hall effect in a quasi-one-dimensional system

PHYSICAL REVIEW B 15 MARCH 1998-IVOLUME 57, NUMBER 11

Hall effect in a quasi-one-dimensional system

A. V. LopatinDepartment of Physics, Rutgers University, Piscataway, New Jersey 08855

~Received 11 August 1997!

We consider the Hall effect in a system of weakly coupled one-dimensional chains with Luttinger interactionwithin each chain. We construct a perturbation theory in the interchain hopping term and find that there is apower-law dependence of the Hall conductivity on the magnetic field with an exponent depending on theinteraction constant. We show that this perturbation theory becomes valid if the magnetic field is sufficientlylarge.@S0163-1829~98!00504-9#

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I. INTRODUCTION

It is well known that one-dimensional interacting eletrons form a non-Fermi-liquid system.1,2 The Green functionacquires an anomalous scaling power 2D, which implies thebreakdown of the basic assumption of the Fermi-liqutheory. However, not all correlation functions have anomlous scaling: the correlators containing only density and crent operators are similar to those of a noninteracting systTherefore most physical quantities are usual and it is inesting to consider those that do have a nontrivial contribufrom the interaction.

In this paper we consider the influence of the electinteraction on the Hall effect in a quasi-one-dimensional stem. To understand the effect of two dimensionality letconsider a system of weakly coupled one-dimensional chas an example of a quasi-one-dimensional system. The inchain hopping termt' can be neglected for electrons wisufficiently large energy. Therefore the system behavesone dimensional on scales much smaller thanl;(eF /t')ax , whereax is the distance between the atomwithin the chains andeF is the Fermy energy. Actually, thanomalous scaling of Green functions leads to a correctiothe above qualitative expression tol;(t' /eF)21/(122D)ax~see Refs. 3–5!. On scales larger thanl the system is essentially two-dimensional and one-dimensional Green functiodo not give even qualitatively the right answer. Thereforea physical effect comes from the length scales smaller thal ,then the answer can still have one-dimensional anomaWe show that the Hall effect in this system is stronglyfected by the anomalous powers and acquires a powerdependence on the magnetic field

s';H2114D.

This result is valid only if the magnetic field is stronenough. Indeed, the Hall effect is due to interferencescalesl H;F0 /ayH, where F0 is a quantum of magneticflux anday is the interchain distance. We want the systembe ‘‘one dimensional’’ on these scales, i.e., we needl @ l H .Substituting the expression forl we get

Haxay

F0@S t'

eFD 1/~122D!

. ~1!

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To find the Hall conductivity we use perturbation theoin t' . This perturbation theory becomes valid when the codition ~1! is satisfied because in this case the effect ofhopping term is small on the scales important for the Heffect.

The main technical difficulty of this problem will be thathe Hall conductivity of the chains with the linear electrospectrum,

e656vF~p7pF!,

is zero due to the particle-hole symmetry. Therefore itnecessary to consider a nonlinear correction to the spect

e656vF~p7pF!1a~p7pF!2. ~2!

We consider the simplest model of spinless electrons. Ialso assumed that the system is not close to half-fillingthat the umklapp processes can be neglected.

So the Hamiltonian of the problem is

H5vFE dx(i

c i†t3~2 i ]x!c i2aE dx(

ic i

†]x2c i

1gE dx(i

c i 1† c i 1c i 2

† c i 2

1t'E dx(^ i , j &

c i†c je

2 i ~e/c!Ai , j , ~3!

wherec is composed from the right- and left-moving eletrons

c5S c1

c2

D ,

t3 is a Pauli matrix,Ai , j5* ijAdl, and we use the Landa

gaugeAy5Hx. The second term in the Hamiltonian is thnonlinear correction to the free electron spectrum. The mo

6342 © 1998 The American Physical Society

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57 6343HALL EFFECT IN A QUASI-ONE-DIMENSIONAL SYSTEM

without the hopping term anda term can be solved exactlyfor example, by the bosonization method. It is also possto bosonize thea term6 but it leads to a cubic interactiobetween the bosons and the model with such an interacbeing not exactly solvable. Therefore we have to consithe a term as a perturbation, too. The nonlinear term inspectrum~2! is small compared with the linear term on scabigger thana/vF;a, hence a perturbation theory ina isvalid if l H@a. This condition can be written as

Haxay

F0!1,

and it is satisfied for any real experimental situation. Thefore a perturbation theory ina is always a good approximation.

