Recently we have used spectroscopic mapping with the scanning tunneling microscope to probe modulations of the electronic density of states in single crystals of the high temperature superconductor Bi2Sr2CaCu2O8+d (Bi-2212) as a function of temperature [C. V. Parker et al., Nature (London) 468, 677 (2010)]. These measurements showed Cu-O bond-oriented modulations that form below the pseudogap temperature with a temperature-dependent energy dispersion displaying different behaviors in the superconducting and pseudogap states. Here we demonstrate that quasiparticle scattering off impurities does not capture the experimentally observed energy- and temperature-dependence of these modulations. Instead, a model of scattering of quasiparticles from short-range stripe order, with periodicity near four lattice constants (4a), reproduces the experimentally observed energy dispersion of the bond-oriented modulations and its temperature dependence across the superconducting critical temperature, Tc. The present study confirms the existence of short-range stripe order in Bi-2212.
Motivated by the recent observation of the time-reversal symmetry broken state in K-doped BaFe$_2$As$_2$ superconducting alloys, we theoretically study the collective modes and the short time dynamics of the superconducting state with $s+is$-wave order parameter using an effective four-band model with two hole and two electron pockets. The superconducting $s+is$ state emerges for incipient electron bands as a result of hole doping and appears as an intermediate state between $s^{\pm}$ (high number of holes) and $s^{++}$ (low number of holes). The amplitude and phase modes are coupled giving ri…
We review briefly several approaches used to investigate the stability of stripe phases in high temperature superconductors, where charge inhomogeneities arise from competing kinetic and magnetic energies. The mechanism of stripe formation, their consequences for the normal state and enhancement of pairing interaction triggered by charge inhomogeneities are briefly summarized. Finally, we demonstrate that orbital degeneracy ($i$) leads to a more subtle mechanism of stripe formation, and ($ii$) plays an important role and determines the symmetry of the superconducting state in pnictides.
Stripe phases are predicted and observed to occur in a class of strongly-correlated materials describable as doped antiferromagnets, of which the copper-oxide superconductors are the most prominent representative. The existence of stripe correlations necessitates the development of new principles for describing charge transport, and especially superconductivity, in these materials.
Charge and spin stripe order is a type of electronic crystal observed in certain layered cuprates associated with high-temperature superconductivity. Quantum-disordered stripes could be relevant for understanding the superconductivity. Here I discuss recent experimental characterizations of the stripe-ordered state in La(1.875)Ba(0.125)CuO(4), and compare them with properties of superconducting compositions.
Resting on the gauge theory of topological quantum melting in 2+1 dimensions, we predict that a superconductor characterized by crystalline correlations on a length scale large compared to the lattice carries a new collective mode: the massive shear photon. This mode is visible in the electrodynamic response and the ramification of our theory is that electron energy loss spectroscopy can be employed to prove or disprove the existence of dynamical stripes in cuprate superconductors.
Using a phenomenological lattice model of coupled spin and charge modes, we determine the spin susceptibility in the presence of fluctuating stripe charge order. We assume the charge fluctuations to be slow compared to those of the spins, and combine Monte Carlo simulations for the charge order parameter with exact diagonalization of the spin sector. Our calculations unify the spin dynamics of both static and fluctuating stripe phases and support the notion of a universal spin excitation spectrum in doped cuprate superconductors.
The magnetic excitations in the stripe phase of high-T_c superconductors are investigated in a model of spin ladders which are effectively coupled via charged stripes. Starting from the effective single-triplon model for the isolated spin ladder, the quasi-one-dimensional spin system can be described straightforwardly. Very good agreement is obtained with recent neutron scattering data on La_(15/8)Ba_(1/8)CuO_4 (no spin gap) and YBa_2Cu_3O_(6.6) (gapped). The signature of quasi-one-dimensional spin physics in a single-domain stripe phase is predicted.
