Spin vortices, skyrmions and the Kosterlitz-Thouless transition in the two-dimensional antiferromagnet
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Spin vortices, skyrmions and the Kosterlitz-Thoulesstransition in the two-dimensional antiferromagnet
Takashi Yanagisawa
Electronics and Photonics Research Institute, National Institute of Advanced IndustrialScience and Technology, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, JapanE-mail: [email protected]
Abstract.
We investigate spin-vortex excitations in the two-dimensional antiferromagnet onthe basis of the nonlinear sigma model. The model of two-dimensional Heisenberg quantumantiferromagnet is mapped onto the (2+1)D nonlinear sigma model. The 2D nonlinear sigmamodel has an instanton (or skyrmion) solution which describes an excitation of spin-vortex type.Quantum fluctuations of instantons are reduced to the study of the Coulomb gas, and the gas ofinstantons of the 2D nonlinear sigma model is in the plasma phase. We generalize this pictureof instanton gas to the (2+1)D nonlinear sigma model. We show, using some approximation,that there is a Kosterlitz-Thouless transition from the plasma phase to the molecular phase asthe temperature is lowered.
1. Introduction
It is important to understand the magnetic structure of two-dimensional antiferromagneticin the light-doping region in the study of cuprate high-temperature superconductors[1]. Thephysics of underdoped region, which is expected to be closely related to the pseudogapstate, is still not clear. The influence of doping holes on the antiferromagnetic state inthe parent materials of cuprate superconductors is one of the most interesting problems instrongly correlated electron systems. It has been experimentally found that the stripe order isstabilized in the underdoped region of La − x − y Nd y Sr x CuO ,[4] La − x Sr x CuO (LSCO)[2], andLa − x Ba x CuO [3]. A checkerboard-like charge density modulation with a roughly 4 a × a period( a is a lattice spacing) has also been observed by scanning tunneling microscopy experiments inBi Sr CaCu O δ (Bi-2212)[5],Bi Sr − x La x CuO δ [6], and Ca − x Na x CuO Cl (NA-CCOC)[7].It has been pointed out that these types of modulated structures can be understood withinthe framework of correlated electrons by using the variational Monte Carlo method for thetwo-dimensional Hubbard model[8, 9, 10].It is expected that there appears a new state in an extremely light-doping region wherethe charge modulated structures such as stripes and checkerboards are instable and only spin-modulations, for example, a spin-vortex state, will be formulated. The purpose of this paper is toinvestigate spin-vortex excitation on the basis of the two-dimensional quantum antiferromagnet.In this paper, we use the mapping of a two-dimensional magnet onto a nonlinear sigma model.The nonlinear sigma model is a nonlinear field theory and its renormalization properties wereapplied to ferromagnets in two dimensions[11, 12, 13]. The O(3) nonlinear sigma model can benterpreted as the continuum limit of an isotropic ferromagnet, and the action is S = 12 g Z d d x ( ∂ µ φ ) , (1)where the three-component scalar field φ ( x ) is under the condition φ · φ = 1.