The plan of the paper is the following: In Sec. II wexpress the Hall conductivity through the single-chain corlation functions. In Sec. III we explain the technique thatwill use to calculate one-dimensional correlators. In Sec.we calculate the Hall conductivity. Finally, we summariour results and discuss their possible applications in Sec

II. EXPRESSION FOR THE HALL CONDUCTIVITY

Let us choose a coordinate frame so that the magnfield points along thez axis and the chains are along thexaxis. The labeli , which denotes the number of the chaincreases in they direction. The electric fieldEx5E0e2 ivt isapplied along the chains (x axis!. In this geometry the Hallconductivitys'(v) relates the electric fieldEx with the cur-rent between the chainsj y ,

j y5s'~v!Ex .

According to the Kubo formula the conductivity is expressthrough the retarded current-current correlator,

s'~v!51

v(iE dxPR~x,i ,v!, ~4!

where

PR~x,i ,v!5E2`

`

dt eivt@ j y~x,i ,t !, j x~0,0,0!#u~ t !. ~5!

In these expressionsj y is the Heisenberg operator of thcurrent between the chains,

j y~x,i ,0!5t'ei~ c i†~x!c i 11~x!e2 i ~e/c!Ai ,i 112H.c.!, ~6!

and j x is the in-chain current operator,

j x~x,i ,0!5e„vFc i†~x!t3c i~x!12ac i

†~x!~2 i ]x!c i~x!….~7!

We will work in the Euclidean space, which correspondsthe Wick rotationt→2 i t , v→ iv ~see, for example, Ref. 7!.The Euclidean Lagrangian of this problem is

le

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e

-

-

V.

tic

o

LE5E dxE dt(i

c i†~2]01 ivFt3]11a]1

2!c i

2gE dxE dt(i

c i 1† c i 1c i 2

† c i 2

2t'E dxE dt(^ i , j &

c i†c je

2 i ~e/c!Ai , j , ~8!

wherec†,c are two-vector anticommuting variables whiccorrespond to the operatorsc†,c of the Hamiltonian formal-ism and ]05]/]t,]15]/]x. Let us define the Euclideancurrent-current correlator,

PE~x,i ,v!5E2`

`

dt eivt^Tt j y~x,i ,t !, j x~0,0,0!&, ~9!

whereTt means time ordering in the ‘‘Euclidean’’ time. FoEuclidean correlators the standard perturbation theory caused. The retarded correlator~5! is the analytical continua-tion of the Euclidean one,

PR~v!52 iPE~2 iv!. ~10!

Considering the last term in Eq.~8! as a perturbation andusing Eqs.~4! and ~10! for the Hall conductivity we get

s'~v!522eit'

2

vGE~2 iv!, ~11!

whereGE(v) is the Euclidean correlator,

GE~v!5E2`

`

dteivtGE~ t !,

with

GE~ t12t3!5(sE dx3E dx2E dt2@^ j ~x3 ,t3!

3cs~x1 ,t1!cs†~x2 ,t2!&^cs~x2 ,t2!cs

†~x1 ,t1!&

1~x1 ,t1↔x2 ,t2!#sinq~x12x2!. ~12!

In this formulaq5eHay /c and the labels51,2. Note thatthe one-dimensional currentj in Eq. ~12! contains a contri-bution from the nonlinear correction to the spectrum,

j 5 j ~0!1 j ~1!, ~13!

where

j ~0!5evFc†t3c, ~14!

j ~1!52eac†~2 i ]1!c. ~15!

The angular brackets on the right-hand side of Eq.~12! rep-resent averaging with respect to the single-chain Lagrang

LE~1!5E dxE dt„c†~2]01 ivFt3]11a]1

2!c

2gc1† c1c2

† c2…. ~16!

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6344 57A. V. LOPATIN

III. BOSONIZATION

Now the problem is to find the one-dimensional corretors in Eq.~12!. The standard way to treat a one-dimensiomodel with the linear spectrum is bosonization. Without ta term the model~16! can be solved exactly. We will treathis term as a perturbation and begin with bosonization ofHamiltonian. Thea term in the fermionic Lagrangian resulin a cubic interaction between the bosons.6 The single-chainHamiltonianH1 in the bosonized form~see Appendix A! is

H15 12 @~]1F!21P2#1

a

3b~]1F!S 3

p

b2 P21b2

p~]1F!2D ,

~17!

where F and P are canonically conjugate Bose operatoandb is a constant that depends on the interaction consg,

b5ApS vF2g/2p

vF1g/2p D 1/4

.

