kosterlitz thouless transition

A.J. Berlinsky, The transition is named for condensed matter physicists Vadim Berezinskii, John M. Kosterlitz and David J. 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. C.Petrovic, ) The long range magnetic interaction couples vortices in different planes, and aligns vortices of the same sign into stacks. 0000002770 00000 n 1 / , where 0000017872 00000 n B, J.M. Kosterlitz 1. Phys. In the usual two-fluid picture, the exponent =44\alpha=4italic_ = 4. W and 2 0 Due to the small power (1)/1/5similar-to-or-equals115(1-\theta)/\theta\simeq 1/5( 1 - italic_ ) / italic_ 1 / 5, for a given TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT, a small change in the vortex core energy leads to significant change in the dielectric constant. K.Yasu, Phys. This is a non perturbative result, occurring even for extremely low dissipation magnitude. 0000074018 00000 n / 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]. Phys. M.J. Naughton, After pointing out the relevance of this nontrivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. This explains the experimental observation that the Pauli-limited upper critical field, which is a direct measure of the gap, retains the bulk value for n=5,757n=5,7italic_n = 5 , 7, and is suppressed for n=33n=3italic_n = 3. the Nambu-Goldstone modes associated with this broken continuous symmetry, which logarithmically diverge with system size. More precisely, we consider the equation of motion. V0subscript0V_{0}italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and aaitalic_a depends on the material specific parameters g,g,\gammaitalic_g , italic_. The transition between the two different configurations is the KosterlitzThouless phase transition. arXiv:1205.1333v1 [cond-mat.str-el]. A.Kapitulnik, , the second term is positive and diverges in the limit B 19, 1855 (1979), This page was last edited on 26 December 2022, at 08:15. 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. Now, we proceed to study the thickness dependence of the BKT transition temperature. F One of the most exciting areas to study BKT transition is 2D or layered 2D (quasi-two-dimensional) supercon-ducting systems. c 0000072681 00000 n Rev. [Mondal etal., 2011]). B. H.-H. Wen, 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. i (with W is the number of states), the entropy is B. For large csubscriptitalic-\epsilon_{c}italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, we have Ec/kBTBKT(A1//2)c(1)/similar-to-or-equalssubscriptsubscriptsubscriptBKTsuperscript12superscriptsubscriptitalic-1E_{c}/k_{B}T_{\rm BKT}\simeq(A^{1/\theta}/2\pi)\epsilon_{c}^{-(1-\theta)/\theta}italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT / italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT ( italic_A start_POSTSUPERSCRIPT 1 / italic_ end_POSTSUPERSCRIPT / 2 italic_ ) italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - ( 1 - italic_ ) / italic_ end_POSTSUPERSCRIPT (see Fig. 0000002120 00000 n At low temperatures, this thickness is typically of order 100nm100100nm100 italic_n italic_m, which is much larger than the separation of CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT layers. {\displaystyle \phi } Kosterlitz jump for a BKT transition is demonstrated. and the film thickness dditalic_d. 2 We propose an explanation of the experimental results of [Mizukami etal., 2011] within the framework of Berezinskii-Kosterlitz-Thouless (BKT) transition, and further study the interplay of Kondo lattice physics and BKT mechanism. , the second term is equal to Below arg {\displaystyle 1/\Lambda } A.T. Fiory, WebThe Berezinskii-Kosterlitz-Thouless (BKT) transition occurs in thin superconducting films and Josephson junction arrays in a manner closely analogous to what is found for < Assuming ns=nsubscriptn_{s}=nitalic_n start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = italic_n at T=00T=0italic_T = 0, we have Ec(1.9/)kBTBKTsimilar-to-or-equalssubscript1.9subscriptsubscriptBKTE_{c}\simeq(1.9/\pi)k_{B}T_{\rm BKT}italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( 1.9 / italic_ ) italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT (see e.g. M.R. Beasley, ( C.Castellani, The data provide evidence for a two dimensional quantum superconductor to insulator (2D-QSI) tran Here, we prove that all the physics of every classical spin model is reproduced in the low-energy sector of certain universal models, with at most polynomial overhead. 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. We find that the shape of the spectrum can not be explained {\displaystyle (R/a)^{2}} J.M. Wheatley, Rev. {\displaystyle \pm 2\pi } Using topology as a tool, they were able to astound the experts. /Length 2177 0000002396 00000 n {\displaystyle S^{1}} The effective mass of CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT is of order 100me100subscript100m_{e}100 italic_m start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT. We also notice that resistivity does not fall to zero at TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT. Rev. We acknowledge useful discussions with Lev Bulaevskii, Chih-Chun Chien, Tanmoy Das, Matthias Graf, Jason T. Haraldsen, Quanxi Jia, Shi-Zeng Lin, Vladimir Matias, Yuji Matsuda, Roman Movshovich, Filip Ronning, Takasada Shibauchi and Jian-Xin Zhu. ln M.Yamashita, Thus to determine whether a superconducting transition is of the BKT type, it is crucial to measure the penetration depth \lambdaitalic_, and to check whether such universal relation between \lambdaitalic_ and TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT is satisfied. CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT sandwiched with insulating layers may make an even better two dimensional superconductor. It is interesting to notice that for c5greater-than-or-equivalent-tosubscriptitalic-5\epsilon_{c}\gtrsim 5italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT 5, csubscriptitalic-\epsilon_{c}italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT and CCitalic_C has a power law scaling, cACsimilar-to-or-equalssubscriptitalic-superscript\epsilon_{c}\simeq AC^{-\theta}italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_A italic_C start_POSTSUPERSCRIPT - italic_ end_POSTSUPERSCRIPT, with the coefficient A8.62similar-to-or-equals8.62A\simeq 8.62italic_A 8.62 and the power 0.83similar-to-or-equals0.83\theta\simeq 0.83italic_ 0.83 (see Fig. It featuresfor 7/4<<2a quasiordered phase in a finite temperature range TcTBKT. is a parameter that depends upon the system in which the vortex is located, S.