Physica A 270 (1999) 335–341www.elsevier.com/locate/physa

Phase transition of directed polymer in randompotentials on 4 + 1 dimensions

Jin Min KimDepartment of Physics and CAMDRC, Soongsil University, Dongjakgu, Seoul, 156-743, South Korea

Received 1 March 1999

Abstract

Finite-temperature-directed polymer in random potentials is described by a transfer matrixmethod. On 4 + 1 dimensions, the evidence for a �nite-temperature phase transition is found atTc ≈ 0:18, where the free energy uctuation grows logarithmically as a function of time t. WhenT. Tc, the uctuation of the free energy grows as t! with ! ≈ 0:156. The phase transitionof the restricted solid-on-solid model, which is closely related to the directed polymer problemthrough the Kardar–Parisi–Zhang equation, is also discussed. c© 1999 Elsevier Science B.V. Allrights reserved.

PACS: 05.40+j; 64.60.Ht; 05.70.Ln

Keywords: Kardar–Parisi–Zhang equation; Directed polymer; Random potential; Restrictedsolid-on-solid model

In the last decade, there have been considerable studies on the problem of directedpolymers (DP) in random potentials [1–6]. It is related through various mapping tomany other physical phenomena such as the Kardar–Parisi–Zhang (KPZ) equation [7],Eden model [8], ballistic aggregation [9], restricted solid on solid (RSOS) model [10–13], and Burgers’ equation [13]. Despite the simple mathematical representation ofthe directed polymer (DP) problem, they are not completely understood except in twodimensions.The �nite-temperature-directed polymer problem on random potentials in d= 1+ 1,

2 + 1 and 3 + 1, was studied in the literature [14]. In this work, we have studied theDP problem specially on d = 4 + 1. We shall usually write d = 5 as d = 4 + 1 toindicate that there is four transverse and one longitudinal directions. For completenesswe describe the model here brie y. Consider a directed polymer with random potential�(x; t) assigned to each point (x; t) where x is the d− 1 dimensional transverse vector

0378-4371/99/$ - see front matter c© 1999 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(99)00182 -X

336 J.M. Kim / Physica A 270 (1999) 335–341

and t is the longitudinal length of the polymer. The polymer starts from 0 and abending energy is given against a transverse bending. The Hamiltonian H of adirected polymer is

H =∫dt

[ (dxdt

)2+ �(x; t)

]: (1)

The random potential �(x; t) is a white noise satisfying

�(x; t)�(x′; t′) = 2D�(t − t′)�d−1(x − x′) ; (2)

where �A is the sample average of A. The partition function Z(x; t) for the polymerending at (x; t) can be written as the path integral

Z(x; t) =∫ (x; t)

(0;0)Dx′(t′) exp

(−1T

∫ t

0dt′[ (dx′

dt′

)2+ �(x′; t′)

]); (3)

where T is temperature. There are two competing terms, one is a bending energyforcing the polymer straight, the other is the quenched impurity �(x; t) which prefersthe polymer deformed through the minimum potentials. Eq. (3) satis�es

@Z(x; t)@t

=[T2

∇2 − 1T�(x; t)

]Z(x; t) : (4)

Through the Cole Hope transform, Eq. (4) can be exactly related to the Kardar–Parisi–Zhang (KPZ) equation [7]

@h(x; t)@t

= �∇2h(x; t) +�2(∇h)2 + �(x; t) ; (5)

where �(x; t) is linearly proportional to �(x; t) and h(x; t) is the surface height. Wede�ne a total partition function Z(t) as

∫dx Z(x; t) and the related free energy

F(t) =−T ln Z(t) : (6)

The free energy F of the directed polymer problem plays the same role as the heightvariable h of the ballistic deposition model [9] or the restricted solid-on-solid model[10]. We are interested in the free energy uctuation

�F(t) = (F2(t)− F(t)2)1=2 ; (7)

which is a similar quantity as the surface width W (t) in the growth models [1,2,4,5].Other interesting quantity is the transverse uctuation of the polymer 〈x2〉, where thesymbol 〈A〉 is the thermal average of a quantity A given by

〈A〉 ≡∑

x A Z(x; t)∑x Z(x; t)

: (8)

These quantities are characterized by the scaling exponents ! and z by

�F(t) ∼ t ! and 〈x2〉 ∼ t2=z : (9)

The exponents !, and z are connected by the relation !+ 1 = 2=z at low-temperaturephase, which comes from the invariance of the Galilean transform of the KPZ

J.M. Kim / Physica A 270 (1999) 335–341 337

equation [7]. So, there is only one exponent that has to be determined. Most of therecent e�orts on this problems have been devoted to verify the exponents. In d=1+1,the exponents are known to be != 1

