Manipulating the Tomonaga-Luttinger exponent by electric field modulation

0.25 MB·English

by  Hiroyuki Shima

Manipulating the Tomonaga-Luttinger exponent by electric field modulation
Hiroyuki Shima∗
Department of Applied Physics, Graduate School of Engineering,
Hokkaido University, Sapporo 060-8628, Japan and
Department of Applied Mathematics 3, LaC`aN, Universitat Polit`echnica de Catalunya (UPC), Barcelona 08034, Spain
Shota Ono
Department of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan
0
1
Hideo Yoshioka
0
Department of Physics, Nara Women’s University, Nara 630-8506, Japan
2
(Dated: January 30, 2010)

n
a We establish a theoretical framework for artificial control of the power-law singularities in
J Tomonaga-Luttinger liquid states. The exponent governing the power-law behaviors is found to
increase significantly with an increase in the amplitude of the periodic electric field modulation
0
applied externally to the system. This field-induced shift in the exponent indicates the tunability
3
of thetransport properties of quasi-one-dimensional electron systems.

]
l PACSnumbers: 73.21.Hb,71.10.Pm
l
a
h
s- I. INTRODUCTION εF and dI/dV become constant.
e
m
Interacting electrons in one-dimensional (1D) met-
. als constitute a highly collective state of matter: the
t
a Tomonaga-Luttinger liquid (TLL) state.1–3 The collec- The TLL exponent α is nonuniversal; it is dependent
m tive nature of the TLL states is what distinguishes them on the interaction strength,3 the geometric shape of the
- from their higher-dimensional counterparts. Interact- system,29 and the position of tunneling.30 In fact, dif-
d ing electrons in two or three dimensions form a Fermi ferent values of α were obtained in carbon nanotube ex-
n
liquid, wherein the only effects of interaction are the periments when each electron tunneled into the end or
o
c modification of their effective mass and the possibility bulk of the system.4,5,31 A further non-trivial shift in α
[ of being scattered. In one dimension, however, even was suggested in multiwalled nanotubes, where α varies

the slightest correlation between electron motions has a in a continuous manner under the application of a high
1
dramatic effect, leading to distinctive features that can- transverse magnetic field.32 Continuous variation in α
v
4 not be explained by the Fermi liquid theory. To date, was also found in a nuclear magnetic resonance study
5 physical consequences of the TLL states have been ex- of CuBr4(C5H12N)2 crystals;33 in this case, an exter-
0 perimentally observed in various systems, including car- nal magnetic field acted as the chemical potential. Such
0 bon nanotubes,4–8 semiconducting quantum wires,9–13 field-induced variations in α can be exploited for achiev-
2. quasi-1D organic conductors,14–16 quantum Hall edge ing artificial control of transport properties in quasi-1D
0 states,17 and other materials having highly anisotropic conductors, which would play a fundamental role in the
0 conductivity.18–25Fromthetheoreticalviewpoint,amore development of next-generation quantum devices.
1 generalTLL theorywithanonlineardispersion26 aswell
:
v asanovelwave-packeddynamicsthroughY-shapedTLL
i junctions27 have been recently suggested.
X
A hallmark of TLL states is a pseudogap in the one- In this paper, we propose a theoretical framework for
r
a particle density of states ν(ε) at the Fermi energy εF. TLL exponent manipulationbased onelectric field mod-
Injection of an additional electron in the TLL ground ulation. An analytical expression of the exponent α for
state disrupts the pre-existing correlation, thus requir- quasi-1Dconductorssubjectedtoastepwiseperiodicpo-
ing excitation of an infinite number of collective modes. tential is established in terms of the potential amplitude
Thisresultsinapower-lawsingularityoftheformν(ε) and period. Under feasible physical conditions, α in-
ε ε α, where α(> 0) is called the TLL exponen∝t. creasessignificantlywithanincreaseinthepotentialam-
F
| − |
The same power-law arises in the case of a differential plitude;thisindicatesthatitispossibletotunethequan-
tunneling current4 dI/dV V α at high bias voltages tum transportproperties of quasi-1Dsystems by manip-
∝ | |
(eV k T) and a temperature-dependent conductance ulating α. For a concise description, we focus our at-
B
G(T)≫ Tα at low voltages (eV k T), although α tention on 1D spinless fermion systems, considering that
B
may ch∝ange due to environment eff≪ects.28 These power- the effects ofspindegreeoffreedomrequiresno substan-
lawbehaviorsareinstrongcontrastwiththebehaviorof tial revision of the present conclusion. This issue will be
Fermi liquids; in the case of Fermi liquids, ν(ε) close to revisited in Sec. IV.
