/Length 3413 Phys. We obtain the superfluid weight and Berezinskii-Kosterlitz-Thouless (BKT) transition temperature for microscopic tight-binding and low-energy continuum models. 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). [Raman etal., 2009] that TcsubscriptT_{c}italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT is only slightly modified.
Suppose that a given field configuration has 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 . , Phys. The BerezinskiiKosterlitzThouless (BKT) theory3,4 associates this phase transition with the emergence of a topological order, resulting from the pairing of vortices with opposite circulation. / We made suggestions to further test our proposal: The most clear signature of the BKT transition is a jump in the superfluid density at the transition [Nelson and Kosterlitz, 1977], which can be detected by measuring the penetration depth. 0000053628 00000 n
In addition, we observe non-Hall-type transverse signal including Vxy 0 , exactly above the possible BKT transition temperature T BKT, pointing to the existence of thermally excited unbound vortices. 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. The transition between the two different configurations is the KosterlitzThouless phase transition. In BKT theory, the vortex system is descibed by the Hamiltonian, where the stiffness K=ns2/4mkBTsubscriptsuperscriptPlanck-constant-over-2-pi24subscriptK=n_{s}\hbar^{2}/4mk_{B}Titalic_K = italic_n start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT roman_ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 4 italic_m italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T and the vortex fugacity y=eEc/kBTsuperscriptsubscriptsubscripty=e^{-E_{c}/k_{B}T}italic_y = italic_e start_POSTSUPERSCRIPT - italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT / italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T end_POSTSUPERSCRIPT obey the renormalization group (RG) equations [Kosterlitz, 1974; Jos etal., 1977]. WebThis transition is called Berezinskii-Kosterlitz-Thouless (BKT) transition and still remains to be a topic of active research. Just below [1] BKT transitions can be found in several 2-D systems in condensed matter physics that are approximated by the XY model, including Josephson junction arrays and thin disordered superconducting granular films. 3b of [Mizukami etal., 2011]. Phys. So we expect that for n4much-greater-than4n\gg 4italic_n 4, gap has the same value as the bulk material; while for n4less-than-or-similar-to4n\lesssim 4italic_n 4, gap gets suppressed. A direct consequence of the reduced proximity effect is an enhanced c axis resistivity, which can be measured directly in experiment. Rev. Web7.4 Kosterlitz-Thouless transition 7.4 Kosterlitz-Thouless transition. For convenience, we work with the universal cover R of ; Zahn et al. 2D XY-model was extensively studied to capture the nature of BKT transition in these systems. Assume a field (x) defined in the plane which takes on values in {\displaystyle \phi } J.M. Kosterlitz, xb```f``b`c``d@ A;SVF7_P: . B 19, 1855 (1979), This page was last edited on 26 December 2022, at 08:15. In the usual two-fluid picture, the exponent =44\alpha=4italic_ = 4. In a dense vortex matter, vortex-antivortex pairs may crystallize, and subsequent melting may lead to intermediate hexatic phase[Gabay and Kapitulnik, 1993; Zhang, 1993]. A.D. Caviglia, {\displaystyle T_{c}} If >2, we find the usual SR phenomenology with a BKT phase transition. B, Y.Matsuda, Rev. Such a topological phase transition has long been sought yet undiscovered directly in magnetic materials. . The vortex core energy can be written as Ec=(Cc/2)kBTBKTsubscriptsubscriptitalic-2subscriptsubscriptBKTE_{c}=(C\epsilon_{c}/2\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. Lett. Physical Review Letters is a trademark of the American Physical Society, registered in the United States, Canada, European Union, and Japan. Expand 7.6 Renormalization group analysis 7.6 Renormalization group analysis. /Length 4 0 R On the other hand, when However, the magnetic field dependence disagree with the current theoretical picture. = 1 J.N. Eckstein, i , where we have switched to the complex plane coordinates for convenience. B, O.T. Valls, However, one finds a low-temperature quasi-ordered phase with a correlation function (see statistical mechanics) that decreases with the distance like a power, which depends on the temperature. R k 0000027382 00000 n
= 0000061748 00000 n
