cv(β, h) = β2 h−Jk2
cosh2 β h−Jk + β2 4π
X
l=s,0
Z π 0
ε2l(k) cosh2 βεl(k)
2
!dk , (6.19)
whereas
εs(k) = Jkcos(k)−h+J⊥+Jk,
ε0(k) = Jkcos(k)−h+Jk. (6.20) Also in this case we notice that the critical value ofhfor singlets is different from the second critical fieldhladc2 of both ladder models. We need therefore to adopt a proper mapping following again the same strategy adopted in Eq. (6.3).
6.2 Analytical results against experiments and DMRG
Exploiting the results we have just obtained we can compute the Grüneisen parameter Γmagas a function of magnetic field (at fixed temperature) and temperature (at fixed magnetic field). Let’s start from the first case, and let’s focus mainly on the chlorine compound.
In Fig. 6.1 we compute the Grüneisen parameter as a function of the magnetic field, for 4 different temperatures, analytically. Dashed lines are obtained via the free fermion effective theory, as explained in subsection 6.1.1. Solid lines are obtained exploiting the refined approximation presented in subsection 6.1.2. The difference between the two approaches becomes more and more important as the temperature increases, and this is expected since the solid lines take into account higher excita-tions thermally activated. In all cases the curves present correctly a change of sign around the critical field, and a divergence seems to set in around that point as the temperature is lowered. By comparing solid lines of Fig. 6.1 and the experimental points of Fig. 3 in Ref. [28], a good agreement is observed. With respect to the bromine compound, the chlorine one is characterized by a smaller ratioJk/J⊥: this reduces the gap to the higher energy triplets making the role of temperature more important.
Let’s now move to the discussion of the temperature dependence of the Grüneisen ratio for fields very close to the critical ones. In Fig. 6.2 experimental results for
|Γmag| (dots with corresponding errorbars) for different temperatures and for both compounds are reported. In order to collect the results in one figure we renormal-ize byJ⊥. The fact that the two compounds possess different energy scales allow us to cover a wide range of energies (temperatures). To make the picture clearer and to emphasize occurring power-law behaviors we chose a log-log scale. Red dots correspond to the bromine compound ath ∼ hc1, green dots to the chlorine compound ath ∼ hc1, blue dots to the chlorine compound at h ∼ hc2. hc2 for the bromine compound is too high and could not be reached by experiments. Solid
6. GRÜNEISEN PARAMETER AND QUANTUM PHASE TRANSITIONS IN
T=320 mK, free fermions (3 species) T=420 mK, free fermions (3 species) T=520 mK, free fermions (3 species) T=620 mK, free fermions (3 species) T=320 mK, free fermions (1 species) T=420 mK, free fermions (1 species) T=520 mK, free fermions (1 species) T=620 mK, free fermions (1 species)
Figure 6.1:Analytical results forΓmag(h)for the HpipCuCl4compound. To obtain the dashed lines, the XXZ Hamiltonian coming from the spin-chain mapping of the ladder is approximated in the limit of extreme diluteness of excitations by an XX one (effective theory). Solid lines are obtained from the strong coupling expansion, taking into account the presence of higher energy excitations which can only be created by temperature.
green and light blue lines are DMRG results for the ladder model at the experimen-tal field obtained by P. Bouillot. The red solid line corresponds to analytical results within the XX approximation for the bromine compound, close tohc1. The dotted black line represents the behavior in temperature at the critical field of the Grüneisen ratio, with a prefactor obtained by fitting the experimental data with a power law aT−1. The correct exponent (-1) is the one computed already in Eq. (6.14). We de-termineda= 0.46. The very good collapse of the results on the top left of Fig. 6.2, obtained for the two compounds and for different theoretical models at low temper-atures, proves the validity of the universal behavior (dotted black line). At higher temperatures (right part of Fig. 6.2) the microscopic structure of the material be-comes important and deviations from the scaling occur. We observe clearly different behavior for the two QCPs for the chlorine compound. This is due to the energies of the higher triplet excitations which shift with the magnetic field. Here the agreement between experimental data with errorbars and the DMRG calculations for the lad-der system (solid lines) is extremely good. The trend we can observe for the lowest available temperatures seems to indicate once more a clear tendency to a collapse onto the universal critical behavior.
In the case of the chlorine compound, deviations from the critical behavior at experimental temperatures are quite remarkable especially for h ∼ hc1. DMRG simulations for the full ladder model can capture these effects. In the following we will adopt our approximation of subsection 6.1.2 and try to compute the same curves.
