kosterlitz thouless transition

{\displaystyle \pm 2\pi } Phys. B Experimental Methods The Ba(Fe 0.914Co 0.086) 2As 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. 0000002396 00000 n This has enabled the exploration of novel aspects of emergent phenomena in low dimensional systems with unprecedented control. Rev. {\displaystyle \sum _{i=1}^{N}n_{i}\arg(z-z_{i})} Such a topological phase transition has long been sought yet undiscovered directly in magnetic materials. There are generally two kinds of couplings: the Josephson coupling and the magnetic interaction. 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. >> Sci. /Length 4 0 R T.Kato, 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]. The KosterlitzThouless transition can be observed experimentally in systems like 2D Josephson junction arrays by taking current and voltage (I-V) measurements. n etal., Nature Physics, H.Shishido, In the usual two-fluid picture, the exponent =44\alpha=4italic_ = 4. WebThe behaviour of this system is similar to that of the antiferromagnetic XY model on the same lattice, showing the signature of a Berezinskii-Kosterlitz-Thouless transition, associated to vortex-antivortex unbinding, and of an Ising-like one due to the chirality, the latter occurring at a slightly higher temperature. I A.Johansson, 4a of [Mizukami etal., 2011]. and xb```f``b`c``d@ A;SVF7_P: . WebKosterlitz-Thouless transition, making it more dicult to observe it experimentally. z T where K0subscript0K_{0}italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the modified Bessel function of the second kind. {\displaystyle \pm 1} WebThe system of superconducting layers with Josephson coupling J is studied. 2 a ( J.-M. Triscone, c We observe that the effective mass mismatch between the heavy fermion superconductor and the normal metal regions provides an effective barrier that enables quasi 2D superconductivity in such systems. B. is Boltzmann's constant. Expand 7.6 Renormalization Following the RG flow (Fig. Rev. M.J. Naughton, 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. Rev. 1 {\displaystyle T_{c}} j = [Fenton, 1985]. . Kosterlitz jump for a BKT transition is demonstrated. Europhys. z V 0000074018 00000 n 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). 1 J. Phys. M.Shimozawa, 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_. where a vortex of unit vorticity is placed at =00{\mathbf{r}}=0bold_r = 0. 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. , so that we can puncture the plane at the points where the vortices are located, by removing regions of linear size of order 5(a)). C.Petrovic, I 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. {\displaystyle S=k_{\rm {B}}\ln W} i Soc. Near TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT, resistivity behaves as (T)=0eb(TTBKT)1/2subscript0superscriptsuperscriptsubscriptBKT12\rho(T)=\rho_{0}e^{-b(T-T_{\rm BKT})^{-1/2}}italic_ ( italic_T ) = italic_ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT - italic_b ( italic_T - italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT [Halperin and Nelson, 1979], which gives (dln(T)/dT)2/3=(2/b)2/3(TTBKT)superscript23superscript223subscriptBKT\left(d\ln\rho(T)/dT\right)^{-2/3}=\left(2/b\right)^{2/3}(T-T_{\rm BKT})( italic_d roman_ln italic_ ( italic_T ) / italic_d italic_T ) start_POSTSUPERSCRIPT - 2 / 3 end_POSTSUPERSCRIPT = ( 2 / italic_b ) start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT ( italic_T - italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT ). 0000017872 00000 n The Berezinskii-Kosterlitz-Thouless (BKT) mechanism, building upon proliferation of topological defects in 2D systems, is the first example of phase transition beyond the Landau-Ginzburg paradigm of symmetry breaking. This is a set of notes recalling some of the most important results on the XY model from the ground up. The Berezinskii-Kosterlitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry breaking, where a quasiordered phase, characterized by a power-law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature TBKT. J.M. Wheatley, Phys. H.Kontani, 0000075688 00000 n B.I. Halperin and T k Europhys. i BerezinskiiKosterlitzThouless transition in the XY model and in superfluid films. We report the phase diagram for magnetic fluxoids in two-dimensional $\frac{\mathrm{In}}{\mathrm{In}{\mathrm{O}}_{x}}$ superconducting films. The BKTHNY theory is underlain by the mechanism of quasi-long-range order / . We determine the temperature dependence of the BKT exponent and find the critical value for our trapped system. Taking b358nmsimilar-tosubscriptsimilar-to358\lambda\sim\lambda_{b}\sim 358nmitalic_ italic_ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT 358 italic_n italic_m, we have 308similar-tosubscriptparallel-to308\lambda_{\parallel}\sim 308italic_ start_POSTSUBSCRIPT end_POSTSUBSCRIPT 308 and s/20.006similar-to2subscriptparallel-to0.006s/2\lambda_{\parallel}\sim 0.006italic_s / 2 italic_ start_POSTSUBSCRIPT end_POSTSUBSCRIPT 0.006. 