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. is Boltzmann's constant. Phys. decomposes into the sum of a field configuration with no punctures, {\displaystyle \sum _{i=1}^{N}n_{i}\neq 0} Conclusions: In conclusion, we have proposed that superconducting transition in the heavy fermion superlattice of Mizukami et al. Rev. ISSN 1079-7114 (online), 0031-9007 (print). B, L.Benfatto, j right below the transition temperature, where 0=hc/2esubscript02\Phi_{0}=hc/2eroman_ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_h italic_c / 2 italic_e is the flux quantum. , the relation will be linear xref
. Thin film growth technology recently has advanced to the point that artificial two-dimensional structures can be fabricated with atomic-layer precision. The energy of a single vortex is Phys. Conditions and any applicable , entropic considerations favor the formation of a vortex. C.Castellani, Finite-size scaling, finite-entanglement scaling, short-time critical dynamics, and finite-time scaling, as well as some of their interplay, are considered. B. More precisely, we consider the equation of motion. 0000076421 00000 n
3 0 obj << At low temperatures and large S.Doniach and T 0000070328 00000 n
In order to determine quantitatively the evolution of the dielectric constant near the QCP, more material specific microscopic calculations are needed. 0000002770 00000 n
5(b)), one can see that, only very close to the transition temperature, the dielectric constant changes substantially with scale. This jump from linear dependence is indicative of a KosterlitzThouless transition and may be used to determine For such systems, one thus has Tc=TBKTsubscriptsubscriptBKTT_{c}=T_{\rm BKT}italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT. =7Q.rc^D -`++.Lt$!DRP>\|I:WgF#2R6PbkfZzbp|T C.A. Hooley, B, A.Serafin, d In the presence of competing orders, the vortex core energy is reduced, Ec=Ec(0)|Ec|subscriptsuperscriptsubscript0subscriptE_{c}=E_{c}^{(0)}-|\delta E_{c}|italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT - | italic_ italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT |. B. H.-H. Wen, At low temperatures with TTc0much-less-thansubscript0T\ll T_{c0}italic_T italic_T start_POSTSUBSCRIPT italic_c 0 end_POSTSUBSCRIPT, (T)\xi(T)italic_ ( italic_T ) is of order 0subscript0\xi_{0}italic_ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, which is about the thickness of four layers of CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT. T.Terashima, . S Europhys. 2 Then, | For cuprates and CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT, it has been found that =22\alpha=2italic_ = 2 [Bonn etal., 1993; Kogan etal., 2009]. over any contractible closed path 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. The unrenormalized 2d carrier density ns2D=ns3Ddsuperscriptsubscript2superscriptsubscript3n_{s}^{2D}=n_{s}^{3D}ditalic_n start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 italic_D end_POSTSUPERSCRIPT = italic_n start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 italic_D end_POSTSUPERSCRIPT italic_d is determined by the 3d carrier density ns3D(T)=ns3D(0)b2(0)/b2(T)superscriptsubscript3superscriptsubscript30superscriptsubscript20superscriptsubscript2n_{s}^{3D}(T)=n_{s}^{3D}(0)\lambda_{b}^{2}(0)/\lambda_{b}^{2}(T)italic_n start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 italic_D end_POSTSUPERSCRIPT ( italic_T ) = italic_n start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 italic_D end_POSTSUPERSCRIPT ( 0 ) italic_ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 0 ) / italic_ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_T ), Rev. Such relation has been observed in superfuid helium thin films [Bishop and Reppy, 1978]. Expand 7.6 Renormalization {\displaystyle 1/\Lambda } Subscription L.P. Kadanoff, Phys. x And, even though the basic details of this transition were worked out in Rev. 3 0 obj << 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. , which is the total potential energy of a two-dimensional Coulomb gas. 0000073683 00000 n
