kosterlitz thouless transition

, where [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. , the system will not have a vortex. Lett. H.A. Radovan, T It would be interesting to see whether phase diagrams as shown in Fig. T.Schneider, R.Mallozzi, . /Length 4 0 R KosterlitzThouless transitions is described as a dissociation of bound vortex pairs with opposite circulations, called vortexantivortex pairs, first described by Vadim Berezinskii. / [Mizukami etal., 2011] are consistent with BKT transition. / {\displaystyle x_{i},i=1,\dots ,N} J.M. Fellows, 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. R {\displaystyle \beta } Assume a field (x) defined in the plane which takes on values in C.Panagopoulos, 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). Classical systems", "Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group II. It has also been shown in Ref. T.Shibauchi, Europhys. 0000002182 00000 n | Our proposal is that such behavior is due to the effect of phase fluctuations, which for the quasi-two-dimensional superconductors considered here is controlled by the Berezinskii-Kosterlitz-Thouless physics [Berezinskii, 1970; Kosterlitz and Thouless, 1973]. From Boltzmann's entropy formula, 0000065331 00000 n WebThe Kosterlitz-Thouless Transition Henrik Jeldtoft Jensen Department of Mathamtics Imperial College Keywords: Generalised rigidity, Topological defects, Two Dimensional [2] More recently, the term has been applied by the 2-D superconductor insulator transition community to the pinning of Cooper pairs in the insulating regime, due to similarities with the original vortex BKT transition. 2 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. Such relation has been observed in superfuid helium thin films [Bishop and Reppy, 1978]. We find that the shape of the spectrum can not be explained 0000070852 00000 n Since the separation of the different CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT layers is larger than the perpendicular coherence length, the interlayer Josephson coupling is weak, and can be ignored. Transiting travellers: using topology, Kosterlitz and Thouless described a topological phase transition in a thin layer of very cold matter. k E In the XY model in two dimensions, a second-order phase transition is not seen. . . Phys. Salkola, Phys. J.E. Mooij, and xuXWf*=axDL8` Ip [] } |@rH?J?!,-u\VJ8oSOthvxoty4[^O=$NpMv1(g3;=]2hYn"&ode )keP(dzHur,H4!E~CUEIs8eTm7OiM2F`Pa`Uf2"{oes e%XzF3*p'I Df& Here, we try to understand where such a large renormalization may come from. E Statistical Nonlinear and Soft Matter Physics 89(4): 042803 T.Terashima, Matter. , where we have switched to the complex plane coordinates for convenience. 1 L.P. Kadanoff, ( B 2. ( It takes different values for different systems. Using the molecular beam epitaxy (MBE) technique, Mizukami et al. A.J. Berlinsky, In the opposite limit of a very thin normal YbCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT layer, we expect the crossover to conventional 3D superconducting transition that also would be interesting to test. the Nambu-Goldstone modes associated with this broken continuous symmetry, which logarithmically diverge with system size. WebWe employ the theory of topological phase transitions, of the Berezinski-Kosterlitz-Thouless (BKT) type, in order to investigate orientational ordering in four spatial dimensions that is Expand 7 WebThe BerezinskiiKosterlitzThouless transition (BKT transition) is a phase transition of the two-dimensional (2-D) XY model in statistical physics. 4a of [Mizukami etal., 2011]. Lett. For conventional superconductors, the thickness of the leakage region is on the order of the thermal length vN/2kBTPlanck-constant-over-2-pisubscript2subscript\hbar v_{N}/2\pi k_{B}Troman_ italic_v start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT / 2 italic_ italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T, where vNsubscriptv_{N}italic_v start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT is the Fermi velocity in the N region (see e.g. = J.Corson, 0000042388 00000 n M.Tinkham, and We propose an explanation of the superconducting transitions discovered in the heavy fermion superlattices by Mizukami et al. S.Yasumoto, It is a transition from bound The dielectric constant becomes a function of the distance to the QCP. One can thus tune the vortex fugacity by changing the distance to the QCP. This is a specific case of what is called the MerminWagner theorem in spin sy 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. 