2
II. ONE-PARTICLE STATES IN 1D PERIODIC
SYSTEMS (a) (b) x0.1
1.0
1.0
A. Eigenenergy analysis
0.5 y]
R
We consider a quasi-1D electron system having a thin f() 0.0 [F
cylindrical shape with length L, which is subjected to a 0.5
periodic external potential field. The cylinder radius d -0.5
is so small (d L) that all electrons reside in the low-
≪ -1.0
est subband, χ, of the transversemotion. Single-particle x0.01 0
wavefunctions thus have the form 0.0 0.5 1.0 1.5 2.0 2.5 0 1.0 2.0
[Ry] u0 [Ry]
Ψ(z,r )=χ(r )ψ(z), (1)
⊥ ⊥
FIG.1: (a)Variationofthecurvef(ε)withanincreaseinthe
where the two-dimensional vector r = (x,y) spans the potentialamplitudefromu0 =0.2(solid)to0.5(dashed),1.0
tchirecuSlcahrr¨ocrdoinssgesrecetqiouna.tioTnhweiatxhiaelffeccotm⊥ivpeomneanstsψm(∗z,) obeys (wdhaesrheeind-daoBtte=d),~2aǫn/d(m2.∗0e2()d.ottTehde) ipnauranmitestoerfsRay ≡=e52./0(ǫaanBd)
b = 1.0 in units of aB are fixed. (b) Slow decay of εF with
~2 d2ψ increasing u0;see text for the definition of εF.
+u(z)ψ(z)=εψ(z), (2)
− 2m∗ dz2
radius a = ~2ǫ/(m∗e2) for length and units of the ef-
where u is a stepwise periodic potential given by B
fective RydbergRy =e2/(ǫa )for energy,whereǫ is the
B
0, (0 z <a) dielectricconstantofthewire. ForGaAs-basedquantum
u(z)=(cid:26)u0 >0, (−b≤≤z <0) (3) wmierVes.,36for instance, we have aB ∼ 10 nm and Ry ∼ 50
and u(z)=u(z+a+b). The periodicity of u(z) implies Figure 1(a) shows the plot of f(ε) corresponding to
that the eigenstates of Eq. (2) are represented by the the lowest energy band of the allowed ε. The poten-
Bloch function tial amplitude u0 ranges from 0.2 to 2.0, and fixed pa-
rameters a = 5.0 and b = 1.0 are used. With increas-
1
ψ(z)= φ (z)eikz, φ (z+a+b)=φ (z), (4) ing u0, the band width shrinks monotonically and the
k k k
√N ground-state energy ε (i.e., the specific ε that gives
grd
f(ε) = 1 in the plot) shifts to the right. Thereafter,
where L ψ(z)2dz = 1 and a+b φ(z)2dz = 1 with we set the Fermi energy ε (measured from ε ) such
0 | | 0 | | F grd
L = N(a+b). Hence, all eigenstates are labeled by the that the electron density n satisfies na = 0.3, which
B
R R
indexk. Itisnoteworthythattheperiodicpotentialssim- gave r (2na )−1 = 1.67. This value is much lower
s B
≡
ilartotheabovecanberealizedbyintroducinggeometric than the critical value of r , viz., r = 36, at which
s s
curvature (instead of field modulation) to quati-1D sys- the Wigner crystal transition takes place37; it is also
tems, and such the curvedsystems may show non-trivial lower than r 2.2, above which the 4k correlation
s F
quantum transport.34,35 in the chargede∼nsity distribution wassuggested.38 Since
The dispersion relationof the system given by Eq. (2) n=(2/π) kF dk,wecanevaluateε foragivenu from
0 F 0
is (see Appendix A) the relation f(ε ) = cos[πn(a+b)]; see Eq. (5). Figure
R F 2
1(b) presents the u -dependencies of ε . It decays with
0 F
f(ε)=cos[k(a+b)]. (5) increasingu but remains of the orderof 10−2 Ry for all
0
u under consideration.
