The uniformly frustrated two-dimensional XY model in the limit of weak frustration
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec The uniformly frustrated two-dimensional
X Y model in the limit of weak frustration
Vincenzo Alba , Andrea Pelissetto and Ettore Vicari Scuola Normale Superiore and INFN, I-56126 Pisa, Italy Dipartimento di Fisica dell’Universit`a di Roma “La Sapienza” and INFN, Sezionedi Roma I, I-00185 Roma, Italy Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Sezione di Pisa, I-56127Pisa, ItalyE-mail:
[email protected], [email protected]
Abstract.
We consider the two-dimensional uniformly frustrated XY model in thelimit of small frustration, which is equivalent to an XY system, for instance a Josephsonjunction array, in a weak uniform magnetic field applied along a direction orthogonalto the lattice. We show that the uniform frustration (equivalently, the magnetic field)destabilizes the line of fixed points which characterize the critical behaviour of the XY model for T ≤ T KT , where T KT is the Kosterlitz-Thouless transition temperature:the system is paramagnetic at any temperature for sufficiently small frustration. Wepredict the critical behaviour of the correlation length and of gauge-invariant magneticsusceptibilities as the frustration goes to zero. These predictions are fully confirmedby the numerical simulations. he uniformly frustrated two-dimensional XY model in the limit of weak frustration
1. Introduction
The uniformly frustrated two-dimensional (2D) XY model is defined by the latticeHamiltonian H = − X h xy i Re ψ x U xy ψ ∗ y = − X h xy i cos( θ x − θ y + A xy ) , (1)where ψ x ≡ e iθ x and U xy ≡ e iA xy . 2D arrays of coupled Josephson junctions in a magneticfield are interesting physical realizations of this model [1]. In this case, the sum C ( P nm )of the variables A xy along the links of an elementary plaquette P nm , C ( P nm ) ≡ A ( n,m ) , ( n +1 ,m ) + A ( n +1 ,m ) , ( n +1 ,m +1) − A ( n,m +1) , ( n +1 ,m +1) − A ( n,m ) , ( n,m +1) , (2)is related to the flux of an external magnetic field applied along an orthogonal direction: C ( P nm ) = a B/ Φ , where a is the lattice spacing, B is the magnetic field and2Φ = hc/e . Hamiltonian (1) depends on A xy through the phases U xy and thusthe relevant physical quantity is the product of the phases around a plaquette, i.e., U ( P ) ≡ exp[ iC ( P )]. If U ( P ) is not 1, H is frustrated. In this paper we assume U ( P )to be independent of the chosen plaquette, i.e., that U ( P ) = e πif , (3)with 0 ≤ f ≤
1, independent of P . Using the invariance of the Hamiltonian under thetransformation ψ x → ψ ∗ x , it is not restrictive to take f in the interval 0 ≤ f ≤ /
2. Wewill work in a finite lattice of size L with periodic boundary conditions. Therefore, wehave Y P U ( P ) = 1 , (4)where the product is extended over all lattice plaquettes. This implies that f L mustbe an integer.Hamiltonian (1) is invariant under the local gauge transformations ψ x → V x ψ x , U xy → V ∗ x U xy V y , (5)where V x is a phase, | V x | = 1. Physical observables must be gauge invariant. Forsuch observables, the choice of the fields A xy is irrelevant: only the value of f isrelevant. In a finite volume, this statement is strictly true only if free boundaryconditions are taken. If one considers periodic boundary conditions, one must alsospecify the value of exp( i P A xy ) along two non-trivial lattice paths that wind aroundthe lattice (they are sometimes called Polyakov loops). For instance, one must also fix P ( m ) = exp( i P n A ( n,m ) , ( n +1 ,m ) ) and P ( m ) = exp( i P n A ( m,n ) , ( m,n +1) ) for some fixedvalue of m . If we require the absence of magnetic circulation along these non-trivialpaths, we must have P ( m ) = P ( m ) = 1 for any m . On a finite lattice of size L ,this condition can be satisfied only if f L is an integer, a condition that will be alwayssatisfied in the numerical simulations that we shall present. he uniformly frustrated two-dimensional XY model in the limit of weak frustration XY models changes dramaticallywith f . For f = 0 the model corresponds to the standard XY model, which is notfrustrated. It shows a Kosterlitz-Thouless transition at T KT [on a square lattice [2] T KT = 0 . ξ diverges as ln ξ ∼ ( T − T KT ) − / for T ∼ > T KT ; the low-temperature phase, T < T KT , is characterized by quasi long-range order—correlation functions decay algebraically—associated with a line of fixedpoints. In the case of maximal frustration, i.e. for f = 1 /
2, the system undergoes twovery close continuous transitions (their critical temperature is T ≈ .