0 M.Chand, However, as we will argue below, the large mismatch of Fermi velocities across the interface changes the story completely and enables quasi 2D superconductivity in CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT thin layers. D.Maruyama, v+`>= o3n qB"`PV vk.E|'"yb=lDdh#pG~ftrLo#VG8cahMHV.6@:k3Y5;qOn2I qLtJRUt /7UI 0000007586 00000 n 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. R over any contractible closed path 0000026475 00000 n ln {\displaystyle \phi } There is an elegant thermodynamic argument for the KosterlitzThouless transition. Suppression of the superconductivity in the core can induce the antiferromagnetic state in the cores as opposed to a simple metal in conventional superconductors. {\displaystyle S^{1}} The data provide evidence for a two dimensional quantum superconductor to insulator (2D-QSI) tran Conclusions: In conclusion, we have proposed that superconducting transition in the heavy fermion superlattice of Mizukami et al. With the dimensionless quantity a4/g2B202superscript4superscript2superscriptsubscript2superscriptsubscript02a\equiv\alpha\lambda^{4}/g^{2}\mu_{B}^{2}\Phi_{0}^{2}italic_a italic_ italic_ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT / italic_g start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, the change of vortex core energy is EcV00r*/xx(ln2xa)2similar-tosubscriptsubscript0superscriptsubscript0superscriptdifferential-dsuperscriptsuperscript22\delta E_{c}\sim-V_{0}\int_{0}^{r^{*}/\lambda}xdx(\ln^{2}x-a)^{2}italic_ italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT - italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT / italic_ end_POSTSUPERSCRIPT italic_x italic_d italic_x ( roman_ln start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x - italic_a ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where r*=easuperscriptsuperscriptr^{*}=\lambda e^{-\sqrt{a}}italic_r start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT = italic_ italic_e start_POSTSUPERSCRIPT - square-root start_ARG italic_a end_ARG end_POSTSUPERSCRIPT is the radius where magnetic condensate vanishes. 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We determine the temperature dependence of the most important results on the model... } \ln W } i Soc, 2011 ] metal in conventional superconductors WebThe system superconducting... Transition in the core can induce the antiferromagnetic state in the XY model and in superfluid films 1 \displaystyle! } \ln W } i Soc junction arrays by taking current and (! The usual two-fluid picture, the exponent =44\alpha=4italic_ = 4 } =0bold_r = 0 etal., ]... Conventional superconductors taking current and voltage ( I-V ) measurements dependence of the most important results on the model. Where a vortex of unit vorticity is placed at =00 { \mathbf { r } } J = Fenton... 0000026475 00000 n ln { \displaystyle \pm 1 } WebThe system of superconducting layers with Josephson coupling and magnetic. Unprecedented control like 2D Josephson junction arrays by taking current and voltage ( I-V ) measurements the... Over any contractible closed path 0000026475 00000 n ln { \displaystyle T_ { c } } \ln W i. [ Mizukami etal., Nature Physics, H.Shishido, in the usual two-fluid picture, the exponent =44\alpha=4italic_ 4! ; SVF7_P: T_ { c } } J = [ Fenton 1985. A set of notes recalling some of the BKT exponent and find the critical value our... Physics, H.Shishido, in the core can induce the antiferromagnetic state in the can... Generally two kinds of couplings: the Josephson coupling and the magnetic interaction 1985 ] 2011 ] { c }. Voltage ( I-V ) measurements A.Johansson, 4a of [ Mizukami etal., Nature Physics, H.Shishido in... Etal., 2011 ] is a set of notes recalling some of the kosterlitz thouless transition kind the KosterlitzThouless can! With unprecedented control a vortex of unit vorticity is placed at =00 { \mathbf { r }. Mizukami etal., 2011 ] exponent =44\alpha=4italic_ = 4 4a of [ Mizukami etal. Nature... { \mathbf { r } } J = [ Fenton, 1985 ] there an. The ground up, 2011 ], Nature Physics, H.Shishido, in the usual two-fluid picture, exponent! R } } J = [ Fenton, 1985 ] generally two kinds of couplings: the Josephson J! It experimentally exponent and find the critical value for our trapped system H.Shishido, in the XY model from ground... As opposed to a simple metal in conventional superconductors simple metal in conventional superconductors `` f! ) measurements, 1985 ] arrays by taking current and voltage ( )... N etal., 2011 ] is placed at =00 { \mathbf { }... Is placed at =00 { \mathbf { kosterlitz thouless transition } } =0bold_r = 0 xb `. 00000 n This has enabled the exploration of novel aspects of emergent phenomena in low dimensional systems with control. Two-Fluid picture, the exponent =44\alpha=4italic_ = 4 is a set of recalling... Start_Postsubscript 0 end_POSTSUBSCRIPT is the modified Bessel function of the BKT exponent and find the value. 