Entanglement in quantum critical phenomena

Quantum phase transitions occur at zero temperature and involve the appearance of long-range correlations. These correlations are not due to thermal fluctuations but to the intricate structure of a strongly entangled ground state of the system. We present a microscopic computation of the scaling properties of the ground-state entanglement in several 1D spin chain models both near and at the quantum critical regimes. We quantify entanglement by using the entropy of the ground state when the system is traced down to $L$ spins. This entropy is seen to scale logarithmically with $L$, with a coefficient that corresponds to the central charge associated to the conformal theory that describes the universal properties of the quantum phase transition. Thus we show that entanglement, a key concept of quantum information science, obeys universal scaling laws as dictated by the representations of the conformal group and its classification motivated by string theory. This connection unveils a monotonicity law for ground-state entanglement along the renormalization group flow. We also identify a majorization rule possibly associated to conformal invariance and apply the present results to interpret the breakdown of density matrix renormalization group techniques near a critical point.Comment: 5 pages, 2 figure

Construction of a matrix product stationary state from solutions of finite size system

Stationary states of stochastic models, which have $N$ states per site, in matrix product form are considered. First we give a necessary condition for the existence of a finite $M$-dimensional matrix product state for any ${N,M}$. Second, we give a method to construct the matrices from the stationary states of small size system when the above condition and $N\le M$ are satisfied. Third, the method by which one can check that the obtained matrices are valid for any system size is presented for the case where $M=N$ is satisfied. The application of our methods is explained using three examples: the asymmetric exclusion process, a model studied in [F. H. Jafarpour: J. Phys. A: Math. Gen. 36 (2003) 7497] and a hybrid of both of the models.Comment: 22 pages, no figure. Major changes: sec.3 was shortened; the list of references were changed. This is the final version, which will appear in J.Phys.

Increased serum neurofilament light chain concentration indicates poor outcome in Guillain-Barré syndrome

BACKGROUND Guillain-Barré syndrome (GBS) is an autoimmune disease that results in demyelination and axonal damage. Five percent of patients die and 20% remain significantly disabled on recovery. Recovery is slow in most cases and eventual disability is difficult to predict, especially early in the disease. Blood or cerebrospinal fluid (CSF) biomarkers that could help identify patients at risk of poor outcome are required. We measured serum neurofilament light chain (sNfL) concentrations from blood taken upon admission and investigated a correlation between sNfL and clinical outcome. METHODS Baseline sNfL levels in 27 GBS patients were compared with a control group of 22 patients with diagnoses not suggestive of any axonal damage. Clinical outcome parameters for GBS patients included (i) the Hughes Functional Score (HFS) at admission, nadir, and discharge; (ii) the number of days hospitalised; and (iii) whether intensive care was necessary. RESULTS The median sNfL concentration in our GBS sample on admission was 85.5 pg/ml versus 9.1 pg/ml in controls. A twofold increase in sNfL concentration at baseline was associated with an HFS increase of 0.6 at nadir and reduced the likelihood of discharge with favourable outcome by a factor of almost three. Higher sNfL levels upon admission correlated well with hospitalisation time (rs = 0.69, p < 0.0001), during which transfer to intensive care occurred more frequently at an odds ratio of 2.4. Patients with baseline sNfL levels below 85.5 pg/ml had a 93% chance of being discharged with an unimpaired walking ability. CONCLUSIONS sNfL levels measured at hospital admission correlated with clinical outcome in GBS patients. These results represent amounts of acute axonal damage and reflect mechanisms resulting in disability in GBS. Thus, sNfL may serve as a convenient blood-borne biomarker to personalise patient care by identifying those at higher risk of poor outcome

Application of the density matrix renormalization group method to finite temperatures and two-dimensional systems

The density matrix renormalization group (DMRG) method and its applications to finite temperatures and two-dimensional systems are reviewed. The basic idea of the original DMRG method, which allows precise study of the ground state properties and low-energy excitations, is presented for models which include long-range interactions. The DMRG scheme is then applied to the diagonalization of the quantum transfer matrix for one-dimensional systems, and a reliable algorithm at finite temperatures is formulated. Dynamic correlation functions at finite temperatures are calculated from the eigenvectors of the quantum transfer matrix with analytical continuation to the real frequency axis. An application of the DMRG method to two-dimensional quantum systems in a magnetic field is demonstrated and reliable results for quantum Hall systems are presented.Comment: 33 pages, 18 figures; corrected Eq.(117