Here, the function f(ε) is given by 0
1 η η
1 0
f(ε) = sin(ζ η )sinh(ζ η ) B. Field-induced shift in Fermi velocity
a 0 b 1
2 η − η
(cid:18) 0 1(cid:19)
+cos(ζaη0)cosh(ζbη1), (6) The group velocity vg(ε) ≡ ~−1∂ε/∂k of the system
is evaluated by differentiating both sides of Eq. (5) with
where η = √ε, η = √u ε, ζ2 = 2m∗a2/~2, and respect to k. The result is
ζ2 = 2m0∗b2/~2. Ac1cording 0to−Eq. (a5), f(ε) must fall in
b
therange 1to1forεtobephysicallyrelevant;i.e.,only a+b 1 f(ε)2
− v (ε)= − , (7)
ε’s that satisfy the condition f(ε) 1 are allowed to g − ~ ∂f/∂ε
| | ≤ p
be the eigenenergiesof the system. Note that Eq.(5) re-
ducestothetrivialrelationε=~2k2/(2m∗)whenu =0, The explicit form of ∂f/∂ε is given in Eq. (A5) in Ap-
0
since Eq. (6) becomes f(ε,u = 0) = cos[(ζ +ζ )√ε]. pendix A. v vanisheswhenε satisfiesf(ε)= 1,i.e., at
0 a b g
±
Throughoutthispaper,weuseunitsoftheeffectiveBohr the lower and upper band edges; between the two edges,
3
where
(a) (b)
0.4 0.3 g4 = LV˜(kF,kF; qz =0), (10)
g = LV˜(k , k ; q =0), (11)
2 F F z
−
0.3 L
v / vg00.2 v / vF0 0.2 g1 = 2+[VV((kFk,−,kkF;; qqz ==−2k2k)F]) (12)
F F z F
−
0.1
0.1 and
V˜(k ,k ,q )= k +q ; k q Vˆ k ; k . (13)
0 0 1 2 z h 1 z 2− z| | 2 1i
0 0.05 0.10 0 1.0 2.0

- grd[Ry] u0 [Ry] The right side of (13) is the matrix element of
the Coulomb interaction between two-electron states
FIG. 2: (a) Profiles of the group velocity vg(ε) in units of rj kj Ψ(rj) given by Eq. (1), where qz describes the
v0 ≡ RyaB/~. Each curve corresponds to the value of u0 hmom| eint≡um transfer in the axial (z) direction. It should
in the same manner as Fig. 1. A concave curve connecting benotedthatinformula(9),wecanprovethatg =g as
4 2
the two end points is the trajectory of the Fermi velocity aconsequenceofthesymmetricpropertyφ (z)=φ∗ (z)
vF ≡vg(ε=εF) obtained through the u0 variation. (b) The ofthe function φ (z)introduced inEq.(4)k. See Eq−.k(22)
k
Fermi velocity vF as a function of u0. for the proof.
v takes the maximum value. The only exception oc-
g B. Fourier representation of Coulomb interaction
curs in the limit of u = 0, in which v (ε) = 2ε/m∗
0 g
increases monotonically with ε.
