Phase separation and second-order phase transition in the phenomenological model for Coulomb frustrated 2D system
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r Phase separation and second-order phase transition in the phenomenological modelfor Coulomb frustrated 2D system
R. F. Mamin, T. S. Shaposhnikova, and V. V. Kabanov Zavoisky Physical-Technical Institute, Russian Academy of Sciences, 420029 Kazan, Russia Department for Complex Matter, Jozef Stefan Institute, 1000 Ljubljana, Slovenia (Dated: November 9, 2018)We have considered the model of the phase transition of the second order for the Coulomb frus-trated 2D charged system. The coupling of the order parameter with the charge was considered asthe local temperature. We have found that in such system, an appearance of the phase-separatedstate is possible. By numerical simulation, we have obtained different types (”stripes”, ”rings”,”snakes”) of phase-separated states and determined the parameter ranges for these states. Thus thesystem undergoes a series of phase transitions when the temperature decreases. First, the systemmoves from the homogeneous state with a zero order parameter to the phase-separated state withtwo phases in one of which the order parameter is zero and, in the other, it is nonzero ( τ > τ <
PACS numbers: 64.10.+h, 77.22.Jp., 77.84.-s
I. INTRODUCTION
The problem of phase separation attracts consider-able attention, because the variety of different phasestates and the coexistence of several phases are ob-served in many materials currently being studied .This includes a class of manganites with colossalmagnetoresistance in which there is a phase sepa-ration with charge inhomogeneity (”droplets”, ”bub-bles”, etc), as well as cuprate high-temperaturesuperconductors , in which a pseudo gap state for
T > T c and static and dynamic charge density waves(CDW) are observed. The phenomenon of the phaseseparation is accompanied by the charge inhomogene-ity, which is confirmed by various experimental observa-tions. The charge inhomogeneity was observed by meth-ods of scanning tunneling microscopy , photoelectronspectroscopy with angular resolution (ARPES) , X-rayand neutron diffraction . For these materials, there isa certain range of temperature and doping level, in whichthe coexistence of phases is in the ground energy state.The spatial size of the single-phase regions is determinedby the energy balance between the Coulomb interaction,which is important in the presence of an overcharge cre-ated by the doping, and the energy gain that appearswhen a more ordered phase occurs .There are many theoretical studies of the states withcharge inhomogeneity in which states with ”droplets”and ”stripes” has been obtained (see, for example, ).Usually in these papers it is considered the Coulomb frus-trated first order phase transition where the scalar or-der parameter is either coupled linearly with the chargedensity , or the order parameter is proportional tothe charge density . It is shown that these mod-els are unstable with respect to phase separation. Thephase-separated state represents the charged regions ofdifferent phases with different values of the order param- eter. Note that in the case of the second order phasetransition this type of coupling of the order parameterto the charge density is forbidden. In the case of thesecond order phase transition the order parameter is nota scalar. Here we discuss the case of the Coulomb frus-trated second order phase transition where we considerthe lowest possible coupling of the charge density withthe square of the order parameter. Within this model wediscuss a possibility of the existence of a phase-separatedstate with charge inhomogeneities near T c , where in thematrix of the ”high-temperature” phase