The fermionic and current operators~see Appendix A! are

c6~x,t !51

A2pde6 i F6~x,t !,

F6~x,t !5bF~x,t !

7p

bE2`

x

dx8P~x8,t !, ~18!

j 52eS vF

bP12aP]1F D , ~19!

whered is the inverse momentum-space cutoff, and the nmal ordering of operators is implied in Eqs.~17! and ~19!.Here we have rescaled the energy units so that the velocithe Bose particlesvB is 1. The second term in the curreoperator is due to thea term in the Hamiltonian. It is moreconvenient to calculate the correlation functions usingfunctional representation. In this formalism we introducefields F,P, which are related to the fermion fieldsc†,c via

c6~x,t !51

A2pde6 iF6~x,t !,

F6~x,t !5bF~x,t !

7p

bE2`

x

dx8P~x8,t ! ~20!

j 52eS vF

bP12aP]1F D . ~21!

The Green function

G~x12x2 ,t12t2!5^Ttc~x1 ,t1!c†~x2 ,t2!&H , ~22!

calculated by the Hamiltonian method, is related toGreen function,

Gf~x12x2 ,t12t2!5^c~x1 ,t1!c* ~x2 ,t2!& f ,

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of

ee

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calculated by the functional method through the followirelation:

G~x12x2 ,t12t2!5sgn~ t12t2!Gf~x12x2 ,t12t2!.~23!

Indeed, due to the fundamental property of the correspdence between the Hamiltonian and functional methods,have

^Tt~boz!c~x1 ,t1!c†~x2 ,t2!&H5^c~x1 ,t1!c* ~x2 ,t2!& f ,

whereTt(boz) means time ordering of the boson operators

cause thec operators are constructed from the boson opetors. But the definition~22! implies time ordering of fermi-ons, and this difference should be corrected by insertsgn(t12t2) into Eq.~23!. So to get an actual ‘‘Hamiltonian’’Green function we should multiply the corresponding ‘‘funtional’’ Green function by sgn(t12t2).

The Lagrangian density corresponding to the single-chHamiltonian~17! in the Euclidean space is

LE~1!5LE~0!1V~P,]1F!, ~24!

where

LE~0!5 iP]0F2 12 ~P21~]1F!2!, ~25!

V~P,]1F!52a

3b~]1F!S 3

p

b2 P21b2

p~]1F!2D .

~26!

We will treat the interactionV as a perturbation. The barcorrelation functions are

^F~x1 ,t1!F~x2 ,t2!& f~0!52d~x,t ! ~27!

E2`

x1dx8E

2`

x2dx9^P~x8,t1!P~x9,t2!& f

~0!52d~x,t !

~28!

E2`

x1dx8^P~x8,t1!F~x2 ,t2!& f

~0!

52i

2psgntS arctan

x

utu1d1

p

2 D , ~29!

where

d~x,t !51

4pln

~ utu1d!21x2

l 2 . ~30!

In the above formulast5t12t2 and x5x12x2. The label(0) in Eqs.~27!–~29! means averaging with respect to thfree-boson LagrangianLE

(0) and l is the length of the chainsNow let us reproduce the well-known result for the on

particle Green function. According to the formula

^eA&5e^A2&/2, ~31!

whereA can be any linear functional of the fieldsP,F, wecan easily find the correlators ofc functionals

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G6, f~0! ~j12j2!5^c6~j1!c6* ~j2!& f

~0!

5sgn~ t12t2!i

2p

1

6x1 i t S d2

x21t2D D

, ~32!

whereD5 14 (p/b21b0

2/p22) andj5(x,t). Here and be-low we setd equal to wherever it does not lead to appardivergences. The appearance of the factor sgn(t12t2) is inaccordance with the formula~23!. Indeed, due to this for-mula the fermion Green function with proper time orderiis

G6~0!~j12j2!5^Ttc6~x1 ,t1!c6

† ~x2 ,t2!&H~0!

