† Service de Physique Théorique, CEA/Saclay - DSM/SPhT, F-91191 Gif sur Yvette Cedex, France 1 ∗∗ Institute for Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
Abstract. In a quantum measurement, a coupling g between the system S and the apparatus A triggers the establishment of
correlations, which provide statistical information about S. Robust registration requires A to be macroscopic, and a dynamical symmetry breaking of A governed by S allows the absence of any bias. Phasetransitions are thus a paradigm for quantum measurement apparatuses, with the order parameter as pointer variable. The coupling g behaves as the source of symmetry breaking. The exact solution of a model where S is a single spin and A a magnetic dot (consisting of N interacting spins and a phonon thermal bath) exhibits the reduction of the state as a relaxation process of the off-diagonal elements of S + A, rapid
DOI: 10.1103/PhysRevLett.105.216804 PACS numbers: 73.43.f, 11.15.q, 11.25.Hf
One of the most challenging problems in the study of quantum many-body systems is to understand transitions between topologically ordered states [ 1 ]. Since topological states cannot be characterized by broken symmetry and local order parameters, we cannot use the conventional Ginzburg-Landau theory. When non-Abelian topological states are involved, the transitions that are currently under- stood are essentially all equivalent to the transition between weak and strong-paired BCS states [ 2 , 3 ]. Over the last ten years, while there has been much work on the subject, there has not been another quantumphase transition in a physi- cally realizable system, involving a non-Abelian phase, for which we can answer the most basic questions of whether the transition can be continuous and what the critical theory is. Here we present an additional example in the context of fractional quantum Hall (FQH) systems.
There are many examples of continuous quantumphasetransitions that are beyond the Landau paradigm. First, one or both phases may have non-Landau order (for instance, they may have topological order). Then, since an order- parameter-based description fails to capture the non- Landau phase, it is not surprising that the critical theory is not within the standard LGW paradigm (see Ref.  for a review). Perhaps more surprisingly, Landau-forbidden continuous phasetransitions may even occur between phases that are Landau allowed themselves. A classic example is the N´eel-to-valence-bond solid-state transition of spin- 1=2 quantum magnets on a 2d square lattice [4 –18] . The theory for this transition is an example of a phenome- non dubbed “deconfined quantum criticality.” The critical field theory is conveniently expressed in terms of “decon- fined” fractionalized degrees of freedom (d.o.f.), though the phases on either side only have conventional “confined” excitations. Now, there are many other proposed examples of deconfined quantum critical points in 2 þ 1 space-time dimensions [19 –39] . Very similar (sometimes equivalent) theories emerge for critical points between trivial and symmetry-protected topological (SPT) phases of bosons in 2 þ 1 space-time dimensions [19,40 –46] . (For a general introduction to SPT phases, see, e.g., Refs. [42,47 –54] .)
DOI: 10.1103/PhysRevLett.107.175302 PACS numbers: 67.85. d, 05.30.Rt, 37.10.Jk, 61.05.a
Ultracold atomic gases are an ideal system in which to study many-body phenomena because of the relative ease with which parameters in the model Hamiltonian can be tuned across a wide range [ 1 , 2 ]. Such studies have resulted in a better understanding of various phasetransitions such as the Berezinskii-Kosterlitz-Thouless transition in two-dimensional systems [ 3 ], the BEC-BCS crossover of interacting fermions [ 4 ], and the superfluid to Mott insulator transition in a three-dimensional lattice [ 5 ]. One major goal of this field is to realize spin phases such as antiferromagnetic states to explore quantum magnetism and its interplay with high-temperature super- conductivity [ 6 ].
Experiments have so far observed FQH plateaus in two- component systems at = 2 /3, 4/5, 4/7, 4/9, 6/5, 6/7, 1/4, etc. 28 – 33 In some cases, these plateaus have been observed in
both bilayer and spin-unpolarized single-layer systems while in others, the plateau has only been observed in one of them. At all of these filling fractions, there exists also one 共or sev- eral兲 candidate Abelian phase共s兲; in most cases, it is assumed that these plateaus are described by one of the Abelian phases. However, the pattern-of-zeros construction also yields many simple non-Abelian states at these filling frac- tions. In some situations, we expect the non-Abelian states to be good candidate states.
