Stochastic processes with Z_N symmetry and complex Virasoro representations. The partition functions
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Stochastic processes with Z N symmetry and complexVirasoro representations. The partition functions Francisco C. Alcaraz , Pavel Pyatov and Vladimir Rittenberg Universidade de S˜ao Paulo, Instituto de F´ısica de S˜ao Carlos, Caixa Postal 369,13560-590 S˜ao Carlos, S˜ao Paulo, BrazilE-mail: [email protected] National Research University Higher School of Economics, Laboratory ofMathematical Physics, 20 Myasnitskaya street, Moscow 101000, Russia & BogoliubovLaboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980Dubna, Moscow Region, RussiaE-mail: [email protected] Physikalisches Institut, Universit¨at Bonn, Nussallee 12, 53115 Bonn, GermanyE-mail: [email protected]
Abstract.
In a previous Letter [1] we have presented numerical evidence that aHamiltonian expressed in terms of the generators of the periodic Temperley-Liebalgebra has, in the finite-size scaling limit, a spectrum given by representations of theVirasoro algebra with complex highest weights. This Hamiltonian defines a stochasticprocess with a Z N symmetry. We give here analytical expressions for the partitionfunctions for this system which confirm the numerics. For N even, the Hamiltonian hasa symmetry which makes the spectrum doubly degenerate leading to two independentstochastic processes. The existence of a complex spectrum leads to an oscillatingapproach to the stationary state. This phenomenon is illustrated by an example. Submitted to:
J. Phys. A: Math. Gen. tochastic processes and complex Virasoro representations Z N symmetric representations of the periodic Temperley-Lieb algebra P T L L ( x ) [2, 3, 4, 5], we have defined a Hamiltonian as a linear combination of thegenerators of this algebra. Taking x = 1, this Hamiltonian gives the time evolutionof a one-dimensional stochastic process. Looking at the finite-size scaling spectra ofthis Hamiltonian, we have obtained numerical evidence for the appearance of Virasororepresentations with complex highest weights. Moreover, the real part of the complexhighest weights is smaller than the real highest weights and hence dominate the largetime behavior of the systems. This observation was a big surprise and was the maincontent of a previous Letter [1]. In the present one, we give an analytic derivation ofthis result and present the partition function for each sector of the model. We alsopresent an application of our results. For N even we show that there is a symmetryin the model which makes the spectrum for any lattice size to be doubly degenerateindicating the presence of a zero fermionic mode. This Letter is basically a continuationof the previous one [1], we did nevertheless our best to make it self-consistent.The P T L L ( x ) algebra has L generators e k , k = 1 , , . . . , L satisfying the relations: e k = xe k , e k e k ± e k = e k , [ e k , e ℓ ] = 0 , | k − ℓ | > , (1)with e k + L = e k . We take L even only. We consider two quotients of the algebra:( AB ) N A = A, (2)where A = L/ Y j =1 e j , B = L/ − Y j =0 e j +1 , (3)and ABA = α A (4)with α = e i πr/N , r = 0 , , . . . , N −
1. One can see that the quotient (4) is a solution ofeq.(2) which defines the first quotient. This observation will be crucial in obtaining thepartition functions mentioned above.In order to get the Z N symmetric representations of (1) and (2), we consider N copies of a one-dimensional periodic system with L sites. Each copy consistsof (cid:0) LL/ (cid:1) configurations of link patterns on a cylinder and n noncontractible loops( n = 0 , , . . . , N −
1) on the same cylinder. This is the vector space in which thegenerators of the
P T L L ( x ) algebra act. It has the dimension N × (cid:0) LL/ (cid:1) . In Fig.1 weshow the 6 configurations for L = 4 and n = 2.An alternative way to label the states in the vector space is to use the spinrepresentation in which the slopes in the arches are used +( − ) for the beginning (ending) tochastic processes and complex Virasoro representations (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) | + , + , − , −i (2) |− , + , + , −i (2) |− , − , + , + i (2) | + , − , − , + i (2) | + , − , + , −i (2) |− , + , − , + i (2) Figure 1.
The six link patterns configurations for L = 4 sites on a cylinder and twocircles without sites (noncontractible loops). The open arches and circles meet behindthe cylinder. The corresponding spin presentations of the same link patterns are givenon the right. of an arch. The number of non-contractible loops is indicated by a supplementary label.This notation is also given in Fig.1.The action of the generators e k on the link patterns for a given copy n is the same asthe one used for the usual (non-periodic) Temperley-Lieb algebra [6] with one exception.If the generator acts on the bond connecting the beginning and the end of an arch havingthe size of the system L , one obtains a configuration of the copy n + 1 (see Fig.2). (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Figure 2.
