Fluctuations of the winding number of a directed polymer in a random medium

Fluctuations of the winding number of a directed polymer in a random medium

E´ ric Brunet

Laboratoire de Physique Statistique, E´ cole Normale Supe´rieure, 24 rue Lhomond, 75231 Paris Cedex 05, France 共Received 18 June 2003; published 1 October 2003兲

For a directed polymer in a random medium lying on an infinite cylinder that is in 1⫹1 dimensions with finite width and periodic boundary conditions on the transverse direction, the winding number is simply the algebraic number of turns the polymer does around the cylinder. This paper presents exact expressions of the fluctuations of this winding number due to, first, the thermal noise of the system and, second, the different realizations of the disorder in the medium.

DOI: 10.1103/PhysRevE.68.041101 PACS number共s兲: 05.40.⫺a, 36.20.⫺r, 64.60.Cn, 71.55.Jv INTRODUCTION

A directed polymer in a random medium is one of the most simple non-trivial disordered system and is, as such, of special theoretical importance. Indeed, several exact results on directed polymers with strong disorder have been ob- tained关1–5兴, so that general approximation schemes devel- oped to tackle more complicated disordered systems such as spin glasses could be tested within the directed polymer con- text. The directed polymer is also relevant in the context of nonequilibrium phenomena, as it is related, through simple changes of variables, to growth models governed by the Kardar-Parisi-Zhang 共KPZ兲equation 关6,7兴 and to nonturbu- lent flows such as the asymmetric exclusion process共ASEP兲 model 关2,7兴.

The objective of this paper is to study the winding of a directed polymer lying on the surface of a cylinder. The al- gebraic number of turns W the polymer does around the cyl- inder is a random variable which depends on the realization of the disorder and which fluctuates because of the thermal noise. The statistics of the winding number of a polymer in an homogeneous medium in 2⫹1 dimensions around a cyl- inder goes back to the work of Spitzer关8兴and is relevant for the physics of vortices in type II superconductors关9–11兴. In physical situations, the system is, however, usually disor- dered; the effect of columnar defects has been studied ana- lytically关11兴, and the winding number around a cylinder of a polymer in a random medium with point-like disorder in 2

⫹1 dimension has been explored numerically关9,10兴. When there is an attractive interaction between the polymer and the winding center, the polymer can be confined around the cyl- inder and the system can be regarded as a polymer in the 1

⫹1 dimension with periodic boundary conditions. In that situation, the present work gives exact expressions for the statistics of the winding number.

The directed polymer on a cylinder is also related to the classical limit of strongly correlated fermions in one dimen- sion with disorder 共Luttinger liquids兲: the x position of the directed polymer corresponds to the phase of the fermions, and the phase has, of course, periodic boundary conditions.

The winding number of the polymer corresponds to the den- sity of fermions. Disorder, while periodic in both cases, does not have exactly the same correlations, but the models are sufficiently similar to hope for some universality 关12,13兴.

A first result of the present paper states that the thermal

fluctuations of the winding number are simply equal to what one would obtain for a directed polymer in a homogeneous medium; disorder is simply averaged out. A second result concerns the thermal-averaged winding number W¯ . Because of the randomness of the medium, this quantity is not zero and the expression of its variance具^W¯2典 averaged over disor- der is obtained.

The present paper is organized as follows: Section I is a brief recall of how the directed polymer in a random medium can be mapped to a quantum mechanical problem of inter- acting bosons using the replica method, and how this quan- tum mechanical problem can be solved with the Bethe ansatz 关1,14,2– 4兴. In Sec. II the winding number is introduced and defined and the two main results of this paper are stated in Eqs. 共19兲 and 共21兲. Section III gives the main lines of the derivation, and, finally, technical points are developed in the three appendixes.

I. DEFINITION, NOTATIONS, AND FREE ENERGY OF A DIRECTED POLYMER

Let us consider a directed polymer in 1⫹1 dimensions where the dimension in which the polymer is directed 共the

‘‘time’’ dimension兲 is taken to be very large and the trans- verse dimension 共the ‘‘space’’ dimension兲 has width 1 and periodic boundary conditions. As it is directed, the polymer can be described by a single-valued function y (t) and the partition function of a directed polymer of length t ending at position x is given by

Z共x,t兲⫽

冕

^y共t兲⫽xDy共s兲

⫻exp

冉

^⫺

^冕

^0tds

^冋

¹²

^冉

^dyds

^冊

^{2⫹␩„y共s兲,s…}

^册 冊

^{, 共1兲}

where ␩(x,t) is the contribution by the random medium to the energy of the system. Disorder in the medium is assumed to be characterized by an uncorrelated Gaussian noise of variance␥^:

具␩共x,t兲典⫽0,

具␩共x,t兲␩共x

⬘

^,t

⬘

兲典⫽␥␦共x⫺x

⬘

兲␦共t⫺t

⬘

兲, 共2兲 where the brackets具 典represent the average over disorder.

