3.2 Tomonaga-Luttinger Liquid (TLL)
3.2.2 Evaluation of TLL parameters
Equations (3.38) and (3.39) are valid only for the XXZ model. Having a procedure able to compute the values of the TLL parametersuandK for various other sys-tems is of crucial importance, since from these two values, as we saw above, one can obtain a lot of information about the behavior of the correlations. In this work we propose a new scheme for the numerical evaluation of these two important pa-rameters. This procedure has been adopted in Ref. [29] and it can be summarized as follows: the ratioK/ucan be evaluated from the static TLL susceptibility, while the productu·Kis related to the variation of the ground state energy with respect to a flux threaded onto the system. Then, the two values ofuandKtrivially follow by recombination of the two previous results.
The ratio K/uis determined from the static TLL susceptibility of the system according to the relations [7]
κ−1=Ld2E0
whereL is the size of the system, E0 is the ground state energy, M is the total magnetization. The second derivative has to be discretized sinceM in a spin-1/2 system can only vary by integer steps (∆M = 1), thus:
K
u(M) = π(∆M)2
L[E0(M+ ∆M) +E0(M−∆M)−2E0(M)]
= π
L[E0(M+ 1) +E0(M −1)−2E0(M)]. (3.45)
3. METHODS
E0can be evaluated at fixed values of magnetization via standard finite size DMRG.
The magnetization, defined at the beginning of the simulation by the initial distri-bution of spins, is thus set as a conserved quantum number. At this point it is also possible to determine the values of the magnetic field corresponding to each possible value of the magnetization of the system. These values ofhwill become useful for the next step of the procedure, and we know that they can be obtained from the curve E0(M)according to the relation:
h(M0) = dE
The productu·Kcan be determined by studying the variation of the ground state energy of the system in response to a variation of a flux through the system [7,54–56].
To be more precise, for a fixed value M of the magnetization (and therefore for the corresponding value ofhpreviously obtained), we have that
uK(M) =πLd2E0(Φ, M)
The flux is represented by twisted periodic boundary conditions,Ψ(L) = Ψ(0)·eiΦ. This condition can be transferred into the Hamiltonian via the following transforma-tion:
S+jS−j+1−→S+jSj+1− ·eiΦL,
Sj−S+j+1−→S−jS+j+1·e−iΦL, (3.48) which distributes homogeneously the total fluxΦalong the chain. For each fixed value ofM we evaluateE0(Φ)|M via an infinite size MPS algorithm for symmetric values ofΦaround zero at finite magnetic field. The resulting ground state energy E0(Φ), close toΦ = 0, can be approximated by a parabola and we fit the points with a second degree polynomial of the formP(Φ) =aMΦ2+bMΦ +cM, where aM, bM andcM are the fit parameters. According to Eq. (3.47), the fit parameteraM
is related to the product
uK(M) = 2πLaM. (3.49)
In Fig.3.3 a curveE0(Φ) for the dimerized model ath = 4.209Jw is reported as an example with the corresponding fit using a 2nd degree polynomial (bM ≈ 0 and therefore not reported). This procedure is adopted in chapters 4 and 5. In the following we show as an example our results foru(m)andK(m) (beingm the magnetization per site) for the dimerized chain compound presented in chapter 2.
The curve for the parameterutends correctly to 0 form → 0andm →0.5. The curve forK, which should tend to 1 for those values ofm, shows on the contrary a downturn in both limits. We attribute these errors to the extremely difficult derivation
3.2 Tomonaga-Luttinger Liquid (TLL)
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
−1.5813125
−1.581312
−1.5813115
−1.581311
−1.5813105
−1.58131
−1.5813095
FluxΦ Energyperbond[unitsofJw]
h= 4.209·Jw
Infinite size MPS
Fit: E(Φ) = 5.781·10−6·Φ2- 1.5813127
Figure 3.3: Numerical result using infinite size MPS for the ground state energy as a function of the fluxΦ, for the dimerized model ath= 4.209Jw(blue stars). The points are fitted using a 2nd degree polynomial (red curve).