3 and z=32 . There are some variations among the

values of the exponents quoted by various authors in higher dimensions [1,2,4,5]. Otherinteresting study is to look for a phase transitions between the high- and low-temperaturephase [14]. In d=2+1 there is no phase transition and only the strong-coupling phaseexists. In d = 3 + 1, z is near 1.67 [10,14] for T.Tc, and z = 2 implying that 〈x2〉is linear to t for T/Tc. The phase transition temperature Tc is near 0:23, where(�F)2 ∼ ln t [14]. There is a controversy about the existence of phase transitionon d = 4 + 1 [15–19]. Some mode coupling calculations of KPZ equation suggestthat the upper critical dimensions is less than or equal to 4 + 1 [15,16]. Above thecritical dimensions, z is expected as two and !=0 for all temperature regime. Recentwork [16] also has claimed that the KPZ equation has an upper critical dimensiondc64 + 1, where the exponents in the strong coupling regime take the values z = 2and ! = 0, equal to those at the roughening transition. However, the numerical studyof the directed polymer problem shows a di�erent phase transition [18], and the simu-lation result of the related problem so-called RSOS model, which belongs to the KPZuniversality class, shows the existence of strong-coupling regime [10,17,19].Here, we present more detailed analysis of the numerical data on both the directed

polymer in random potential problem specially on d=4+1. We �nd a phase transitionat Tc = 0:18 in DP problem. At low temperature, the uctuation of free energy growsas t 0:156. At the transition temperature, �F grow logarithmically as a function of time.Above Tc we obtain z = 2 and ! = 0. We also study the generalized RSOS growthmodel and �nd the same phase transition on d = 4 + 1 where the surface width Wshows the similar behaviors of �F in the directed polymer problem.Consider a directed polymer on a discrete “hyper-pyramid” structure with random

potential �(x; t) assigned to each site (x; t). The polymer starts from 0 and its path isrestricted by |x(t)− x(t+1)|=0 or 1. There is a bending energy against a transversejump |x(t) − x(t + 1)| = 1. Then the partition function Z(x; t) for the polymer endingat (x; t) can be obtained recursively. For example in d= 1 + 1

Z(x; t) = Z(x; t − 1) exp(−�(x; t − 1)

T

)

+Z(x − 1; t − 1) exp(−�(x − 1; t − 1) +

T

)

+Z(x + 1; t − 1) exp(−�(x + 1; t − 1) +

T

): (10)

For simplicity, we choose a continuous random number in between 0 and 1 with equalprobability for �(x; t).At in�nite temperature, the random potential has no e�ect so the left–right symmetry

is restored with 〈x〉=0 and z=2, where all paths contribute equally (high-temperaturephase). However, at low temperature, the polymer is attracted by local minimumpotentials and 〈x2〉 becomes super-di�usive with z¡ 2 (low-temperature phase).

338 J.M. Kim / Physica A 270 (1999) 335–341

Fig. 1. g(T; t) as a function of t for T = 0:30, 0.25, 0.18, 0.10, and 0.05 from the bottom to the top. g(t)remains almost constant at T = 0:18.

To look at the transition between these two phases, we consider a dimensionlessquantity [14]

g(T; t) ≡ 〈x〉2=〈x2〉 ; (11)

which depends on T and t. At T = ∞, since the random potential term becomesirrelevant, we expect g = 0. On the other hand, at T = 0, only one path is dominantthat 〈x2〉= 〈x〉2 and g becomes unity. There is an exact relation [20]

〈x2〉 − 〈x〉2 ∼ t (12)

from Eq. (3). If z is smaller than two, these two quantity 〈x2〉 and 〈x〉2 should behavesame way as t goes to in�nity.We monitor g numerically in d=4+1 by the transfer matrix method as described in

Eq. (10). A graph of g(T; t) versus t with the system size L=40 for various temperaturesis shown in Fig. 1. g increases as a function of t for T ¡ 0:17 and decreases with t forT ¿ 0:19. Near T = 0:18, g is independent of the length t that the critical temperatureTc is 0:18 ± 0:01 on d = 4 + 1. This value is quite far from zero, so that the uppercritical dimension might be much higher than 4 + 1 dimensions.The numerical data of the transverse uctuation 〈x2〉 is shown as a function of time

t for various temperatures in Fig. 2. For T =0:30 (which is larger than Tc=0:18), we�nd z = 2 implying that 〈x2〉 is linearly proportional to t. At T = 0:05 (.Tc), 〈x2〉increases as a power law t1:14 with z ≈ 1:75 (2=z = 1:14). So, the value of z is lessthan two in the low-temperature phase. However, the measured transverse displacement uctuation 〈x2〉 is proportional to t both above Tc and at Tc [22].To determine ! the exponent governing the growth rate of the free energy, we mea-

sure �F as a function of polymer length for a system size L=40. At low temperature,�F grows as t ! as shown in Fig. 3. For T = 0:05, we obtain

!= 0:156± 0:01 : (13)

This value is a little bit smaller than 16 but consistent with the conjecture 1=(d+1) [10]

within the error bar. At T = 0:3(¿Tc), �F is almost constant for large t. Above Tc,

J.M. Kim / Physica A 270 (1999) 335–341 339

Fig. 2. 〈x2〉 as a function of t in log–log plot. The data are for T = 0:30, 0.25, 0.18, 0.10, and 0.05 fromthe top to the bottom. The slope of the data represents 2=z. The slope of the upper guide line is one (z=2)and that of the lower guide line is 1.14 (z ≈ 1:75 for 5¡t¡ 25).