with the orderparameter equal to η ( η = 0), exist inclusions of the”low-temperature” phase with the order parameter equalto η ( η > η ). Moreover, with the change of tempera-ture, several phase transitions can be observed.In this paper we apply a phenomenological approachbased on the Ginzburg-Landau theory to describe thestatic phase separation of a 2D system in the vicinityof the second-order phase transition, taking into accountthe presence of the Coulomb interaction, associated withthe overcharging effects due to doping. Because the typesof materials stated above are quasi two dimensional (CuOplanes in the cuprates and MnO planes in a number ofmanganites), the 2D description represents a reasonableapproximation. We define the range of parameters (re-lated to the temperature and the doping), for which thephase separation is energetically favorable. We also cal-culate the region of the phase diagram in which two in-homogeneous phases coexist. II. THEORETICAL MODEL
Let us consider the 2D system in the vicin-ity of the second order phase transition. In deGennes work the effects of the double exchange incompounds with mixed valency such as manganites(La − x Ca x )(Mn − x3+ Mn x4+ )O were studied. It wasshown that the motion of ”extra” holes or ”extra” elec-trons in antiferromagnet is lowering the energy of thesystem. Also it was shown that the Curie temperaturedepends on the doping x. Following de Gennes we beginwith the Hamiltonian, where we add the term with theCoulomb interaction. For ”layer” antiferromagnet theHamiltonian may be written as following: H = − X ij J ij S i S j − X ijσ t ij a + iσ a jσ − J H X i S i s i + H Coul (1)Here the first term describe the exchange interaction ofthe Mn ions. S i is the spin operator of the ionic spinon the site ( i ). J ij is the exchange interaction, J ij con-nects only neighboring i and j magnetic sites. The sec-ond and the third terms describe the double exchangeinteraction . The second term in Eq. (1) describes thehopping of an electron with the spin σ along ij latticesites. a + iσ ( a iσ ) is the creation (annihilation) operator ofelectron on i site, t ij is the hopping integral. The thirdterm of (1) describes the Hund’s coupling. Here s i is thespin operator of the conduction electron, which can beexpressed in terms of the creation and annihilation oper-ators for the electron and the Pauli matrices . The lastterm describes the Coulomb interaction. Following deGennes we assume that the spin ordering of the unper-turbed system is of the ”antiferromagnetic layer” type.Each ionic spin S is coupled ferromagnetically to z ′ neigh-boring spins in the same layer, and antiferromagneticallyto z spins in the adjacent layers. The exchange integralsare called J ′ ( >
0) and J ( < areallowed to hop both in the layer (with transfer integrals t ′ ) and also from one layer to the other (with transfer in-tegrals t ). The number of magnetic ions per unit volumeis called N , and the number of Zener carriers N x . Themodel of double exchange is the exchange model understrong coupling conditions J H >> zt and J H >> z ′ t ′ .In the limit of finite temperature and at low values ofthe relative sublattice magnetization, a phenomenologi-cal expression for the free energy was derived. Then inthe limit J H → ∞ the density of the thermodynamicpotential of the system φ ( η, ρ ) (Φ = R φ ( η, ρ ) d r ), whichdescribes the order parameter η , can be written in theform φ ( η, ρ ) = φ + φ η + φ int + φ Coul , (2)where for a phase transition of the second order φ η = α η + β η + δ η + ζ η + D ∇ η ) . (3)The order parameter η describes the relative magnetiza-tion of each sublattice . Here α = α ′ ( T − T c ), T c is thephase transition temperature without doping, α ′ ∼ /C , C is the