Site-dependent NMR relaxation time $T_1({\bf r})$ is calculated in the vortex state using the Bogoliubov-de Gennes theory, taking account of possible "field-induced stripe'' states in which the magnetism arises locally around a vortex core in d-wave superconductivity. The recently observed huge enhancement $T_1^{-1}({\bf r})$ below $T_c$ at a core site in Tl$_2$Ba$_2$CuO$_6$ is explained. The field-induced stripe picture explains consistently other relevant STM and neutron experiments.
A summary is given of the main outcomes of the quantum-critical-point scenario for high-$T_c$ superconductors, developed in the last few years by the Rome group. Phase separation, which commonly occurs in strongly correlated electronic systems, turns into a stripe instability when Coulomb interaction is taken into account. The stripe phase continuously connects the high doping regime, dominated by charge degrees of freedom to the low doping regime where spin degrees of freedom are most relevant. Dynamical stripe fluctuations enslave antiferromagnetic fluctuations at high doping. Critical fluct…
Recent neutron diffraction studies have yielded evidence that, in a particular cuprate family, holes doped into the CuO(2) planes segregate into stripes that separate antiferromagnetic domains. Here it is shown that such a picture provides a quantitatively consistent interpretation of the spin fluctuations measured by neutron scattering in La(1.85)Sr(0.15)CuO(4) and YBa(2)Cu(3)O(6+x).
A prevailing description of the stripe phase in underdoped cuprate superconductors is that the charge carriers (holes) phase segregate on a microscopic scale into hole rich and hole poor regions. We report resonant elastic x-ray scattering measurements of stripe-ordered La$_{1.475}$Nd$_{0.4}$Sr$_{0.125}$CuO$_4$ at the Cu $L$ and O $K$ absorption edges that identify an additional feature of stripe order. Analysis of the energy dependence of the scattering intensity reveals that the dominant signature of the stripe order is a spatial modulation in the energies of Cu 3d and O 2p states rather tha…
We present a minimal model of a doped Mott insulator that simultaneously supports antiferromagnetic stripes and d-wave superconductivity. We explore the implications for the global phase diagram of the superconducting cuprates. At the unrestricted mean-field level, the various phases of the cuprates, including weak and strong pseudogap phases, and two different types of superconductivity in the underdoped and the overdoped regimes, find a natural interpretation. We argue that on the underdoped side, the superconductor is intrinsically inhomogeneous -- striped coexistence of of superconductivit…
We calculate electronic single-particle properties of short-range ordered stripe states, using Monte-Carlo simulations of collective charge-density wave (CDW) order parameters coupled to fermions on a 2d square lattice. For superconducting bond-centered stripes with a d-wave form factor, we find a valence-bond "glass" which coexists with low-energy quasiparticles featuring interference phenomena, in agreement with recent scanning-tunneling-microscopy (STM) measurements on underdoped BSCCO and CaNaCuOCl. Together with earlier work, our calculations provide a link between CDW features seen in ST…
The universal behaviour of the magnetic excitations in high-temperature superconductors is described in a model with static stripes retaining only the localized spin degrees of freedom. The stripes are represented by a model of coupled two-leg spin ladders. We start from the results obtained previously by continuous unitary transformations for an isolated spin ladder. A quantitative description of neutron scattering data is reached, using a model for a single cuprate layer with well established values of the exchange coupling constants. The neutron resonance peak is explained in terms of a sad…
We investigate the upper critical field in a stripe--phase and in the presence of a phenomenological pseudogap. Our results indicate that the formation of stripes affects the Landau orbits and results in an enhancement of $H_{c2}$. On the other hand, phenomenologically introduced pseudogap leads to a reduction of the upper critical field. This effect is of particular importance when the magnitude of the gap is of the order of the superconducting transition temperature. We have found that a suppression of the upper critical field takes place also for the gap that originates from the charge--den…