Computation of the TLL exponent α requires evalu-
p
Figure 2(a) shows the ε-dependence of vg in units of ations of V˜(k1,k2,qz) at the specific values of k1,k2,qz
v0 RyaB/~. u0 varies from 0.2 to 2.0 in the same givenabove. TheexplicitformofV˜(k ,k ,q )isobtained
≡ 1 2 z
manner as in Fig. 1. With increasing u , the maximum
0 by assuming the screened Coulomb potential
value of v decreases and the peak position shifts to a
g
lower ε. A concave curve connecting the two end points e−κ|rij|
is the trajectory of the Fermi velocity vF ≡ vg(ε = εF) hri,rj|Vˆ|rj,rii=V(rij) = cǫ rij , (14)
obtained through the u variation. The value of v de- | |
0 F
creases monotonically as u0 increases, and it then con- where cǫ = e2/(4πǫ) and κ is the inverse of the screen-
verges to the origin in the limit of u0 . Figure 2(b) ing length. Equation (14) has an alternative expression,
→∞
shows the plot of vF vs. u0. Note that vF/v0 0.1 given as
corresponds to v 105 m/s if we employ the ma∼terial
F
∼
constants of GaAs.36 1 4πc
V(r )= dq ǫ eiq·rij. (15)
ij (2π)3 q 2+κ2
Z | |
From Eqs. (13) and (15), we obtain (see also Eq. (B6))
III. BOSONIZATION OF 1D PERIODIC
SYSTEMS λ dq′ dq′
V˜(k ,k ,q )= z ⊥
1 2 z L 2π (2π)2
A. TLL exponent in 1D periodic system Z Z
4πc F(q′ )2
ǫ| ⊥ | G(q ,q′,k )G (q ,q′,k ),(16)
×(q′)2+ q′ 2+κ2 z z 1 U z z 2
Here we discuss the effect of field modulation on the z | ⊥|
TLLstates. AnimportantindicatorofTLLstaterealiza- where λ = a+b, q = (q ,q ) is the transverse compo-
⊥ x y
tionisapower-lawsingularityoftheone-particledensity
nent of the three-dimensional wavevectorq, and
of states ν(ε) near ε represented by3
F
F(q′ )= dr χ(r )2e−iq⊥′ ·r⊥, (17)
ν(ε) ε εF α, α= 1 + 1 1. (8) ⊥ Z ⊥| ⊥ |
∝| − | 2(cid:18)K K(cid:19)− G(qz,qz′,k1)= dz1φ∗k1+qz(z1)φk1(z1)e−i(qz−qz′)z1,(18)
Here, α is the TLL exponent, and the parameter is Z
defined as (see Appendix B) K GU(qz,qz′,k2)= dz2φ∗k2−qz(z2)φk2(z2)ei(qz−qz′)z2.(19)
ZU
= 2π~vF +(g4−g2+g1) 1/2, (9) The subscript U in GU and U in Eq. (19) indicates in-
K (cid:20)2π~vF +(g4+g2−g1)(cid:21) tegration within the unit celRl domain z2 ∈ [0,λ]. It is
4
fixed the pamameters a = 5.0, b = 1.0, and κa = 10−4
(a) 0.6 (b) so as to satisfy the condition κ k in accoBrd to the
F
≪
2.0 bosonization procedure.3
0.5
The most important observation in Fig. 3(a) is the
monotonic increase in α with increasing u . In partic-
0.4 1.5 0
g1 ular, the data of α for d = 1.0 exhibits a pronounced
0.3 g , 4 1.0 ebnyhaansciemmpelentm. eSchuachnistmhebua0s-eddriovnenthsehibftehinavαioriss ocfaucosend-
0.2 stituents v , g (= g ) and g contained in formula (9)
F 4 2 1
0.5 of . First, it follows from Fig. 3(b) that g4 and g1
0.1 areKalmost independent of the change in u and satisfy
0
the inequality g > g . Hence, a decrease in v causes
0 0.0 4 1 F
0 2 4 6 8 10 0 5 10 a decrease in defined by Eq. (9). Second, the above
K
u0 [Ry] u0 [Ry] inequality implies < 1, and thus the decrease in
K K
results in an increase in α as understood from Eq. (8).