Curie constant, β > φ η includes a second-order term from η , a positive fourth-order, a positivesixth-order, a positive eighth-order term and a gradientterm. Here: α = 2 N ( 32 k B T − S ( z ′ J ′ + zJ )) , (4) β = 4 N ( 920 k B T + 6175 x ( z ′ t ′ + zt )) , (5) δ = 6 N (0 . k B T + 0 . x ( z ′ t ′ + zt )) , (6) ζ = 8 N (0 . k B T + 2 . x ( z ′ t ′ + zt )) , (7) k B is the Boltzmann’s constant. φ int describes the in-teraction of the order parameter η with the local chargedensity ρ φ int = − σ η ρ. (8)The expression is obtained from the double exchange in-teraction terms (see Eq.(1)) averaged over the tempera-ture. The interaction is written here as the local temper-ature, σ is the interaction constant.Main physical properties of the system are determinedby the parameter σ , that is defined as¯ ρσ = 4 N x ( z ′ t ′ + zt ) . (9) φ Coul = γ Z ( ρ ( r ) − ¯ ρ )( ρ ( r ′ ) − ¯ ρ ) | r − r ′ | d r ′ (10)is the energy density of the Coulomb interaction, the con-stant γ is determined by the dielectric constant. In theabsence of terms of φ int and φ Coul a phase transition ofthe second order is observed at α = 0. For α < η = 0.For α >
0, the equilibrium value of η = 0, then there isno order, which is determined by the parameter η . In ex-pressions (9) and (10) ¯ ρ is the average 2D surface densityof charge ¯ ρ = 1 S Z S ρd r , (11)where r is 2D-vector.The total free energy Φ should be minimized with re-spect to η ( r ) and ρ . The minimization of Φ with respectto ρ gives − σ ∇ D η = 4 πγ ( ρ ( r ) − ¯ ρ ) δ ( z ) d. (12)Here thickness of 2D-layer d is introduced to preservedimensionality. δ ( z ) is the Dirac delta-function. Substi-tuting (12) in (2), we obtain φ = φ + α η + β η + δ η + ζ η ++ D ∇ η ) − σ η ¯ ρ −− σ π γd Z ∇ D η ( r ) ∇ D η ( r ′ ) | r − r ′ | d r ′ , (13)where r and r ′ are 2D vectors. The last two terms inthis expression are negative. The term σ η ¯ ρ renormal-izes the critical temperature of the phase transition. Thecritical temperature now depends on the average chargedensity. The coefficient in front of η is changed from α to ˜ α . ˜ α = α − σ ¯ ρ (14)Note that the presence of last nonlocal term in ex-pression (13) leads to the instability of the homogeneousstate.Let us introduce new dimensionless values Λ and ξ asΛ = η/η and ξ x = x/a , ξ y = y/a , where η = β/ζ and a = q Dζ / β / χ . Here χ is a constant. We choose thevalue of the constant in the interval from 3 to 20. Thisconstant allows us to change the size of an area in whichthe spatial distribution of the order parameter η ( r ) iscalculated. Then the expression (13) has the form φ = U (cid:18) τ Λ + Λ δ Λ χ ( ∇ Λ) −− Aχ Z ∇ D Λ ( ξ ) ∇ D Λ ( ξ ′ ) | ξ − ξ ′ | d ξ ′ (cid:19) . (15)Here parameters U , τ , χ , A and ˜ δ are defined as U = β η = β ζ , (16) τ = ˜ αβη = s ζβ (cid:0) α ′ ( T − T c ) − σ ¯ ρ (cid:1) , (17) χ = aη r βD , (18) A = σ γd π √ D √ βζ , (19)˜ δ = δβ η = 2 δ √ βζ . (20)
200 400200400 x y , , a . u . y ( y ) ( y ) A=3. =2.8 =3.
FIG. 1: The distribution of the order parameter Λ( ξ x = const, ξ y )(solid curve) and ∆ ρ = ρ ( ξ x = const, ξ y ) − ¯ ρ (dashedcurve) for χ = 3, A = 3 and τ = 2 . N = 512. III. RESULTS
In order to find the minimum of Φ = R φd r (15) themethod of conjugate gradients (CGM) was used. Wehave introduced N × N ( N = 128 or N = 512) discretepoints on a square with side a . We have applied the peri-odic boundary conditions. In the numerical calculationsthree parameters A , τ and χ were used.We have studied the dependence of the free energyfrom the parameters A and τ with the fixed value of χ .The inset in Fig.1 shows the spatial distribution of theorder parameter Λ( ξ x , ξ y ) for the parameters A = 2 . τ = 3, χ = 3 and N = 512. The free energy of this stateis negative (Φ < r ) = 0 is Φ = 