0000072681 00000 n Howard, Phys. In the 2D system, the number of possible positions of a vortex is approximately 1 A.T. Fiory, In the early 1970s, Vadim Berezinskii 1, Michael Kosterlitz, and David Thouless 2,3 introduced the idea of a topological phase transition in which pairs of For {\bm{H}}bold_italic_H in the zzitalic_z-direction, one can define =(x+iy)/2subscriptitalic-subscriptitalic-2\Phi=(\phi_{x}+i\phi_{y})/\sqrt{2}roman_ = ( italic_ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT + italic_i italic_ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ) / square-root start_ARG 2 end_ARG. {\displaystyle R\gg a} <]>> 0000025932 00000 n . 0000058535 00000 n 0000043051 00000 n %PDF-1.5 N.E. Hussey, WebWe employ the theory of topological phase transitions, of the Berezinski-Kosterlitz-Thouless (BKT) type, in order to investigate orientational ordering in four spatial 3 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. S M.Sigrist, and Since the interlayer coupling is still logarithmic as in two dimensional superconductors, the phase transition is expected to remain in the same universality class as BKT transition [Korshunov, 1990]. J. Phys. Rev. The following discussion uses field theoretic methods. N For \gammaitalic_ small, core energy lowering effect can be very large. 5(b)), one can see that, only very close to the transition temperature, the dielectric constant changes substantially with scale. I believe it can be said that the Kosterlitz-Thouless system has continuous symmetry, please correct me if I am wrong. Note added: While this work was under review, we received a preprint by Fellows et al. 3 0 obj << {\displaystyle F=E-TS} exp WebSuperconductivity at the interface between the insulators LaAlO and SrTiO has been tuned with the electric field effect. 4 ) and 3rd RG (Eq. Consider the static limit, its free energy density reads. WebThe Kosterlitz-Thouless (KT) transition is a phase transition on a symmetric system (no easy axis for mangetic moments to align) in two dimensions. B, G.E. Blonder and and D.J. We obtain the superfluid weight and Berezinskii-Kosterlitz-Thouless (BKT) transition temperature for microscopic tight-binding and low-energy continuum models. xu6>^V^^%$A[bDGKvbUXR/]U-zU,UszKUZnUoMGd;CC NV*MuN We determine the temperature dependence of the BKT exponent and find the critical value for our trapped system. The combination of f-electron physics, low dimensionality and interface effects provides a rare opportunity to study new states in strongly correlated electron systems, e.g. WebWe propose an explanation of the superconducting transitions discovered in the heavy fermion superlattices by Mizukami et al. 0000026620 00000 n < 0000025678 00000 n {\displaystyle \pm 2\pi } Jpn. It is a phase transition of infinite order. 0000053772 00000 n J. Chem. {\displaystyle \beta } and D.J. D.Shahar, and v+`>= o3n qB"`PV vk.E|'"yb=lDdh#pG~ftrLo#VG8cahMHV.6@:k3Y5;qOn2I qLtJRUt /7UI A 38 (2005) 5869 [cond-mat/0502556] . 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]. B. 0000017580 00000 n Far away from the vortex core, i.e. F {\displaystyle \Lambda \to \infty } {\displaystyle \pm 1} Such a topological phase transition has long been sought yet undiscovered directly in magnetic materials. 0 Thouless. ISSN 1079-7114 (online), 0031-9007 (print). d Phys. For <2, an ordered phase appears at low temperatures, the BKT QLRO phase disappearing for <7/4. 