FIG. 3: (a) The TLL exponent α as a function of the field
Asaconsequence, αcanbe raisedbyenhancingthe field
modulationamplitudeu0. Thecylinderradiusdofthequasi-
modulationamplitudeu . Foru 10.0,αconvergesto
1D system is taken to be d=0.1 (dashed), d=0.3 (dashed- alimiting value determin0edby α0=≫( + −1)/2 1and
dotted) and d = 1.0 (solid) in units of aB. The pamameters K K −
a = 5.0, b = 1.0, κ = 10−4 are fixed for all curves. (b) u0- K=[g1/(2g4−g1)]1/2, since vF vanishes asymptotically.
dependences of g4(= g2) (thick curves) and g1 (thin) under The upper limit of α is dependent of the geometric pa-
thenumerical conditions same as in (a). rametersd, a,b;therefore,itiscrucialtosetappropriate
valuesofthethreeparametersinordertoobtainoptimal
controllability of the exponent α.
natural to assume that χ(r⊥) has a Gaussian form such It should be remarked that we have discussed spinless
as35 fermionsystems;theyarerealizedinspin-polarizedultra-
cold fermionic gases39 and in spin-1/2 antiferromagnetic
χ(r )= 2 1/2exp |r⊥|2 , (20) ladders,40–42 whereasmanyquasi-1Dsystemsinvolvethe
⊥ πd2 − d2 effectsofthespindegreesoffreedom. Whenwetakeinto
(cid:18) (cid:19) (cid:18) (cid:19)
accountthespindegreesoffreedom,Eqs.(8)and(9)can
which gives F(q′ )=exp q′ 2d2/2 . As a result, the be rewritten because of the presence of SU(2) symmetry
⊥ −| ⊥|
integral term with respect to q′ in Eq. (16) is rewritten as3
(cid:0) ⊥ (cid:1)
as
1 1 1
Z dq⊥′ (qz′)e2−+|q|⊥′q|⊥2′d|22/2+κ2 =Z0∞dtt+ qz′π2e+−tκ2 d2 α = 42(cid:18)πK~vρ++Kρg(cid:19)−+2g, g g +g 1/(223)
=−πe(qz′2+κ2)d2 Ei − qz′2+κ2 d2 ,(cid:0) (cid:1)(21) Kρ = "2π~vFF +(cid:0)g44kk+g44⊥⊥−+g22kk−+g22⊥⊥−g11kk(cid:1)# (2.4)
h (cid:16) (cid:17) i
where Ei(x) is the exponential-integral function defined (cid:0) (cid:1)
Here,g andg (i=1,2,4)expressthematrixelements
by Ei(x) = ∞t−1e−tdt. Finally, we obtain the ex- ik i⊥
− −x of interaction between electrons with parallel spin and
plicit form
those with antiparallel spin, respectively. In the usual
R
Coulombinteraction,thematrixelementsdonotdepend
c λ
V˜(k ,k ,q ) = ǫ dq′G(q ,q′,k )G (q ,q′,k ) on the spin degree of freedom; consequently, g = g
1 2 z −4πL z z z 1 U z z 2 ik i⊥
Z is expected. Therefore, we anticipate that the sizeable
e(qz′2+κ2)d2 Ei q′2+κ2 d2 , (22) shift in α demonstrated in Fig. 3 should be qualitatively
× − z correct even if we take into account the spin degree of
h (cid:16) (cid:17) i
freedom. Details including the actual calculation will be
by which we can compute g , g , g in Eqs. (10)-(12) for
4 2 1
shown elsewhere.
a given k .
F
IV. RESULTS AND DISCUSSIONS V. SUMMARY
We demonstrate belowthat the value ofα canbe tun- WehavetheoreticallyshownthattheTLLexponentα
ableartificiallybyimposinganappropriatemagnitudeof canbeartificiallycontrolledbyapplyingaperodicexter-
the periodic external field. Figure 3(a) shows the field- nal electric field to the system. The bosonization proce-
induced change in the TLL exponent α for different val- dure has been used to obtain the analytic expression of
ues of the cylinder radius: d = 0.1 (dashed), d = 0.3 theexponentαasafunctionoftheamplitudeu andpe-
0
(dashed-dotted) and d = 1.0 (solid) in units of a . We riod λ of the external field modulation u(z). The result
B
5
indicates that α increases significantly with the poten- for Q = 0, the determinant of matrix M given by Eq.
