We start from the result for the time ordered, onsite,+−correlations in real time.
It can be obtained directly from Eq. (C.55) in Ref. [7] with the obvious substitution 2K→ 2K1 andK→4K1 (Kis the dimensionless TLL parameter): αis a small cutoff,βis the inverse temperature anduis the sound velocity (the other TLL parameter). Using the relation connecting the retarded correlations with the time-ordered ones, and Eq. (C.61) in Ref. [7], we get that
χR+−(x, t) =−2Θ(t)Im χT+−(x, t) whereΘ(t)is the step function. Now,
χR+−(x= 0, ω) =
A.2 Derivation of the result in Eq. (4.11)
To solve the integral we make use of Eq. (C.65) in Ref [7] to get (with some simple algebra) whereB(. . .) is the Euler beta function. Let’s now then recall the first line of Eq. (4.11) and introduce in it the previous result:
1 Since we are working in the low energy limit, ω is small and one can develop B(. . . , . . .)with respect to the first argument: This result makes the limit in Eq. (A.12) easily computable, and we get
1
which is exactly Eq. (4.11) except for an overall factor of2Axwhich has to be added if one takes into account the proper definition of the operatorsS+andS−.
A. TECHNICAL ASPECTS OF CHAPTER 4
APPENDIX B
Extrapolation Method for Spin-Lattice Relaxation Rate
As discussed in Section 4.2, the numerical results for correlations are only available up to a certain timetmax. Since in principle the time integral of these correlations should be performed up to∞, one needs to find a way to approximate the value of the extended integral and to associate an error bar to it. In order to do this we study the behavior of these integrals as a function of1/tmaxfor1/tmax→0. We compute the integrals for the 20 largest available values oftmax, then we perform a linear fit and we extrapolate the result for1/tmax = 0. If the value of the integral still shows a considerable trend, we associate to the extrapolated value a one-sided error bar corresponding to the difference between the extrapolated value and the value of the integral for the maximumtmaxavailable. An example is shown in Fig. B.1 for the dimerized chain in the TLL phase.
For the case where the integral oscillates around a certain value and no clear trend is visible, we choose to associate a symmetric error bar with semi-amplitude equal to the distance between the extrapolated value itself and the value of the integral for the maximumtmax available. An example is shown in Fig. B.2 for the dimerized chain in the gapped phase. In both cases, the extracted error bars should give a (most probably pessimistic) estimate of the uncertainty on the value of the integral.
B. EXTRAPOLATION METHOD FOR SPIN-LATTICE RELAXATION RATE
0 0.005 0.01 0.015 0.02 0.025 0.03 2.1
2.15 2.2 2.25 2.3 2.35 2.4
¯h/(tmaxJw) (Jw/¯h)·R+tmax −tmaxdtS−+(t)
0 10 20 30 40
0 0.2 0.4
tJw/¯h S−+(t)
Figure B.1:Integral over time from−tmaxto+tmaxof the onsiteS−+correlations as a function of~/(tmaxJw)ath≈3.02Jw &hc1, at the temperaturekBT ≈0.082Jwfor the dimerized model. The extrapolation is shown as a solid (red) line. The extrapolated point is reported with its error bar. The inset shows the correlations as a function of tJw/~. Picture taken from Ref. [29].
0 0.005 0.01 0.015 0.02
−0.04
−0.02 0 0.02 0.04
¯h/(tmaxJw) (Jw/¯h)·R+tmax −tmaxS−+(t)
0 20 40 60
−0.5 0 0.5
tJw/¯h S−+(t)
Figure B.2:Integral over time from−tmaxto+tmaxof the onsiteS−+correlations as a function of~/(tmaxJw)ath= 0, at the temperaturekBT ≈0.082Jwfor the dimerized model. The extrapolation is shown as solid (red) line. The extrapolated point is reported with its error bar. The inset shows the correlations as a function oftJw/~. Picture taken from Ref. [29].