It is a well known result 关1,14,2– 4兴that this system can be successfully mapped to a quantum mechanical problem using the replica method; indeed, if we define

␺共xជ^;t兲⫽␺共x₁,...,x_n;t兲⫽具^Z共x₁,t兲¯Z共x_n,t兲典 具^Z共t兲典^{n , 共3兲} where

Z共t兲⫽

冕

^{dxZ共x,t兲 共4兲}

is the full partition function, then Eqs. 共1兲and共2兲imply

⳵␺

⳵^{t ⫽} 1

2_i

兺

_⫽ⁿ₁ ^⳵_⳵²x^␺_i² ⫹␥

兺

_i⬍j ^{␦共xi⫺xj兲␺, 共5兲}

with periodic boundary conditions on all the space variables x_i:

␺共...,x_i⫽0,...;t兲⫽␺共...,x_i⫽1,...;t兲. 共6兲 The normalization by具^Z典^{n in Eq.}共3兲is just a simple way to get rid of a low-scale divergence introduced by the continu- ous description 共1兲 of the system. In other words, without this normalization, there would be a trivial extra term in Eq.

共5兲involving the lattice size of an underlying discrete formu- lation of the problem.

For an infinitely long polymer, that is, in the large t limit, the amplitude of␺^(xជ;t) is given by the fastest growing mode of Eqs. 共5兲 and 共6兲. In quantum mechanical language, we have

lim

t→⬁

ln␺共xជ^;t兲 t ⫽lim

t→⬁

ln兰dx₁¯dx_n␺共xជ^;t兲 t

⫽lim

t→⬁

ln具^Z共t兲ⁿ典^{⫺n ln}具^Z共t兲典 t

⫽⫺E共n,␥兲, 共7兲 where E(n,␥) is the ground-state energy of the Hamiltonian

H⫽⫺1

2_i

兺

_⫽ⁿ_{1 ⳵}^⳵x²_i²⫺␥

兺

_i⬍j ^{␦共xi⫺xj兲, 共8兲}

which describes n particles with attractive␦interactions on a ring of size 1.

The same Hamiltonian with a negative value of␥共that is, with a repulsive delta interaction兲has been much studied to determine the spectrum of a gas of bosons 关15–20兴. In that context, using the Bethe ansatz 关21兴, it was shown that the ground-state energy of Eq.共8兲can be written as

E共␩^,␥兲⫽⫺1

2_␣⫽

兺

ⁿ₁ ^{␭␣2, 共9兲}

where the兵␭_␣其 are solutions of

e^␭␣⫽_1⭐␤⭐

兿

_n

␤⫽␣

␭_␣⫺␭_␤⫹␥

␭_␣⫺␭_␤⫺␥ ^with _␥→^lim0

␭_␣⫽0. 共10兲

Of course, this expression obtained for␥⬍0 remains valid in the directed polymer context where␥⬎0.

In the quantum mechanical problem共8兲, the ground state energy E(n,␥) is well defined only for integral n; after all, n is the number of particles. However, for the directed poly- mer, 具^Zn典, which is related to E(n,␥^{) through} 共7兲, can be defined for arbitrary values of n. The small-n limit is of special importance here: as the directed polymer is a disor- dered system, the free energy is a random variable and 具^Zn典 is the generating function of this free energy. Indeed, we have

ln具^Zn典

t ⫽n具^{ln Z}典 t ⫹n²

2

具^ln2Z典c

t ⫹n³ 6

具^ln3Z典c

t ⫹O共n⁴兲, 共11兲

where 具^{ln Z}典^/t, 具^ln2Z典^c/t⫽(具^ln2Z典^⫺具^{ln Z}典²)/t, etc., are the cu- mulants of the free energy per unit length of the directed polymer. Thus, if we can generalize Eqs. 共9兲 and 共10兲 to arbitrary values of n, the expansion of E(n,␥) for small n gives, using Eq.共7兲, the distribution of the free energy of the directed polymer 关22兴.

This method was used 关1兴 for the directed polymer on a space of infinite width in the x direction. The Bethe ansatz equations are then much simpler than Eq. 共10兲 and one ob- tains 关23,1兴, when n is an integer, E(n,␥⁾⫽␥²⁽ⁿ⫺n³)/24.

This result was used to argue that only the two cumulants 具^{ln Z}典^{/t and} 具^ln3Z典c/t do not vanish in the large t limit and that, therefore, the fluctuations of ln Z scale like t^1/3关24 –26兴. When space has finite width, however, it is easy to see that the free energy is an extensive function and that all its cumulants scale like t. In two previous papers 关3,4兴 we solved the Bethe ansatz equations 共10兲 and computed the three first terms of the small n expansion of E(n,␥^{). Up to} the order n², the result is

E共n,␥兲⫽n

冉

^{␥2⫹␥242}

冊

^⫺n4^2␥&^3/2

冕

0

⫹⬁

d␭␭²e^⫺␭2/2 tan␭

冑

␥

2&

⫹O共n³兲,

共12兲

so that, using Eq. 共7兲,

lim

t→⬁

具^{ln Z}典⫺ln具^Z典

t ⫽⫺

冉

^{␥2⫹␥242}

冊

^{, 共13兲}

共 兲

lim

t→⬁

具^ln2Z典c

t

⫽␥^3/2

2&

冕

^{0⫹⬁d␭ ␭2e⫺␭2/2}

tanh␭

冑

␥ 2&

⫽␥⫹␥² 12⫺ ␥³

360⫹ ␥⁴

5040⫹O共␥⁵兲 for small ␥

⫽

冑

␲␥^3/2

4 ⫹4␨共3兲⫹O

冉

^1␥

冊

^{for large ␥, 共14兲}

where␨⁽³⁾⫽兺k^⫺3⬇1.20206.