M. Hasenbusch, The Two dimensional XY model at the transition temperature: A High precision Monte Carlo study, J. Phys. We present a theoretical study of the Berezinskii-Kosterlitz-Thouless transition of a two-dimensional superfluid in the presence of an externally imposed c 111With smuch-less-thansubscriptparallel-tos\ll\lambda_{\parallel}italic_s italic_ start_POSTSUBSCRIPT end_POSTSUBSCRIPT, the transition temperature now reads Tc=(/2)s(1s2)subscript2subscript12subscriptparallel-toT_{c}=(\pi/2)\rho_{s}(1-\frac{s}{2\lambda_{\parallel}})italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = ( italic_ / 2 ) italic_ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( 1 - divide start_ARG italic_s end_ARG start_ARG 2 italic_ start_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG ), where ssitalic_s is the layer spacing, subscriptparallel-to\lambda_{\parallel}italic_ start_POSTSUBSCRIPT end_POSTSUBSCRIPT is the in-plane penetration depth, and s=02s/(1632)subscriptsuperscriptsubscript0216superscript3superscriptsubscriptparallel-to2\rho_{s}=\Phi_{0}^{2}s/(16\pi^{3}\lambda_{\parallel}^{2})italic_ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = roman_ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_s / ( 16 italic_ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_ start_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) is the in-plane superfluid stiffness, which can be measured directly. ( R [Kogan, 2007; Benfatto etal., 2009]). P.Ziemann, Itbeginswiththediscoveryofpossibleeldcongurationsthatone T.Shibauchi, 0000075688 00000 n
For two dimensional systems with continuous Abelian symmetry, despite the lack of broken symmetry due to strong fluctuations, there exists a finite temperature phase transition mediated by topological defects, e.g. 0000062403 00000 n
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_. Phys. HvzsuperscriptsubscriptH_{v}^{z}italic_H start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT is a superpostion of the magnetic fields generated by vortices at different locations, Hvz()=iniH0(i)superscriptsubscriptsubscriptsubscriptsubscript0subscriptH_{v}^{z}(\mathbf{r})=\sum_{i}n_{i}H_{0}({\mathbf{r}}-{\mathbf{R}}_{i})italic_H start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT ( bold_r ) = start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( bold_r - bold_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ), with nisubscriptn_{i}italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT the vorticity. Lett. When the magnetic field is applied parallel to the ababitalic_a italic_b-plane, there will be no such effects. 0000017872 00000 n
WebMy parents, Hans Walter and Johanna Maria Kosterlitz (Gresshner) had fled Hitlers Germany in 1934 because my father, a non-practicing Jew, came from a Jewish family and was forbidden to marry a non-Jewish woman like my mother or to be paid as a medical doctor in Berlin. The transition is named for condensed matter physicists Vadim B n At the transition, the renormalized penetration depth satisfies the relation [Nelson and Kosterlitz, 1977] kBTBKT=02d/3222subscriptsubscriptBKTsuperscriptsubscript0232superscript2superscript2k_{B}T_{\rm BKT}=\Phi_{0}^{2}d/32\pi^{2}\lambda^{2}italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT = roman_ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d / 32 italic_ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT (Eq. Europhys. The behavior of gap and TcsubscriptT_{c}italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT for different number of CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT layers is shown in Fig. and F"$yIVN^(wqe&:NTs*l)A;.}: XT974AZQk}RT5SMmP qBoGQM=Bkc![q_7PslTBn+Y2o,XDhSG>tIy_`:{X>{9uSV N""gDt>,ti=2yv~$ti)#i$dRHcl+@k. .lgKG7H}e
Jm#ivK%#+2X3Zm6Dd;2?TX8 D}E^|$^9Ze'($%78'!3BQT%3vhl.YPCp7FO'Z0\ uC0{Lxf? All rights reserved. WebRemarkably, a Berezinskii-Kosterlitz-Thouless transition with TBKT 310 mK is revealed in up to 60 nm thick flakes, which is nearly an order of magnitude thicker than the rare examples of two-dimensional superconductors exhibiting such a transition. J.-M. Triscone, 0000072221 00000 n
Howard, Phys. L Rev. B. M.Mondal, Here we elaborate on the understanding of the dielectric constant csubscriptitalic-\epsilon_{c}italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT. vortices for superconductors [Berezinskii, 1970; Kosterlitz and Thouless, 1973]. {\displaystyle \phi _{0}} {\displaystyle x_{i},i=1,\dots ,N} The KosterlitzThouless transition can be observed experimentally in systems like 2D Josephson junction arrays by taking current and voltage (I-V) measurements. i Lett. After pointing out the relevance of this nontrivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. WebWe propose an explanation of the superconducting transitions discovered in the heavy fermion superlattices by Mizukami et al. G.Orkoulas and Including the effect of screening, KKitalic_K changes with the scale rritalic_r. , The transition is named for condensed matter physicists Vadim Berezinskii, John M. Kosterlitz and David J. 60 0 obj<>