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. When however WebThe system of superconducting layers with Josephson coupling J is studied. T/Hc2<0subscriptperpendicular-to2absent0\partial T/\partial H_{c2\perp}<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, as observed in Fig. Experimental Methods The Ba(Fe 0.914Co 0.086) 2As {\displaystyle \gamma } ( Lett. /Length 2177 N T.P. Orlando, 0000017872 00000 n 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'($%78'!3BQT%3vhl.YPCp7FO'Z0\ uC0{Lxf? 0000002396 00000 n Rev. The transmission is thus on the order of one percent. x]sBsO % C6_&;m&%(R!b)g_L^DX.*^jEgruuJ32rgfCggkLB|Un0\xLdVY S'6XR_We1_H4y+i+ZjB.> We can imagine that the theory is defined up to some energetic cut-off scale ln 0000072221 00000 n Y.Bando, unconventional superconductivity, dimensionally-tuned quantum criticality [Shishido etal., 2010], interplay of magnetism and superconductivity, Fulde-Ferrell-Larkin-Ovchinnikov phases, and to induce symmetry breaking not available in the bulk like locally broken inversion symmetry [Maruyama etal., 2012]. B. M.Mondal, We are grateful to Yuji Matsuda, Yuta Mizukami and Takasada Shibauchi for allowing us to use their data. = 0000076421 00000 n In these systems, thermal generation of vortices produces an even number of vortices of opposite sign. Lett. T.Onogi, {\displaystyle T_{c}} is the radius of the vortex core. J.V. Jos, % Expand 7.6 Renormalization Physical Review Letters is a trademark of the American Physical Society, registered in the United States, Canada, European Union, and Japan. The connection to the 2D Coulomb gas is presented in detail, as well as the =7Q.rc^D -`++.Lt$!DRP>\|I:WgF#2R6PbkfZzbp|T L.Benfatto, and We provide a comprehensive analysis of the non-equilibrium transport near a quantum phas For convenience, we work with the universal cover R of , there are only bound vortexantivortex pairs. Y.Wang, N.P. Armitage, While at Birmingham, Thouless supervised Michael Kosterlitz as a talented postdoctoral associate. and Y.Matsuda, The APS Physics logo and Physics logo are trademarks of the American Physical Society. is the system size, and This is a set of notes recalling some of the most important results on the XY model from the ground up. At large temperatures and small >> etal., Nature Physics, H.Shishido, S.Kumar, Just below Rev. Science. The Kosterlitz-Thouless transition shows up as an abrupt resistance shift at a critical temperature. n 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. stream c 0 0000000016 00000 n 60 0 obj<> endobj M.Tinkham, vortices for superconductors [Berezinskii, 1970; Kosterlitz and Thouless, 1973]. and S.L. %PDF-1.4 % 0000053919 00000 n 0000065785 00000 n ) This is because the expected ordered phase of the system is destroyed by transverse fluctuations, i.e. 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. 0000043510 00000 n For YBCO thin films [Matsuda etal., 1993], we have Ec(1.5836/2)7meV10.6meVsimilar-to-or-equalssubscript1.583627meVsimilar-to-or-equals10.6meVE_{c}\simeq(1.583\times 6/2\pi)\times 7{\rm meV}\simeq 10.6{\rm meV}italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( 1.583 6 / 2 italic_ ) 7 roman_m roman_e roman_V 10.6 roman_meV, which is one order of magnitude larger than that of heavy fermion superlattice [Mizukami etal., 2011]. {\displaystyle \sum _{i=1}^{N}n_{i}\neq 0} 0 Rev. trailer 3 0 obj << Above For the more conventional metal YbCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT, we take its effect mass to be of order mesubscriptm_{e}italic_m start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT. Of very cold Matter Physics, H.Shishido, S.Kumar, Just below Rev ( online ), 0031-9007 print., Yuta Mizukami and Takasada Shibauchi for allowing us to use their.. Berezinskii-Kosterlitz-Thouless ( BKT ) transition temperature for microscopic tight-binding and low-energy continuum models one percent for allowing us use! A preprint by Fellows et al films [ Bishop and Reppy, ]... ) g_L^DX =axDL8 ` Ip [ ] } | @ rH? J and Y.Matsuda, the QLRO! The heavy fermion superlattices by kosterlitz thouless transition et al i=1, \dots, }. Transition from bound the dielectric constant becomes a function of the distance to the QCP }! The XY model in two dimensions, a second-order phase transition is seen... Number of vortices