Exact results for the O(N) model with quenched disorder
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Exact results for the O ( N ) model with quenched disorder Gesualdo Delfino , and Noel Lamsen , SISSA – Via Bonomea 265, 34136 Trieste, Italy INFN sezione di Trieste
Abstract
We use scale invariant scattering theory to exactly determine the lines of renormalizationgroup fixed points for O ( N )-symmetric models with quenched disorder in two dimensions.Random fixed points are characterized by two disorder parameters: a modulus that vanisheswhen approaching the pure case, and a phase angle. The critical lines fall into three classesdepending on the values of the disorder modulus. Besides the class corresponding to thepure case, a second class has maximal value of the disorder modulus and includes Nishimori-like multicritical points as well as zero temperature fixed points. The third class containscritical lines that interpolate, as N varies, between the first two classes. For positive N , itcontains a single line of infrared fixed points spanning the values of N from √ − N -independent)along this line. For N = 2 a line of fixed points exists only in the pure case, but accountsalso for the Berezinskii-Kosterlitz-Thouless phase observed in presence of disorder. aining theoretical access to the critical properties of disordered systems with short rangeinteractions has been a challenging problem of statistical mechanics. For weak randomness,the Harris criterion [1] relates the relevance of disorder to the sign of the specific heat criticalexponent of the pure system. If this sign is positive weak disorder drives the system towards anew (“random”) fixed point of the renormalization group, responsible for new critical exponentsthat in some limits can be computed perturbatively (see e.g. [2]). In the regime of strongdisorder, a relevant role is played by the gauge symmetry [3] exhibited by systems such asthe Ising model with ± J bond randomness. This allows, in particular, the idenfitication of amulticritical point along the phase boundary separating the ferromagnetic and the paramagnetic(or spin glass, if present) phases in the temperature-disorder plane. For the rest, the study ofcritical properties at strong disorder has essentially relied on numerical methods.Particularly noticeable has been the absence of exact results in two dimensions, to the pointthat one could legitimately wonder whether random fixed points of planar systems possess theinfinite-dimensional conformal symmetry [4, 5] that yielded the exact critical exponents in thepure case. Progress has been achieved recently [6] extending to the random case the ideaof implementing conformal invariance within the basis of the underlying particle excitations[7, 8]. It was explicitly shown in [6, 9] for the q -state Potts model with quenched disorder howthe method yields exact equations for the scattering amplitudes whose solutions correspond torandom fixed points. One of the remarkable emerging properties is the presence of superuniversal(i.e. symmetry independent) sectors able to shed light on longstanding numerical and theoreticalpuzzles for critical exponents.In this paper we consider two-dimensional disordered systems with O ( N ) symmetry that re-duce to the N -vector ferromagnet in the pure limit. It is known that weak disorder is marginallyirrelevant at N = 1 (Ising) [10], and becomes relevant for N <
1. This means that slightly below N = 1 an infrared random fixed point can be found through a perturbative approach similar tothat used in [11, 12] for the q → + Potts model. This perturbative study was performed in [13],where the one-loop beta function was used to argue that the line of infrared fixed points spansan interval N ∈ ( N ∗ , N ∈ (0 , N ∗ ) the system flows directly to a strongdisorder regime; the estimate N ∗ ≈ .
26 was obtained within the one-loop approximation. The O ( N ) model with a specific bimodal distribution of bond disorder was then studied in [14] withina numerical transfer matrix approach. In particular, this study confirmed the presence of thelower endpoint N ∗ for the line of infrared fixed points originating at N = 1, and obtained theestimate N ∗ ≈ .