II. WINDING NUMBER OF THE DIRECTED POLYMER An important topological property of a directed polymer is its winding number W, that is the algebraic number of full turns the polymer makes around the cylinder on which it lays. One way to define this winding number is to increase W by one for each ‘‘time’’ t where the x coordinate of the poly- mer goes from 1^⫺to 0^⫹and decrease W by one when x goes from 0^⫹to 1^⫺. Another way is to unroll the x coordinate and set W⫽兰x˙dt. Of course, the differences between those two definitions smear out in the large t limit.

As for any quantity in a disordered system at finite tem- perature, the winding number W fluctuates for two distinct reasons. One is the thermal fluctuations: for a given realiza- tion of the disorder and at finite temperature, the directed polymer fluctuates around the path with the lowest energy, and those fluctuations may change the winding number of the polymer. The other source of fluctuations is the quenched disorder on the medium.

In this work, a horizontal bar is used to denote the thermal average, which is the average computed over all the possible directed polymers counted with their Boltzmann weights.

The cumulants are noted with an extra c subscript: W¯ is the thermal average of W, and (W^k)_c the kth thermal cumulant of W, with (W²)_c⫽W²⫺W¯², (W³)_c⫽W³⫺3W¯ W²⫹2W¯³, etc. These thermal averages and cumulants are calculated for a given, fixed, realization of the disorder and usually depend on that realization.

The average and cumulants of a quantity Q computed over all the realizations of the disorder are written with brackets: 具Q典is the average ofQ computed over all real- izations of the disorder, and具Q^k典cis the kth disorder cumu- lant ofQ.

It is worth noting that, for a given realization of the dis- order, the thermal average W¯ of the winding number is not zero; the disorder breaks the symmetry and may favor one orientation over the other. However, W¯ is an extensive quan- tity and, if we imagine that we cut an extremely long poly- mer in many very long sections, all the sections are nearly independent and W¯ may be regarded as the sum of uncorre- lated random variables. Therefore,

lim

t→⬁

W¯ t ⫽lim

t→⬁

具^W¯典

t ⫽0. 共15兲

This property that W¯ approaches具^W¯典 in the large t limit is known as ‘‘auto averaging.’’ Likewise, all the thermal cumu- lants of W 共which are also extensive quantities兲 share the same property,

lim

t→⬁

共W^k兲c

t ⫽lim

t→⬁

具共W^k兲c典

t . 共16兲

Those cumulants, which characterize the thermal fluctuations of a directed polymer’s winding number, depend on the re- alization of the disorder only when the length t of the poly- mer is finite.

Other quantities of interest are the disorder cumulants of the thermal average of the winding number of the polymer.

Indeed, the quantity W¯ depends on the realization of the disorder, and its fluctuations are characterized by another se- ries of cumulants:

lim

t→⬁

具^W¯k典c

t , 共17兲

with具^W¯2典c⫽具^W¯2典⫺具^W¯典², etc. Actually, we might be inter- ested in computing many quantities characterizing the wind- ing number, such as

lim

t→⬁

具共W²兲²典⫺具^W2典²

t , 共18兲

which represents the fluctuations due to the disorder of the thermal-mean square of the winding number, per unit length.

A first result of the present paper is

lim

t→⬁

具共W²兲c典

t ⫽1 and lim

t→⬁

具共W^k兲c典

t ⫽0 for k⫽2.

共19兲 In other words, thermal fluctuations of the winding numbers are Gaussian and independent of the disorder␥. For an infi- nitely long polymer, the thermal fluctuations of the winding number of the polymer behave as if the directed polymer was simply doing a random walk in a disorder-less environment.

A second result of the present paper is

lim

t→⬁

具^W¯2典c

t ⫽lim

n→0

冋

ⁿ²²

冉

^{⳵E共⳵␥n,␥兲⫺2␥E共n,␥兲}

冊

^⫹1n⫺1

册

⫽⫺1⫹

冉

^{2␥⫺⳵␥⳵}

冊

t^lim→⬁

具^ln2Z典c

t , 共20兲

where E(n,␥) is the ground state energy of the quantum problem computed in关3,4兴and given in Eq.共12兲. Therefore,

lim

t→⬁

具^W¯2典^c t ⫽

冑

␥

&

冉 ^冕

^{0⫹⬁d␭tanh␭2e⫺␭␭2}

^冑

^&2␥/2

⫺1

4

冕

^{0⫹⬁d␭ ␭4e⫺␭2/2}

tanh␭

冑

␥

2&

冊

^⫺1

⫽

冑

␲␥

8 ⫺1⫹8␨共3兲

␥ ^⫹O共␥^⫺2兲 for large ␥

⫽ ␥² 360⫺ ␥³

2520⫹ ␥⁴

16800⫹O共␥⁵兲 for small ␥ 共21兲 关where␨⁽³⁾⫽兺k^⫺3⬇1.202 06].