5i

2p

1

6x1 i t S d2

x21t2D D

, ~33!

which is the right answer.It will be convenient for us to introduce the followin

generating functional:

Z6~ f 0 , f 1!5^c6~j1!c6* ~j2!e*d2j@ f 0~j!P~j!1 f 1~j!]1F~j!#& f~0! .~34!

This functional can be calculated using the formula~31!,

Z6~ f 0 , f 1!5G6, f~0! ~j1 ,j2!

3eF6~ f 0 , f 1!11/2*d2j1d2j2f T~j1!D~j1 ,j2! f ~j2!.

~35!

In this formula G6, f(0) is the Green function~32!, f

5( f 0 , f 1)T is a two vector constructed fromf 0 , f 1, andF6( f 0 , f 1) is a linear functional off 0 , f 1,

F6~ f 0 , f 1!5E d2j~ f 0~j!J06~j1 ,j2 ,j!

1 f 1~j!J16~j1 ,j2 ,j!!, ~36!

where

J06~j1 ,j2 ,j3!5

1

2p

2 ip

b~x12x3!7b~ t12t3!

~ t12t3!21~x12x3!2

2~x1 ,t1↔x2 ,t2!, ~37!

J16~j1 ,j2 ,j3!5

1

2p

6 ib~x12x3!1p

b~ t12t3!

~ t12t3!21~x12x3!2

2~x1 ,t1↔x2 ,t2!. ~38!

Finally, D is a matrix Green function that in the momentuspace is

D~v,p!5p2

p21v2F 1 iv

p

iv

p1G . ~39!

t

IV. CALCULATION OF THE HALL CONDUCTIVITY

To calculateG we need to find the correlators

K6~j1 ,j2 ,j3!5E dx3^ j ~j3!c6~j1!c6† ~j2!&

and

G6~j!5^c~j!c†~0!&,

where the averaging is done with respect to the single-chLagrangian~16!. Let us apply the technique developed in tprevious section to calculate these correlators. First of allshould find these correlators to the zeroth order ina. TheGreen functionG6

(0) was already calculated in the previousection. The correlatorK6 to the zeroth order ina can becalculated using the generating functional~34,35!,

K6~0!52

evF

bsgn~ t12t2!E dx3

d

d f 0Z6~ f 0 , f 1! u f 05 f 150

52evF

b E dx3G6~0!~j1 ,j2!J0~j1 ,j2 ,j3!.

Calculating the integral overx3 we get

K6~0!5E dx3^ j 0~j3!c6~j1!c6

† ~j2!&~0!

571

2evFG6

~0!~j12j2!@sgn~ t32t1!2sgn~ t32t2!#.

~40!

Substituting these expressions and the Green function~33!into G @Eq. ~12!# we get zero due to the fact thatG6

(0)(x1

2x2 ,t12t2) is odd under the transformationx1 ,t1↔x2 ,t2

andK6(0) is even. Indeed, the model with the linear spectru

(a50) should give zero Hall effect due to the particle-hosymmetry. To the first order ina we can conveniently represent the product of these two correlators as

^ j cc†&^cc†&5^ j 0cc†&^cc†&1^ j 1cc†&~0!^cc†&~0!,~41!

where j 0 and j 1 are defined in Eq.~13!. The first term in Eq.~41! contains the current to the zeroth order so that the nlinear corrections come from the Lagrangian. The secoterm contains noa corrections from the Lagrangian becauj 1 is already proportional toa. Fortunately the first termgives no contribution when substituted into the formula forG~see Appendix B! and, therefore, we should calculate onthe contribution from the second term. The correlator

E dx3^ j 1~j3!c6~j1!c6† ~j2!&~0!,

where the current correctionj 1 in the bosonized form is

j 1522eaP ]1F,

can be calculated using the formulas~34,35!,

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E dx3^ j 1~j3!c6~j1!c6† ~j2!&~0!

522eaE dx3J06~j1 ,j2 ,j3!J1

6~j1 ,j2 ,j3!G6~0!~j1 ,j2!,

where it was used that*dx1D(j1 ,j2)50 ~ see Appendix B!.Taking the integral overx3 we get

E dx3^ j 1~j3!c6~j1!c6† ~j2!&

5aei

2pG6

~0!~j12j2!1

~x12x2!21~ t12t2!2

3@62p i ~ t12t2!2„~p/b!21b2…~x12x2!#

3@sgn~ t32t1!2sgn~ t32t2!#. ~42!