6
tial amplitude u , whose magnitude is within the realm (A2) is
0
of the existing experiments. The significant variation in
α is attributed to the field-induced shift in the Fermi detM
velocity v of the single-particle state. The present re-
F 1 1 1 1 1 1
sults indicate that the quantum transport properties of − Qω −
= Qqω iK iK Q + iK iK Q
quasi-1D systems can be tuned by manipulating α. Ex- (cid:12) − − (cid:12) ρ (cid:12) − (cid:12)
(cid:12) σ 1/σ ω/ρ(cid:12) (cid:12) σ 1/σ ρω (cid:12)
perimentalconfirmationwillprovideanovelapproachto (cid:12) − (cid:12) (cid:12) − (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
manipulating 1D quantum systems. (cid:12)1 1 1 (cid:12) (cid:12) 1 1 1(cid:12)
iK (cid:12) − − (cid:12) (cid:12) − −(cid:12)
iK Q Q iKσ iK Q Q
− σ (cid:12) − (cid:12)− (cid:12)− − (cid:12)
(cid:12) σ ω/ρ ρω (cid:12) (cid:12) 1/σ ω/ρ ρω (cid:12)
(cid:12) − − (cid:12) (cid:12) − − (cid:12)
Acknowledgments (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
= 4iK(cid:12)Q 1+ω2 2ωco(cid:12)sKacosh(cid:12) Qb (cid:12)
−
We are grateful to K. Yakubo, S. Iwabuchi, J. Onoe, +4i K(cid:0)2 Q2 ωsinKasinhQb, (cid:1) (A3)
T. Ito, and Y. Toda for their helpful comments prior −
to commencement of this study. HS is thankful for the and thus(cid:0)detM =0(cid:1)reads as
financial support from the Kazima Foundation and for
1 1 K2 Q2
the assistance provided by M. Arroyo at the UPC facil-
ω+ =cosKacoshQb − sinKasinhQb.
ity. This work is supported by a Grant-in-Aid for Sci- 2 ω − 2KQ
(cid:18) (cid:19)
entific ResearchfromMEXT,Japan,andNaraWomen’s (A4)
University Intramural Grant for Project Research. Nu- ThisresultisequivalenttothesetofEqs.(5)and(6). In
merical simulations were carried out in part using the addition, the derivative ∂f/∂ε is obtained by differenti-
facilities of the Supercomputer Center, ISSP, University atingtherightsideofEq.(6)withrespecttoε. Straight-
of Tokyo. forward calculation yields
∂f ζ η2 2ζ
= b 0 a 1 sin(ζ η )cosh(ζ η )
∂ε 4η η2 − ζ − a 0 b 1
Appendix A: Dispersion relation of 1D periodic 0(cid:18) 1 b (cid:19)
systems ζ η2 2ζ
+ b 1 b 1 cos(ζ η )sinh(ζ η )
4η η2 − ζ − a 0 b 1
1(cid:18) 0 a (cid:19)
Here, we derive the dispersion relation of Eq. (5) for 1 η2 η2
a 1Dsystemsubjected to the stepwiseperiodic potential − 4η η η12 + η02 +2 sin(ζaη0)sinh(ζbη1).(A5)
given by Eq. (3). It follows from Eqs. (2) and (3) that 0 1(cid:18) 0 1 (cid:19)
ψ(z) behaves as Specifically, when Q = 0 (i.e., ε = u ), the continuity
0
conditions of ψ and dψ/dz at z = a yield Ka = ℓπ and
AeiKz +Be−iKz, (0 z <a) k(a+b)=ℓπ with aninteger ℓ. Given u >0, therefore,
ψ(z)= ≤ (A1) 0
( CeQz +De−Qz, ( b z <0) the eigenstate exactly at ε=u0 exists only if there is an
− ≤ integer ℓ that satisfies ~2/(2m∗) (ℓπ/a)2 = u . Other-
0
·
wise, no eigenstate lies at ε=u .