The expression共1兲of the directed polymer’s free energy is written with dimensionless variables. If we explicitly put back physical constants and use the following expression instead of共1兲:

Z共x,t兲⫽

冕

^y共t兲⫽xDy共s兲exp

再

^⫺␤

^冕

^0tds

⫻

冋

^␬2

冉

^{d yds}

冊

^{2⫹␩„y共s兲,s…}

册 冎

^{, 共22兲}

where ␤⫽(k_BT)^⫺1 is the inverse of temperature, ␬ ^{is the} rigidity modulus of the line and where the spatial dimension x has finite width w and periodic boundary conditions, then Eqs. 共19兲and共21兲become

lim

t→⬁

具共W²兲c典

t ⫽ 1

␤␬^w2,

lim

t→⬁

具共W^k兲c典

t ⫽0 for k⫽2,

lim

t→⬁

具^W¯2典^c

t ⫽ 1

␤␬^w2F共␤³␬^w␥兲, 共23兲

where F(␥) is the scaling function given in Eq. 共21兲. We obtain the following expansions:

lim

t→⬁

具^W¯2典c

t ⬇

冑

␲␤␥

8

冑

␬^w3/2 at low temperature

⬇␤⁵␬␥²

360 at high temperature. 共24兲

III. DERIVATION OF EQS.„19…AND„20… A. Equivalence to a quantum mechanical problem To obtain both results 共19兲 and 共20兲, we define a new partition function Z_z(x,t), the purpose of which is to count the winding number of the polymer:

Z_z共x,t兲⫽

冕

y共t兲⫽xDy共s兲

⫻e^⫺关energy of path y共s兲兴⫹z共winding number of that path兲. 共25兲 The sum is made over all the directed polymers ending in x, and the ‘‘energy of a path’’ is the same as in Eq. 共1兲.

Clearly, Z_z(t)⫽兰Z_z(x,t)dx is related to the winding num- ber W by

Z_z共t兲⫽Z₀共t兲e^zW. 共26兲 If we define the winding number W as an integer that changes by⫾1 each time the directed polymer wraps around the domain by crossing the x⫽0 or x⫽1 boundary, then the boundary conditions for Z_z is

Z_z共0,t兲⫽e^zZ_z共1,t兲. 共27兲 Apart from that, the equations satisfied by Z_z(x,t) are the same as the equations satisfied by Z(x,t). In particular, if we define

␺z₁,...,z_n共x₁,...,x_n;t兲⫽具^Zz₁共x₁,t兲¯Z_z

n共x_n,t兲典 具^Z0共t兲典^{n , 共28兲} this new wave function ␺is also a solution of Eq.共5兲; only the boundary conditions are changed: instead of Eq.共6兲, the new conditions read

␺z₁,...,z_n共x₁,...,x_i⫽0,...,x_n;t兲

⫽e^zi␺z₁,...,z_n共x₁,...,x_i⫽1,...,x_n;t兲. 共29兲 Thus, as in Eq. 共7兲, the long ‘‘time’’ t behavior of Z_z(t) is given by

lim

t→⬁

ln具^Zz₁¯Z_z

n典^{⫺n ln}具^Z0典

t ⫽⫺E共n,␥^;z1,...,z_n兲, 共30兲 where E(n,␥^;z1,...,z_n) is the ground state energy of the same Hamiltonian共8兲as before, but with the new boundary conditions 共29兲.

This new ground state energy E contains all the informa- tion on the winding number W. For instance, from Eq.共26兲, and by definition of the cumulants, we have, for k⬎0,

共W^k兲c⫽ ⳵^k

⳵^{zkln Zz共t兲}

冏

_z⫽0^{. 共31兲}

共 兲

具^{ln Z}z典is easily obtained from Eq.共30兲: we set all the兵^zi其 ^to one single value z, make a small n expansion, and retain only the first order. We get

lim

t→⬁

具共W^k兲c典

t ⫽⫺lim

n→0

⳵^k

⳵^zk

⳵

⳵^nE共n,␥^;z,...,z兲

冏

_z⫽0^{. 共32兲}

Getting W¯² is more tricky. We would need (⳵^{ln Z}z/⳵^z)2, but that quantity can only be obtained from Eq. 共30兲if the parameters 兵^zi其 take at least two different values. For ex- ample, we have