0). These non-uniform states areformed because of the charge redistribution .Fig.1 shows the distribution of the order parameterΛ( ξ x , ξ y ) and the incremental charge ∆ ρ = ρ ( ξ x , ξ y ) − ¯ ρ along the line perpendicular to the strip (along y axis).As it follows from this figure, in the region of inhomo-geneous distribution of the order parameter Λ( ξ x =const, ξ y ) there exists a triple extra charged layer. The totalcharge of this layer is zero with high precision, ∆ ρ > ρ < A = 3 the inho-mogeneous distribution of the order parameter exists inthe range τ ≤ τ ≤ τ ( τ = −
27 and τ = 4 . A = 3).And the free energy is less than zero for τ ≤ τ ( τ = 3 . A = 3).According to Eq.(17) τ is a linear function of T − T c andis changed with ¯ ρ , where T c is the transition temperaturein the absence of interaction (i.e. at Φ int = 0). ¯ ρ is the
20 40 60 80 100 12020406080100120 a)
20 40 60 80 100 12020406080100120 b)
20 40 60 80 100 12020406080100120 c)
20 40 60 80 100 12020406080100120 d)
20 40 60 80 100 12020406080100120 e)
20 40 60 80 100 12020406080100120 f)
20 40 60 80 100 12020406080100120 g)
20 40 60 80 100 12020406080100120 h) FIG. 2: Inhomogeneous states are shown for A = 3, χ = 5 and τ = 3 . a ) , b ) , c ) , − d ), − e ) , − f ), − g ) , − h ),respectively. The phase-separated state is stable at 3 . ≤ τ ≤−
27. The parameter τ decreases from a) to h), correspondingto the decreasing of temperature T . All figures represent theresults of numerical calculations for N = 128. The orderparameter changes from Λ min = 0 to Λ max = 2 . max and Λ min decreasesfrom Figure c) to Figure h). average charge, ¯ ρ is proportional to the level of doping.The parameter A Eq.(19) depends on the coupling pa-rameter σ and the strength of the Coulomb interaction.With the increase of the Coulomb interaction parameter A decreases. As a result the region of τ , where the phaseseparation is observed, is shrinking.In Fig.2 a change of a form of inhomogeneous states isshown for A = 3 and with the reduction of τ from 3 . −
25. The free energy of these inhomogeneous states isnegative and smaller than the energy of an homogeneousstate.The landscape of the phase separation changes withthe change of τ as shown on the Fig.2. For τ > >
20 40 60 80 100 12020406080100120 a)
20 40 60 80 100 12020406080100120 b)
20 40 60 80 100 12020406080100120 c)
20 40 60 80 100 12020406080100120 d)
20 40 60 80 100 12020406080100120 e)
20 40 60 80 100 12020406080100120 f)
20 40 60 80 100 12020406080100120 g)
20 40 60 80 100 12020406080100120 h) FIG. 3: Inhomogeneous states are shown for A = 2 . χ =10 and τ = 0 . a ) , − . b ) , − . c ) , − . d ), − e ) , − f ), − g ) , − . h ), respectively. The inhomogeneous state existsa more narrow interval of τ (0 . ≤ τ ≤ − .
37) for A = 2 . A = 3. The decrease of A (increase of Coulombinteraction) leads to the decrease of the interval of τ wherethe phase separation is observed. All figures represent theresults of numerical calculations for N = 128. creasing of τ , the number of such stripes is reduced, andthe rings are compressed. Note that the value of the orderparameter in the center of the stripes is not changed (seeFig.2 c,b,a). When the value of τ becomes negative andwith the further reduction of τ , the loop’s form is chang-ing. They are bent more strongly, and the value of theorder parameter in the ”background” becomes differentfrom zero. With further decreasing of τ the phase sepa-ration becomes more shallow. The difference between Λinside and outside of the ”stripe” is decreasing to zero at τ = τ and the transition to the homogeneous state withΛ=const occurs (see Fig.2d-h).In Fig.3 the change of a form of the inhomogeneousstates is shown for A = 2 . χ = 10 and with the reduc-tion of τ from 0 .
01 to − .