Delta-Function Bose Gas Picture of S=1 Antiferromagnetic Quantum Spin Chains Near Critical Fields

We study the zero-temperature magnetization curve (M-H curve) of the S=1 bilinear-biquadratic spin chain, whose Hamiltonian is given by $H=\sum_{i} S_i S_{i+1}+\beta (S_iS_{i+1})^2 with$0 \leq \beta <1$. We focus on validity of the delta-function bose-gas picture near the two critical fields: the saturation field$H_s$and the lower critical field$H_c$associated with the Haldane gap. Near$H_s$, we take ``low-energy effective S-matrix'' approach, which gives correct effective bose-gas coupling constant$c$, different from the spin-wave value. Comparing the M-H curve of the bose gas with the product-wavefunction renormalization group (PWFRG) calculation, excellent agreement is seen. Near$H_c$, comparing the PWFRG result with the bose-gas prediction, we find that there are two distinct regions of$\beta$separated by a critical value$\beta_c(\approx 0.41)$. In the region$0<\beta<\beta_c$, the effective coupling$c$is positive but rather small. The small value of$c$makes the ``critical region'' of the square-root behavior$M\sim \sqrt{H-H_c}$very narrow. Further, we find that in the$\beta \to \beta_c-0$, the square-root behavior transmutes to a different one,$M\sim (H-H_c)^{1/4}$. In the region$\beta_c<\beta <1$, the square-root behavior is rather distinct, but the effective coupling$c$ becomes negative.Comment: 6 pages, RevTeX, 8 ps figure

Corner Transfer Matrix Algorithm for Classical Renormalization Group

We report a real-space renormalization group (RSRG) algorithm, which is formulated through Baxter's corner transfer matrix (CTM), for two-dimensional (d = 2) classical lattice models. The new method performs the renormalization group transformation according to White's density matrix algorithm, so that variational free energies are minimized within a restricted degree of freedom m. As a consequence of the renormalization, spin variables on each corner of CTM are replaced by a m-state block spin variable. It is shown that the thermodynamic functions and critical exponents of the q = 2, 3 Potts models can be precisely evaluated by use of the renormalization group method.Comment: 20 pages, 10 ps figures, JPSJ style files are include

The density-matrix renormalization group

The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly acquired the status of method of choice for numerical studies of one-dimensional quantum systems. Its applications to the calculation of static, dynamic and thermodynamic quantities in such systems are reviewed. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and non-equilibrium statistical physics, and time-dependent phenomena is discussed. This review also considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by DMRG.Comment: accepted by Rev. Mod. Phys. in July 2004; scheduled to appear in the January 2005 issu

Circulating adrenomedullin estimates survival and reversibility of organ failure in sepsis: the prospective observational multinational Adrenomedullin and Outcome in Sepsis and Septic Shock-1 (AdrenOSS-1) study

Background: Adrenomedullin (ADM) regulates vascular tone and endothelial permeability during sepsis. Levels of circulating biologically active ADM (bio-ADM) show an inverse relationship with blood pressure and a direct relationship with vasopressor requirement. In the present prospective observational multinational Adrenomedullin and Outcome in Sepsis and Septic Shock 1 (, AdrenOSS-1) study, we assessed relationships between circulating bio-ADM during the initial intensive care unit (ICU) stay and short-term outcome in order to eventually design a biomarker-guided randomized controlled trial. Methods: AdrenOSS-1 was a prospective observational multinational study. The primary outcome was 28-day mortality. Secondary outcomes included organ failure as defined by Sequential Organ Failure Assessment (SOFA) score, organ support with focus on vasopressor/inotropic use, and need for renal replacement therapy. AdrenOSS-1 included 583 patients admitted to the ICU with sepsis or septic shock. Results: Circulating bio-ADM levels were measured upon admission and at day 2. Median bio-ADM concentration upon admission was 80.5 pg/ml [IQR 41.5-148.1 pg/ml]. Initial SOFA score was 7 [IQR 5-10], and 28-day mortality was 22%. We found marked associations between bio-ADM upon admission and 28-day mortality (unadjusted standardized HR 2.3 [CI 1.9-2.9]; adjusted HR 1.6 [CI 1.1-2.5]) and between bio-ADM levels and SOFA score (p &lt; 0.0001). Need of vasopressor/inotrope, renal replacement therapy, and positive fluid balance were more prevalent in patients with a bio-ADM &gt; 70 pg/ml upon admission than in those with bio-ADM ≤ 70 pg/ml. In patients with bio-ADM &gt; 70 pg/ml upon admission, decrease in bio-ADM below 70 pg/ml at day 2 was associated with recovery of organ function at day 7 and better 28-day outcome (9.5% mortality). By contrast, persistently elevated bio-ADM at day 2 was associated with prolonged organ dysfunction and high 28-day mortality (38.1% mortality, HR 4.9, 95% CI 2.5-9.8). Conclusions: AdrenOSS-1 shows that early levels and rapid changes in bio-ADM estimate short-term outcome in sepsis and septic shock. These data are the backbone of the design of the biomarker-guided AdrenOSS-2 trial. Trial registration: ClinicalTrials.gov, NCT02393781. Registered on March 19, 2015

