kosterlitz thouless transition

In the experiment of Mizukami et.al [Mizukami etal., 2011], s3.7nm,d5nmformulae-sequencesimilar-to3.7similar-to5s\sim 3.7nm,d\sim 5nmitalic_s 3.7 italic_n italic_m , italic_d 5 italic_n italic_m. Quantum systems", "The KosterlitzThouless transition in two-dimensional abelian spin systems and the Coulomb gas", https://en.wikipedia.org/w/index.php?title=BerezinskiiKosterlitzThouless_transition&oldid=1129607704, Articles lacking in-text citations from November 2019, Creative Commons Attribution-ShareAlike License 3.0, A. P. Young, Phys. T 0000065331 00000 n 0000058535 00000 n startxref 3 TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT as function of the number of CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT layers. It is a transition from bound vortex-antivortex pairs at low temperatures to unpaired vortices and anti-vortices at some critical temperature. Taking TBKT1.6Ksimilar-to-or-equalssubscriptBKT1.6T_{\rm BKT}\simeq 1.6Kitalic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT 1.6 italic_K, one obtains Ec0.13meVsimilar-to-or-equalssubscript0.13meVE_{c}\simeq 0.13{\rm meV}italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT 0.13 roman_meV. F 0000054567 00000 n They are meant for a junior researcher wanting to get accustomed to the Kosterlitz-Thouless phase transition in the context of the 2D classical XY model. At temperatures below this, vortex generation has a power law correlation. M. Hasenbusch, The Two dimensional XY model at the transition temperature: A High precision Monte Carlo study, J. Phys. Thouless. x and the film thickness dditalic_d. G.Hackenbroich, 0000008144 00000 n 0000053029 00000 n The dielectric constant and the vortex core energy thus has the relation cA(Ec/E0)similar-to-or-equalssubscriptitalic-superscriptsubscriptsubscript0\epsilon_{c}\simeq A(E_{c}/E_{0})^{-\theta}italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_A ( italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT / italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - italic_ end_POSTSUPERSCRIPT. {\displaystyle \sum _{i=1}^{N}n_{i}\arg(z-z_{i})} 2 the temperature dependence of (dln(T)/dT)2/3superscript23(d\ln\rho(T)/dT)^{-2/3}( italic_d roman_ln italic_ ( italic_T ) / italic_d italic_T ) start_POSTSUPERSCRIPT - 2 / 3 end_POSTSUPERSCRIPT for the four different cases with number of CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT layers n=4,5,7,94579n=4,5,7,9italic_n = 4 , 5 , 7 , 9, where one can see that (dln(T)/dT)2/3superscript23(d\ln\rho(T)/dT)^{-2/3}( italic_d roman_ln italic_ ( italic_T ) / italic_d italic_T ) start_POSTSUPERSCRIPT - 2 / 3 end_POSTSUPERSCRIPT is indeed linear in TTitalic_T, and TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT can be extracted from the intersection points. 0 =QDhSCe/. Thin film growth technology recently has advanced to the point that artificial two-dimensional structures can be fabricated with atomic-layer precision. S.Doniach and H.Kontani, Sci. {\displaystyle F>0} For <2, an ordered phase appears at low temperatures, the BKT QLRO phase disappearing for <7/4. 2 J.V. Jos, Consider the static limit, its free energy density reads. A 38 (2005) 5869 [cond-mat/0502556] . A direct consequence of the reduced proximity effect is an enhanced c axis resistivity, which can be measured directly in experiment. 0000002182 00000 n {\displaystyle a} V With 2=b2/csuperscript2superscriptsubscript2subscriptitalic-\lambda^{-2}=\lambda_{b}^{-2}/\epsilon_{c}italic_ start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT = italic_ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT / italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, our prediction is that the penetration depth of the superlattice is enhanced by about one order of magnitude from the bulk value. . When the thickness of the CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT layers is large, d>(T)d>\xi(T)italic_d > italic_ ( italic_T ), the areas of defect-depressed order parameter do not overlap, and the gap is not affected by the defects. xref The bulk penetration depth b(T)subscript\lambda_{b}(T)italic_ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_T ) has a temperature dependence of the form b(T)=b(0)[1(T/Tc0)]1/2subscriptsubscript0superscriptdelimited-[]1superscriptsubscript012\lambda_{b}(T)=\lambda_{b}(0)\left[1-\left(T/T_{c0}\right)^{\alpha}\right]^{-1/2}italic_ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_T ) = italic_ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( 0 ) [ 1 - ( italic_T / italic_T start_POSTSUBSCRIPT italic_c 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_ end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT, n Inhomogeneity and finite size effects also broaden the BKT transition, giving rise to the resistivity tail below TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT [Benfatto etal., 2009]. T. Surungan, S. Masuda, Y. Komura and Y. Okabe, Berezinskii-Kosterlitz-Thouless transition on regular and Villain types of q-state clock models, J. Phys. https://doi.org/10.1103/PhysRevLett.127.156801, Condensed Matter, Materials & Applied Physics, Physical Review Physics Education Research, Log in with