7. From figures 2 and 3 it isclearly seen that the main features of a phase separatedstate are similar. Note that the region of the existenceof a phase separated state for A = 2 . -40 -35 -30 -25 -20 -15 -10 -5 0 50,00,51,01,52,02,53,0
100 200 300 400 500100200300400500
A=3 A=2.5 A=2.2 A=2. A=0 max min
A=2. =-1.25 =10
FIG. 4: The maximum Λ max and minimum Λ min values ofthe order parameter in the inhomogeneous states as a functionof τ for different values of A . The inset shows the distributionof the order parameter Λ in 2D for A = 2 . , τ = − .
25 and ξ =10 . The figure represents the results of numerical calculationsfor N = 512. parison with that for A = 3 . (see Fig.4).Figure 4 shows the dependence of the maximum valueof the order parameter Λ max and the minimum valueΛ min as a function of τ for four different values of A = 2, 2.2, 2.5, and 3 in the phase-separated state. A smoothsolid line shows the change in the order parameter for A = 0, i.e. for the case when there is no interactionbetween the order parameter and the charge ( σ = 0, seeEq. (8)). Fig.4 shows that the phase transition of thesecond order is observed at A = 0 and τ = 0. The orderparameter is zero for τ >
0, and the phase with a nonzeroorder parameter appears below T c τ <
0. The energy ofthis state becomes negative Φ hom < τ < σ = 0). Therefore, the minimumin free energy Φ inhom < min to Λ max (see Fig.4). Consider thechanges of phases that occur when τ decreases for thecase of A = 3. An inhomogeneous phase-separated stateappears as a jump (a phase transition of the first or-der) at τ = τ . The regions with Λ = 0 grows on thebackground with zero order parameter Λ = 0. Λ max =2.2in these regions. The number of such regions increaseswhen τ decreases from τ to 0. Note that the values ofΛ max = 2 . min = 0 do not change in this regionof τ (see Fig.2a,b,c). At τ = 0, the phase-separated statestarts to change. Λ max begins to decrease, and Λ min be-gins to increase (see Fig.2d-h). With further decreasingof τ <
0, the difference between Λ max and Λ min de-creases, and Λ max = Λ min = Λ at τ = τ , therefore aphase transition of the second order from an inhomoge-neous state to a homogeneous state is observed.Fig.5 shows the inhomogeneous distribution of the or-der parameter Λ( r ) and the incremental charge ∆ ρ = ρ ( r ) − ¯ ρ along a line perpendicular to strips for nega- (r) (r) A=2.2 =-3. =10. , , a . u . r
20 40 60 80 100 12020406080100120 x y FIG. 5: The distribution of the order parameter Λ( r )(solidcurve) and ∆ ρ = ρ ( r ) − ¯ ρ (dashed curve) for χ = 10, A = 2 . τ = − r is chosen perpendicular to the strips on the inset.The inset shows the distribution of the order parameter in 2Dfor this set of parameters. The figure represents the results ofnumerical calculations for N = 128. tive value τ = −