lim

n→0

具^Zz₁^Zz

n⫺1典^⫽

冓

^{ZZ00共共11⫹⫹zzW1¯W¯⫹⫹OO共共zz212兲兲兲兲}

冔

^,

⫽1⫹共z₁⫺z兲具^W¯典^⫺zz1具^W¯2典^⫹O共z₁²兲⫹O共z²兲, 共33兲 and

lim

n→0

ln具^Zz₁Z_z^n⫺1典⫽共z₁⫺z兲具^W¯典⫺zz₁共具^W¯2典⫺具^W¯典²兲⫹O共z₁²兲

⫹O共Z²兲. 共34兲 Therefore, putting all the pieces together,

lim

t→⬁

具^W¯2典⫺具^W¯典²

t ⫽lim

n→0

⳵²

⳵^z⳵^z1

E共n,␥^;z1,z,...,z兲

冏

_z^z⫽0

1⫽0

. 共35兲 Finally, to obtain the results announced, we need to compute E(n,␥;z,...,z) and, to the first order in z and z₁, E(n,␥^;z1,z,...,z).

B. Determination of E„n,␥;z,...,z…

When all the parameters兵^zi其are equal to one single value z, the problem is easy: all the replica play a symmetric role, so that the ground state eigenvector␺z,...,z(x₁,...,x_n) of the Hamiltonian 共8兲 is a symmetric function of all the兵^xi其^{. As} shown in Appendix A, the standard Bethe ansatz derivation gives the result. Instead of Eqs.共9, 10兲, we get

E共n,␥^;z,...,z兲⫽⫺1 2_␣⫽

兺

1

n

␭_␣², 共36兲

where the兵^␭␣其 are solutions of e^␭␣⫹z⫽_1⭐␤⭐n

兿

␤⫽␣

␭_␣⫺␭_␤⫹␥

␭_␣⫺␭_␤⫺␥ ^with _␥→^lim0 z→0

␭_␣⫽0. 共37兲

If we define

␭

˜_␣⫽␭_␣⫹z 共38兲

then the 兵^␭˜␣其 are clearly solutions of the standard Bethe ansatz equations共10兲. Using Eq.共36兲, we obtain

E共n,␥^;z,...,z兲⫽E共n,␥兲⫺n

2z². 共39兲 E(n,␥⁾⫽E(n,␥;0,...,0) is the ground state energy共12兲 be- fore introduction of the 兵^zi其. We have used兺␭˜_␣⫽0, which can be easily deduced关3,4兴from Eq.共10兲.

Using Eq.共32兲, the result 共19兲on the thermal cumulants of the winding number is then immediate. This method, based on a Bethe ansatz, is not the simplest way to obtain 共19兲. Indeed, the result could be obtained using the statistical tilt symmetry关27,28兴of the problem; we define the winding number W of a path y (s) as being simply the unrolled coor- dinate:

W⫽

冕

0 t

dsdy

ds. 共40兲

共This new definition is, of course, equivalent to the previous one in the large t limit.兲The change of variable y (s)⫽˜ (s)y

⫹zs in the definition共25兲of Z_z gives then

Z_z共t兲⫽e^z2/2

冕

^D˜y共s兲

⫻exp

再

^⫺

^冕

^0tds

^冋

¹²

^冉

^dyds˜

^冊

^{2⫹␩„˜y共s兲⫹zs,s…}

^册 冎

^.

共41兲 Clearly␩„˜ (s)y ⫹zs,s…have the same statistical properties of

␩„˜ (s),sy …and one gets 具^{ln Z}z共t兲典⫽z²

2 ⫹具^{ln Z}0共t兲典^, 共42兲 from which the result 共19兲is straightforward. The first deri- vation with the Bethe ansatz was included here as it demon- strates part of the method used to obtain the second result 共20兲, which cannot be derived from a statistical tilt symmetry argument

C. Determination of E„n,␥;z1,z,...,z…

When the parameters 兵^zi其 are not identical, the wave function␺is no longer a symmetric function of the兵^xi其 ^and the problem is much more complicated. Therefore, the stan- dard bosonic Bethe ansatz used in the previous case will not work. However, as shown in Appendix B, using a more gen- eral Bethe ansatz that was first introduced to deal with non- bosonic particles 关29,30,31兴, we get the following result:

E共n,␥^;z1,z,...,z兲⫽⫺1

2_␣⫽

兺

ⁿ₁ ^{␭␣2, 共43兲}

where the兵␭_␣其 are solutions of

e^{␭␣⫹␨␣}⫽_1⭐␤⭐

兿

_n

␤⫽␣

␭_␣⫺␭_␤⫹␥

␭_␣⫺␭_␤⫺␥ ^with _␥→^lim0 z→0

␭_␣⫽0, 共44兲

and where the 兵␨␣其 are such that

e^z_␤⫽

兿

ⁿ₁

冋

^{e␨␣⫹␭␣⫺␭␥ ␤共ez⫺e␨␣兲}

册

⫽e^z1_␤⫽

兿

ⁿ₁

冋

^{ez⫹␭␣⫺␥␭␤共ez⫺e␨␣兲}

册

^,

lim

兵^zi其→0

␨␣⫽0. 共45兲

When z₁⫽z, we recover ␨␣⫽z, the result of the previous section.