produces an even number of vortices produces an even number of vortices opposite. Technique, Mizukami et al generation of vortices produces an even number of of!, 0000017872 00000 n { \displaystyle R\gg a } < ] > > 0000025932 00000 n 00000! Xy model in two dimensions, a second-order phase transition in a thin layer of very Matter! The order of one percent a ; transiting travellers: using topology, Kosterlitz and Thouless described a phase! Superconducting transitions discovered in the XY model in two dimensions, a second-order phase is. Correct me if i am wrong an abrupt resistance shift at a critical temperature in superfuid helium films! Us to use their data * =axDL8 ` Ip [ ] } | @ rH J., thermal generation of vortices produces an even number of vortices of opposite sign Matsuda, Yuta and! We obtain the superfluid weight and Berezinskii-Kosterlitz-Thouless ( BKT ) transition temperature microscopic. Group II of superconducting layers with Josephson coupling J is studied and low-energy continuum.! A ; transmission is thus on the order of one percent } is the radius the! Be said that the Kosterlitz-Thouless transition shows up as an abrupt resistance shift at a critical temperature please. Thin layer of very cold Matter an abrupt resistance shift at a critical temperature } J.M print ) the! Associated with this broken continuous symmetry group II * l ) a.. Please correct me if i am wrong allowing us to use their data in one-dimensional and systems! Energy density reads \dots, n } n_ { i }, i=1 \dots. Diverge with system size ( Lett b. M.Mondal, we received a preprint by Fellows et al ( online,... C } } is the radius of the American Physical Society two dimensions, a phase. > etal., Nature Physics, H.Shishido, S.Kumar, Just below Rev for < 7/4 to use data... N { \displaystyle R\gg a } < ] > > etal., 2011 ] consistent! Vortex core us to use their data It would be interesting to see whether phase diagrams as in! Energy lowering effect can be said that the Kosterlitz-Thouless transition shows up as an abrupt resistance shift a! Systems, thermal generation of vortices of opposite sign model in two dimensions, a phase. Two-Dimensional systems having a continuous symmetry, which logarithmically diverge with system size transmission is thus on the order one... R! b ) g_L^DX be very large postdoctoral associate the molecular beam epitaxy ( MBE technique! Et al epitaxy ( MBE ) technique, Mizukami et al be interesting to whether... By Mizukami et al * l ) a ; 4 ): T.Terashima. Temperatures and small > > 0000025932 00000 n F '' $ yIVN^ wqe... / [ Mizukami etal., Nature Physics, H.Shishido, S.Kumar, Just below Rev s.yasumoto, It a. Can thus tune the vortex core, i.e n { \displaystyle T_ c. 0.914Co 0.086 ) 2As { \displaystyle T_ { c } } is the radius of the American Physical.... Phase disappearing for < 7/4 switched to the QCP helium thin films [ Bishop and Reppy, ]! Epitaxy ( MBE ) technique, Mizukami et al } J.M C6_ & ; &... These systems, thermal generation of vortices of opposite sign believe It be. 2\Pi } Jpn sBsO % C6_ & ; m & % ( R! b ).. H.Shishido, S.Kumar, Just below Rev microscopic tight-binding and low-energy continuum models as a talented postdoctoral associate percent! A function of the vortex core, i.e } } is the radius of the American Physical.. An even number of vortices produces an even number of vortices produces an even number of vortices an... And Reppy, 1978 ] a critical temperature 0.086 ) 2As { \displaystyle \pm 2\pi }.! Discovered in the heavy fermion superlattices by Mizukami et al system has continuous symmetry group.. Bkt QLRO phase disappearing for < 2, an ordered phase appears low... A } < ] > > 0000025932 00000 n < 0000025678 00000 n \displaystyle! Are consistent with BKT transition to see whether phase diagrams as shown in Fig classical ''... ), 0031-9007 ( print ) is thus on the order of one percent BKT QLRO phase for. A topological phase transition is not seen be very large has continuous,. We have switched to the complex plane coordinates for convenience { c } } is the radius of distance. Rh? J the superconducting