45 on the squarelattice), respectively in the Ising and Kosterlitz-Thouless universality classes, see, e.g.,[3, 4] and references therein. The critical behaviour for other values of f is even morecomplex, see, e.g., [5, 6, 7, 8, 9, 10, 11, 12], and [13] for experiments. There may beseveral transitions, whose nature is not clear in most of the cases. Even the structureof the ground state is only partially understood [14, 15, 16]. For f = 1 /n , where n is aninteger number, if T c is the critical temperature where the paramagnetic phase ends, T c decreases with increasing n ; for example, [9] T c ∼ < .
22 for f = 1 / T c ∼ < . , . n = 30 and 56, respectively. These studies suggest that T c vanishes [8, 7] as T c ∼ /n when n → ∞ . The critical behaviour for irrational values of f is even less clear, see,e.g., [11, 12]. In this case, there are some indications that the system is paramagneticfor any T and that a glassy transition occurs at zero temperature [12].The above-mentioned works studied the critical behaviour as a function of thetemperature T , while keeping the uniform frustration f fixed. In this paper weinvestigate a different critical limit, i.e. we consider the limit f → T in theregion T ≤ T KT . In other words, we investigate the effect of a small uniform frustrationon the low-temperature XY critical behaviour. We show that a uniform frustration isa relevant perturbation at the fixed points that occur in the XY model for T ≤ T KT .As soon as f is non-vanishing, the correlation length becomes finite and the system isparamagnetic.The critical behaviour for small values of f can be understood within the Coulomb-gas picture [17]. If one considers the Villain Hamiltonian corresponding to (1), one canwrite the partition function as Z Villain = Z Y x dθ x e − β H = Z SW X { n x } exp (2 πβ H CG ) , (6)where [17] Z SW is the spin-wave contribution and H CG is the Coulomb-gas Hamiltonian: H CG = 12 X ij ( n i − f ) V ( r i − r j )( n j − f ) , (7)where n i is an integer (vorticity) defined at the site i of the dual lattice and V ( r ) isthe lattice Coulomb potential. In (6) the sum over n x is restricted to configurationssatisfying the neutrality condition [17] P i ( n i − f ) = 0. For f = 0 and T < T KT this representation allows one to show that correlations functions decay algebraically.The two-point correlation function is the product of a spin-wave contribution, whichdecays algebraically, and of a vortex contribution. For T < T KT charged vortices are he uniformly frustrated two-dimensional XY model in the limit of weak frustration f > f , in the temperatureinterval f T KT < T < T KT , there are unbounded particles with n = 0 and charge − f ,which screen the Coulomb interaction among the vortices of charge n − f ≈ n , n = 0.The Debye screening length can be easily computed. Consider a vortex of charge 1,surrounded by particles of charge − f . Since there is one charge − f for each lattice site,complete screening is achieved when these charges occupy a circle of area A , such that Af = 1. Thus, the screening length ξ should be proportional to f − / . In this picture,for f →