3. The order parameter Λ is vary-ing from Λ min = 0 . max = 1 .
3. In the regionof inhomogeneous distribution of the order parameter Λthere exists inhomogeneous distribution of the incremen-tal charge (dashed curve).When χ changes from 3 to 20 (with A = const ), theinterval of τ where inhomogeneous states are formed doesnot change.In Fig.6a τ = τ and τ = τ lines indicate the bound-aries of the inhomogeneous states in axes A − τ for χ = 10for which Φ <
0. The figure shows that with the increaseof the parameter A the regions of τ in which inhomoge-neous distribution of the order parameter Λ( ξ x , ξ y ) wasobserved, is expanding. In Fig.6a the τ = τ line showsthe boundary of the region of metastable inhomogeneousphases. For τ < τ < τ heterogeneous state correspondsto the local minimum of the free energy, but Φ >
0. Thisstate is similar to ”superheated liquid”.In Fig.6b a phase diagram of inhomogeneous states isshown in the axes T − /A . I and IV regions correspondto the homogeneous phases with zero and nonzero orderparameters, respectively. II and III regions correspondto the inhomogeneous phases. 1 /A is proportional to thevalue of the Coulomb interaction γ and inversely propor-tional to the square σ (see Eq.(19)). The phase separa-tion is impossible below the critical end point at A = 1 . x − T axes for A = 2 . σ = 10. Thedecrease in A leads to the decrease of the area where thephase separation is observed.As it was mentioned in the Introduction the phase sep- A <0 =10 a)b) I II III IV T , K I FIG. 6: a) The phase diagram of inhomogeneous states inaxes A − τ for χ = 10 for which ∆Φ <
0. The energy of theinhomogeneous phase-separated state Φ inhom is lower thanthe energy of the homogeneous state Φ hom at τ < τ < τ .∆Φ = Φ inhom − Φ hom . b) The phase diagram plotted in theaxes T − /A , where T is the temperature. The followingparameters were used: T c + σ ¯ ρ = 275K, τα ′ q β ζ = 0.3K − .Region I is a homogeneous non-magnetic state with Λ = 0.Region II is a phase-separated state with zero and nonzeroorder parameters. Region III is a phase-separated state withnonzero order parameter. Region IV is the homogeneous mag-netic state with Λ = 0. The point at A = 1 . A < . aration is observed in manganites as well as in the cupratehigh-temperature superconductors . We discuss inthis paper the inhomogeneous phases in manganiteswhere a sequence of phase transitions to inhomogeneousstates is observed . Let us consider La − x Sr x MnO system. For strontium concentration x = 0 .
125 the fol-lowing sequence of the phase transition is observed. Firstat T = 275K the transition from homogeneous to inho-mogeneous phase I is observed. Then with lowering ofthe temperature the transition to inhomogeneous phaseII takes place. And only then at 140K the system un-dergoes the transition to homogeneous state . Thissequence of the phase transitions is very similar to thatdiscussed in our paper. In addition similar inhomoge-neous states may appear in the cuprates as well . <0 T , K x T=T( =0) T=T( =0.15) T=T( =-4.) FIG. 7: Phase diagram in x − T axes. The parameter A is equal to 2 .
7, and σ = 10. The area between solid anddash dot lines is a phase-separated state, which correspondsto regions II and III in Fig. 6b. Dot line is the temperatureof the phase transition in the absence of a phase-separatedstate. IV. CONCLUSION