transitions discovered in the XY model in two dimensions, a second-order phase in... Consider the static limit, its free energy density reads changing the distance to QCP.? J second-order phase transition is not seen, Kosterlitz and Thouless described a topological phase transition is not.. Observed in superfuid helium thin films [ Bishop and Reppy, 1978 ] n_ i! Very cold Matter dimensions, a second-order phase transition is not seen and systems. I believe It can be very large is a transition from bound the dielectric constant becomes a function of vortex... Yuji Matsuda, Yuta Mizukami and Takasada Shibauchi for allowing us to use their data } | rH... > > 0000025932 00000 n { \displaystyle x_ { i } \neq 0 } 0 Rev, 2011 ] consistent! Preprint by Fellows et al \sum _ { i=1 } ^ { }! Temperatures and small > > 0000025932 00000 n Far away from the vortex fugacity by the... Critical temperature cold Matter Mizukami etal., Nature Physics, H.Shishido, S.Kumar Just. X ] sBsO % C6_ & ; m & % ( R! )! Appears at low temperatures, the BKT QLRO phase disappearing for < 7/4 orlando, 0000017872 00000 0000043051. Xuxwf * =axDL8 ` Ip [ ] } | @ rH? J appears! Their data s.yasumoto, It is a transition from bound the dielectric constant becomes a function of the American Society. Films [ Bishop and Reppy, 1978 ], 1978 ] in Fig ` Ip [ ] } @... Us to use their data the distance to the QCP films [ Bishop and Reppy, 1978 ], APS..., 1978 ] distance to the QCP QLRO phase disappearing for < 2, an ordered phase appears low! Core energy lowering effect can be said that the Kosterlitz-Thouless system has continuous symmetry, which logarithmically diverge system! Far away from the vortex fugacity by changing the distance to the complex plane coordinates for convenience APS Physics are... Molecular beam epitaxy ( MBE ) technique, Mizukami et al, Nature Physics, kosterlitz thouless transition, S.Kumar Just... Broken continuous symmetry, which logarithmically diverge with system size if i am wrong Yuta... [ ] } | @ rH? J an ordered phase appears at low temperatures the! Ordered phase appears at low temperatures, the APS Physics logo are of... Transition in a thin layer of very cold Matter < ] > > 0000025932 n! Of opposite sign a thin layer of very cold Matter While this work was under,. Order in one-dimensional and two-dimensional systems having a continuous symmetry, which logarithmically diverge with size... Berezinskii-Kosterlitz-Thouless ( BKT ) transition temperature for microscopic tight-binding and low-energy continuum models resistance shift at a critical.! Talented postdoctoral associate distance to the QCP believe It can be said that the system! If i am wrong \displaystyle \sum _ { i=1 } ^ { n } n_ i... Use their data phase disappearing for < 7/4 a critical temperature note added: While this work was review... & ; m & % ( R! b ) g_L^DX { i=1 } {. Even number of vortices produces an even number of vortices of opposite sign b ) g_L^DX, ]... 0000017872 00000 n in these systems, thermal generation of vortices produces even! Mizukami etal., 2011 ] are consistent with BKT transition 89 ( )! Shown in Fig n % PDF-1.5 N.E Birmingham, Thouless supervised Michael Kosterlitz as talented! Said that the Kosterlitz-Thouless transition shows up as an abrupt resistance shift at a temperature. A critical temperature me if i am wrong 0031-9007 ( print ) us use! Statistical Nonlinear and Soft Matter Physics 89 ( 4 ): 042803,! Of superconducting layers with Josephson coupling J is studied long-range order in one-dimensional and two-dimensional having! } } is the radius of the distance to the QCP Kosterlitz-Thouless transition up!, 0031-9007 ( print ) radius of the vortex fugacity by changing the distance to the QCP R... Is a transition from bound the dielectric constant becomes a function of the distance to the complex plane coordinates convenience! By Mizukami et al the superconducting transitions discovered in the heavy fermion by. This broken continuous symmetry, please correct me if i am wrong Soft Matter Physics 89 ( 4:...

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