichwere determined via Eqs. (A6) and (35) respectively). InYBCO, BSCCO and TBCCO, the interplane distance c is taken as the mean value. FIG. 3: Comparison of CLB order parameter to square rootof c -axis superfluid density from Ref. [41]. Model parametersare α = 10 − , c/a = 0 . κ ≡ ω ab /J = 150. Data wastaken on BSCCO with T c = 87 K. Dashed line is d-wave BCSenergy gap, given for comparison. V. EXPERIMENTAL PROBES OF Φ( T ) In cuprates, the BCS relation, (2), does not hold, sincethe maximal gap ∆ pg is weakly temperature dependent[15, 42], while Φ( T ) vanishes at T c . Here we proposeexperimental probes to measure Φ( T ) / Φ(0). A. c -axis superfluid density Since the zero temperature interlayer pair tunneling isweak, the layered system can be treated as a one di-mensional array of Josephson junctions. Within a vari-ational approximation, the order parameter can be ex-tracted from the temperature dependence of the c -axissuperfluid density, ρ cs ( T ) = ρ cs (0) | Φ( T ) | . (36)Indeed, as seen in Fig. 3, agreement between theoreticalcurves Φ( T ) and values extracted from electrodynami-cal data of BSCCO [41] are quite good, except near thetransition. B. ARPES In d -wave BCS theory the quasiparticle spectrum is givenby Eq. (33). Above T c , ∆ BCS = 0, and the full Fermisurface should be detected as zero energy crossings of theARPES quasiparticle peaks. However, in underdopedcuprates as temperature is raised above T c , only finiteFermi arcs appear around the nodal directions. The gapin the anti-nodal directions ∆ pg survives to much higher Compound a [A] c [A] T c [K] ~ ω ab [eV] ~ ω c [meV] Ω [meV] λ ab [ µ m] λ c [ µ m] α [10 − ] ReferencesYBa Cu O − δ Sr CaCu O δ − δ Sr δ CuO Ba CaCu O δ a , mean interplane distances, c , critical temperatures, T c , planar and interplaneplasma frequencies, ω ab , ω c , energy scales Ω( α, a, c ) of Eq. (13), magnetic field penetration depths, λ ab , λ c and anisotropyfactors, α , at zero temperature. All quantities except for Ω( α, a, c ) and α were obtained experimentally, while Ω and α wereobtained via Eqs. (A6) and (35) respectively. Some quantities depend on doping (e.g. λ ab , λ c are diminished with doping) andvalues for each compound correspond to similar dopings.FIG. 4: Boson-Fermion model for the transverse quasiparticleexcitations below and above T c . θ is the azimuthal coordinatetransverse to the nodal direction. Above T c (red color online),vanishing of E k on a finite ’arc’ reflects the inner edge of thehole pocket. The pseudogap ∆ pg is the hole fermions energyat the antinodal wavevectors ( π, , (0 , π ), which has no directbearing on the superconducting properties. Below T c (bluecolor online), the Andreev coupling of hole fermions to hole-pair bosons yields a d -wave gap with a node at θ = 0. Thetransverse nodal velocity v ⊥ ( T ) is a direct measurement ofΦ( T ). The break in the curve at the arc edge is consistentwith ’two gaps’ phenomenology [48]. temperatures [15, 42]. In contrast to ∆ pg , the transversenodal velocity v ⊥ vanishes abruptly at T c [21, 22]. Below T c , v ⊥ introduces a singularity | k ⊥ | in the electronic prop-agator, which translates to an infinite correlation lengthin real space.A microscopic connection between v ⊥ ( T ) and Φ( T ) canbe provided by an effective Boson-Fermion hamiltonianwith small hole pockets, described below. C. Boson-Fermion theory