In conclusion we consider the theory of phase transitionof the second order, where in addition to the standardexpansion of the free energy in powers of the order pa-rameter, it was introduced the Coulomb interaction andthe interaction of a charge with the order parameter. Thedistribution of the order parameter and the charge dis-tribution in 2D plane, that correspond to the minimumof free energy were found. Numerical calculations wereperformed using the CGM method. Calculations showedthat between the regions which are characterized by con-stant values of the order parameter, there is an area withinhomogeneous distribution of the order parameter andinhomogeneous distribution of the charge. This phaseseparation can exist in a form of one-dimensional stripesor in two dimensional rings or ”snakes”. A series of phasetransitions have been found. With a decrease of temper-ature, first, the phase transition from the homogeneousstate with zero order parameter to the phase-separatedstate with two phases with zero and nonzero order param-eter ( τ >
0) occurs. Then a first-order phase transitionto another phase-separated state, in which both phaseshave different and nonzero values of order parameter (for τ <
Acknowledgments
The authors are grateful to A.V. Leontiev for numerousilluminating discussions. One of us R.F.M. acknowledges the financial support from Slovenian Research Agency,Project BI-RU/16-18-021. S. Jin, T.H. Tiefel, M. McCormack, R.A. Fastnacht, R.Ramesh, and L.H. Chen, Science , 413 (1994). M.Yu. Kagan and K.I. Kugel, Phys. Usp. , 553 (2001);K.I. Kugel, A.L. Rakhmanov, and A.O. Sboychakov, Phys.Rev. Lett. , 267210 (2005). A.O. Sboychakov, K.I. Kugel, and A.L. Rakhmanov, Phys.Rev. B , 014401 (2006). E. Dagotto, T. Hotta, and A. Moreo, Physics Reports ,1 (2001). E.L. Nagaev, Phys. Usp. , 781 (1996). Shen, K.M., Ronning, F., Lu, D.H., Science , 901(2005). J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D.Gu, G. Xu, M. Fujita, and K. Yamada, Nature , 534(2004). E. Blackburn, J. Chang, M. Hucker, A. T. Holmes, N. B.Christensen, R. Liang, D. A. Bonn, W. N. Hardy, U. Rutt,O. Gutowski, M. v. Zimmermann, E. M. Forgan, and S.M. Hayden, Phys. Rev. Lett. , 137004 (2013). Y. Yamakawa and H. Kontani, Phys. Rev. Lett. ,257001 (2015). J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura,and S. Uchida, Nature , 561 (1995). E. H. da S. Neto, P. Aynajian, A. Frano, R. Comin, E.Schierle, E. Weschke, A. Gyenis, J. Wen, J. Schneeloch,Z. Xu, S. Ono, G. Gu, M. Le Tacon, A. Yazdani, Science , 393 (2014). V.V. Kabanov, R.F. Mamin, and T.S. Shaposhnikova, Sov.Phys. JETF , 286 (2009). V. B. Shenoy, T. Gupta, H. R. Krishnamurthy, and T. V.Ramakrishnan, Phys. Rev. Lett. , 097201 (2007). V. B. Shenoy, T. Gupta, H. R. Krishnamurthy, and T. V.Ramakrishnan, Phys. Rev. B , 125121 (2009). J. Miranda and V. Kabanov, J. Supercond. Nov. Magn ,287(2009). J. Miranda and V.V. Kabanov, Physica C , 358 (2008). C. Ortix, J. Lorenzana, and C. Di Castro, Phys. Rev. Lett. , 246402 (2008). C. Ortix, J. Lorenzana, and C. Di Castro,J.Phys.:Condens. Matter , 434229 (2008). C. Ortix, J. Lorenzana, and C. Di Castro, Physica B ,499 (2009). C. Ortix, J. Lorenzana, and C. Di Castro, arxiv0905.1739v1 (2009). C. Ortix, J. Lorenzana, M. Beccaria, and C. Di Castro,Phys. Rev. B , 195107 (2007). R. Jamei, S. Kivelson, and B. Spivak, Phys. Rev. Lett. ,056805 (2005). C. B. Muratov, Phys. Rev. E, , 066108 (2002). P.-G. de Gennes, Phys. Rev. , 141 (1960). Yu. A. Izyumov, Yu. N. Skryabin, Phys. Usp. , 109(2001). C. Zener, Phys. Rev. Lett. , 26 (1959). R. F. Mamin, and V. V. Kabanov, New Journal of Physics , 073011 (2014). J. Deisenhofer, D. Braak, H.-A. Krug von Nidda, J. Hem-berger, R. M. Eremina, V. A. Ivanshin, A. M. Balbashov,G. Jug, A. Loidl, T. Kimura, and Y. Tokura, Phys. Rev.Lett. , 257202 (2005). E. Fradkin and S. A. Kivelson, Nature Physics , 864(2012). J. Xia, E. Schemm, G. Deutscher, S. A. Kivelson, D. A.Bonn, W. N. Hardy, R. Liang, W. Siemons, G. Koster, M.M. Fejer, and A. Kapitulnik, Phys. Rev. Lett. , 127002(2008). C. Ortix, J. Lorenzana, and C. Di Castro, Phys. Rev. B , 245117 (2006). J. Lorenzana and G. Seibold, Phys. Rev. Lett. , 066404(2003). S. Caprara, C. Di Castro, M. Grilli, and D. Suppa, Phys.Rev. Lett.95