In Fig. 6.3 analytical results within our approximation, for the chlorine compound
6.3 Summary & conclusions of the chapter
log 𝑘
𝐵𝑇/J
⊥log Γ
𝑚𝑎𝑔∙ J
⊥/𝑔 𝜇
𝐵Figure 6.2: |Γ(T)|renormalized byJ⊥ in order to deal with unitless quantities, in log-log scale. Results for HpipCuBr4are available only ath∼hc1. Red dots are exper-imental data and the red line is the analytics within the XX approximation. Results for HpipCuCl4are reported close to both critical fields (green and blue dots are experimen-tal data, solid lines are DMRG results for the full ladder model). The black dotted line is the critical behavior at criticalityΓ∝T−1with prefactor determined with a fit of the experimental points. Picture taken from Ref. [28].
at both critical fields, are compared to the DMRG ones for the full ladder model.
DMRG results already showed a very good agreement with experimental data in Fig. 6.2. The new approximation is able to capture the deviations from the critical behavior at the experimental temperatures. The presence of higher energy excitations (triplets "0" and "-") seems to be the main responsible for the bending of the curve ath∼ hc1. The results coming from this approximation remain pretty good if we go back to the bromine compound and compare against DMRG results, as it can be seen in Fig. 6.4.
6.3 Summary & conclusions of the chapter
In this chapter we have focused on two compounds whose microscopic structure can be described by a two-leg spin-1/2 ladder Hamiltonian: by applying and increasing a magnetic field to the system, two phase transitions take place at respectivelyh=hc1
(from gapped spin liquid to Tomonaga-Luttinger liquid) andh=hc2 > hc1(from Tomonaga-Luttinger liquid to gapped, fully polarized).
6. GRÜNEISEN PARAMETER AND QUANTUM PHASE TRANSITIONS IN SPIN-1/2 LADDERS
−1 −0.8 −0.6 −0.4
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6
log(kBT /J⊥) log(|Γmag|J⊥/gµB)
DMRG (ladder), h=4.375 T
Free fermions (3 species), h=4.375 T DMRG (ladder), h=1.75 T
Free fermions (3 species), h=1.75 T Γmag=0.46⋅ T−1
Figure 6.3: |Γ(T)|renormalized byJ⊥ in order to deal with unitless quantities, in log-log scale, computed analytically and numerically for the HpipCuCl4at both critical fields. Solid lines are analytical results obtained within the refined approximation (free fermions, 3 species). Crosses are DMRG data for the full ladder model obtained by P. Bouillot. The black dotted line is the critical behavior at criticalityΓmag ∼0.46T−1 with prefactor previously determined with a fit of the experimental points.
By studying the field dependence of the Grüneisen parameter we have shown analytically the development of a divergence as the temperature is lowered, and a change of sign, around the criticality. By studying the temperature dependence, we have shown how at low enough temperature and at criticality, results for both com-pounds collapse onto what is the expected asymptotic behavior. Close to criticalities, free fermions effective theory are able to describe in a satisfactory way the physics of the ladder models only at very low temperature. By retaining more terms from the strong coupling expansion of the model, including the possibility of having higher energy excitations thermally activated, we have managed to describe well the devia-tions from the critical behavior at intermediate temperatures.
The study performed in this chapter has shown once more the model character of the two "Hpip" compounds. The Grüneisen parameter can be successfully used as a tool to detect and characterize quantum phase transitions in spin systems. These result open the path to the investigation of more complicated systems, for example the disordered ones. Samples with randomly alternated bromine and chlorine bonds can be now investigated to understand the effects of disorder in low-dimensional
6.3 Summary & conclusions of the chapter
−1 −0.8 −0.6 −0.4
−1.5
−1
−0.5 0 0.5 1
log(kBT /J⊥) log(|Γmag|J⊥/gµB)
DMRG (ladder), h=14.4996 T
Free fermions (3 species), h=14.4996 T DMRG (ladder), h=7 T
Free fermions (3 species), h=7 T Γmag=0.46⋅ T−1
Figure 6.4: |Γ(T)|renormalized byJ⊥ in order to deal with unitless quantities, in log-log scale, computed analytically and numerically for the HpipCuBr4at both critical fields. Solid lines are analytical results obtained within the refined approximation (free fermions, 3 species). Crosses are DMRG data for the full ladder model obtained by P.
Bouillot. The black dotted line is the critical behavior at criticalityΓmag∼0.46T−1, the prefactor is the same used in Fig. 6.3.
antiferromagnets.