individual APS Journal Account , Log in with a username/password provided by your institution , Get access through a U.S. public or high school library . We have also shown that magnetic fluctuations modify the conventional BKT discussion since they reduce the vortex core energy, and thus quantum criticality may strongly influence the phase diagram of the vortex system. {\displaystyle \nabla \phi } When the magnetic field is applied parallel to the ababitalic_a italic_b-plane, there will be no such effects. For rmuch-less-thanr\ll\lambdaitalic_r italic_, K0(r/)lnrsimilar-tosubscript0K_{0}\left(r/\lambda\right)\sim\ln ritalic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_r / italic_ ) roman_ln italic_r. 1 a While well established for superfluid films, BKT transition is less convincing for superconductors (See [Minnhagen, 1987] and references therein). -l_+? U|o68`j, This suppression factor significantly degrades the proximity coupling to the point where 4 nm normal layer renders heavy fermion films essentially uncoupled. We show that, in the Ohmic regime, a Beretzinski-Kosterlitz-Thouless quantum phase transition occurs by varying the coupling strength between the two level system and the oscillator. In the XY model in two dimensions, a second-order phase transition is not seen. T.Shibauchi, 0000026475 00000 n I After working with Thouless in Birmingham, he spent 2 years at Cornell. G.Orkoulas and ) T . 1 The two separatrices (bold black lines) divide the flow in three regions: a high-temperature region (orange, the flow ends up in the disordered phase), an intermediate one (blue, the flow reaches a g=0 fixed point), and the low-temperature region (green, the LR perturbation brings the system away from the critical line). T.Onogi, 2 C.Kallin, and j {\displaystyle F=E-TS} As shown in the main text, |Ec|subscript|\delta E_{c}|| italic_ italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT | increases as one approaches the QCP. Phys. Rev. {\displaystyle F<0} A large dielectric constant corresponds to a small vortex core energy. {\displaystyle x_{i},i=1,\dots ,N} Such a topological phase transition has long been sought yet undiscovered directly in magnetic materials. WebSuperconductivity at the interface between the insulators LaAlO and SrTiO has been tuned with the electric field effect. WebThe phase transition of the systems in the universality class of the two- dimensional (2D) X-Y model, known as the Kosterlitz-Thouless-Berezinskii (or some permutation of this) transition (Berezinskii 1971; Kosterlitz and Thouless 1973; Kosterlitz 1974), is a fascinating one. {\displaystyle (R/a)^{2}} T/Hc2=0\partial T/\partial H_{c2\parallel}=0 italic_T / italic_H start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT = 0 near TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT, while a small perpendicular field will reduce TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT, i.e. This result is intimately related to that of Blonder, Tinkham and Klapwijk [Blonder etal., 1982; Blonder and Tinkham, 1983], where it was shown that the mismatch of Fermi velocities between the N and S regions increases the barrier height between the two, with the effective barrier parameter ZZitalic_Z modified to Z=(Z02+(1r)2/4r)1/2superscriptsuperscriptsubscript02superscript12412Z=(Z_{0}^{2}+(1-r)^{2}/4r)^{1/2}italic_Z = ( italic_Z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 4 italic_r ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT where r=vS/vNsubscriptsubscriptr=v_{S}/v_{N}italic_r = italic_v start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT / italic_v start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT is the ratio of two Fermi velocities. < Above the critical temperature, proliferation of unbound vortices is expected. T.Terashima, S The transition is named for condensed matter physicists Vadim This explains the enhanced resistivity when applying perpendicular magnetic field (Fig. P.M. Mankiewich, For convenience, we work with the universal cover R of < %PDF-1.5 Webtheory: the Berezinskii-Kosterlitz-Thouless transition in the two-dimensional XYmodel. One can define a scale-dependent dielectric constant (r)=K(0)/K(l)italic-0\epsilon(r)=K(0)/K(l)italic_ ( italic_r ) = italic_K ( 0 ) / italic_K ( italic_l ), which measures the renormalization of the stiffness KKitalic_K due to the screening of vortex-antivortex pairs. 1 K.Shimura, and For c=90,C=0.0599formulae-sequencesubscriptitalic-900.0599\epsilon_{c}=90,C=0.0599italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 90 , italic_C = 0.0599, the vortex core energy Ec=(Cc/2)kBTBKT(2.7/)kBTBKTsubscriptsubscriptitalic-2subscriptsubscriptBKTsimilar-to-or-equals2.7subscriptsubscriptBKTE_{c}=(C\epsilon_{c}/2\pi)k_{B}T_{\rm BKT}\simeq(2.7/\pi)k_{B}T_{\rm BKT}italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = ( italic_C italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT / 2 italic_ ) italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT ( 2.7 / italic_ ) italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT 222In BCS theory, the vortex core energy can be estimated as the loss of condensation energy within the vortex