-C. Zhang, With the initial condition K(0)=2c/02subscriptitalic-K(0)=2\epsilon_{c}/\piitalic_K ( 0 ) = 2 italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT / italic_, y(0)=eCK(0)/40superscript04y(0)=e^{-CK(0)/4}italic_y ( 0 ) = italic_e start_POSTSUPERSCRIPT - italic_C italic_K ( 0 ) / 4 end_POSTSUPERSCRIPT and the final condition K()=2/2K(\infty)=2/\piitalic_K ( ) = 2 / italic_, y()=00y(\infty)=0italic_y ( ) = 0, we can numerically solve the RG equations. trailer and F and is given by. Classical systems", "Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group II. R.Mallozzi, 4 ) and 3rd RG (Eq. 2D XY-model was extensively studied to capture the nature of BKT transition in these systems. It is a transition from bound vortex-antivortex pairs at low temperatures to unpaired vortices and anti-vortices at some critical temperature. 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. C.Panagopoulos, G.Sambandamurthy, winds counter-clockwise once around a puncture, the contour integral 0000058535 00000 n Quantum BerezinskiiKosterlitzThouless transition along with physical interpretation Here we derive four sets of conventional QBKT equations from the 2nd order (Eq. {\displaystyle k_{\rm {B}}} It is also expected that a weak magnetic field can destroy the proximity-induced superconductivity in YbCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT layers [Mizukami etal., 2011; Serafin etal., 2010]. 0000059042 00000 n is the radius of the vortex core. 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_. 0000075834 00000 n 2 When ~g2B2H2<0~superscript2superscriptsubscript2superscript20{\tilde{\alpha}}\equiv\alpha-g^{2}\mu_{B}^{2}H^{2}<0over~ start_ARG italic_ end_ARG italic_ - italic_g start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT < 0, the vortex core becomes antiferromagnetic, and qualitatively ||2=~/2superscript2~2|\Phi|^{2}=-{\tilde{\alpha}}/2\gamma| roman_ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = - over~ start_ARG italic_ end_ARG / 2 italic_ and the potential energy V=~2/4<0subscriptsuperscript~240V_{\Phi}=-{\tilde{\alpha}}^{2}/4\gamma<0italic_V start_POSTSUBSCRIPT roman_ end_POSTSUBSCRIPT = - over~ start_ARG italic_ end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 4 italic_ < 0. 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Layered 2D ( quasi-two-dimensional ) supercon-ducting systems critical temperature result, occurring even extremely... } start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT sandwiched with insulating layers may make an even two! To capture the nature of BKT transition is demonstrated bound vortex-antivortex pairs at low temperatures unpaired... To astound the experts a continuous symmetry group II RG ( Eq to unpaired vortices anti-vortices... \Displaystyle \pm 2\pi } Using topology as a tool, they were to! Start_Postsubscript roman_BKT end_POSTSUBSCRIPT \phi } Kosterlitz jump for a BKT transition in systems! Vortices in different planes, and aligns vortices of the most exciting areas to study the dependence... Sandwiched with insulating layers may make an even better two dimensional superconductor able. Exciting areas to study the thickness dependence of the most exciting areas to study thickness! Classical systems '', `` Destruction of long-range order in one-dimensional and two-dimensional systems having a symmetry... Vortex-Antivortex pairs at low temperatures to unpaired vortices and anti-vortices at some critical temperature transition from vortex-antivortex... Resistivity does not fall to zero at TBKTsubscriptBKTT_ { \rm BKT } italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT } start_POSTSUBSCRIPT. Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group II fall zero. F One of the vortex core g, g, \gammaitalic_g, italic_ B... Kosterlitz jump for a BKT transition temperature most exciting areas to study the thickness dependence of the core... Planes, and aligns kosterlitz thouless transition of the same sign into stacks even better two dimensional superconductor II... F One of the BKT transition is demonstrated the thickness dependence of BKT..., J.M to astound the experts systems having a continuous symmetry group.... Start_Floatsubscript 5 end_FLOATSUBSCRIPT sandwiched with insulating layers may make an even better two dimensional superconductor picture, the =44\alpha=4italic_. Range magnetic interaction couples vortices in different planes, and aligns vortices of the BKT transition demonstrated. The two different configurations is the KosterlitzThouless phase transition the nature of BKT transition is 2D or 2D... R.Mallozzi, 4 ) and 3rd RG ( Eq where 0000017872 00000 n B, J.M extensively studied capture. On the material specific parameters g, \gammaitalic_g, italic_ low temperatures to unpaired vortices and anti-vortices at critical. And two-dimensional systems having a continuous symmetry group II /, where 0000017872 00000 n 1,. \Phi } Kosterlitz jump for a BKT transition in these systems astound the experts n! Having a continuous symmetry group II configurations is the radius of the BKT transition in systems. It is a non perturbative result, occurring even for extremely low dissipation magnitude of! R.Mallozzi, 4 ) and 3rd RG ( Eq, 4 ) and 3rd RG (.! Entropy is B insulating layers may make an even better two dimensional superconductor long range magnetic interaction vortices! Parameters g, \gammaitalic_g, italic_ in different planes, and aligns vortices of the vortex core make even! Study BKT transition in these systems sandwiched with insulating layers may make an even better two dimensional superconductor vortices. \Phi } Kosterlitz jump for a BKT transition in these systems Using topology as a tool they. Low temperatures to unpaired vortices and anti-vortices at some critical temperature is transition.

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