Fig. 3. (�F)2 as a function of t in log–log plot for T = 0:05 (top) and T = 0:3 (bottom).

�F becomes saturated as t increases (high-temperature regime), implying ! = 0. Atthe critical temperature T =0:18; (�F)2 grows as ln t as shown in Fig. 4. The similarbehavior is observed in both directed polymer problem [14] and the surface width ofa growth model on d= 3 + 1 also [21].As mentioned above, the DP in random potential problem is described by the KPZ

equation. It is known that the RSOS growth model follows the KPZ equation verywell [10,23]. In DP problem the temperature is e�ectively related to � of the KPZequation (Eq. (5)). In the RSOS model � can be varied by adjusting the depositionrate P+ and evaporation rate P− [11]. We describe the generalized RSOS model [11]here brie y. The growth rule of the model is that select a site randomly on d − 1substrate dimensions and deposit a particle on the site with the probability P+ (orevaporate a particle with the probability P−) if the height con�guration satis�es theheight restriction after the deposition (or evaporation) that the height di�erence betweenthe nearest neighbor is less than or equal to one [10]. Otherwise ignore the deposition(or evaporation) process and repeat the procedure. The sum of P+ and P− is one.

340 J.M. Kim / Physica A 270 (1999) 335–341

Fig. 4. (�F)2 as a function of t in semi-log plot for T = 0:05, 0.18, and 0.30 from the top to the bottom.The free energy uctuation (�F)2 grows as ln t at Tc.

Fig. 5. W 2 as a function of t in semi-log plot on the generalized RSOS growth model for P+ = 0:95, 0.89and 0.85 from the top to the bottom. The surface height uctuation W 2 grows as ln t as a function of timefor P+ = 0:89. The oscillation of W 2 is due to the discrete treatment of the height.

By adjusting P+ we can control �. It is generally believed that � is proportional to(P− − P+) approximately. If P+ = 1, it is the case of the RSOS growth model (puredeposition model). When P+ = P−, the model is expected to belong to the EdwardsWilkinson universality class [24]. To con�rm the phase transition, we simulate thegeneralized RSOS model and monitor the surface width W as a function of time forvarious value of P+, where W is the standard deviation of the surface height. Thesurface width in the growth model is exactly related to �F . In 1. t.Lz regime,we would expect W ∼ t! for large |�| and W being �nite (almost independent oftime) for P+¡Pc where Pc is a critical deposition probability. The simulation dataare given in Fig. 5 as a function of P+. At P+ = 1, we obtain ! ≈ 0:16, which isclose to the value in DP problem [3] and the recent numerical calculations of the samemodel [19]. Near P+ = 0:89, W 2 shows logarithmic behaviors. When P+ is smallerthan 0.89, W 2 becomes saturated. The same behavior is shown in Fig. 4 of the DPproblem that (�F)2 grows logarithmically as a function of time at Tc and it becomes

J.M. Kim / Physica A 270 (1999) 335–341 341

saturated above Tc. The logarithmic behavior was expected at the transition point [22]as was seen in d=3+1 growth model. So there exist a phase transition at P+ =0:89,where the related � is around −0:2. 1We have studied the directed polymer problem in random potentials on d = 4 + 1

and have found a phase transition. We obtain z ≈ 1:75, ! ≈ 0:156 at low temperatureT =0:05 (.Tc), and z ≈ 2, ! ≈ 0 for T¿Tc. The obtained value of ! is very similarto 0.16 obtained from the restricted solid-on-solid model in the largest simulation [19]and it is consistent with the conjecture 1=(d + 1) within the error bar. The criticaltemperature Tc=0:18±0:01 in d=4+1 is close to 0.23 of the critical temperature ond=3+1 [14] and far from zero temperature implying that the upper critical dimensionsis higher than 4+1 dimensions. The study of the RSOS model on d=4+1 also supportsthe existence of the phase transition. The existence of the upper critical dimensionsremains unsolved. More large simulations on higher dimensions are required.

This work is supported by the KRF (No. 98-015-D00091).

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1 We have measured � by tilt dependent growth velocity.