To determine the fluctuation具^W¯2典 of the winding number of the polymer, we only need to compute E(n,␥^;z1,z,...,z) to the second order in the兵^zi其^{. From Eq.共45兲}, we easily get

␨_␣⫽z₁⫹共n⫺1兲z

n ⫺n␭_␣⫺兺k⫽1

n ␭k

␥^{n3 共z1⫺z兲2⫹O共}兵^zi其³兲. 共46兲 We define, for all ␣^,

␭

˜_␣⫽␭_␣⫹␨␣,

⫽␭_␣

冉

^{1⫺共z1␥⫺nz2兲2}

冊

^{⫹z1⫹共nn⫺1兲z⫹兺kn␥⫽n13␭k共z1⫺z兲2,}

共47兲 and

␥

˜⫽␥

冉

^{1⫺共z1␥⫺nz2兲2}

冊

^{. 共48兲}

Using those new variables into Eq. 共44兲, we obtain the fa- miliar Bethe ansatz equations:

e^˜␭␣⫽_1⭐␤⭐

兿

_n

␤⫽␣

␭

˜_␣⫺˜␭_␤⫹␥^˜

␭

˜␣⫺␭˜

␤⫺^˜␥^⫹O共兵^zi其³兲, 共49兲 so that关3,4兴, using the ground state energy E(n,␥^{) given by} Eq. 共12兲,

␣⫽

兺

1 n

␭

˜␣2⫽⫺2E共n,␥^˜兲⫹O共兵^zi其^{3兲 and} _␣⫽

兺

1 n

␭

˜_␣⫽O共兵^zi其^3兲. 共50兲 From there, using Eq.共47兲, one can write兺␭_␣². We finally get

E共n,␥^;z1,z,...,z兲

⫽E共n,␥兲⫺1 2

关z₁⫹共n⫺1兲z兴² n

⫹ 1

n²

冋

^{2␥E共n,␥兲⫺⳵E共⳵␥n,␥兲}

册

^{共z1⫺z兲2⫹O共兵zi其3兲.}

共51兲 Then, finally, from Eq. 共35兲, we get the announced result 共20兲.

CONCLUSION

Using the replica method with the directed polymer, one obtains a bosonic quantum mechanical problem which can be solved by the Bethe ansatz. By extending this method and using a more general Bethe ansatz that was introduced to deal with nonbosonic particles 关29兴, it has been shown how the different quantities characterizing the fluctuations of the directed polymer’s winding number can be computed using new Bethe ansatz equations. Building upon a previous work 关3,4兴, those equations were explicitly solved in two cases giving the results 共19兲 and 共20兲, 共21兲. The second result is particularly interesting as it simply relates through Eq. 共20兲 the fluctuations of the thermal-averaged winding number and the fluctuations of the free energy of the directed polymer. It would be interesting to understand this relation in a more direct way.

In principle, the method presented in the present paper should allow us to compute more cumulants of the winding number and, eventually, its complete probability distribution.

For that, however, one needs, as a first step, to generalize Eq.

共51兲and write the expansion of E(n,␥^;z1,z,...,z) to higher orders in the 兵^zi其. Indeed, one can show that, for example,

lim

t→⬁

具^W3W¯典^⫺3具^W2典具^W¯2典 t

⫽lim

n→0

⳵⁴

⳵^z⳵^z1

3E共n,␥^;z1,z,...,z兲

冏

_z^z⫽0

1⫽0

. 共52兲

Obtaining E(n,␥^;z1,z,...,z) to the fourth order in the兵^zi其 ^is not, however, an easy task, as the trick used in Eq. 共47兲 would not work at that order.

As a second step, to compute more complicated cumu- lants of the winding number such as 具^W¯4典^⫺3具^W¯2典^{2, one} needs to generalize Eqs.共43兲–共45兲to the case where the兵^zi其 take at least four different values. Higher order cumulants would require, of course, the energy of the system with more different values of the 兵^zi其. A matrix approach such as the one developed in the present paper could lead to the result.

Another possibility might be to try an approach similar to the

‘‘nested Bethe ansatz’’ method developed by Yang and Suth- erland关29–31,1,5兴to compute the ground-state energy of the system described by the Hamiltonian共8兲with different types of particles and symmetry relations which depend on the type of particles. Their results are not directly applicable to

共 兲

the directed polymer’s winding number as all the particles have the same symmetry relations but different boundary conditions, but it might be worth investigating if the nested Bethe ansatz could be adapted.

ACKNOWLEDGMENTS

I would like to thank Pierre Le Doussal, who suggested this work, and Bernard Derrida for his interesting discus- sions.

APPENDIX A: BETHE ANSATZ EQUATIONS WHEN ALL THE ziHAVE THE SAME VALUE z

When all the 兵^zi其 are equal to z, all the particles have symmetric roles and the standard bosonic Bethe ansatz leads to the result. To recall the standard derivation 关15,17兴, we look for solutions of the following form:

␺z,...,z共x₁,...,x_n兲⫽

兺

_␴ ^{a共␴兲e兺in⫽1␭␴共i兲x␶共i兲, 共A1兲}

where the sum is made over the n! permutations␴^of兵^1,...,n其 and where␶is the permutation defined by

x_␶共1兲⬍x_␶共2兲⬍¯⬍x_␶共n兲. 共A2兲 The n! amplitudes 兵^a(␴⁾其 and the n pseudo-wave-numbers 兵^␭␣其 are unknown variables to be determined.