The action of the e generator acting the bond between the sites 2 and 3which are the end of an arch of the size of the system L = 4. A new circle is createdon the cylinder and one moves from the copy n = 2 to the copy n = 3. In order to obtain the Z N representation of the P T L L ( x ) algebra with the quotient(2) one identifies the copy n = N with the copy n = 0.In what follows, we take the parameter x in the P T L L ( x ) algebra equal to one.With this choice the Hamiltonian H = L X k =1 (1 − e k ) (5)gives the time evolution of a stochastic process. We want to stress that the propertiesof the spectra which are going to be discussed below, stay valid for any value of x .Notice that for N even the Z N symmetric representation decomposes into a pair ofidentical N × (cid:0) LL/ (cid:1) -dimensional irreps. ‡ To see this, consider two linear transformations ‡ For N odd the Z N representation is irreducible. tochastic processes and complex Virasoro representations Z N space.The transformation X acts diagonally in a following way. We start consideringconfigurations with no arches hidden in the back of the cylinder (the first andfifth link patterns in Fig.1). These configurations get a factor of ( − n . Theconfigurations translated with one lattice unit get a factor ( − ( n +1) (the second andsixth configurations in Fig.1). The next translated configurations get again the factor( − n and so on and so forth.Transformation X permutes copies as follows: | . . . i (2 k ) ↔ | . . . i (2 k +1) , k = 0 , , . . . , N − . The two transformations constitute the algebra:( X ) = ( X ) = id , X X + X X = 0 . (6)This algebra has only 2-dimensional equivalent irreducible representations. Since X and X commute with the action of the P T L L ( x ) on the Z N symmetric link patterns,it follows that for N even, the spectrum of H is doubly degenerate. To illustrate thisobservation, let us take L = 4 and N = 2. The Hamiltonian splits into two stochasticHamiltonians having each six states. The first one having the states | + + − −i (0) , | − − + + i (0) , | + − + −i (0) , | − + + −i (1) , | + − − + i (1) and | − + − + i (1) , the secondone having the six states in which the copies 0 and 1 are permuted.We are interested in the spectra of H in the finite-size scaling limit. Let uskeep in mind that in a stochastic process, the energies coincide with the energy gapssince the ground-state energy is zero for any system size. Since H is invariant undertranslations ( e k e k +1( mod L ) ) and the cyclic rotations Z N ( | . . . i ( n )
7→ | . . . i ( n +1 (mod N ) ) ),one has N × L sectors labeled by p = 0 , ± , ± , . . . corresponding to the momenta P = 2 πp/L and by r = 0 , , . . . , N −
1, labeling the irreps of Z N . If E rp ( q ), q = 1 , , . . . , are the energy levels in the sector ( p, r ), the scaling dimensions x rp ( q ) are given bylim L →∞ ( E rp ( q ) L ) = 2 πv s x rp ( q ), with the sound velocity v s = 3 √ / H for N = 3. We went up to L = 30 andlooked at the lowest excitations. In the r = 0 sector we confirmed the expected value x (1) = 0 .