6. GRÜNEISEN PARAMETER AND QUANTUM PHASE TRANSITIONS IN SPIN-1/2 LADDERS
CHAPTER 7
General Conclusions & Perspectives
In this work we have performed a study, mainly numerical, of the thermal effects in low-dimensional spin systems. This thesis is composed by three main projects, corresponding to chapters 4-5-6. In the following we will briefly summarize the results:
1. In chapter 4, using an MPS technique for real time dynamics at finite temper-ature, we have performed a numerical quantitative study of the temperature dependence of the NMR1/T1 relaxation rate for spin-1/2 chains (XXZ and dimerized). We were able to reproduce in all cases the expected behavior in the low energy (temperature) limit and at criticality, we have shown interesting deviations from these analytical results at intermediate temperatures. We com-puted the TLL parameters numerically using an infinite size MPS algorithm.
2. By means of the same numerical method, in chapter 5 we have investigated the excitation spectra of the dimerized spin-1/2 chain, for different temperatures and applied magnetic field. We have characterized and explained qualitatively the structures seen in the three phases of the model by resorting to the single strong bond picture (gapped phases) or to the TLL theory (gapless phase). We have shown how temperature comes into play by asymmetrically broadening the spectra and by redistributing the intensity of the structures. We also dis-cussed the band narrowing effect for the triplon excitations, and its interplay with the broadening of the spectrum.
3. In chapter 6, by combining analytical results, DMRG and experimental mea-surements performed on spin-1/2 two-leg ladder models under magnetic field,
7. GENERAL CONCLUSIONS & PERSPECTIVES
we demonstrated the validity of the Grüneisen parameter as a tool to detect and study quantum phase transitions. We discussed its field and temperature dependence at criticality via several approximations of the true model (spin-chain mapping, strong coupling expansion). We have noticed once more the crucial role of the temperature in experiments.
In the first two parts we have successfully demonstrated how recent developments in DMRG/MPS techinques allow for the simulation of the dynamics of one-dimensional spin systems at finite temperature. This offers new opportunities for direct compar-isons for instance with NMR and neutron scattering experiments, since now by com-bining analytical and numerical methods it is possible to cover the whole range of temperatures experimentally accessible with quantitative predictions. We have seen that for temperatures lower than the energy scale of the system (which is typically the couplingJ), the algorithm is extremely powerful: real time evolution at finite temperature can be performed efficiently and with accuracy up to long times. Dif-ferently, in regions where temperature is comparable toJ or higher, it is not easy to achieve a satisfactory resolution in frequency combined with a good precision. In this regime the number of retained states needed to perform efficiently the time evo-lution grows extremely fast, causing a substantial increase of the computational time (which is proportional to the cube of the number of retained states) also on powerful machines. This of course leaves space for future optimizations of the algorithm.
In this work we have computed the dynamics at finite temperature of purely 1D systems. A natural extension would be the exploration of quasi-1D systems, like for example spin-1/2 two-leg ladders for which, as we have seen in chapters 2 and 6, excellent experimental realizations exist. This extension is already quite challenging since it would need an enlargement of the local Hilbert space, and we also know that the computational time grows exponentially fast with the increase of the transverse direction. These preliminary considerations show the complexity of the scenario for the purely 2D problem. Nevertheless, in the last years, several attempts have been made in this direction for the computation of static and thermodynamic quantities in two dimensions [90]: the 2D DMRG [91], a natural extension of the MPS method known as Projected Entangled Pair State (PEPS) [92] or the Multiscale Entanglement Renormalization Ansatz (MERA) [93].
APPENDIX A
Technical Aspects of Chapter 4
A.1 Equality between the time integrals of onsite S
+−and S
−+correlations
In this section we show that Z +∞
−∞
dt
Sj+(t)Sj−(0)
= Z +∞
−∞
dt
Sj−(t)S+j(0)
. (A.1)
Let’s start by the Lehmann representations of the two correlations inside the integrals S+j(t)S−j(0)
=X
m,n
e−βEnei(En−Em)thn|Sj+|mi hm|Sj−|ni, (A.2) S−j (t)Sj+(0)
=X
m,n
e−βEnei(En−Em)thn|Sj−|mi hm|Sj+|ni=
inverting m and n...=X
m,n
e−βEmei(Em−En)thn|Sj+|mi hm|Sj−|ni. (A.3) Comparing Eqs. (A.2)-(A.3) we see that the time integral of each of the two correla-tion reduces to sums of integrals of complex exponentials. Since
Z +∞
−∞
dt ei(En−Em)t= Z +∞
−∞
dt ei(Em−En)t= 2πδ(En−Em), (A.4)
A. TECHNICAL ASPECTS OF CHAPTER 4
one clearly sees that the equality (A.1) is true. Now, in this work we are not per-forming integrals up to infinite time but to a finite timet0, therefore we need to show that first approximation one sees that the two members of the identity to prove differ by multiplicative factors of ordere−β/t0 ∼1in all cases considered in this work.