0, the system is equivalent to a dilute gas (the density is proportional to f / ) ofneutral particles interacting by means of a screened Coulomb potential V sc ( r ). We canthus perform a standard virial expansion to predict that the vortex-vortex correlationfunction is proportional to V sc ( r ), hence decays exponentially with a rate controlled bythe Debye screening length. This argument indicates that, for sufficiently small f andany T < T KT , the system is paramagnetic with a correlation length that scales as ξ ∼ f − / , (8)for f → f and T , consider a real-space renormalisation-group (RG) transformation.Eliminate lattice sites obtaining a lattice with a link length that is twice that of theoriginal lattice. In lattice units we have ξ ′ = ξ/
2, where we use a prime for quantitiesthat refer to the decimated lattice. Analogously, we obtain f ′ = 4 f for the frustrationparameter. It follows ξ ′ f ′ / = ξf / . This quantity is therefore constant under RGtransformations, i.e. ξf / = c . Under the RG transformation, the Hamiltonianparameters also change. In particular, the transformation induces a temperature change T → T ′ . However, for small f , one is close to the XY line of fixed points and thuswe expect T ′ ≈ T . Thus, the condition ξf / = c holds at (approximately) fixedtemperature and f →
0. Therefore, it implies (8).In this paper we wish to verify numerically (8) and study the critical behaviour ofgauge-invariant susceptibilities (they will be defined in the next section). Note that, in asense, at fixed T ≤ T KT , the magnetic flux f plays the role of the reduced temperature,with an associated correlation-length exponent ν = 1 /
2. Definitions and general scaling properties
In order to check prediction (8), we consider two different gauge-invariant correlationfunctions: G sq ( x ; y ) ≡ |h ψ x ψ ∗ y i| , he uniformly frustrated two-dimensional XY model in the limit of weak frustration G Γ ( x ; y ) ≡ h Re ψ x U [Γ x ; y ] ψ ∗ y i . (9)Here Γ x ; y is a path that connects sites x and y and U [Γ x ; y ] is a product of phasesassociated with the links that belong to Γ x ; y . More precisely, if a link h wz i belongs tothe path, w and z have coordinates w = ( w , w ) and z = ( z , z ), such that z − w ≥ z − w ≥
0, we define R wz = U wz if point w occurs before point z while movingalong the path; otherwise, we set R wz = U ∗ wz . The phase U [Γ x ; y ] is the product of allthe phases R wz associated with the links belonging to the path.The definition (9) of G Γ ( x ; y ) depends on a family of paths Γ = { Γ x ; y } . We assumethis family to be translationally invariant: the path Γ x ; y is obtained by rigidly translatingthe path Γ y − x that connects the origin to y − x . In this case, the correlation function G Γ ( x ; y ) is uniquely defined by specifying the paths from the origin to any point x .Because of the presence of the gauge field, the Hamiltonian is not translationallyinvariant, nor is it symmetric under the symmetry transformations of the lattice.Nonetheless, there are generalized symmetries of the Hamiltonian that also involve gaugetransformations. For instance, if Lf is an integer, the Hamiltonian is invariant underthe generalized translations ψ ′ ( n,m ) = ψ ( n +1 ,m ) U ∗ ( n,m ) , ( n +1 ,m ) e − πimf ,ψ ′ ( n,m ) = ψ ( n,m +1) U ∗ ( n,m ) , ( n,m +1) e πinf . (10)Gauge-invariant correlation functions are invariant under these transformations. Thisimplies that they do not depend on x and y separately, but only on the difference y − x .This invariance can be understood intuitively if one notes that gauge-invariant quantitiesshould only depend on the value of the flux through a plaquette, i.e., U ( P ), and of thePolyakov correlations P ( m ) and P ( m ). In our model U ( P ) is independent of P and, if Lf is an integer, P ( m ) and P ( m ) do not depend on m : hence, translation invarianceholds.Analogously, the Hamiltonian is invariant under generalized transformations thatinvolve lattice symmetries and gauge transformations. For instance, in infinite volumethe Hamiltonian is invariant under the generalized reflection transformations ψ ′ ( n,m ) = ψ ∗ ( − n,m ) K ∗ m | n |− Y k =0 [ U ( k,m ) , ( k +1 ,m ) U ∗ ( − k − ,m ) , ( − k,m ) ] , (11)where K m = m − Y k =0 U ,k ) , (0 ,k +1) for m ≥ m = 0, − m − Y k =0 U ∗ ,k + m ) , (0 ,k + m +1) for m ≤ −
1. (12)Under these symmetries G sq ( x ; y ) transforms covariantly. If T is a lattice symmetry, G sq ( x ; y ) = G sq ( T x ; T y ). These relations do not hold in general for G Γ ( x ; y ) since alattice symmetry also changes the path family. he uniformly frustrated two-dimensional XY model in the limit of weak frustration G Γ ( x ; y ) and G sq ( x ; y ), we define the corresponding susceptibilities χ Γ ≡ X y G Γ ( x ; y ) , χ sq ≡ X y G sq ( x ; y ) , (13)where the sums are extended over all lattice points y . Because of translational invariance, χ sq and χ Γ do not depend on the point x . Of course, χ Γ depends on the family of pathsΓ = { Γ x ; y } . Then, for any gauge-invariant correlation function G ( x ; y ) we define on afinite lattice of size L F ≡ X y ≡ ( y ,y ) cos[ q min ( y − x )] G ( x ; y ) (14)where x ≡ ( x , x ) and q min ≡ π/L . The correlation length is defined by ξ ≡
14 sin ( q min / χ − FF . (15)Note that an equally good definition of F is F ≡ X y ≡ ( y ,y ) cos[ q min ( y − x )] G ( x ; y ) . (16)For the correlation function G sq ( x ; y ), one can show that these two definitions of F areequivalent, but this is not generically the case of G Γ ( x ; y ), since this quantity is notsymmetric under lattice transformations. In the following we use definition (14) for F .In the introduction we derived a prediction for the correlation length, ξ ∼ f − / .We wish now to obtain a similar result for the susceptibilities. In order to predict theirscaling behaviour, let us note that, for f = 0 and T ≤ T KT , h ψ ψ ∗ x i decays algebraically,i.e., h ψ ψ ∗ x i ∼ x − η ( T ) . The critical exponent η ( T ) depends on T and varies between η (0) = 0 and η ( T KT ) = 1 /
4. For f = 0, it is natural to assume that χ Γ ∼ Z x<ξ d x x − η ( T ) ∼ ξ − η ( T ) ∼ f − η ( T ) / ,χ sq ∼ Z x<ξ d x x − η ( T ) ∼ ξ − η ( T ) ∼ f − η ( T ) . (17)In particular, these equations predict χ Γ ∼ f − / and χ sq ∼ f − / at T = T KT .The check of the previous prediction for χ sq does not present conceptual difficulties.Instead, when considering χ Γ , one shoud keep in mind that this quantity depends on apath family. Thus, there is a natural question that should be considered first. Given apath family Γ ( f ) for a given value f = f of the frustration parameter, we must specifywhich path family Γ ( f ) must be considered for f = f = f . Only if Γ ( f ) is chosenappropriately, does the relation χ Γ ( f χ Γ ( f ≈ (cid:18) f f (cid:19) − η ( T ) / (18)hold for f , f →
0. A naive choice would be Γ ( f ) = Γ ( f ) . As we now discuss, thischoice is not correct: different path families should be chosen for different values of f .To clarify this issue, let us imagine we are working in the continuum. For each f , let us consider a family of paths Γ ( f ) = { Γ ( f ) x ; y } . Because of translation invariance, he uniformly frustrated two-dimensional XY model in the limit of weak frustration y y f f =f /4 Figure 1.