We discuss the problem of competition between a superconducting (SC) ordered state with a charge density wave (CDW) state in stripe phases of high $T_c$ superconductors. We consider an effective model for each stripe motivated by studies of spin-gapped electronic ladder systems. We analyze the problem of dimensional crossover arising from inter-stripe SC and CDW couplings using non-Abelian bosonization and renormalization group (RG) arguments to derive an effective $O(4)$-symmetric nonlinear $σ$-model in $D=2+1$ for the case of when both inter-stripe couplings are of equal magnitude as well as…
Motivated by the recent observations of small Fermi energies and comparatively large superconducting gaps, present also on bands not crossing the Fermi energy (incipient bands) in iron-based superconductors, we analyse the doping evolution of superconductivity in a four-band model across the Lifshitz transition including BCS-BEC crossover effects on the shallow bands. Similar to the BCS case we find that with hole doping the phase difference between superconducting order parameters of the hole bands changes from $0$ to $π$ through an intermediate $s+is$ state breaking time-reversal symmetry. T…
We present a short account of the present experimental situation of stripes in cuprates followed by a review of our present understanding of their ground state and excited state properties. Collective modes, the dynamical structure factor, and the optical conductivity of stripes are computed using the time-dependent Gutzwiller approximation applied to realistic one band and three band Hubbard models, and are found to be in excellent agreement with experiment. On the other hand, experiments like angle-resolved photoemission and scanning tunneling microscopy show the coexistence of stripes at hi…
We examine the temperature dependence of the electronic states in the stripe phase of high-Tc cuprates by using the t-J model with a potential that stabilizes vertical charge stripes. Charge and spin-correlation functions and optical conductivity are calculated by using finite-temperature Lanczos method. At zero temperature, the antiferromagnetic correlation between a spin in a charge stripe and that in a spin domain adjacent to the stripe is weak, since the charge stripe and the spin domain are almost separated. With increasing temperature, the correlation increases and then decreases toward …
Based on a two-dimensional model of coupled two-leg spin ladders, we derive a unified picture of recent neutron scattering data of stripe-ordered La_(15/8)Ba_(1/8)CuO_4, namely of the low-energy magnons around the superstructure satellites and of the triplon excitations at higher energies. The resonance peak at the antiferromagnetic wave vector Q_AF in the stripe-ordered phase corresponds to a saddle point in the dispersion of the magnetic excitations. Quantitative agreement with the neutron data is obtained for J= 130-160 meV and J_cyc/J = 0.2-0.25.
This article is devoted to a discussion of stripe and electron-nematic order and their connection to electronic properties in the pseudogap regime of copper-oxide superconductors. We review basic properties of these symmetry-breaking ordering phenomena as well as proposals which connect them to quantum-oscillation measurements. Experimental data indicate that these orders are unlikely to be the cause of the pseudogap phenomenon, implying that they occur on top of the pseudogap state which itself is of different origin. Specifically, we discuss the idea that the non-superconducting pseudogap gr…
We define a distinct phase of matter, a "pair density wave" (PDW), in which the superconducting order parameter $ϕ$ varies periodically as a function of position such that when averaged over the center of mass position, all components of $ϕ$ vanish identically. Specifically, we study the simplest, unidirectional PDW, the "striped superconductor," which we argue may be at the heart of a number of spectacular experimental anomalies that have been observed in the failed high temperature superconductor, La$_{2-x}$ Ba$_x$CuO$_4$. We present a solvable microscopic model with strong electron-electron…
One type of order that has been observed to compete with superconductivity in cuprates involves alternating charge and antiferromagnetic stripes. Recent neutron scattering studies indicate that the magnetic excitation spectrum of a stripe-ordered sample is very similar to that observed in superconducting samples. In fact, it now appears that there may be a universal magnetic spectrum for the cuprates. One likely implication of this universal spectrum is that stripes of a dynamic form are present in the superconducting samples. On cooling through the superconducting transition temperature, a ga…