3.1.3 Fermi surface
The strongly correlated f electron systems have attracted a lot of interest among the scientific community and this especially due to the low energy phenomena, such as heavy fermion or quantumphase transition, that were some of the main topics. The analysis of the Fermi surface (FS) of these elements is a crucial source of information to understand the Physics sustaining this phenomena. Among the different tools used these past few decades in the investigation of FS, the de Hass van Alphen (dHvA) effect has been the most intensively used method to explore a wide range of f electron systems. In the heavy fermion family, CeRu 2 Si 2 has been the subject of numerous studies to explore its FS. The interest for this compound was motivated by a simple but not yet clear question of whether the f electrons becomes localized or not after the pseudo-metamagnetic transition that occurs at H m = 7.8 T. The volume of the FS was commonly used to ascribe the itinerant or localized nature of the electrons. Nevertheless, this belief has felt into ruins with the discovery that such a modification in FS is not enough sufficient by itself to conclude in the localization of the electrons [30, 31]. Moreover, it has been argued that even thought the detected volume of the FS is near to what is usually qualified of small FS, the term localized could not be used in some cases . In the field region under H m , the FS has been well determined from both an experimental and theoretical perspectives [23, 35]. CeRu 2 Si 2 is a case where the experimental measured frequencies are extraordinarily consistent with the band calculations [36, 37]. The same remark cannot be done on the region above H m where the FS has only been partially discovered. However, based on the existing similarities of the observed frequencies and their angular dependences between CeRu 2 Si 2 and LaRu 2 Si 2 above H m , it has been reported that the La-doped case could be representative of the CeRu 2 Si 2 behaviour in the polarized paramagnetic phase . The results obtained from the band structure calculation, considering that the f electrons belong to the conduction electrons, are represented in the figure 3.7 (a). On the FS are also reported the orbits of the measured dHvA oscillations when a field is applied along the c-axis. Figure 3.7 (b) represents the FS of LaRu 2 Si 2 , that is proposed to be the nearest picture of what could be the FS of CeRu 2 Si 2 above H m . The tables on the figure 3.8 and 3.9 give the corresponding frequencies, and effective masses, of the dHvA orbits below and above H m respectively.
In the past, the experimental study of quantum spin dynamics in 1D has been largely limited to microscopic spins in condensed-matter systems. It has been demonstrated that such quasi-1D spin materials as LiHoF 4 and CoNb 2 O 6 can be
continuously tuned across the quantumphase transition (QPT) [ 11 – 13 ]. Though these works opened up new vistas in the studies of transverse ﬁeld Ising model, the experimental realization of 1D quantum spin models in well-controllable and tunable systems remains a challenge. Indeed, as an experimental tool, the quasi-1D spin systems in solids are limited in several respects: (i) the inter-chain interactions are not negligibly weak, and, thus, these systems are inevitably quasi-1D, (ii) the exchange interactions between the nearest-neighbor spins cannot be varied, (iii) the exchange interactions are the same for all pair of spins, which does not allow for exploring the effect of disorder andphase boundaries without adding a signiﬁcant amount of impurities, and (iv) the available experimental tool for these systems – scattering of neutrons – interacts only with a narrow class of excitations. Flexibility in the design of artiﬁcial spin systems, which are free from these limitations, facilitates bridging the gap between the theoretical study of ideal spin chains and the experimental investigation of bulk magnetic samples. In particular, this ﬂexibility allows one to address an important issue of the effects of disorder on the statics and dynamics of transverse ﬁeld spin models. Recently, the transverse-ﬁeld Ising model was realized in the chain of artiﬁcial and fully-controllable spins – eight ﬂux qubits with tunable spin – spin couplings [ 10 ]. We pursue a similar approach using specially designed one-dimensional Josephson ladders.
(X-STM) measurements are performed on the (110) natural cleavage plane. These QDs have been grown by solid-source molecular beam epitaxy under the same conditions (3ML deposited at 580 °C), already reported in details in previous studies [ 20 , 29 ], except that a GaP capping layer is used for cross-sectional STM (X-STM), while QDs are uncapped for the plan-view imaging. From the plan-view image of Fig. 1(a) , the morphology of the unburied QD can easily be defined at the atomic scale, analyzing angles of reconstructed facets with the nominal (001) plan, and QD edge direction angles with the [1–10] and  directions that is fixed by the dimer orien- tation on the (001) nominal surface [the wetting layer (WL), here]. Resulting geometry and crystallographic orientations are illustrated in Fig. 1(c) . During the GaP capping process, the morphology of QDs may change due to the (In,Ga)As/GaP interplay. In this regard, a typical X-STM profile image taken on the (110) natural cleavage plane is shown in Fig. 1(b) . In this image, the lateral dimension of the QD and the angles of the facets are similar to what was observed in plan-view imaging. We therefore assume in the following that the facets previously defined and the in-plane footprint of the QD remain unchanged during the capping process. On the other hand, the capped QD observed in cross-sectional STM has a slightly smaller height, with a smoother edge at the apex, as compared to the uncapped QD, due to mass transport during the capping process [ 37 ]. Therefore, the QD is truncated at 3.4 nm in the calculations, as illustrated in Fig. 1(c) . A thin WL is also observed in Fig. 1(b) ; therefore, a 1-monolayer-(ML)-thick WL is added in the definition of the QD geometry for calculations.