Substituting Eqs.~42! and~33! into the expression forG andshifting the variablesx2→x21x1 ,t2→t21t1 for GE(v) weget

GE~v!5ea i

2p3 d4DE dt8E dt eivt@sgnt2sgn~ t81t !# f ~ t8!,

~43!

where

f ~ t !5E dx1

~x1 i t !2

1

~x21t2!112D @22p i t 1x„~p/b!2

1b2…#sinqx.

Taking the integral overt in Eq. ~43! we get

GE~v!52ea

vp3 d4DE0

`

dt~cosvt21! f ~ t !. ~44!

Introducing cylindrical coordinatesx5r cosf,t5r sinf wecan integrate overf obtaining

GE~v!5ea

p2vd4DE

0

` dr

r2~112D!F ~p/b

2b!2q

Av21q2J1~rAv21q2!2~p/b

1b!2q~q223v2!

~v21q2!3/2 J3~rAv21q2!2~v50!G ,

~45!

whereJ1 andJ3 are the Bessel functions@not to be confusedwith J functions~37,38!#. Calculating the integral overr andusing Eq.~11! for the Hall conductivity we get

s'~v!522e2t'

2 a

p S d

2D 4D G~122D!

G~212D!

3q

v2F ~q22v2!2Dq21v2

q22v2 2q4DG , ~46!

which is the main result of this paper. Expanding inv we getthe dc conductivity,

s'524

p

G~122D!

G~212D!~12D!

t'2 e2a

vB2 S eHayvB

c D 124DS d

2vBD 4D

,

~47!

wherevB was restored. One can see thats' depends on themagnetic field asH4D21. Note that the formula~47! does notwork for D. 1

2. @The integral overr in Eq. ~45! does notconverge in this case.# The fact thatD5 1

2 is a critical value isnot surprising because it was shown8 that in the caseD. 1

2

the interchain hopping term in the Hamiltonian is irrelevain the renormalization group~RG! sense. To avoid confusionwe note that the irrelevance of the hopping term in the cD. 1

2 should be understood in the straightforward dimesional RG sense. Actually, considering two-particle tunning processes, one can see that the hopping term is releeven forD. 1

2 ~See Refs. 4,5,9!. As was mentioned abovethis theory is valid for high magnetic fields. To find the aplicability criterion we should consider the next nonzero oder in t' . It is hard to calculate it but from dimensionaanalysis it follows that the correction should have the for

s'→s'S 11constt'2

S eHayvF

c D 2S eHayd

c D 4DD .

Indeed, expanding to the next nonzero order int' one getsfour additionalc operators that give the factord4D and theremaining factors can be restored from dimensionality. Tcorrection is small if

H@F0

axayS t'eF

D 1/~122D!

,

whereF0 is a quantum of magnetic flux. It agrees with oexpectation~1! in the Introduction.

V. DISCUSSION AND CONCLUSIONS

We considered the Hall effect in a quasi-one-dimensiointeracting electron system. It was found that in high manetic fields,

H@F0

axayS t'eF

D 1/~122D!

, ~48!

there is a power-law dependence of the Hall conductivitythe magnetic field,

s';H2114D, ~49!

where 2D is the anomalous exponent of the one-dimensioGreen function~33!. This formula can be applied forD, 1

2.This result was obtained for the zero-temperature case.for nonzero temperatures much lower thaneHaxvF /c itshould still hold because the temperature will changecorrelation functions only in the low-energy region, whichirrelevant in the case of high magnetic fields. Therefore tresult can be applied for nonzero temperatures if

T

eF!

Haxay

F0. ~50!

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We considered the simplest model of spinless electrons.more realistic case of the Hubbard model the single-chGreen function has a more complicated form

G~x,t !5i

2p

1

~6x1vri t !1/2~6x1vsit !1/2S d2

x21~vrt !2D D

,

~51!

~in the Euclidean space!, wherevs and vr are the spin andcharge velocities. Using dimensional analysis one can arthat the result~49! should still hold. Indeed, comparing thformula for G ~12! with the final answer, one can see theveryc operator gives rise to an additional factordD in thefinal answer, while the currentj gives no additional powersbecausej is a local function of the Bose fields. Therefore texponent of the magnetic field in the final answer shouldthe same withD defined by Eq.~51!.