2. (2+1)D nonlinear sigma model
The two-dimensional quantum Heisenberg antiferromagnet is mapped to a (2+1) dimensionalnonlinear sigma model. The (2+1)D nonlinear sigma model is[14] S = 12 g Z /k B T dτ Z d d r h ( ∇ ϕ ) + 1 c (cid:18) ∂ϕ∂τ (cid:19) i , (2)where ϕ is a three-component scalar field on the sphere ϕ ( τ, r ) = 1. The parameters g and c are g = a d − /J S and c = 2 √ dJ Sa where where J is the coupling constant of theantiferromagnetic nearest-neighbor interaction, S is the magnitude of the spin and a is the latticespacing. In this paper we consider the two-dimensional case d = 2. We change the scale of τ by c τ → τ , and then the action is S = 12 g Z β dτ Z d d r h ( ∇ ϕ ) + (cid:18) ∂ϕ∂τ (cid:19) i . (3)Here, we defined in two-space dimensions d = 2[15], g = g c = 2 √ dS Λ − , β = gt , t = k B TJ S , (4)for Λ = 1 /a . To be more precisely, S should be replaced by p S ( S + 1). Although g hasdimension in this expression, g can be dimensionless by scaling the field variables r and τ .This model describes the Nambu-Goldstone mode, that is, the spin-wave mode of the quantumantiferromagnetic model. The renormalization of the spin-wave mode was investigated using theWilson-Kogut renormalization method[14, 15].In the case of hole doping, when the doping rate is small, the nonlinear sigma model isexpected to be still relevant if we use the coupling constant g renormalized by the doping effect.It is plausible that we can use the nonlinear sigma model for the extremely light-doping case.The 2D nonlinear sigma model has an instanton solution [16, 17, 18, 19] (being interpretedas skyrmion or meron[20, 21]) which describes an excitation of spin-vortex type. Quantumfluctuations of instantons were computed and found to be reduced to the study of the Coulombgas[22, 23, 24]. According to this study, the gas of instantons of the 2D nonlinear sigma modelis in the plasma phase. This means that spin-vortex excitations are independent each other andnever form dimers. This is the case for the two-dimensional classical Heisenberg model. Then,we necessarily have a question such as: is this picture still correct for the (2+1)D nonlinearsigma model? To investigate this, we generalize the instanton gas approximation to the (2+1)Dnonlinear sigma model. We use the parametrization of the field ϕ in terms of a single complexfield w : ( ϕ , ϕ , ϕ ) = w + ¯ w | w | , − i w − ¯ w | w | , | w | −
11 + | w | ! , (5)where ¯ z is the complex conjugate of w . Then, using the complex variable z = x + iy for r = ( x, y ),the action is written as S = 2 πg Z β dτ Q ( τ ) + 1 g Z β dτ d r | w | ) (cid:16) | ∂ τ w | + 4 | ∂ ¯ z w | (cid:17) . (6)ere, Q is Q ( τ ) = 1 π Z d r | w | ) (cid:16) | ∂ z w | − | ∂ ¯ z w | (cid:17) . (7)A solution that satisfies ∂ ¯ z w = 0 is called instanton solution. A typical instanton solution is, foran integer k , w = ( z − a )( z − a ) · · · ( z − a k )( z − b )( z − b ) · · · ( z − b k ) . (8) a i and b i ( i = 1 , · · · , k ) indicate positions of vortex-like excitations in the antiferromagnetic spinorder. This each local structure can be interpreted as a skyrmion. In general, a i and b i have τ -dependence: a i = a i ( τ ), b i = b i ( τ ) ( i = 1 , · · · , k ). When a i and b i are constants, Q is theinteger quantum number equal to k [16]. We assume that a i ( τ ) and b i ( τ ) are continuous functionof τ , then Q ( τ ) is also a continuous function of τ and Q ( τ ) = k since Q is an integer. For thisinstanton solution, the action is given as S = 2 πkg β + 1 g Z β dτ d r | w | ) | ∂ τ w | . (9)
3. Fluctuation effect