One the left we report two paths connecting the origin to y and y ,respectively. On the right, we report the corresponding paths connecting the origin to2 y and 2 y . The figure on the left correspond to a frustration parameter f = f , thaton the right to f = f = f / we can limit ourselves to paths going from the origin to any point y . These paths canbe parametrised in terms of a function X ( f ) ( t ; y ) such that X ( f ) (0; y ) = 0 for all y , X ( f ) (1; y ) = y . The path from the origin to y is given by x = X ( f ) ( t ; y ) t ∈ [0 , . (19)To determine the relation between Γ ( f ) and Γ ( f ) , one should remember that x/ξ shouldbe kept fixed in the critical limit. Thus, we expect the path family to be invariant onlyif all lengths are expressed in terms of ξ . In other words, set ¯ x = x/ξ f , ¯ y = y/ξ f , andrewrite (19) as¯ x = 1 ξ f X ( f ) ( t ; ¯ yξ f ) t ∈ [0 , , (20)where ξ f is the correlation length for the system with frustration parameter f . Thenatural requirement is therefore that the right hand side be independent of f , that is1 ξ f X ( f ) ( t ; ¯ yξ f ) = 1 ξ f X ( f ) ( t ; ¯ yξ f ) . (21)Since we expect ξ f ∼ f − / , we obtain the relation X ( f ) ( t ; ry ) = rX ( f ) ( t ; y ) , r = (cid:18) f f (cid:19) / . (22)In Fig. 1 we report an example corresponding to f = 4 f . The paths from the originto y and y which belong to Γ ( f ) completely fix the paths to 2 y and 2 y belonging toΓ ( f ) . Of course, on the lattice it is impossible to ensure (22) exactly. However, notethat the relevant scale is fixed by the correlation length and thus, violations at the levelof the lattice spacing are irrelevant in the critical limit.In the following we shall consider the path families Γ n ≡ { Γ n ;0; x } , which are specifiedby a non-negative integer n . They are defined as follows (see Fig. 2). The path Γ n ;0; x connecting the origin to the point x ≡ ( x , x ) consists of three segments: the first oneconnects the origin to ( − n, − n,
0) to ( − n, x ); the last he uniformly frustrated two-dimensional XY model in the limit of weak frustration x n Figure 2.
The path connecting the origin to the point x which belongs to the pathfamily Γ n . one is horizontal, from ( − n, x ) to point x . We indicate with χ n ( f ) the correspondingsusceptibilities and with ξ n ( f ) the corresponding correlation lengths. These families ofpaths behave simply under the transformation (22). If we consider the path Γ n ;0; x for f = f , the mapping (22) implies that, for f = f , one should consider the path Γ rn ;0; rx between the origin and the point rx . This implies that, if we take the path family Γ n for f = f , we must consider Γ rn for f = f . As a consequence, χ n and ξ n scale correctlyonly if we consider the limit n → ∞ , f → nf / . Thus, we predict the scalingbehaviours χ n = f − η ( T ) / F χ ( nf / ) ,ξ n = f − / F ξ ( nf / ) , (23)where F χ ( x ) and F ξ ( x ) are appropriate scaling functions. In the next Section, we verifythese predictions.
3. Numerical results
We perform simulations for various values of f = 1 /m , m integer, and T in the interval T ≤ T KT , where T KT is the critical temperature of the XY model, T KT = 0 . L , where L is a multiple of 1 /f , and periodic boundaryconditions for the spins. Since we perform MC simulations in a gapped phase, boundaryconditions are expected to be irrelevant in the thermodynamic limit. Cluster algorithmscannot be used in the presence of frustration and thus we use an overrelaxed algorithm,which consists in performing microcanonical and Metropolis updates. Predictions (8)and (17) hold in the thermodynamic limit, i.e. for sufficiently large values of the ratio he uniformly frustrated two-dimensional XY model in the limit of weak frustration n f ξ n f / f=1/20f=1/25f=1/30 Figure 3.
Scaling plot for the correlation length ξ n at T KT . For each f we reportthe data satisying n ≤ / (2 f ). L/ξ , where finite-size effects are negligible. We find numerically that size effects aremuch smaller than our statistical errors for Lf ∼ > A xy = 0 if y = x + ˆ1 , (24) A xy = 2 πf x if y = x + ˆ2 , which is consistent with (3) and with P ( m ) = P ( m ) = 1, as long as L is an integermultiple of 1 /f . With this gauge choice the computation of the susceptibilities χ n andof the corresponding correlation lengths ξ n is quite simple. Indeed, U [Γ n ; x ; y ] = 1 for any y if the first component of x is n , i.e., if x = ( n, m ), m arbitrary. Thus, if we choose x = ( n, m ) in definition (13), we can compute χ n without taking into account the phases U xy . In practice, we have determined χ n by using χ n = 1 L X m X y h Re ψ ( n,m ) ψ ∗ y i . (25)An analogous expression holds for the correlation lengths.In Figs. 3 and 4 we plot the correlation lengths ξ n and the susceptibilities χ n at T = T KT for several values of f and n . In this case η ( T ) = 1 / χ n should scaleas f − / . It is easy to show that χ n = χ n +1 /f , χ n = χ /f − n , (26)so that in (23) one must restrict oneself to data satisfying 0 ≤ n ≤ / (2 f ). The resultsreported in the figures show the scaling behaviour (23) quite precisely, confirming thetheoretical arguments. Note that the scaling function F χ ( x ) apparently goes to zero as x increases. This behaviour will be confirmed below by the analysis of a non-gauge-invariant correlation function. he uniformly frustrated two-dimensional XY model in the limit of weak frustration n f χ n f / f=1/20f=1/40f=1/80f=1/100f=1/120 Figure 4.