Quantum Impurity Entanglement

Entanglement in J_1-J_2, S=1/2 quantum spin chains with an impurity is studied using analytic methods as well as large scale numerical density matrix renormalization group methods. The entanglement is investigated in terms of the von Neumann entropy, S=-Tr rho_A log rho_A, for a sub-system A of size r of the chain. The impurity contribution to the uniform part of the entanglement entropy, S_{imp}, is defined and analyzed in detail in both the gapless, J_2 <= J_2^c, as well as the dimerized phase, J_2>J_2^c, of the model. This quantum impurity model is in the universality class of the single channel Kondo model and it is shown that in a quite universal way the presence of the impurity in the gapless phase, J_2 <= J_2^c, gives rise to a large length scale, xi_K, associated with the screening of the impurity, the size of the Kondo screening cloud. The universality of Kondo physics then implies scaling of the form S_{imp}(r/xi_K,r/R) for a system of size R. Numerical results are presented clearly demonstrating this scaling. At the critical point, J_2^c, an analytic Fermi liquid picture is developed and analytic results are obtained both at T=0 and T>0. In the dimerized phase an appealing picure of the entanglement is developed in terms of a thin soliton (TS) ansatz and the notions of impurity valence bonds (IVB) and single particle entanglement (SPE) are introduced. The TS-ansatz permits a variational calculation of the complete entanglement in the dimerized phase that appears to be exact in the thermodynamic limit at the Majumdar-Ghosh point, J_2=J_1/2, and surprisingly precise even close to the critical point J_2^c. In appendices the relation between the finite temperature entanglement entropy, S(T), and the thermal entropy, S_{th}(T), is discussed and and calculated at the MG-point using the TS-ansatz.Comment: 62 pages, 27 figures, JSTAT macro

Tensor network states and geometry

Tensor network states are used to approximate ground states of local Hamiltonians on a lattice in D spatial dimensions. Different types of tensor network states can be seen to generate different geometries. Matrix product states (MPS) in D=1 dimensions, as well as projected entangled pair states (PEPS) in D>1 dimensions, reproduce the D-dimensional physical geometry of the lattice model; in contrast, the multi-scale entanglement renormalization ansatz (MERA) generates a (D+1)-dimensional holographic geometry. Here we focus on homogeneous tensor networks, where all the tensors in the network are copies of the same tensor, and argue that certain structural properties of the resulting many-body states are preconditioned by the geometry of the tensor network and are therefore largely independent of the choice of variational parameters. Indeed, the asymptotic decay of correlations in homogeneous MPS and MERA for D=1 systems is seen to be determined by the structure of geodesics in the physical and holographic geometries, respectively; whereas the asymptotic scaling of entanglement entropy is seen to always obey a simple boundary law -- that is, again in the relevant geometry. This geometrical interpretation offers a simple and unifying framework to understand the structural properties of, and helps clarify the relation between, different tensor network states. In addition, it has recently motivated the branching MERA, a generalization of the MERA capable of reproducing violations of the entropic boundary law in D>1 dimensions.Comment: 18 pages, 18 figure