APPENDIX C
Consistency Test Using the XX Model
To test the accuracy of the numerical calculations, we derived some results for the XX model under a magnetic field along thezdirection:
H= J
We focused on two specific cases,h= 0(gapless phase) andh= 5J(gapped phase), and onSzzcorrelations. In particular, we determine for different temperatures the ratio1/T1forSzzcorrelations, which we define here as:
1
We compare our numerical results obtained using our MPS code against exact an-alytically results [7]. In the limit of an infinite-size system, the exact result for the onsite correlations at a temperatureTandh= 0is given by
Sjz(t)Sjz(0)
C. CONSISTENCY TEST USING THE XX MODEL
whereJ0(. . .)is the 0th-order Bessel function of the first kind,iis the imaginary unit,λk =Jcos (k), wherekis the dimensionless momentum, and
fk(β) = 1
1 +eβλk (C.4)
is the Fermi function, whereβis the inverse temperature.
0 1 2 3 4 5 analytical results correspond to infinite system size andtmax = 20~/J. The numerical results are obtained forL= 100andtmax = 20~/J. The agreement is excellent at all temperatures. Picture taken from Ref. [29].
Forh6= 0one obtains
Hereλ0k =Jcos (k)−hand
fk(β) = 1
1 +eβλ0k. (C.6)
Simulations are performed for a chain ofL = 100spins. Onsite correlations are measured in the center of the chain (j = 50) to minimize boundary effects. Imagi-nary time evolutions are performed using the following parameter set: minimal trun-cationεβ = 10−20, retained states maximum 400 and temperature step kBδT = 0.01J. Real time evolutions are performed using: minimal truncationεt = 10−10, a retained states maximum of 800, maximal truncated weight10−6, and time step δt= 0.05~/J up totmax = 20~/J. Results of the comparison theory-numerics are reported in Fig. C.1. The agreement between the analytical and the numerical results is extremely good at all temperatures.
C. CONSISTENCY TEST USING THE XX MODEL
APPENDIX D
Dynamics of a Single Excitation in the Strong Dimerization Limit
In the strong dimerization limit, we can restrict ourselves to the single strong bond picture. The four-dimensional Hilbert space on a single strong bond is spanned by the four states|si,|t+i,
t0
and|t−i. If we put ourselves in a gapped regime, which meansh= 0orh > hc2, the ground state of the system within this picture presents a singlet or a triplet “+”, respectively, on each strong bond. The application of a spin operator (S+,S−orSz) on a site may induce a transition on the corresponding strong bond to an excited state. This excitation can then propagate along the chain since the strong bonds are not totally disconnected from each other (Jwfinite). In the following we will do simple calculations for the two cases (h= 0orh > hc2) which make the interpretation of the correlation plots easier in those regions.
D.1 h = 0
In absence of magnetic field the ground state of the system in the strong dimerization limit is made of singlets on each strong bond. The application of any of the three operators indicated above induces a transition from a singlet to one of the triplets.
Taking into account that we always act on the first site of a strong bond we get:
S1+|si −→ −√1
2|t+i, S1−|si −→ √1
2|t−i, S1z|si −→ 12
t0 .
(D.1)
D. DYNAMICS OF A SINGLE EXCITATION IN THE STRONG DIMERIZATION LIMIT
Let us now consider a couple of strong bonds connected by the couplingJw, and let’s try to see how one of these excitations can hop from one strong bond to the next one. We start by|ϕαi =|tαi |si, beingα= +,0or−, and we applyHw = JwSj,2Sj+1,1, where the two spin operators act respectively on the second spin of the first bond, and on the first spin of the second bond. What one gets is
The most interesting resulting terms are the three highlighted in bold, which express the hopping of the excitations from one strong bond to the next one. The other terms represent higher energy processes and therefore will be neglected. From these results it can be easily seen that the spectrum of the system in all the three cases can be approximated by the one of an isolated particle (the triplet) moving with a tight binding Hamiltonian with hopping matrix element−Jw/4.