First, semen suspensions are far more heterogeneous than many other types of biological suspensions, such as bac- terial suspensions grown under controlled incubation conditions and using selected strains. In the case of semen, the suspension constituents as well as the fluid suspension are uncontrolled. This is incidentally one motivation of this study to assess semen quality from collective motion. The cell composition (ratio of mobile/immobile cells, cell concentration, motility heterogeneity within the mobile cell population), the plasma biochemical composition as well as its rheology are variable from one semen sample to another, even if the sample comes from the same subject. Plasma rheo- logy has been previously studied  showing viscosity variations as large as 50%. Two main parameters might be responsible for the observed variability: the individual cell velocity and the percentage of immobile cells. Concerning indi- vidual cell velocity variability we report in table 1 various measurements obtained for five different males, where the intrasample variability is also given. One can observe that intrasample variability has an average value of 54%, whereas inter-individual variability has an average of 58%. So, intra- sample variability is comparable with inter-individual variability. The immobile cell proportion could be estimated by CASA measurements. For the samples analysed in table 1, the ratio of non-mobile/mobile sperm can vary between 20% and 40%, adding again another important contribution to the variability of collective behaviour. Finally, one should also con- sider that more complex behaviour may influence collective behaviour, such as the ability of sperm cells to synchronize their flagellum beating with their neighbours, but such feature is very difficult to quantify experimentally. One could wonder if more reproducible results could have been obtained by harvesting the cells from many samples and resuspending them in a fluid of known rheology mimicking semen. However, this is a highly challenging experiment to perform. Indeed, collective motility lasts no longer than half an hour that leaves a very short time window. Additionally, centrifuging has a known negative and non-controllable impact on individual motility. Finally, concentrating cells generates many cellular degradations and a large percentage of cell deaths.
A SCINTAG diffractometer (Model XDS 2000) equipped with a graphite monochromator and a Cu KR radiation source was used to analyze the semicrystalline samples. The experi- ments were carried out at room temperature, and a scan rate of 4°/min was chosen for all measurements. Thermal analysis was done using a TA Instruments DSC 2910. All samples were scanned at 10 °C/min in a dry nitrogen environment. In- situ measurements of CO 2 -induced glass transition and crys- tallization were carried out using a Setaram DSC 121. Unlike conventional DSCs, this machine has a rather massive furnace with two symmetrical cavities containing the reference and sample vessels. A sample size up to 100 mg can be used, and the vessels are rated to 100 atm. The experimental details, calibration of the temperature and energy scales, and the technique used for making high-pressure calorimetric mea- surements are described elsewhere. 39 Briefly, after installing the sPS film, the system was evacuated for 1 h, and then both reference and sample sides were pressurized with CO 2 to the desired value and held at 35 °C for 1 h to allow the polymer- gas system to reach equilibrium. While still in contact with the gas, the sample was scanned from 20 to 200 °C for nonisothermal measurements or rapidly brought to 122 °C and held at this temperature for enough time to record the heat flow signal associated with isothermal crystallization kinetics. Results and Discussion
Using a low-energy approach, when the chains are weakly coupled, we find that all these quantumphasetransitions are described by an SU(2)N CFT with central charge c = 3N/(N + 2). This result generalizes the findings of Ref. 46 for N = 2 where a SU(2) 2 emerging quantum criticality has been revealed by means of a Majorana-fermion approach, which is very specific to the N = 2 case. In this respect, we have shown that the quantumphasetransitions, driven by four-spin exchange interactions, belong to the SU(2)N universality class when N = 2,3,4. For N > 4, a relevant perturbation is likely to be generated and to drive the system away from criticality. A first-order phase transition is then expected for N > 4 although further work is clearly called for to confirm or infirm this result. In close parallel to this field-theory analysis, we have investigated numerically the phasetransitions for N = 2,3,4 by means of ED and DMRG approaches. As we have seen, these techniques give a precise location of the quantumphasetransitions for all signs of K. The position of the transition does not vary too much as a function of N when J ⊥ > 0, and seems to be independent of N when J ⊥ < 0. Surprisingly enough, the estimate of the transition obtained within the low-energy approach is in rather good agreement with the numerical results. The nature of the quantumphasetransitions was then obtained by extracting numerically the central charge c and the scaling dimension x of the lowest primary field of the SU(2)N CFT. Let us emphasize that while DMRG is the tool of choice to measure the central charge c, the presence of logarithmic correction prevents a reliable determination of x; on the contrary, x can be extracted accurately from ED data as shown in our study. In all cases, we found a very good agreement with the prediction of the low-energy approach according to which the quantumphasetransitions should belong to the SU(2)N universality class for N 4. In this respect, the N = 3 case is intriguing due to the prediction of a transition between the standard gapless c = 1 phaseand the dimerized phases (SD and UD phases) that is not of the BKT type with central charge c = 1, as for the J 1 − J 2 Heisenberg spin-1/2 chain, but rather an SU(2) 3 transition with the emergence of nontrivial critical modes with fractional central charge c = 9/5. All our results are summarized in Table I .