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Generated on Sat Dec 17 01:38:46 2022 by, Y.Mizukami, 0000058535 00000 n
0000007586 00000 n
A salient feature of the heavy-fermion superconductor CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT is the proximity to an antiferromagnetic quantum critical point (QCP). The BKTHNY theory is underlain by the mechanism of quasi-long-range order Thus the vortex core energy is significantly reduced due to magnetic fluctuations. H.Ikeda, / WebThe BerezinskiiKosterlitzThouless transition (BKT transition) is a phase transition of the two-dimensional (2-D) XY model in statistical physics. This is because the expected ordered phase of the system is destroyed by transverse fluctuations, i.e. i Sketch of the possible phases of the model: ordered with magnetization (solid black), BKT QLRO (dashed light gray), disordered (dashed dark gray). This is a non perturbative result, occurring even for extremely low dissipation magnitude. Sign up to receive regular email alerts from Physical Review Letters. At large temperatures and small (Nature Physics 7, 849 (2011)) in terms of At temperatures below this, vortex generation has a power law correlation. This enables us to measure the phase correlation function, which changes from an algebraic to an exponential decay when the system crosses the Berezinskii-Kosterlitz-Thouless (BKT) transition. {\displaystyle 2\pi } {\displaystyle S=2k_{\rm {B}}\ln(R/a)} The following discussion uses field theoretic methods. . ii) Then we extract from the resistivity data the transition temperature TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT. Phys. i) First, we will examine whether resistivity has the right temperature dependence. Natl. DOI:https://doi.org/10.1103/PhysRevLett.127.156801. E.D. Bauer x 2c in [Mizukami etal., 2011]). Information about registration may be found here. 0000073805 00000 n
One can also see that a small parallel field will not change TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT, i.e. We can imagine that the theory is defined up to some energetic cut-off scale 3 It would be interesting to look for such phases in systems close to a magnetic QCP, where vortex core energy can be substantially reduced. A.T. Fiory, 4 can be observed experimentally. S.Adachi, 7.5 Interaction energy of vortex pairs 7.5 Interaction energy of vortex pairs. WebThe Kosterlitz-Thouless Transition Henrik Jeldtoft Jensen Department of Mathamtics Imperial College Keywords: Generalised rigidity, Topological defects, Two Dimensional this distance increases, and the favoured configuration becomes effectively the one of a gas of free vortices and antivortices. x F It is therefore desirable to have a well-controlled, readily-tunable system to investigate the BKT physics. 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. More extensive numerical studies of proximity effect in N/S junctions have been carried out recently [Valls etal., 2010], where it was shown that proximity effect is substantially suppressed with moderate mismatch of Fermi energies. And, even though the basic details of this transition were worked out in J.V. Jos, H0()subscript0H_{0}({\mathbf{r}})italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( bold_r ) can be obtained from its Fourier transform H0()=0/(1+2k2)subscript0subscript01superscript2superscript2H_{0}(\mathbf{k})=\Phi_{0}/(1+\lambda^{2}k^{2})italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( bold_k ) = roman_ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT / ( 1 + italic_ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ), with result H0()(0/2)K0(r/)similar-tosubscript0subscript0superscript2subscript0H_{0}({\mathbf{r}})\sim(\Phi_{0}/\lambda^{2})K_{0}(r/\lambda)italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( bold_r ) ( roman_ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT / italic_ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_r / italic_ ), . Without screening, KKitalic_K takes the bulk value K(0)=02d/163b2(T)kBT0superscriptsubscript0216superscript3subscriptsuperscript2bsubscriptK(0)=\Phi_{0}^{2}d/16\pi^{3}\lambda^{2}_{\rm b}(T)k_{B}Titalic_K ( 0 ) = roman_ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d / 16 italic_ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT ( italic_T ) italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T, with bsubscriptb\lambda_{\rm b}italic_ start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT the bulk penetration depth.