Charge ordering is one of the most intriguing and extensively studied phenomena in correlated electronic materials because of its strong impact on electron transport properties including superconductivity. Despite its ubiquitousness in correlated systems, the occurrence of charge ordering in iron-based superconductors is still unresolved. Here we use scanning tunneling microscopy to reveal a long-range charge-stripe order and a highly anisotropic dispersion of electronic states in the ground state of stoichiometric FeTe, the parent compound of the Fe(Te, Se, S) superconductor family. The forma…
This article will give an overview on both theoretical and experimental developments concerning states with lattice symmetry breaking in the cuprate high-temperature superconductors. Recent experiments have provided evidence for states with broken rotation as well as translation symmetry, and will be discussed in terms of nematic and stripe physics. Of particular importance here are results obtained using the techniques of neutron and x-ray scattering and scanning tunneling spectroscopy. Ideas on the origin of lattice-symmetry-broken states will be reviewed, and effective models accounting for…
Quantum oscillations have revealed the presence of a small pocket in the Fermi surface of the cuprate superconductor YBCO, whose nature and origin are the subject of much debate. Interpretations include electron and hole pockets; scenarios include Fermi-surface reconstruction by antiferromagnetism, d-density-wave order, and stripe order. Here we report quantum oscillations in the Seebeck and Nernst coefficients of YBCO and show, from the magnitude and sign of the Seebeck coefficient, that they come from an electron pocket. Using measurements of the Seebeck coefficient as a function of hole dop…
The Nernst effect in metals is highly sensitive to two kinds of phase transition: superconductivity and density-wave order. The large positive Nernst signal observed in hole-doped high-Tc superconductors above their transition temperature Tc has so far been attributed to fluctuating superconductivity. Here we show that in some of these materials the large Nernst signal is in fact caused by stripe order, a form of spin / charge modulation which causes a reconstruction of the Fermi surface. In LSCO doped with Nd or Eu, the onset of stripe order causes the Nernst signal to go from small and negat…
The phase diagram of superconducting copper oxides is calculated as a function of doping based on a theory of dynamic stripe induced superconductivity. The two major conclusions from the theory and the numerical analysis are that T* (the pseudogap onset) and Tc (superconductivity onset) are correlated through the pseudogap, which induces a gap in the single particle energies that persists into the superconducting state. By decreasing the doping the pseudogap Delta* increases and T* increases, but when Delta* exceeds a certain critical value the superconducting transition temperature Tc is supp…
A pseudogap (PG) was introduced by Mott to describe a state of matter which has a minimum in the density of states at the Fermi level, deep enough for states to become localized. It can arise either from Coulomb repulsion between electrons, or due to an incipient charge or spin order, or a combination of the two. These states are rapidly fluctuating in time with random phase, so they are hard to observe experimentally. Here we present the first coherent quench measurements of the dynamical transition to the pseudogap state in the prototype high temperature superconductor Bi$_{2}$Sr$_{2}$CaCu$_…
Based on the newly developed real-space spin-parallel pairing and superconducting theory, we explore a simple explanation for the observed checkerboard patterns in hole-doped Ca{2-x}Na{x}CuO{2}Cl{2} cuprate superconductor. At a hole concentration x=1/8=0.125, the analytical results show that there exists a phase competition between non-superconducting 4a*4b (a=b) checkerboard phase and 8a*2c superconducting vortex phase, where a, b and c are the lattice constants of the superconductor. At a lower hole concentration x=1/16=0.0625, it is revealed that the metastable 4\sqrt{2}a*4\sqrt{2}a checker…
The details of the pseudogap origin and other gap related properties discussed earlier for cuprates, in the framework of the paired cluster (PC) model, using three dimensional (3D) electronic density of states (DOS), are shown to remain valid even when a two dimensional (2D) cuprate electronic DOS is used. Similarly the stripe phase description is also shown to be similar for the 3D and 2D cases. These results confirm the PC model's capability in explaining the high-Tc superconductivity properties.