and ψ(z+a+b)=ψ(z)eik(a+b), where ε=~2K2/(2m∗) 0
Once an eigenenergy ε and the corresponding k are
and u ε =~2Q2/(2m∗). The continuity conditions of
0− obtained, we introduce them into the equation Mv = 0
ψ anddψ/dz atboth z =0andz =a leadto the matrix
to evaluate the coefficients A,B,C,D of the eigenfunc-
equation Mv =0, in which v =[A,B,C,D]T and
tion ψ(z) that belongs to ε. The explicit forms of the
coefficients are
1 1 1 1
− −
iK iK Q Q B Q+iK eik(a+b)e−Qb eiKa
M = − − , (A2) = − , (A6)
σ 1/σ ω/ρ ρω A −Q iK · eik(a+b)e−Qb e−iKa
 − −  − −
iKσ iK/σ Qω/ρ Qρω  C Q+iK isinKa
 − −  = , (A7)
  A Q · eik(a+b)e−Qb e−iKa
with σ = eiKa, ρ = eQb, and ω = eik(a+b). Solving the −
secular equation detM = 0, we can obtain the desired andD/A=1+(B/A) (C/A);Aisuniquelydetermined
−
result of the dispersion relation (5). by the normalization condition of ψ(z).
In general, the determinant detM of a square ma-
trix M consisting of n2 elements a is represented
ij
by detM = n ( 1)i+ja C wit{h an}arbitrary fixed Appendix B: Proof of Eq. (9)
i=1 − ij ij
value of j, where C is the determinant of the (n 1)
ij
(n 1)matrixPthatisobtainedfromM byerasing−itsit×h TheTLLparameter introducedinEq.(9)isdeduced
row−and jth column (see Ref.43 for the proof). Hence, from the momentum rKepresentation of the two-electron
6
interactionHamiltonian ˆ foraquasi-1Dperiodicsys- momentum conservation that requires relabeling of the
int
tem. In the real-space reHpresentation, ˆ is given by set (k ,k ,k ,k ) by (k + q,k q,k ,k ). Introduc-
int 1 2 3 4 1 2 2 1
H −
ing the assumption into Eq. (B5) and performing sum-
Hˆint = 12 dr1 dr2V (|r1−r2|) mV˜ation with resipsegcitvetnobly, we find that V˜(k1,k2; q) ≡
Z Z k1+q,k2−q;k2,k1
Ψˆ†(r )Ψˆ†(r )Ψˆ(r )Ψˆ(r ), (B1)
1 2 2 1
×
1
V˜(k ,k ; q)= r dz dz V (r r )
where 1 2 ⊥ 1 2 1 2
N D | − |
Z Z ZU
Ψˆ(ri)= cˆkψk(zi)χ r⊥i (B2) ×φ∗k1+q(z1)φk1(z1)e−iqz1φ∗k2−q(z2)φk2(z2)eiqz2(.B6)
k
X (cid:0) (cid:1)
InEq.(B6),thesubscriptU originallyattachedto dz
with a fermionic annihilation operator cˆ . Substituting U 1
k vanishes so that z ranges from 0 to L.