We hope that experimentally this effect can be obserin one-dimensional organic conductors~see Refs. 10 and 11for review!. Actually, the possibility of observing this effecdepends on the value ofD in a particular material. If we takeaxay55310215 cm2 and t' /eF51/10, which is typical forthese materials, then we can rewrite the condition~48! in theform

H@~103 . . . 104!S 1

10D1/~122D!

T.

It can be satisfied from the experimental point of view ifD isnot far from 1

2. Note that one-dimensional organic conductoat low temperatures typically exhibit phase transitions toperconducting, charge-density wave or spin-density wstates and many properties such as Hall effect, magnetortance, etc. are very unusual at these temperatures. Of coour result cannot be applied in these cases because the eof the complicated ground state were not considered. In owords, for this effect to be observed the materials must bthe metallic state.

ACKNOWLEDGMENT

I am indebted to L. B. Ioffe for the idea of this work anvery useful discussions.

APPENDIX A: BOSONIZATION OF THE SINGLE-CHAINHAMILTONIAN

In this section we will follow the approach describedRef. 2. Our task is to bosonize the single-chain Hamilton

H15E dx~2 ivFc†t3]1c2ac†]12c1gc1

† c1c2† c2!,

The product of twoc operators can be written as

c6† ~x!c6~x1!56

1

2p i ~x12x!:e7 i „F6~x!2F6~x1!…:,

where

ain

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d

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erin

n

c6~x,t !51

A2pde6 i F6~x,t !,

F6~x,t !5ApS F~x,t !7E2`

x

P~x8,t !dx8D ,

and : : means normal ordering of operators. Here we hrescaled the energy units so thatvB51. Expanding the ex-ponent to the third order in (x12x), we get

c6† ~x!c6~x1!5:6

1

2p i$6 i F68 ~x!2 1

2 ~x12x!@„F68 ~x!…2

7 i F69 ~x!#1 16 ~x12x!2@6 i F698~x!

23F68 ~x!F69 ~x!7 i „F68 ~x!…3#%:, ~A1!

where all singularc number terms are ignored and thprimes denote differentiation with respect tox. Now it isstraightforward to bosonize all operators that we need.the terms of the Hamiltonian we have

2 ivFE dxc†t3]1c51

2E dxS vB~]1F!211

vBP2D vF ,

gE dxc1† c1c2

† c25g

4pE dxS vB~]1F!221

vBP2D ,

E dxc†]12c52

1

3

p

AvBE dx ~]1F!„3P21vB

2~]1F!2…,

wherevB was restored for a reason that will be seen beloNormal ordering of operators is implied hereafter. In tderivation of the above expressions the terms in Eq.~A1!that are total derivatives were neglected. The sum of theand the second terms can be written as the usual free-bHamiltonian if we renormalize the fieldsF→Fb/Ap,P→Pb/Ap so that

2 ivFE dxc†t3]1c1gE dxc1† c1c2

† c2

5E dx12 „vB

2~]1F!21P2…,

where

b45p2

vF2g

2p

vF1g

2p

,

vB5AvF22S g

2p D 2

.

Knowing the relation betweenvB and vF we can setvBequal to 1 again. Note that the above relations depend ocutoff procedure and are universal only to the first powerg. So the single-chain Hamiltonian expressed throughnew renormalized fields becomes

m

tngst

m


H15 12 @~]1F!21P2#1

a

3b]1FS 3

p

b2P21b2

p~]1F!2D .

~A2!

Finally, the current operator expressed through the safields is

j 52eS vF

bP12aP]1F D .

APPENDIX B

Let us prove that to the first order ina there is no contri-bution toG from the first term in Eq.~41!,

^ j 0~j3!c6~j1!c6* ~j2!&^c6~j2!c6* ~j1!&. ~B1!

To the zeroth order ina it was already proven that this termgives no contribution toG. Therefore we should show thathe first-order corrections give no contribution as well. Usithe formula~35! we can find the Green function to the firorder ina,

^c6~j1!c6* ~j2!&

5G6~0!~j12j2!S 11 K E d2jV~g01J0

6 ,g11J16!jL

gD ,

~B2!

e

ti-

e

whereV is the interaction term in the Lagrangian~24!

V~g0 ,g1!52a

3b g1S 3p

b2 g021

b2

pg1

2D ,

and for convenience we redefined the fieldsP,]1F asg0 ,g1, correspondingly. The subscriptj in Eq. ~B2! repre-sents the arguments of theg fields andJ functions, for ex-ample,

V~g01J06 ,g11J1

6!j[V„g0~j!1J06~j1 ,j2 ,j!,g1~j!