In the former study, the excitations were probed by THz spectroscopy at much higher energy and only at the ZC point ~Q=(0, 0, 0). As the phase transition was believed to be an order-disorder transition, and thus a paramagnetic phase at high field, it has led to a misunderstanding concerning the nature of the excitations. Indeed the explanation proposed by Wang et al. was a deconfinement of the spinons as the interchain interaction would not be effective in the high field phase compared to the applied magnetic field. The problem here was the forgetting of the effective staggered magnetic field which is the key ingredient in this phase transition. In the other study, the excitations were probed by means of neutron scattering experiments. Contrary to the THz study, the staggered field was here taken into account. Both elastic and inelastic neutron scattering experiments have been done and the experimental results are rather similar to ours. However Matsuda et al. did not use polarized neutrons and their theoretical analysis was more limited than in our study. Moreover they were focusing essentially on the dispersion spectrum in the high field phase, rather than on the field dependence of the excitations which was the key to understand the topological nature of the transition.
where t s is the sweep duration and θ(τ ) is the Heavyside step function.
In order to solve numerically the master equation (2.3), we have expressed the time- dependent density matrix in the basis of Fock number states |ni. Convergence of the results have been carefully checked by increasing the cutoﬀ number of photons. Using this numerical method, we can explore regimes with a number of photons of up to a few tens. Figure 2.4 compares the steady-state results obtained from Eqs. (2.7) and (2.11) (noted respectively MF and SS) with the results obtained by numerical integration of the master equation (2.3) with a time-dependent drive amplitude (2.13). The top panel of Fig. 2.4 reports the number of photons as a function of the drive amplitude. The yellow and green curves represent the steady-state values respectively for the mean-ﬁeld and the exact solutions. The mean-ﬁeld solution exhibits the characteristic "S" shape due to the existence of three solutions of Eq.(2.7). The two other curves of this panel are the numerical results of the time-dependent master equation (2.3), showing that when the triangular modulation is applied a dynamical hysteresis appears. The largest hysteresis loop is the result of a faster pump modulation (t s /∆F = 10/γ 2 ) while the smaller one
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S i S j (2)
on the CaVO lattice [ 8 ], a 1=5th depleted square lattice (see Fig. 1 ). The first sum runs over nearest-neighbor spins on 0 bonds (solid lines in Fig. 1 ) while the second is over bonds (dashed lines). This lattice structure can be found in the AF compound CaV 4 O 9 (hence the lattice name), even though the interactions are more complex in the real com- pound [ 9 ]. Varying the coupling allows the occurrence of two QPTs separating an intermediate Ne´el-ordered AF phase from, respectively, a low- plaquette and a high- dimer phase [ 8 , 10 ]. H conserves the total spin of the system and, in particular, AF interactions > 0 lead to a singlet GS.
transition de Curie entre la phase ferroélectrique quadratique et la phase cubique paraélectrique, cependant le pic P 2 peut être relié à la transition morphotropique entre les phases ferroélectriques
rhomboédrique et quadratique. Les évolutions des deux pics P 1 et P 2 sont en bon accord avec les
différents modèles existant dans la littérature et expliquant les transitions de phase du premier ordre. L’introduction des dopants potassium et niobium change la nature du processus de la transition de phase.
exhibits a second order transition: the overlap is continuous, only its derivative has a discontinuity.
In both Fig. 2 and 3 we also compare to the performance of the principal component analysis (PCA) performed on the matrix X. PCA is a standard spectral method to solve data clustering, one computes r leading singular vectors of X and instead of clustering m points in n dimensions, one concatenates the singular vectors into m r-dimensional vectors and clusters in the r-dimensional space which is much simpler. The overlap reached with the PCA clustering follows from a more general theory of low-rank perturbations of random matrices , but it can also be derived from the state evolution analysis of AMP as we present in the appendix. In this case of GMM with equal-size clusters the phase transition observed in PCA coincides with the phase transition of AMP ρ c . Concerning the performance