1
Eq. (B2) into Eq. (B1), we have R
We arereadyto evaluatethe TLL parameter . From
K
Eq. (B6) and Eq. (B3), it follows that
1
ˆ = V˜ cˆ† cˆ† cˆ cˆ , (B3)
Hint 2 k1,k2;k3,k4 k1 k2 k3 k4
1
k1,kX2,k3,k4 ˆ = V˜(k ,k ; q)cˆ† cˆ† cˆ cˆ . (B7)
Hint 2 1 2 k1+q k2−q k2 k1
and k1X,k2,q
V˜ = r dz dz V (r r ) Substituting ki = pikF +κ into Eq. (B7) under the as-
k1,k2;k3,k4 D ⊥ 1 2 | 1− 2| sumption that of κ k and p= 1, we get
Z Z Z | |≪ F ±
ψ∗ (z )ψ∗ (z )ψ (z )ψ (z ), (B4)
× k1 1 k2 2 k3 2 k4 1 1
in which the integration operator r Hˆint = 2 V˜(pkF,pkF; 0)cˆ†p,κ+qcˆ†p,κ′−qcˆp,κ′cˆp,κ
dr1 dr2 χ r1 2 χ r2 2 acts on VD(r⊥ r ≡) Xp κX,κ′,qh
sRince⊥Rri d⊥ep(cid:12)en(cid:0)ds⊥(cid:1)o(cid:12)n(cid:12)r⊥i(cid:0). ⊥I(cid:1)n(cid:12) the meantimeR, w|e1d−eri2v|e + V˜(pkF,−pkF; −2pkF)cˆ†p,κ+qcˆ†−p,κ′−qcˆp,κ′cˆ−p,κ
aalnloawltsertnoaotib(cid:12)vteaifnortmh(cid:12)eo(cid:12)fmV˜okm1,ekn2;(cid:12)tku3,mk4r(eip.er.e,sEenqt.a(tBio6n))ofwhˆich + V˜(pkF,−pkF; 0)cˆ†p,κ+qcˆ†−p,κ′−qcˆ−p,κ′cˆp,κ , (B8)
int i
H
(i.e., Eq. (B11)) relevant to the formula of . where cˆ = cˆ . Since the time reversal symmetry
BearinginmindtheperiodicityofasystemKwithperiod ofthesyps,tκemaspskuFr+eκsthatV˜ =V˜ ,
λ = a+b, we replace the variables z ,z (0 z L, k1,k2;k3,k4 −k1,−k2;−k3,−k4
i = 1,2) in Eq. (B4) with z1 + l′λ a1nd2z2 +≤(l′i−≤l)λ iwmephliaesve V˜(k1,k2; q) = V˜(−k1 − q,−k2 + q; q) that
(0 < z < λ, i = 1,2), respectively; the integers l and l′
i
are independent of each other and vary from0 to N 1.
Applying the periodic boundary condition of ψk(z−) = V˜(kF,kF; 0) = V˜(−kF,−kF; 0), (B9)
ψk(z+L) for an arbitrary k, we obtain V˜(kF,−kF; 0) = V˜(−kF,kF; 0). (B10)
V˜ = 1 r dz dz As a consequence, we obtain the result
k1,k2;k3,k4 N2 D ⊥ 1 2
V (rX1l Xrl2′+Zlλez )φ¯Z(Uz1,z2Z)U Hˆint = 2gL4 cˆ†p,κ+qcˆ†p,κ′−qcˆp,κ′cˆp,κ
× | − | p κ,κ′,q
e−i(k1−k4)z1e−i(k2−k3)z2 X X
× e−i(k1+k2−k3−k4)l′λei(k2−k3)lλ, (B5) + g22−Lg1 cˆ†p,κ+qcˆ†−p,κ′−qcˆ−p,κ′cˆp,κ,(B11)
× p κ,κ′,q
X X
where φ¯(z ,z ) φ∗ (z )φ∗ (z )φ (z )φ (z ) and
1 2 ≡ k1 1 k2 2 k3 2 k4 1 U
symbolizes the integrationwithin the unit cell z =[0,λ]; where g4, g2, and g1 are defined by Eqs.(10)-(12).
e is a unit vector in the axial direction. SummatiRon Through the diagonalization of ˆ in Eq. (B11), we fi-
z int
H
over l′ in Eq. (B5) yields a term Nδ (we con- nally obtain the TLL parameter given in Eq. (9); this
k1+k2,k3+k4 K
sider only normal scattering contributions, omitting the parameterdepends onthe two coefficientsg andg g
4 2 1
−
Umklapp scattering processes). We further assume a prior to the summations in Eq. (B11).
∗
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