The Boson-Fermion (BF) model, which arises by a con-tractor renormalization of the square lattice Hubbardmodel [17], describes spin half fermion holes f k ,s of charge e , coupled to the CLB as H bf = H clb + X k ,s ( ǫ h k − µ ) f † k s f k s + g X r , r ′ e iϕ r d ( r − r ′ ) f r , ↑ f r ′ , ↓ + h.c. . (37)The last Andreev coupling term, describes disintegrationof hole pairs into single spin-half hole fermions. In ourversion of the BF model, the fermion and boson densities,measured with respect to half filling, obey n h + 2 n b = x, (38)where x is the total concentration of doped holes. Thehole dispersion ǫ k has minima near ( ± π/ , ± π/ small pockets of area fraction n h / T c , the small wave vector sides of the pockets ap-pear as the celebrated Fermi ’arcs’ [47].The pseudogap isgiven by the quasiparticle excitation energy at the anti-nodal wavevectors ∆ pg = ǫ ( π, − µ. (39)In the superconducting phase Φ( T ) = h cos( ϕ ) i . Thehole fermions acquire the Dirac cone dispersion near thenodes: E k = ± q ( v F ( k k − k F )) + (2 g Φ( T ) k ⊥ ) , (40)that is depicted in Fig. 4. Thus, the transverse velocitydirectly measures the order parameter, v ⊥ ( T ) = 2 g Φ( T ) . (41)In the underdoped regime, the transverse velocity issmaller than the pseudogap scale ∆ pg d ′ ( k ). This is seenas a ’break’ in E k at the Fermi arcs angles, as shown inFig. 4. Such behavior has been observed in ARPES [48]and found consistent with a ’two gaps’ phenomenology.In Fig. 5 we compare the CLB order parameter to thetransverse nodal velocities measured on three samples ofBSSCO by two groups [21, 22]. The agreement is rea-sonable, although the sharp break in the curves is notclearly confirmed. A comparison to the d -wave BCS ex-pression shows a systematic trend of all the data beinghigher than BCS theory would predict. FIG. 5: Comparison of CLB to transverse nodal velocity, mea-sured on different samples by ARPES. BSSCO samples with T c noted in the figure. Experimental data, including errorbars, are (1) from Ref.[21], (2) and (3) from Ref. [22]. Thezero temperature normalization is chosen by the lowest tem-perature data points. Theoretical curves for several values of α are drawn. using c/a = 0 . κ ≡ ω ab /J = 150. Dashedline is d-wave BCS energy gap, given for comparison. VI. DISCUSSION
This paper calculated the order parameter of cupratesusing a bosonic model of hole pairs. The model includescrucial features of layered cuprates: long range Coulombinteractions and very small anisotropy ratio. It ignoreseffects of fermionic particle hole excitations which areargued to be small for Φ( T ). The calculation predicts atrapezoidal temperature dependence in the small α limit,which is distinct from both BCS theory and the classical xy model. The theoretical curves are compared to datawhere the order parameter is extracted by additional the-oretical assumptions: the c -axis superfluid density (usinga variational argument) and the transverse nodal veloc-ity (using a BF model of small hole pockets). We haveselectively chosen data of BSCCO where α = 10 − , andthe ’trapezoidal’ temperature dependence is most pro-nounced. In other cuprates, with larger values of α , andlarger vortex core energies [25] the shift T c − T BKT islarger, and the curve should be more rounded (less trape-zoidal) and similar to the BCS curve.Additional probes to Φ( T ) could be devised. The criti-cal current of a c -axis Josephson junctions with a higher T c material might be investigated. The transverse nodalvelocity, which we have related to Φ by the BF theory,determines the low energy tunneling spectra and Ramanscattering [49]. In addition, it has been theoretically re-lated to the linear slope of the superfluid density dρ abs /dT [28], and to thermal conductivity.Further comparisons to experiments are warranted.Their success or failure may shed light on the applicabil-ity of the quantum lattice bosons description of cuprates both below and above T c . This would help us resolvesome of the other mysteries of the pseudogap phase. VII. ACKNOWLEDGEMENTS
We thank Ehud Altman, Thierry Giamarchi, AmitKanigel, Amit Keren and Christos Panagopoulous, foruseful advice and information. AA acknowledges sup-port from the Israel Science Foundation, and is gratefulfor the hospitality of Aspen Center for Physics wheresome of the ideas were conceived.