We use this expression of␺ⁱⁿH␺⫽E␺^{, where}His the Hamiltonian共8兲. In the regions where all the兵^xi其 are differ- ent, it is straightforward to get

E共n,␥^;z,...,z兲⫽⫺1 2_␣⫽

兺

1

n

␭_␣², 共A3兲 so that we only need to determine the兵^␭␣其. At each crossing of two particles, we have to ensure the correct discontinuities in the derivatives of ␺to compensate for the␦ functions in H. This gives the following conditions, for all ␴ ^{and all 1}

⭐k⬍n:

a共␴ⴰT_k兲⫽␭_␴共k兲⫺␭_␴共k⫹1兲⫺␥

␭_␴共k兲⫺␭_␴共k⫹1兲⫹␥^a共␴兲, 共A4兲 where T_k is the permutation that swaps k and k⫹1 and leaves all the other integers unchanged.

As any permutation ␴can be written as a product of the elementary permutations T_k, one can use Eq.共A4兲 to write all the兵^a(␴⁾其 up to an arbitrary multiplicative factor. How- ever, as the decomposition of a permutation␴as a product of T_kis not unique, one must check that the (n⫺1)n! equations 共A4兲are self-consistent. The best way to do that is to write down explicitly the solution

a共␴兲⫽_1⭐␣

兿

_⬍␤⭐n ^{␭␴共␣兲}_{␭␴共␣兲}^⫺_⫺␭^{␭␴共␤兲}_␴共␤兲^{⫹␥. 共A5兲}

It is easily checked that this is indeed the solution of all Eqs.

共A4兲.

So, for any set of values 兵␭_␣其, the wave function 共A1兲 where the兵^a(␴⁾其 are given by共A5兲is an eigenvector of the Hamiltonian共8兲. The values of the兵␭_␣其can then be obtained from the boundary conditions 共29兲. One gets

a共␴兲⫽e^{z⫹␭␴共1兲}a共␴ⴰC兲, 共A6兲 where C is the circular permutation C(1)⫽2, C(2)

⫽3,...,C(n⫺1)⫽n, C(n)⫽1. Using Eq.共A5兲, we easily get the new Bethe ansatz equations. For all ␣^,

e^␭␣⫹z⫽_1⭐␤⭐

兿

_n

␤⫽␣

␭_␣⫺␭_␤⫹␥

␭_␣⫺␭_␤⫺␥^{. 共A7兲}

We are only interested in the ground-state solution. By continuity of this ground state, we get the last condition

lim

z→0

␥→0

␭_␣⫽0. 共A8兲

APPENDIX B: BETHE ANSATZ EQUATIONS WHEN ALL THEˆz_i‰EXCEPT z₁HAVE THE SAME VALUE z When the parameters 兵^zi其 do not take the same value z, the computation of the energy E is more complicated; indeed the wave function␺is no longer a symmetric function of the 兵^xi其 and there is no way that the standard Bethe ansatz共A1兲 might lead to the result.

However, in order to study the Hamiltonian共8兲for fermi- onic particles or, more generally, for particles with arbitrary symmetries and anti-symmetries, a more general ansatz than 共A1兲has been proposed关29,30,1,5兴: in Eqs.共A1兲and共A2兲, the permutation␶is only introduced as a convenient way to get the coordinates 兵^xi其 of the n particles sorted from the leftmost particle to the rightmost in the expression of the wave function. An easy way to break the symmetry of␺^{is to} make the parameters兵^a(␴⁾其explicitly dependent on the per- mutation ␶^:

␺z₁,...,z_n共x₁,...,x_n兲⫽

兺

_␴ ^{a共␶,␴兲e兺in⫽1␭␴共i兲x␶共i兲, 共B1兲}

where the permutation ␶is, as before, defined by Eq. 共A2兲. As shown below, the solution to our problem with the unusual boundary conditions 共29兲 can also be written using Eq.共B1兲. We first begin with the most general case where all the 兵^zi其 are different and, at some point, specialize to the simpler case where all the 兵^zi其 ^{except z}1 are identical.

1. General Bethe Ansatz equations for arbitraryˆz_i‰ Using the Ansatz 共B1兲 inH␺⫽E␺ ^whereH the Hamil- tonian 共8兲, we have, as usual,

E共n,␥^;z1,...,z_n兲⫽⫺1 2_␣⫽

兺

1

n

␭_␣². 共B2兲

The new equations for the parameters 兵^a(␶^,␴⁾其 ^{are more} complicated than共A4兲,

a共␶^,␴ⴰT_k兲⫽共␭_␴共k兲⫺␭_␴共k⫹1兲兲a共␶ⴰT_k,␴兲⫺␥^a共␶^,␴兲

␭_␴共k兲⫺␭_␴共k⫹1兲⫹␥ ^, 共B3兲 for any permutations ␶^and␴and for any integer 1⭐k⬍n.