25. The surprise came when we looked at the r = 1 sector where we found: x (1) = 0 . . i, x (2) = 0 . − . i, (7)i.e. complex values. For N even we found E rp ( q ) = E r + N/ p + L/ ( q ) which is a consequence ofthe symmetry (6).We present now our new results. In order to obtain the partition function in eachsector r , we use the fact that the Z N representation of the algebra with the quotient (2)can be decomposed into N representations of the quotient (4) [3]. The representationsof the quotient (4), in the link patterns vector space, are obtained by considering a singlecopy but changing the action of the generators when they act on a bond connecting the tochastic processes and complex Virasoro representations L like in Fig.2. Instead of addinga non-contractible loop, one multiplies the state in the right hand side of the figureby a fugacity α = exp(2 πir/N ). It was shown [2] that the quotient (4) admits also arepresentation in the standard spin 1 / e k = σ + k σ − k +1 + σ − k σ + k +1 + 14 (1 − σ zk σ zk +1 ) + i √
34 ( σ zk +1 − σ zk ) , k = 1 , , . . . , L − ,e L = e i πφ σ + L σ − + e − i πφ σ − L σ +1 + 14 (1 − σ zL σ z ) + i √
34 ( σ z − σ zL ) . (8)The twist φ is related to the parameter α by the relation: α = 2 cos( πφ ) . (9)The vector space of the (cid:0) LL/ (cid:1) link patterns configurations corresponds to the S z = P Lk =1 σ zk = 0 sector of the spin vector space.The Hamiltonian (5),(8) is integrable using the Bethe Ansatz and the scalingdimensions (highest weights of Virasoro representations) are known [7]. They are givenby the Gaussian model. In the S z = 0 sector they are: x = 34 ( s + φ ) −
112 + m + m ′ , p = m − m ′ , (10)where s, m, m ′ = 0 , ± , ± , . . . .From (9) we see that for N even α = e i πr/N and α = e i π ( r + N/ /N = − e i πr/N give the same value for the twist φ and therefore the spectrum of the Hamiltonian (5)is doubly degenerate, in agreement with our previous observation. Moreover, for thesectors r not equal to 0 or N/ φ obtained from (9) are complex, henceforththe scaling dimensions (critical exponents) (10) are complex too.As a check, taking N = 3 and r = 1, from (9) one gets: φ = − . − . i from which we get using (10) with s = 0 and 1 x (1) = 0 . . i ; x (2) = 0 . − . i (11)in excellent agreement with the values (7).We have to stress that although the spectrum of the Z N symmetric Hamiltoniansplits into N sectors, the stochastic process doesn’t. The condition of positivity of thewave function describing the probability distribution function, mixes the sectors.The existence of complex scaling dimensions has consequences on the time behaviorof various correlators showing oscillatory phenomena, more so since their real part issmaller than real scaling dimensions of the r = 0 sector. To illustrate the phenomenon,we looked at the density of ”peaks” d ( n ) ( t, L ) in different copies. Those are + − pairs tochastic processes and complex Virasoro representations z -0.0500.05 d ( n ) (t , L ) - d n=0 n=1n=2 Figure 3.
The density of ”peaks” d ( n ) ( z ) in the copy n as a function of z = 2 πv s t/L for N = 3, and various lattice sizes L = 40 , , , , , d , thevalue of the density in the stationary state, is subtracted. which in the Dyck paths picture of the link patterns [8] correspond to peaks in thepaths. This local observable is measured easily in Monte Carlo simulations. The timedependence of the ”peaks” in various sectors is determined by the initial conditions. Forlarge values of t and large lattice sizes we expect d ( n ) ( t, L ) to be a function of t/L : d ( n ) ( t, L ) = d + X k A k cos [Im( x ( n )0 ( k )) z ] e − Re ( x ( n )0 ( k )) z , z = 2 πv s tL , (12)where d is the density of ”peaks” in the stationary state which is the same for each copy n and the A k ’s are dependent on the initial conditions. We have computed d ( n ) ( t, L ) − d using Monte Carlo simulations in the case N = 3 for different lattice sizes. The initialstate was the configuration | + , − , + , − , . . . , + , −i (0) in the copy n = 0. The results areshown in Fig.3. One can see that, as expected, the densities are dependent on z only.Encouraged by this observation, we did a fit to the n = 0 data (see Fig.4) usingthe parameterization d (0) ( z ) − d = e − az cos b ( z − z )cos bz (13) tochastic processes and complex Virasoro representations z d ( ) (t , L ) - d L=40 L=80 L=2000 Fit L=4000
Figure 4.
A fit to d (0) ( z ) − d as a function of z , using the parameterization (13)and (14). Monte Carlo data for lattice sizes 40 , , and obtain: a = 0 . , b = 0 . , z = 0 . . (14)These values are compatible with x (2) given in (11). To our knowledge, it is forthe first time that expressions like (13) appear in a conformal invariant theory.Before closing this Letter, we would like to notice that in a seminal paper Saleurand Sornette [9] have suggested the possible existence of complex critical exponents innon unitary conformal field theories. We have shown that they indeed exist. Acknowledgements
This work was supported in part by the joint DFG and RFBR grants no.RI 317/16-1and no.12-02-5133-NNIO, by FAPESP and CNPq (Brazilian Agencies), and a grant ofthe Heisenberg-Landau program. PP was also supported by the RFBR grant 14-01-00474 and by the Higher School of Economics Academic Fund grant 14-09-0175. FCAthanks the Bethe Center of the Bonn University for partial financial support. tochastic processes and complex Virasoro representations References [1] Alcaraz F C, Ram A and Rittenberg V 2014
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