0 0.1 0.2 0.3 0.4 0.5 0
0.5 1 1.5 2
Magnetization per site m
TLL parameter ’u’ [K]
0 0.1 0.2 0.3 0.4 0.5 0.7
0.75 0.8 0.85 0.9 0.95 1
Magnetization per site m
TLL parameter ’K’ (unitless)
Original data Interpolated data
Figure 3.4: Numerical results for the TLL parametersu(left) andK(right), as a func-tion of the magnetizafunc-tion per sitem, for the dimerized chain copper nitrate compound.
Results for the parameterKhave been interpolated close tom= 0andm= 0.5using the constraintsK(m= 0) =K(m= 0.5) = 1to reduce the numerical error. In those regionsKis difficult to obtain by recombining the productuKand the ratioK/u, since uK→0whileK/u→ ∞.
3. METHODS
ofKin those regions. SinceK= q
uK·Ku, and since the two quantities under the square root sign tend respectively to zero and to infinity formgoing to0 or 0.5, any small numerical error can heavily influence the result. We decided therefore to discard the last points at the two ends of the curve forK, and to interpolate for the corresponding values ofm, adding the two additional constraints K(0) = 1and K(0.5) = 1.
CHAPTER 4
NMR Relaxation Rate
In this chapter we focus on our MPS results for the so calledspin-lattice relaxation rate1/T1. Its inverse, the spin-lattice relaxation timeT1, is one of the important time-scales of NMR measurements. After a brief introduction about this quantity and its physical meaning, we will discuss how to connect it with what we compute numerically. Then, we also presents some available analytical expressions which we will use to benchmark our data. Finally, we present and comment our results for the two models considered (XXZ and dimerized spin-1/2 chains).
The content of this chapter is based on Ref. [29]. Figures and parts of the text are taken from there. Recently, a similar analysis was carried out for the DTN compound by Capponi, Dupont and Laflorencie [57].
4.1 Spin-lattice relaxation rate 1/T
1In NMR experiments the nuclear spins of the sample, previously polarized by an applied magnetic field, are perturbed using an electromagnetic pulse. The term re-laxationdescribes how the signal re-emitted by the sample deteriorates with time.
This deterioration is usually studied by considering two processes, each of them characterized by its own specific time constant. The first process, calledspin-lattice relaxation, is responsible of the loss of signal’s intensity and it is characterized by the time constantT1. It describes how the component of the magnetization along the direction of the applied magnetic field (denoted here byz) reaches thermodynamic equilibrium with its surroundings (the lattice) after the perturbation [16, 17]. The
4. NMR RELAXATION RATE
evolution of the nuclear magnetization alongzis given by:
Mz(t) =Mz,eq
1−e−t/T1
. (4.1)
The second process, calledspin-spin relaxationand characterized by another time constantT2, is related to the broadening of the NMR signal and to the decay of the magnetization component perpendicular to the applied field.
In a solid the ratio1/T1(spin-lattice relaxation rate) can be related directly to the spin-spin correlations of the electronic system via the Redfield equation [17]:
1
whereγn is the nuclear gyromagnetic ratio of the measured nuclear spin,A⊥ and Akare the longitudinal and transverse components of the hyperfine tensor,Sαα(ω0) withα=x, y, zare the local spin-spin correlation functions at the nuclear Larmor frequencyω0, which can be obtained via a Fourier transform of the onsite correla-tions in real time:
In Eq. (4.4), the expectation valueh. . .iT denotes the thermal and quantum average defined as
h· · · iT =Tr[e−βH· · ·]
Tr[e−βH] . (4.6)
Note that in formula (4.2) it is implicitly assumed that the hyperfine coupling term is essentially q independent. This covers a large number of cases, for example the ones in which the relaxation is measured on the site carrying the electronic spin. There are also interesting cases for which the q dependence of the hyperfine term can filter some modes, for example the modes atq = πif the relaxation is measured mid-point between two neighboring sites. This leads to different formulas and interesting properties [58–60]. Note that techniques similar to the ones used here but computing the finite temperature, space and time dependent spin correlations allow to treat this problem as well. We leave this more complicated case for further studies, and focus here to the generic case for which the local spin-spin correlation is sufficient.
Although the principle of what is measured by the ratio1/T1is simple, the theo-retical determination is far from trivial. Very often various schemes of approximation