5. At N = N ∗ the infrared line was observed to join a line of strong randomnessmulticritical points extending for N > N ∗ , and the universal properties of the point at N = 1 onthis line were found in quantitative agreement with those of the Nishimori multicritical point.Below we will use the scattering formalism to exactly determine the lines of renormalizationgroup fixed points for systems with O ( N ) symmetry in presence of quenched disorder. Wewill show, in particular, that these critical lines belong to three different classes depending onthe values of a disorder modulus ρ , one of two parameters associated to disorder. The threeclasses are: solutions for the pure systems ( ρ = 0), strongly disordered solutions ( ρ = 1),1 Ferro Para M Figure 1: Qualitative phase diagram and expected fixed points for the two-dimensional Isingmodel with ± J disorder. 1 − p is the amount of disorder and M indicates the multicritical(Nishimori) point.and solutions with values of ρ interpolating between 0 and 1. For positve N , the latter classcontains a single line of infrared fixed points, extending from N = 1 (where ρ = 0) down to N = N ∗ = √ − . .. (where ρ = 1). At N ∗ this line joins one of the solutions in theclass ρ = 1, which are defined for any N . The class with ρ = 1 contains fixed points that donot merge a fixed point of the pure system in any limit. Typical examples in this class are themulticritical points of Nishimori type and those encountered flowing from such a multicriticalpoint towards lower temperatures.We start recalling that the random bond N -vector model is defined by the lattice Hamiltonian H = − X h i,j i J ij s i · s j , (1)where s i is a N -component unit vector located at site i , the sum runs over nearest neighboringsites, and J ij are bond couplings drawn from a probability distribution P ( J ij ). The average overdisorder is taken on the free energy, F = X { J ij } P ( J ij ) F ( J ij ) . (2)The well known replica method exploits the fact that F is related to the partition function Z = P { s i } e −H /T as F = − ln Z , so that the identity F = − ln Z = − lim m → Z m − m (3)maps the problem onto that of m → P ( J ij ) = pδ ( J ij −
1) + (1 − p ) δ ( J ij + 1).2 i a i a i a i a i a i b j b j b j b j b j a i b i b i a i a i a i a i a i b i b i b i b i b j Figure 2: Scattering processes corresponding to the amplitudes S , S , S , S , S , S , in thatorder. Time runs upwards, indices i and j correspond to different replicas.When approaching a fixed point of the renormalization group the correlation length divergesand the universal properties of the system can be studied directly in the continuum, withinthe field theoretical framework. For the case we consiser, in which homogeneity of the systemis restored by the disorder average, the field theory in question is rotationally invariant, andcorresponds to the analytic continuation to imaginary time of a relativistically invariant quantumfield theory. We study these field theories within their basis of particle excitations, relying onlyon symmetry and restricting our attention to fixed points.As observed in [17] for the off-critical pure case, O ( N ) symmetry is implemented adoptinga vector multiplet representation of the particle excitations. In our scale-invariant case, theseparticles are left- and right-movers with momentum and energy related as p = ± E . Moreover,such excitations exist in each of the m replicas and will be denoted as a i , where a = 1 , , . . . N , i = 1 , , . . . , m . When considering the scattering of a right-mover with a left-mover, the infinitelymany conservation laws implied by conformal symmetry in two dimensions allow only for finalstates with a left-mover and a right-mover [7]. The scattering amplitudes are energy independentby scale invariance, and the product of two vectorial representations yields the six possibilitiesdepicted in Fig. 2. They correspond to transmission and reflection within the same replica( S and S , respectively) or in different replicas ( S and S ); two identical particles can alsoannihilate producing another pair within the same replica ( S ) or in a different replica ( S ).Crossing symmetry [18] then relates amplitudes under exchange of space and time directions as S = S ∗ ≡ ρ e iφ , (4) S = S ∗ ≡ ρ , (5) S = S ∗ ≡ ρ e iθ , (6) S = S ∗ ≡ ρ , (7)where we introduced parametrizations in terms of ρ and ρ non-negative, and ρ , ρ , φ and θ ρ + ρ = 1 , (8) ρ ρ cos φ = 0 , (9) N ρ + N ( m − ρ + 2 ρ ρ cos φ + 2 ρ cos 2 φ = 0 , (10) ρ + ρ = 1 , (11) ρ ρ cos θ = 0 , (12)2 N ρ ρ cos( φ − θ ) + N ( m − ρ + 2 ρ ρ cos θ + 2 ρ ρ cos( φ + θ ) = 0 . (13)We notice that the superposition P a,i a i a i scatters into itself with amplitude S = N S + S + S + ( m − N S , (14)which must be a phase by unitarity. Similarly, the combinations a i b i + b i a i and a i b j + b j a i scatterinto themselves with phases Σ = S + S , (15)¯Σ = S + S , (16)respectively.The solutions of equations (8)–(13) correspond to renormalization group fixed points charac-terized by O ( N ) invariance and permutational symmetry of the m replicas. Equations (8) and(12) can be used to express ρ and ρ in terms of ρ and ρ , which take values in the interval[0 , ρ , to which we refer as disorder modulus, gives a meausre of the disorderstrength at the fixed point, since for ρ = 0 the replicas decouple ( S = S = 0, S = ±
1) andEqs. (8)–(10) are those for the pure case ( m = 1). The interacting solutions for this pure caseare [7] ρ = 1 , ρ = 0 , − φ = N ∈ [ − , , (17)and ρ = q − ρ , cos φ = 0 , N = 2 ; (18)the latter is a line of fixed points parametrized by ρ that accounts for the Berezinskii-Kosterlitz-Thouless (BKT) phase of the XY model [19].Coming to random fixed points ( ρ = 0, m = 0), Eq. (10) shows that they have ρ = 0, and(11), (12) show that they fall into two classes. The first class has cos θ = 0 and disorder modulusvarying with N , while the second class has fixed (actually maximal) disorder modulus ρ = 1.Considering the class with varying ρ , we look for the line of fixed points that approaches thepure Ising point as N →
1. Then (17) excludes cos φ = 0 for any N , so that (9) implies ρ = 0,and we finally obtain ρ = 1 , ρ = cos θ = 0 , cos φ = − N + 1 , ρ = (cid:12)(cid:12)(cid:12)(cid:12) N − N + 1 (cid:12)(cid:12)(cid:12)(cid:12) r N + 2 N . (19)4igure 3: Projection in the parameter subspace ρ -cos φ of the lines of renormalization groupfixed points of the disordered O ( N ) model, for N ∈ (0 , ρ varies from0 (pure case) to the maximal value 1. Merging occurs at N = 1 for ρ = 0 and N ∗ = √ − ρ = 1.For positive N this solution is defined for N ≥ √ −
1, and has ρ → N →
1, as expected.Notice that for this solution the phase (14) becomes S = 2 cos φ = − N = 1, in agreementwith the fact that the pure Ising model in two dimensions is a free fermionic theory (scatteringon the line involves position exchange); actually, this has been used to fix the sign of cos φ in(19). We know from Harris criterion that the branch with N < N ∗ = √ −
1. At this point thesolution (19) has ρ = 1 and reaches the subspace of fixed points with maximal disorder modulus(Fig. 3). In this subspace there exists and is unique a solution coinciding with (19) at N ∗ ; it isdefined for any N and reads ρ = ρ = 1 , ρ = 0 , cos φ = − √ , cos θ = − N + 2 N − √ N + 1) . (20)The subspace with ρ = 1 contains another solution defined for positive N , and actually for any N ; it differs from (20) for having cos θ = cos φ , and is then completely N -independent.The fixed point pattern of Fig. 3 allows a discussion of the renormalization group flowsbetween ρ = 0 and ρ = 1. First of all we know that weak disorder is relevant for N ∈ (0 , N ∈ (1 , N ∈ (0 , N ∗ ) the flow goes directly fromthe pure model to the strong disorder solution (20), while for N ∈ ( N ∗ ,
1) there are flows fromthe pure model and the solution (20) towards the infrared fixed line (19). For N ∈ (1 , ρ = 1, and we expect this pattern to extend to the region N >
N > The scaling dimension X ε of the energy density operator in the pure model becomes smaller than 1 for N < X ε , is relevant. For
N > N ∗ the solution (19) approaches ρ = 1 only in the asymptotic limit N → ∞ . = S = 0, S = ±
1, consistently with the fact that the pure model with
N > ρ = 1 between solutions differing for the value of thesecond disorder parameter θ . Taking this into account, for values of N inside the interval ( N ∗ , N = 0 . θ will be discussed in more detailin [20], where we will also give the solutions of the fixed point equations for finite number ofreplicas, and will discuss the case N = 0, relevant for polymers in a disordered environment.It is interesting to notice that in [14] the phase diagram was also numerically explored for N = 8, with results that might appear not completely consistent with what we found for theregime N >