Scaling plot for the susceptibilities χ n at T KT . For each f we report thedata satisying n ≤ / (2 f ). ln ln χ Figure 5.
Critical behaviour of χ vs 1 /f at T = 0 .
4. The line is the results of a fitto χ = af − ε , which gives ε = 0 . η ( T = 0 .
4) = 0 . Good agreement is also found at
T < T KT . We check the behaviour of χ n =0 (in this case, the same path family can be used for all values of f ) upto T = 0 .
2. At T = 0 . , . , . , . , .
8, a fit of χ to af − η ( T ) / gives η = 0 . , . , . , . , . η = T / (2 π ), and the MCestimates [19] η = 0 . , . , . , . , . χ at T = 0 .
4, together with the result of the fit. The datashow a clear power-law behaviour in perfect agreement with (17).We also investigated the critical behaviour of χ sq , which is expected to scale as f − / . he uniformly frustrated two-dimensional XY model in the limit of weak frustration ln χ w χ w f Figure 6.
MC results for the non-gauge-invariant susceptibility χ w and for theproduct f / χ w vs ln 1 /f at T = T KT . For 1 /f = 40, 60, 80, we obtain χ sq = 9 . f / χ sq clearly converges to a constant as f → / ln(1 /f ), as in the XY model at T KT ): we have f / χ sq = 0 . f .Finally, we mention that correlation functions which are not gauge invariant showa different behaviour. For example, one may consider the susceptibility χ w associatedwith the two-point function h Re ψ x ψ ∗ y i in the gauge (25): χ w = 1 L X x,y h Re ψ x ψ ∗ y i . (27)At T KT it shows a power-law behaviour χ w ∼ f − ε as well, but with a power ε ≈ . .
875 of the gauge-invariant definition. This resultcan be derived analytically. Indeed, we can rewrite χ w = 1 L L − X n =0 χ n , (28)where χ n is defined in (25). Using the properties (26) of the susceptibilities χ n , (28) canbe rewritten as χ w ≈ f / (2 f ) X n =0 χ n . (29)In this range of values of n , as is clear from Fig. 4, we can use the scaling behaviour(23) and write χ w ∼ f × f − / Z / (2 f )0 dn F ( nf / ) he uniformly frustrated two-dimensional XY model in the limit of weak frustration ∼ f − / Z / (2 f / )0 dx F ( x ) ∼ f − / Z ∞ dx F ( x ) . (30)Thus, provided that F ( x ) is integrable (we already noted that the MC data for χ n are consistent with F ( x ) → x → ∞ ), we predict χ w ∼ f − / = f − . , which isconsistent with the MC data (see Fig. 6).Note that the critical behaviour of χ w depends on the chosen gauge. If we use thegauge A xy = − πf x if y = x + ˆ1 , (31) A xy = πf x if y = x + ˆ2 , the susceptibility χ w does not diverge and approaches a constant as f → XY model for T ≤ T KT . As soon as f is different from zero, the systembecomes paramagnetic. The critical behaviour ξ ∼ f − / can be predicted by simpleCoulomb-gas and scaling arguments. Our numerical simulations fully confirm thisprediction. Also the scaling behaviour (17) for the magnetic susceptibilities is fullyconsistent with the numerical results. References [1] Fazio R and van der Zant H 2001
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