Fluctuation effect to the partition function was evaluated for the 2D nonlinear sigma modelas[23] Z k = const .e − πk/g k !) Z Y i da i db i dcc k e − V ( a i ,b i ) , (10)where V ( a i , b i ) = X ij ln | a i − b j | − X i 4. One-pair skyrmion state Let us consider one-instanton state w ( z ) = ( z − a ) / ( z − b ). If we set b = − a ∈ R , for simplicity,then we obtain ϕ = (0 , , − 1) at z = a , ϕ = (0 , , 1) at z = − a and ϕ = ( − , , 0) at z = 0.In the limit | z | → ∞ , ϕ = (1 , , z = ± a . There is a spin disorder in the regionthat includes z = ± a .For w = ( z − b ) / ( z − a ), the action in eq.(9) is given as S = 2 πβg + 1 g Z β dτ G (cid:16) ˙ a ∗ ˙ a + ˙ b ∗ ˙ b − ˙ a ∗ ˙ b − ˙ b ∗ ˙ a (cid:17) , (13)here G is a constant that depends on | a − b | , and a ∗ and b ∗ are complex conjugates of a and b . We replace g by g/ Λ so that g is dimensionless and add the fluctuation effect to this actionto obtain S Λ = 2 πβg + Z β dτ h g G | ˙ a − ˙ b | − ln | a ( τ ) − b ( τ ) | i . (14)This is the model of single particle in the logarithmic potential if we set u = a − b . We define u = re iθ , then the energy is determined by the angular momentum r ˙ θ and the potential ln r ,which is shown in Fig.1. The particle dynamics corresponds to the oscillation of the size ofspin-vortex pair (Fig.2). r ! E ! r max r min Figure 1. Energy of one-pair skyrmion state. r max r ! r min ! Figure 2. Oscillation of one-pair skyrmionstate. This figure shows the oscillation of thedistance of bounded spin-vortex pair, namely,the size of region of disordered antiferromagneticspin. 5. Skyrmion gas We generalize the one-pair skyrmion to many-skyrmion gas. The action is written as S k Λ = 2 πkg β + 1 g Z β dτ X ij G ij ˙ ψ ∗ i ( τ ) ˙ ψ j ( τ ) + Z β dτ h X i 6. Summary We have formulated skyrmion (instanton) gas picture for the two-dimensional quantumantiferromagnet on the basis of the (2+1)D nonlinear sigma model. We have shown that there is aKosterlitz-Thouless transition from the plasma phase to the molecular phase as the temperatureis lowered by using a static approximation to the instanton gas. In the molecular phase twoinstantons form a bound state, namely, a dipole of two spin vortices. We expect that this isrelated to the pairing mechanism of superconductivity in doped antiferromagnets. References [1] K. H. Bennemann and J. B. Ketterson Eds.: The Physics of Superconductors (Springer-Verlag, Berlin, 2004).[2] T. Suzuki, T. Goto, K. Chiba, T. Shinoda, T. Fukase, H. Kimura, K. Yamada, M. Ohashi, and Y. Yamaguchi:Phys. Rev. B (1998) 3229.[3] M. Fujita, H. Goka, T. Adachi, Y. Koike, and K. Yamada: Physica C - (2005) 257.[4] J. M. Tranquada, J. D. Axe, N. Ichikawa, Y. Nakamura, S. Uchida, and B. Nachumi: Phys. Rev. B (1996)7489.[5] J. E. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, H. Eisaki, S. Uchida, and J. C. Davis: Science (2002) 466.[6] W. D. Wise, M. C. Boyer, K. Chatterjee, T. Kondo, T. Takeuchi, H. Ikuta, Y. Wang, and E. W. Hudson:Nature Phys. (2008) 696.[7] T. Hanaguri, C. Lupien, Y. Kohsaka, D. H. Lee, M. Azuma, M. Takano, H. Takagi, and J. C. Davis: Nature (2004) 1001.[8] T. Yanagisawa, S. Koike, M. Miyazaki, and K. Yamaji: J. Phys. Condens. Matter (2002) 21.[9] M. Miyazaki, T. Yanagisawa, and K. Yamaji: J. Phys. Soc. Jpn. (2004) 1643; T. Yanagisawa, M. Miyazakiand K. Yamaji: J. Phys. Soc. Jpn. (2009) 013706.[10] M. MIyazaki, K. Yamaji, T. Yanagisawa, and R. Kadono: J. Phys. Soc. Jpn. (2009) 043706.[11] A. M. Polyakov: Phys. Lett. B (1975) 79.[12] E. Brezin and J. Zinn-Justin: Phys. Rev. B (1976) 3110.[13] W. A. Bardeen, B. W. Lee, and E. R. Shrock: Phys. Rev. D (1976) 985.[14] S. Chakravarty, B. I. Halperin and D. R. Nelson: Phys. Rev. B (1989) 2344.[15] T. Yanagisawa: Phys. Rev. B (1992) 13896.[16] A. A. Belavin and A. M. Polyakov: JETP Lett. (1975) 245.[17] A. Jevicki: Nucl. Phys. B (1977) 125.[18] A. D’adda, M. L¨uscher, and P. Di Vecchia: Nucl. Phys. B (1978) 63.[19] A. M. Polyakov: Gauge Fields and Strings (Harwood Academic Publisher, Switerland, 1987).[20] H. Koizumi: cond-mat/0506747.[21] T. Morinari: Phys. Rev. B (2006) 064504.[22] I. V. Frolov and A. S. Schwarz: JETP Lett. (1978) 249.[23] V. A. Fateev, I. V. Frolov and A. S. Schwarz: Nucl. Phys. B (1979) 1.[24] B.Berg and M. L¨uscher: Commun. Math. Phys.69