2. The point can be illustrated for the pure case, where only a fixed point with Z symmetry was observed in [14], while we saw that no such a fixed point is allowed by O ( N )symmetry. The explanation is in the fact that the study of [14] is made for the loop model onthe hexagonal lattice. It is well known that the partition function of the N -vector spin modelcan be rewritten as a sum over loop configurations [21, 2]. If this is done on the hexagonallattice [22], the loops cannot intersect. As originally observed in [17], the loop paths correspondin the scattering picture to the particle trajectories, and non-intersection in the pure modelcorresponds to S = 0 (see Fig. 2). It follows that the hexagonal lattice loop model yieldsthe fixed points of the pure O ( N ) spin model in the interval N ∈ ( − , S = 0 (seeEq. (17)), but not in the regime N >
2, where there is no reflection at all. For
N > Z -symmetric fixed point associated to the specificlattice symmetry rathen than to O ( N ) symmetry [23, 14].For N = 2 the equations (8)–(13) admit a line of fixed points only in the pure case ρ = 0;this is the line (18) that, as we already pointed out, accounts for the BKT phase of the puremodel. On the other hand, since the flow from ρ = 1 can end in the infrared onto any pointof the line (18), also the disordered model should exhibit a BKT phase, and this is confirmedby numerical studies (see e.g. [24, 25]). The phase diagram observed in these studies is similarto that of Fig. 1, with the ferromagnetic phase replaced by the BKT phase . On the otherhand, numerical studies still disagree on the values of critical exponents along the portion of thephase boundary going from the multicritical point M to the critical point of the pure model: aconstant magnetic exponent η = 1 / η was found in [25].It can be checked that the scattering phase (14) is N -independent for the solution (19), Notice that the model studied in [24, 25] is the random phase XY model, for which s i = (cos α i , sin α i ), thenearest neighbor interaction is − cos( α i − α j + A ij ), and A ij are the random variables drawn from a distribution P ( A ij ) ∝ e − A ij /σ ; σ replaces 1 − p in Fig. 1. Relying only on symmetry, our formalism applies also to this typeof disorder. N dependence disappears only in the limit m = 0 corresponding to quenched disorder.This means that the symmetry sector of the superposition P a,i a i a i , to which the energy densityoperator belongs, becomes superuniversal along this line of fixed points. An analogous resultobtained in [6] and further discussed in [9] accounts for the accumulated evidence [26, 27, 28,29, 30, 31, 32, 33, 34] that the correlation length critical exponent ν in the random bond q -statePotts ferromagnet does not show any appreciable deviation from the Ising value up to q infinite.On the other hand, the spin operator does not belong to the superuniversal sector and its scalingdimension is expected to vary along the solution (19). This scaling dimension was measuredin [14] at N = 0 .
55 on the infrared fixed line and found to be consistent with the two-loopperturbative result of [13]. We also observe that the phase amplitude (15) is straightforwardlyseen to be N -independent along the solution (20).In summary, we used scale (as well as conformally) invariant scattering theory to exactly de-termine the lines of renormalization group fixed points in O ( N ) invariant models with quencheddisorder. We showed that random fixed points are characterized, in particular, by two disorderparameters: a modulus ρ and a phase angle θ . The critical lines fall into the three classeswith ρ = 0 (pure case), ρ = 1 (containing Nishimori-like multicritical points as well as zerotemperature fixed points), and ρ interpolating between 0 and 1 as N varies. The pattern offixed points allowed us to deduce, in particular, that weak disorder drives the system to ρ = 1for values of N in the interval (0 , N ∗ = √ − N ∗ , N >
1. The exact result N ∗ = 0 . .. is not farfrom the numerical estimate N ∗ ≈ . N ∗ ,