Boltzmann approah to transport in weakly interating, nearintegrable, 1-d fermioni systemsChristian Bartsh, Johen GemmerUniversity of Osnabrük,Quantum E�ienyFreiburg, Nov. 22., 2011C. Bartsh, J.Gemmer Boltzmann for 1-d

ContentsThe model and some of its transport propertiesHow to set up a (linearized) Boltzmann equation?Transition rates and sattering operatorsDi�usion oe�ientsThe model without next nearest neighbor hopping: integrableSpetrum of deay rates of the urrent orrelation funtionSummary

C. Bartsh, J.Gemmer Boltzmann for 1-d

The model and some of its transport propertiesspinless fermions Hubbard-type tight-binding model plus next nearest neighborhoppingH = J NXn=1 h − 12 (a†nan+1 + b · a†nan+2 + h..) + ∆ · a†nana†n+1an+1iWhat is known (more or less for sure)?for b = 0 and any ∆ the model is �integrable�: Bethe ansatz applies,in�nitely many loal onserved quantities, no Wigner-Dyson levelstatistis, et.linear response roughly states that transport is ballisti if PnTr(ĴQ̂n) 6= 0⇒ integrable systems tend do be ballisti (≈ feature in�nite ondutivity,di�usion onstants, et.)Even for b = 0 the transport properties may depend on interation ∆, �lling fand temperature T .for ∆ > 1 the system is �gapped� and non-ballisti, i.e., di�usive for all(most) �llings and temperaturesfor ∆ < 1 the system is most likely ballisti for all ases exept thefollowing:for ∆ < 1, half �lling f = 0.5 and �nite temperature T 6= 0 is urrentlyunder debate C. Bartsh, J.Gemmer Boltzmann for 1-d

How to set up a (linearized) Boltzmann equation ?We intend to investigate transport properties in the regime ∆

How to set up a (linearized) Boltzmann equation ?...something that may be interpreted as the linearized sattering operator of aBoltzmann equation:̇di (t) = Xk 6=i Rik (t)dk(t) − Xk 6=i Rki (t)di (t)where the orresponding rates (T = ∞, f = 0.5) are given byRk,i(t) = 12N2~2 Xl (Re(W (i− k)) − Re(W (k− l)))2tsin[ 1~ (εi + εl − εk − εi−k+l)t]−12 (Re(W (l− i)) − Re(W (k− l)))2tsin[ 1~ (εk + εi − εi+k−l − εl)t]In the limit of large times and large system sizes the rates should beometime-independent (for non-pathologial ases, i.e., no multiple zeroes in thesin-argument) C. Bartsh, J.Gemmer Boltzmann for 1-d

Rates and sattering operators 0

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5,50

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tabove:time evolution of one rate forb = 1, ∆ = 0.01,T = ∞,N = 600right:sattering operator Rb = 1, ∆ = 0.01,T = ∞,N = 200 -2e-05 0 2e-05 4e-05 6e-05 8e-05 0.0001

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C. Bartsh, J.Gemmer Boltzmann for 1-d

Di�usion oe�ientsFor regular linear Boltzmann equations (at T = ∞) the di�usion oe�ientreads D = − 1N vlR−1lk vk veloities : vk = ∂ǫ∂k�instantanous di�usion oe�ients� 4

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D(t

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b=1.0b=0.8b=0.6b=0.5b=0.4b=0.3b=0.2

�long time di�usion oe�ients� 1

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D(b

) ∆2

/Jb

4.0/b1/2

It appears that with b → 0:the timesale on whih the sattering operator beomes onstant divergesthe di�usion oe�ient diverges as D ∝ b−1/2C. Bartsh, J.Gemmer Boltzmann for 1-d

Anisotropi hain without next nearest neighbor hoppingTaking a loser look at b → 0 we �nd thatthe argument in the sin funtion in the rate-expression indeed appraohesmultiple zeroesespeially for b = 0,N → ∞, t → ∞, t → ∞ the sattering operator anbe found analytiallyR(j , k) ∝ (os(k − j) + os(k + j))2|sin(k) − sin(i)|-0.0001

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200��nite time sketh�At t → ∞the sattering operator divergesat some transitions:j + k = π/2, j + k = −π/2the sattering operator hasat least one zero eigenvalue sine

R R(j , k)(δ(k − π/2) − δ(k + π/2))dk = 0C. Bartsh, J.Gemmer Boltzmann for 1-d

Spetrum of deay rates of the urrent-orrelation funtionThe b = 0 sattering operator has zero but also in�nitely large (negative)eigenvalues. Whih dominate the urrent-orrelation funtion? Within thisframework all urrents relax multi-exponentially:ĵ = Xk vka†kak 〈̂j(t )̂j(0)〉 := Xe O(e)2exp{−et}�spetrum of deay rates� 0

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a substantial fration of theurrent relaxes arbitrarily slowthis graph appears to beidependent of the �oarsegraining�one may alulate a timedependent di�usion oe�ientfromD(t) = Z t0 dτ 〈̂j(τ )̂j(0)〉Proeeding this way yields for large times D(t) ∝ ln(t)C. Bartsh, J.Gemmer Boltzmann for 1-d

SummaryWe investigated the di�usion oe�ient of a 1-d (spinless) fermioni systemwith a nearest neighbor hopping of J = 1, a next-nearest neighbor hopping ofb, and a nearest neighbor interation of ∆.We �nd in the limit of small ∆ for the inverse di�usion oe�ient(∝ resistivity): D−1 ≈ 0.25 √b ∆2+?∆4 + · · ·Within this approah transport oe�ients sale at least like ∆−4 for nonext-nearest neighbor hopping.All of this has been done for in�nte temperature, T = ∞, however, the overallbehavior is expeted to remain unaltered at �nite temperaturesC. Bartsh, J.Gemmer Boltzmann for 1-d

Thanks !If you are interested in Refs. just ask me.We thank you, the audiene, for your interest and patiene!

C. Bartsh, J.Gemmer Boltzmann for 1-d