0 0.2 0.4 0.6 0.8 1 1.2
0 0.5 1 1.5 2 2.5
kBT /Jw
Jw(1/T1)−+/¯h
MPS Fit:a·T−b
Analytics+fit:c·T−0.38(2)+d Analytics+fit:g·T−0.5+l
Figure 4.7:(1/T1)−+as defined in Eq. (4.7) multiplied byJw/~to obtain a dimension-less quantity, as a function ofkBT /Jw, for the isotropically dimerized spin-1/2 chain ath≈3.02Jw. Dots with errobars are MPS results, solid lines are fits or combination of fit and analytics. The fit parameters area ≈0.62(1),b ≈0.63(2),c ≈ 1.42(2), d≈ −0.84(1),g≈0.87(1), andl≈ −0.25(1). Picture taken from Ref. [29].
to the TLL predictions.
The black line represents the fit using the separately determined exponent2K1 −1.
A constant offset has been added since the behavior is not entirely dominated by the divergence. Deviations between the comparison of the analytical prediction and the numerical calculation are seen. We attribute these deviations to the proximity of the quantum critical point. In this regime, the TLL behavior is valid only for very low temperatureskBT . h−hc1. From our numerical results only the lowest tem-perature point lies within this region. To verify the influence of the quantum critical point, a fit using the critical power law shifted by a constant offset is performed. This fit leads to good results for the intermediate temperature points (see the green curve).
A fit in which also the exponent is a fit parameter leads to an even larger exponent of b≈0.63(red curve).
4.4 Summary & conclusions of the chapter
Exploiting MPS techinques, we computed in this chapter the spin-lattice relaxation rate1/T1 for a wide range of temperatures, for different Hamiltonians and for dif-ferent quantum phases. In particular, we have considered the XX, Heisenberg, and XXZ Hamiltonians, plus the isotropically dimerized case. For the non-dimerized
4. NMR RELAXATION RATE
cases we have performed a detailed study of the gapless phase, the gapped phase and also of the quantum critical point. Numerical results were in very good agreement with analytical results available in the low-temperature limit. We have shown the de-viation from the low-T law at finite temperature and we have swept through quantum critical points, situations in which theoretical results are more difficult to obtain. Our calculations prove that the MPS method can be successfully used to obtain the NMR relaxation rate in regimes in which the field theoretical asymptotic results would not be applicable. The overlap between the regimes in which the numerical methods are applicable and the regimes covered by the field theoretical asymptotic methods allows essentially a full description of the NMR behavior for the accessible regime of temperatures. Having a method which can quantitatively compute the NMR re-laxation time for a given microscopic Hamiltonian, rather than simple asymptotic expressions, should allow to test that the Hamiltonian which is supposed to describe the real system does not miss an important term, and to fix the various coefficients by comparing the computed temperature dependence with the experimentally mea-sured one. This is similar in spirit to what was achieved by the comparison of the computed neutron spectra with the measured ones for DIMPY [78].
One interesting perspective is the investigation of the behavior of the relaxation mechanism of the spin excitations close to the quantum critical point. Indeed the nature of the relaxation mechanism is potentially different depending on whether one considers theSzzterm or theS±∓ones. For 3D systems a self energy analysis of the transverse part of1/T1was suggesting [74] a behavior1/T1∝e−3∆g/kBT due to the necessity of making three magnon excitations to be able to scatter a magnon and get a finite lifetime. Results for theSzzpart leads, as shown in the present paper, to1/T1 ∝e−∆g/kBT. Our numerical results which are able to correctly determine the exponential decay in the controlled cases of the longitudinal excitations are thus potentially able to address this issue and potentially make contact on the experiments on that point [79]. Such a study clearly going beyond the scope of the present work, is thus left for future projects.
We have seen that the present method works efficiently if the systems are one dimensional. An important challenge on the theoretical level is to extend the present analysis to the case of higher dimensional systems. A first natural extension would be the application to quasi-one dimensional systems like for example spin-1/2 two-leg ladders. In this direction some existing experiments measuring NMR in "Hpip"
compounds (see chapter 2) could be of potential interest [66, 80]. More in general, although other methods such as quantum Monte-Carlo exist, the dynamical corre-lations in real time are still a challenge for which the MPS methods could bring useful contributions. Indeed the (numerically) rather complete knowledge of the one-dimensional correlation functions allow to incorporate them into approximation schemes such as RPA to capture large part of the higher dimensional physics. An-other route is to solve clusters of one dimensional structures, which allows to at least incorporate part of the transverse fluctuations.