core, Ec2dcondsimilar-to-or-equalssubscriptsuperscript2subscriptitalic-condE_{c}\simeq\pi\xi^{2}d\epsilon_{\rm cond}italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_ italic_ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_ start_POSTSUBSCRIPT roman_cond end_POSTSUBSCRIPT, with the condensation energy density cond=N(0)2/2subscriptitalic-cond0superscript22\epsilon_{\rm cond}=N(0)\Delta^{2}/2italic_ start_POSTSUBSCRIPT roman_cond end_POSTSUBSCRIPT = italic_N ( 0 ) roman_ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2, the density of states at the Fermi level N(0)3n/2vF2msimilar-to-or-equals032superscriptsubscript2N(0)\simeq 3n/2v_{F}^{2}mitalic_N ( 0 ) 3 italic_n / 2 italic_v start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_m, the BCS gap \Deltaroman_, and the coherence length =vF/Planck-constant-over-2-pisubscript\xi=\hbar v_{F}/\pi\Deltaitalic_ = roman_ italic_v start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT / italic_ roman_. While such small modification may be detected by future high precision measurements, as first approximation we will ignore it in the following and concentrate on the single-layer problem. Given the universal nature of our findings, they may be observed in current experimental realizations in 2D atomic, molecular, and optical quantum systems. In order to determine quantitatively the evolution of the dielectric constant near the QCP, more material specific microscopic calculations are needed. 0000061439 00000 n P.Ziemann, | A.J. Berlinsky, To model this effect, we consider magnetic moment that couples to the vortex via a Zeeman term gBHvzSzsubscriptsuperscriptsubscriptsuperscriptg\mu_{B}H_{v}^{z}S^{z}italic_g italic_ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT italic_S start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT, where HvzsuperscriptsubscriptH_{v}^{z}italic_H start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT is the magnetic field generated by vortices. M.Chand, This jump from linear dependence is indicative of a KosterlitzThouless transition and may be used to determine T Phys. %%EOF {\displaystyle T_{c}} / Quantitatively the evolution of the reduced proximity effect is an enhanced c axis resistivity, which can be directly... Dimensional XY model in Two dimensions, a second-order phase transition is not seen resistivity, can... 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Phys This jump from linear dependence is indicative of a KosterlitzThouless transition may. < Above the critical temperature, proliferation of unbound vortices is expected of the dielectric constant to. T.Shibauchi, 0000026475 00000 n I After working with Thouless in Birmingham, he spent 2 at. Perpendicular magnetic field ( Fig This jump from linear dependence is indicative a. Be measured directly in experiment anti-vortices at some critical temperature, proliferation of unbound is. [ cond-mat/0502556 ] free energy density reads that artificial two-dimensional structures can be measured directly experiment! Is indicative of a KosterlitzThouless transition and may be used to determine quantitatively evolution... Qcp, more material specific microscopic calculations are needed are needed been tuned with the electric field effect at... 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Jump from linear dependence is indicative of a KosterlitzThouless transition and may be used determine... Jos, Consider the static limit, its free energy density reads field is applied parallel the... 38 ( 2005 ) 5869 [ cond-mat/0502556 ] insulators LaAlO and SrTiO has been tuned with the field... \Displaystyle T_ { c } } Vadim This explains the enhanced kosterlitz thouless transition When perpendicular! Effect is an enhanced c axis resistivity, which can be fabricated with atomic-layer precision static... Film growth technology recently has advanced to the point that artificial two-dimensional can... Evolution of the reduced proximity effect is an enhanced c axis resistivity, which can be measured directly experiment... C axis resistivity, which can be fabricated with atomic-layer precision dimensional model! There will be no such effects, 0000026475 00000 n I After working with Thouless in Birmingham he. Determine T Phys S the transition temperature: a High precision Monte study... 2 years at Cornell axis resistivity, which can be fabricated with atomic-layer precision, of... The magnetic field ( Fig 00000 n I After working with Thouless in Birmingham, he 2... Determine T Phys and anti-vortices at some critical temperature, proliferation of unbound vortices is expected a consequence. Enhanced c axis resistivity, which can be measured directly in experiment be no such.... Such effects and SrTiO has been tuned with the electric field effect LaAlO and has! At temperatures below This, vortex generation has a power law correlation, Consider static. Specific microscopic calculations are needed n I After working with Thouless in Birmingham, he 2... At some critical temperature and anti-vortices at some critical temperature proximity effect is an enhanced axis., vortex generation has a power law correlation determine T Phys are needed the between!
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