1J16~j1 ,j2 ,j!…

and the symbol &g means averaging with the functional

e2~1/2!*d2j1d2j2gT~j1!D21~j12j2!g~j2!, ~B3!

whereD21(j) is the Green function that in the momentuspace is the inverse Green function~39!,

D21~v,p!5F 1 2 iv

p

2 iv

p1G .

Analogously, for the correlator containingj 0, to the first or-der in a we get

^ j 0~j3!c6~j1!c6* ~j2!&52evF

bG6

~0!~j12j2!J06~j1 ,j2 ,j3!S 11 K E djV~g01J0

6 ,g11J16!jL

gD 2

evF

bG6

~0!~j12j2!

3K E djV~g01J06 ,g11J1

6!jg0~j3!Lg

, ~B4!

m,

r--

where the arguments ofJ0 ,J1 are implied according to thesame rule as above. Note that the functionsJ0 ,J1 are oddunder the exchange of the argumentsj1 and j2. Thereforethe term

E dj^V~g01J06 ,g11J1

6!j&g

[E dj^V„g0~j!1J06~j1 ,j2 ,j!,g1~j!

1J16~j1 ,j2 ,j!…&g

is odd under this exchange too.~One can always changg0 ,g1→2g0 ,2g1 because the functional~B3! is invariantunder this transformation.! One can see that after the substution of Eqs.~B2,B4! into Eq. ~B1! the correction in Eq.~B2! and the correction of the same kind in Eq.~B4! canceleach other~to the first order! and Eq.~B1! becomes

^ j 0~j3!c6~j1!c6* ~j2!&g^c6~j2!c6* ~j1!&g

52evF

bG6

~0!~j12j2!G6~0!~j22j1!J0

6~j1 ,j2 ,j3!

2evF

bG6

0 ~j12j2!G60 ~j22j1!

3 K E djV~g01J06 ,g11J1

6!jg0~j3!Lg

. ~B5!

The first term in this expression is the zeroth-order terwhich gives no contribution toG. The second term giveszero when integrated overx3. To see this we need the corelation functions of theg fields. Taking the Fourier transformation of the Green function~39! one gets

ee


^g1~j1!g1~j2!&g5^g0~j1!g0~j2!&g

51

4pS 1

~ t1 ix !2 11

~ t2 ix !2D ,

^g1~j1!g0~j2!&g5^g0~j1!g1~j2!&g

521

4pS 1

~ t1 ix !2 21

~ t2 ix !2D .

m

n-

ev

From the form of theg correlators written above one can sthat

E dx1^g~j1!g8~j2!&50,

whereg andg8 are any fields from theg1 ,g2 fields. There-fore there is no contribution toG from the term~B1!.

1E. Fradkin, Field Theories of Condensed Matter Syste~Addison-Wesley, Redwood City, CA, 1991!.

2R. Shankar,Low-Dimensional Quantum Field Theories for Codensed Matter Physicists~World Scientific, Singapore, 1992!, p.353.

3H. J. Schulz, Int. J. Mod. Phys. B5, 57 ~1991!.4S. Brazovskii and V. Yakovenko, Zh. E´ ksp. Teor. Fiz.89, 2318

~1985! @Sov. Phys. JETP62, 1340~1985!#.5D. Boies, C. Bourbonnais, and A.-M. S. Tremblay, Phys. R

Lett. 74, 968 ~1995!.

s

.

6F. D. M. Haldane, J. Phys. C14, 2585~1981!.7M. E. Peskin and D. V. Schroeder,An Introduction to Quantum

Field Theory~Addison-Wesley, Reading, MA, 1995!.8X. G. Wen, Phys. Rev. B42, 6623~1990!.9V. M. Yakovenko, Pis’ma Zh. E´ ksp. Teor. Fiz.56, 523 ~1992!

@JETP Lett.56, 510 ~1992!#.10D. Jerome and H. J. Schulz, Adv. Phys.31, 299 ~1982!.11L. P. Gor’kov, Usp. Fiz. Nauk.144, 381 ~1984! @Sov. Phys. Usp.

27, 809 ~1984!#.

hall effect in a quasi-one-dimensional system

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