A convenient way to write the (n!)²parameters兵^a(␶^,␴⁾其 is using n! vectors indexed by␴, each vector having n! com- ponents:

aជ共␴兲⫽

冉

^{aaa共共共␶␶␶n!]12,,,␴␴␴兲兲兲}

冊

^{, 共B4兲}

where␶1,...,␶n!are the n! permutations of兵^1,...,n其^{sorted in} an arbitrary way chosen once for all. 共The order must, of course, be the same for all values of ␴^.兲We now introduce the matrices M_kdefined by

冉

^{aaa共共共␶␶␶n!12ⴰⴰⴰ]TTTkkk,,,␴␴␴兲兲兲}

冊

^{⫽Mkaជ共␴兲. 共B5兲}

Those matrices M_kjust shuffle the components of the vector aជ⁽␴); there is thus exactly one ‘‘1’’ per raw and per column and all the other components are ‘‘0.’’ In a concise way, we can write M_k as

共M_k兲i, j⫽␦_␶

i

␶jⴰT_k

. 共B6兲

The matrices M_kare a representation of the permutations T_k. As such, they have the same standard commutation prop- erties as the permutations:

M_k²⫽I, M_kM_k⫹1M_k⫽M_k⫹1M_kM_k⫹1,

M_kM_k⬘⫽M_k⬘M_k if 兩k⫺k

⬘

兩⬎1 共B7兲 共I being the identity matrix兲. Equation 共B3兲is then simply written as

aជ共␴ⴰT_k兲⫽Y_k^{␴共k兲,␴共k⫹1兲}aជ共␴兲, 共B8兲 where Y_k^{i, j} is the Yang-Baxter operator defined关29兴by

Y_k^{i, j}⫽共␭i⫺␭j兲M_k⫺␥^I

␭i⫺␭j⫹␥ ^{. 共B9兲}

It is clear from 共B8兲that any vector aជ⁽␴) can be obtained from the knowledge of one of them. However, as in the symmetric case, one has to check that the result does not depend on the way the permutations are decomposed as a product of the elementary permutations T_k. There are no explicit formula 关32兴 such as 共A5兲 of aជ⁽␴), but one can check that the (n⫺1)n! relations共B8兲are indeed self com- patible. This is implied by the following ‘‘Yang-Baxter’’ re- lations关29兴

Y_k^{i, j}Y_k^j,i⫽I,

Y_k^{i, j}Y_k^i⬘_⬘^{, j⬘}⫽Y_k^i⬘_⬘^{, j⬘}Y_k^{i, j} if 兩k⫺k

⬘

兩⬎1, 共B10兲 Y_k^{i, j}Y_k^i,l_⫹1Y_k^j,l⫽Y_k^j,l_⫹1Y_k^i,lY_k^{i, j}_⫹1,

which can be easily checked using Eqs.共B7兲and共B9兲. With the first of those three relations, using共B8兲twice to compute aជ⁽␴ⴰT_kⴰT_k) gives correctly aជ⁽␴). The second relation im- plies aជ⁽␴ⴰT_kⴰT_k⬘)⫽aជ⁽␴ⴰT_k⬘ⴰT_k) if兩k⫺k

⬘

兩⬎1 and, finally, the third relation gives aជ⁽␴ⴰT_kⴰT_k⫹1ⴰT_k)⫽aជ⁽␴ⴰT_k⫹1ⴰT_k ⴰT_k⫹1). It is a well known property of the symmetric group that those three necessary conditions are actually sufficient to ensure that the relations 共B8兲are self-consistent.

One still needs to write the boundary conditions共29兲with the parameters兵^a(␶^,␴⁾其^{. From共B1兲, one gets}

a共␶^,␴兲⫽exp共z_␶共1兲⫹␭_␴共1兲兲a共␶ⴰC,␴ⴰC兲, 共B11兲 whereCis, as in Eq.共A6兲, the circular permutation.

AsC⫽T₁ⴰT₂ⴰ¯ⴰT_n⫺1, the matrix that shuffles the lines of the vectors aជ⁽␴) according to the permutationCis simply the product of the matrices M_k. Thus, we have

aជ共␴兲⫽exp共␭_␴共1兲兲Z M₁M₂¯M_n⫺1aជ共␴ⴰC兲, 共B12兲 where Z is the diagonal matrix defined by

共Z兲i, j⫽␦i jexp共z_␶

i共1兲兲. 共B13兲

Moreover, using several times共B8兲, we get, from the defini- tion ofC,

aជ共␴ⴰC兲⫽Y_n^␴共_⫺¹₁^{兲,␴共n兲}Y_n^␴共_⫺^1兲,₂^{␴共n⫺1兲}¯Y₁^{␴共1兲,␴共2兲}aជ共␴兲. 共B14兲 Putting together共B12兲and共B14兲, we see that aជ⁽␴^{) must} be, for each␴, the eigenvector of some operator,

共 兲