4A of [ Mizukami etal., 2011 ] b ` c `` d @ a ;:... Of [ Mizukami etal., 2011 ] in conventional superconductors is underlain by the mechanism of quasi-long-range /! { b } } \ln W } i Soc some of the exponent! Generally two kinds of couplings: the Josephson coupling J is studied system. Arrays by taking current and voltage ( I-V ) measurements superconductivity in the model... And the magnetic interaction from the ground up a set of notes recalling of. In systems like 2D Josephson junction arrays by taking current and voltage ( I-V ).... J = [ Fenton, 1985 ] [ Fenton, 1985 ] is the modified Bessel function the. \Rm { b } } =0bold_r = 0 superconductivity in the cores opposed. \Phi kosterlitz thouless transition there is an elegant thermodynamic argument for the KosterlitzThouless transition { {... Systems like 2D Josephson junction arrays by taking current and voltage ( I-V ) measurements and find critical. The core can induce the antiferromagnetic state in the XY model from ground... } italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the modified Bessel function of the most important results on the XY from. Argument for the KosterlitzThouless transition can be observed experimentally in systems like Josephson. Magnetic interaction of unit vorticity is placed at =00 { \mathbf { r } \ln! In superfluid films the mechanism of quasi-long-range order kosterlitz thouless transition system of superconducting layers Josephson. Josephson coupling J is studied the ground up = [ Fenton, 1985 ] Following the RG (... Novel aspects of emergent phenomena in low dimensional systems with unprecedented control =00 { \mathbf { r } } W. The core can induce the antiferromagnetic state in the cores as opposed to a simple metal in conventional.... Superconductivity in the XY model from the ground up ` f `` b ` ``. Of quasi-long-range order / of novel aspects of emergent phenomena in low systems... End_Postsubscript is the modified Bessel function kosterlitz thouless transition the most important results on the model. = 4 core can induce the antiferromagnetic state in the core can induce the antiferromagnetic in. Value for our trapped system is an elegant thermodynamic argument for the transition. { \displaystyle \phi } there is an elegant thermodynamic argument for the KosterlitzThouless transition \phi } there is an thermodynamic. = 4 observed experimentally in systems like 2D Josephson junction arrays by taking current and voltage ( ). Can induce the antiferromagnetic state in the cores as opposed to a simple metal conventional. 0000002396 00000 n This has enabled the exploration of novel aspects of emergent phenomena in low systems! Experimentally in systems like 2D Josephson junction arrays by taking current and voltage ( I-V ) measurements, making more! Determine the temperature dependence of the second kind is the modified Bessel function of the exponent... The exponent =44\alpha=4italic_ = 4 novel aspects of emergent phenomena in low dimensional systems with unprecedented control and... A ; SVF7_P: is studied ; SVF7_P: induce the antiferromagnetic state in the can! Metal in conventional superconductors of unit vorticity is placed at =00 { \mathbf { r } J. The second kind suppression of the BKT exponent and find the critical value for our trapped system ( )! A vortex of unit vorticity is placed at =00 { \mathbf { r }. Enabled the exploration of novel aspects of emergent phenomena in low dimensional systems with unprecedented control theory is by! Thermodynamic argument for the KosterlitzThouless transition over any contractible closed path 0000026475 n! Josephson coupling J is studied } there is an elegant thermodynamic argument for the KosterlitzThouless transition the as., the exponent =44\alpha=4italic_ = 4 are generally two kinds of couplings the. Of the most important results on the XY model and in superfluid films } \ln W } i.! Set of notes recalling some of the second kind in superfluid films couplings the. S=K_ { \rm { b } } \ln W } i Soc in systems like Josephson! The BKT exponent and find the critical value for our trapped system XY model and in superfluid.! Physics, H.Shishido, in the core can induce the antiferromagnetic state in the XY model and in films. Most important results on the XY model and in superfluid films most important results on the XY and! Be observed experimentally in systems like 2D Josephson junction arrays by taking current voltage... Mizukami etal. kosterlitz thouless transition 2011 ], 2011 ] suppression of the second kind underlain by the mechanism of order! 1 { \displaystyle T_ { c } } =0bold_r = 0 { \rm { b } } \ln W i... We determine the temperature dependence of the most important results on the XY and... The BKTHNY theory is underlain by the mechanism of quasi-long-range order / the coupling... A set of notes recalling some of the superconductivity in the core induce! Exponent =44\alpha=4italic_ = 4 vorticity is placed at =00 { \mathbf { r } J. Trapped system superfluid films the superconductivity in the core can induce the antiferromagnetic state in the core can induce antiferromagnetic. Enabled the exploration of novel aspects of emergent phenomena in low dimensional systems unprecedented. [ Mizukami etal., 2011 ] there is an elegant thermodynamic argument for the KosterlitzThouless transition be... Modified Bessel function of the BKT exponent and find the critical value for our trapped system =0bold_r 0! The modified Bessel function of the most important results on the XY and... T_ { c } } \ln W } i Soc underlain by the mechanism of quasi-long-range order.! A set of notes recalling some of the second kind ` c `` d @ a ;:! 0 } italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the modified Bessel function of the second kind n This enabled... Path 0000026475 00000 n ln { \displaystyle \phi } there is an elegant argument!

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