CHAPTER 5
Dynamical Correlations at Finite Temperature of a Dimerized Spin-1/2 Chain
As discussed previously, DMRG/MPS methods has allowed to obtain the frequency and momentum resolved spin-spin correlation functions for several one-dimensional and quasi-one-dimensional systems, see for example [10]. For integrable models Bethe ansatz gives nowadays the possibility to access momentum and frequency re-solved correlations [18–20]. However, up to very recently, these predictions were limited to zero temperature and only finite temperature calculations of the thermo-dynamics were accessible. In addition, the analytical treatment of temperature ef-fects beyond the bosonization limit turns out to be particularly difficult. For gapped systems some semi-classical approximations [81] or form factor expansions [82] can be made, but no complete treatment existed. Recently, full dynamical calculations at finite temperatures in the framework of DMRG/MPS methods [23, 24] have been performed in simple cases, opening the path to the study of temperature effects on spectral functions for more complicated models.
In this chapter we use our MPS algorithm for the dynamics at finite temperature to make an analysis of the properties of the spin-1/2 dimerized chain under magnetic field, focusing also on the thermal effects. The model we refer to and the corre-sponding real material are presented in chapter 2 (see in particular section 2.5 and Fig. 2.5). This dimerized system shows in absence of magnetic field a ground state that is a spin liquid with a finite spin gap. Application of the magnetic field closes the gap and transforms the problem into a TLL [7] in a similar way than for ladder systems [73].
In the following we first give a brief introduction about what we exactly compute
5. DYNAMICAL CORRELATIONS AT FINITE TEMPERATURE OF A DIMERIZED SPIN-1/2 CHAIN
and its relation with inelastic neutron scattering measurements. We then move to the results for dynamical correlations for different values of temperature and field and we give an interpretation of the different structures seen. Finally we show how temperature affects the characteristics of the spectra. The content of this chapter is based on Ref. [30]. Figures and parts of the text are taken from there.
5.1 What we compute: definitions
As already stated in the title of this chapter, we are interested in dynamical spin-spin correlation functions at finite temperature for the dimerized spin-1/2 chain. Numeri-cally we compute objects of the form
Sjλ(tn)Slµ(0)
T =Sλµ(d, tn), (5.1) where(λ, µ)can be(±,∓)or(z, z),tn =nδtis the discretized time at which the correlation is evaluated (nis an integer),j, lare the site indexes andd=j−l. The chain has a size ofL = 130. Also here, the expectation valueh. . .iT denotes that the correlation is evaluated at finite temperature T. Since the unit cell of the original lattice corresponds tod= 2(see Fig. 2.2 for clarity), we choose to focus on objects of the form:
D
Sλj0,r1(tn)Slµ0,r2(0)E
T = Srλµ1r2(d0, tn), (5.2) wherej0 and l0 are now the strong bond indexes. In addition we have attached indexesr1andr2indicating the positions of the spins on each strong bond: 1 for left, 2 for right. With this notation we have for each couple(λ, µ)4 possible species of correlations which we will name 11, 12, 21 and 22 following the choice forr1, r2 indexes. d0 =j0−l0 ranges from−L−1˜2 toL−1˜2 , withL˜ =L/2. In the(z, z)case we substractm2(mis the magnetization per site). In this work we consider only11 correlations (L= 130 ⇒65points), since they contain already the most interesting physics of the model and of the phases considered. 21correlations can be obtained in the same run.
Our data then undergo a double Fourier transform to move to momentum and frequency space. In order to reduce artificial oscillations and other numerical arti-facts, due to finite system length and finitetmaxreached by simulations, we apply a gaussian filter to the correlations before Fourier transforming, see Eqs. (3.33)-(3.34).
For the double Fourier transform we use the same conventions adopted in Ref. [10]
and detailed already in Eq. (3.31) which we recall here for clarity:
S11λµ(q, ω)≈δt
Nt
X
n=−Nt+1
L−1˜ 2
X
d0=−L−1˜2
ei(ωmtn−qkd0)S11λµ(d0, tn).