APPENDIX A: FITTING PHASEFLUCTUATIONS
We define q ≡ q c + q ab , η ≡ ( ω c /ω ab ) = αγ and γ ≡ c/a . For ease of numerical integration Eq. (11) maybe simplified as follows h ϕ l i i = 1 Z Z D ϕ ϕ l i e −S (2) [ ϕ ] (A1)= v Z d q (2 π ) V q ~ ω p ( q ) (cid:18) sinh( β ~ ω p ( q ))cosh( β ~ ω p ( q )) − (cid:19) ≈ γ ~ ω ab π J γπ Z dz π Z dr r ( z + r ) ε (cid:0) η, zr (cid:1) × sinh (cid:2) ε (cid:0) η, zr (cid:1) /T (cid:3) cosh (cid:2) ε (cid:0) η, zr (cid:1) /T (cid:3) − , (A2)where the last expression was obtained in cylindrical co-ordinates. The dispersion is thus parametrized by ε (cid:16) η, zr (cid:17) ≡ ~ ω ab s η ( z/r ) z/r ) . (A3)At extremely low temperatures, T ≪ ~ ω c , all thermalphase fluctuations are frozen out. However, due to thelarge anisotropy, and poor screening, there is a wide sep-aration of energy scales between the interplane plasmagap, ~ ω c , and the planar gap, ~ ω ab and it turns out that ~ ω c ≪ T c ∼ J ≪ ~ ω ab . (A4)At low temperatures, the integral in Eq. (11) may beparametrized as h ϕ i ≈ ATJ e Ω /T , (A5)where the energy scale, Ω( α, γ ), and the coefficient A ( α, γ ) may be parametrized byΩ( α, γ ) ≈ ~ (cid:18) . √ γ + 0 . √ γ (cid:19) √ ω ab ω c , (A6)and A ( α, γ ) ≈ A ( γ ) − A ( γ ) ln( α ) A ( γ ) ≈ . γ + 0 . γ A ( γ ) ≈ . γ − . γ . (A7)The low temperature magnetization, Φ hpf ( T ), is givenby Φ hpf ( T ) ∝ e − h ϕ i = C ( T ) α A T J exp(Ω /T ) , (A8)where the coefficient C ( T ) is given by C ( T ) = e − A T J exp(Ω /T ) . (A9)Notably, γ is of order unity and the energy scale, Ω( α, γ ),in Eq. (A5) is proportional to the geometric average ofthe interplane and planar plasma energies. APPENDIX B: GINZBURG’S CRITERION FORINTERLAYER MEAN FIELD THEORY
One would like to know, in which regime can we trust theIMFT near the transition temperature. Here we estimatethe critical region using the standard Ginzburg Criterion. At small α , we see that the magnetization only variesrapidly below T c , in the narrow region of width ∆ T c givenby Eq. (26). Within that region, Φ imft ( T ) drops fromΦ hpf ( T KT ), as given by the harmonic phase fluctuations(A8), to zero at T c , with a mean field behavior,Φ imft ∼ Φ hpf ( T KT ) (cid:18) | T − T c | ∆ T c (cid:19) β , β = 12 . (B1)Ginzburg’s criterion [50, 51], estimates the temperatureregion below T c , where critical 3D fluctuations becomeimportant and IMFT breaks down. This is where orderparameter fluctuations averaged over a correlation vol-ume of size V ξ = ξ ab ξ c exceed their average, i.e. h (∆Φ) i V ξ = S ( q = 0 , T ) V ξ = cξ c ( T ) ≥ Φ imft ( T ) . (B2)Using the mean field estimation of ξ c ∼ c ( | T − T c | /T c ) − and Eq. 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