Recent experiments indicate that static stripe-like charge order is generic to the hole-doped copper oxide superconductors and competes with superconductivity. Here we show that a similar type of charge order is present in La$_{5/3}$Sr$_{1/3}$CoO$_4$, an insulating analogue of the copper oxide superconductors containing cobalt in place of copper. The stripe phase we have detected is accompanied by short-range, quasi-one-dimensional, antiferromagnetic order, and provides a natural explanation for the distinctive hourglass shape of the magnetic spectrum previously observed in neutron scattering …
In the stripe-ordered state of a strongly-correlated two-dimensional electronic system, under a set of special circumstances, the superconducting condensate, like the magnetic order, can occur at a non-zero wave-vector corresponding to a spatial period double that of the charge order. In this case, the Josephson coupling between near neighbor planes, especially in a crystal with the special structure of La_{2-x}Ba_xCuO_4, vanishes identically. We propose that this is the underlying cause of the dynamical decoupling of the layers recently observed in transport measurements at x=1/8.
In the copper-oxide parent compounds of the high-transition-temperature superconductors, the valence electrons are localized, one per copper site, due to strong intraatomic Coulomb repulsion. A symptom of the localization is antiferromagnetism, where the spins of localized electrons alternate between up and down. The superconductivity appears when mobile 'holes' are doped into this insulating state, and it coexists with antiferromagnetic fluctuations. In one approach to the coexistence, the holes are believed to self-organize into 'stripes' that alternate with antiferromagnetic (insulating) re…
Recent transport experiments in the original cuprate high temperature superconductor, {La$_{2-x}$Ba$_x$CuO$_4$}, have revealed a remarkable sequence of transitions and crossovers which give rise to a form of dynamical dimensional reduction, in which a bulk crystal becomes essentially superconducting in two directions while it remains poorly metallic in the third. We identify these phenomena as arising from a distinct new superconducting state, the "striped superconductor," in which the superconducting order is spatially modulated, so that its volume average value is zero. Here, in addition to …
Recently, a strongly enhanced xy anisotropy of magnetic excitations was observed in YBa_2Cu_3O_y (YBCO_y) with y=6.45 and Tc=35 K [Science 319, 597 (2008)]. Unlike the observation in YBCO_6.6 and YBCO_6.85, the anisotropy grows to be pronounced at lower temperature and at lower energy, and is not suppressed by the onset of superconductivity. We propose that the effect of singlet pairing is substantially reduced in YBCO_6.45. This reduction concomitantly enhances an order competing with singlet pairing, a strong tendency of the so-called d-wave Pomeranchuk instability, leading to the magnetic e…
Based on the time-dependent Gutzwiller approximation for the extended Hubbard model we calculate the energy and momentum dependence of spin excitations for striped ground states. Our starting point correctly reproduces the observed doping dependence of the incommensurability in La-based cuprates and the dispersion of magnetic modes in the insulating parent compound. This allows us to make quantitative predictions for the doping evolution of the dispersion of magnetic modes in the stripe phase including the energy and intensity of the resonance peak as well as the velocity of the spin-wave like…
The coupling of spin, charge and lattice degrees of freedom results in the emergence of novel states of matter across many classes of strongly correlated electron materials. A model example is unconventional superconductivity, which is widely believed to arise from the coupling of electrons via spin excitations. In cuprate high-temperature superconductors, the interplay of charge and spin degrees of freedom is also reflected in a zoo of charge and spin-density wave orders that are intertwined with superconductivity. A key question is whether the different types of density waves merely coexist …
The success of models of coupled two-leg spin ladders in describing the magnetic excitation spectrum of La_[2-x]Ba_xCuO_4 has been widely interpreted as evidence for bond-centered stripes. Here, we determine the magnetic coupling induced by the charge stripes between bond- or site-centered spin stripes modeled by two- or three-leg ladders, respectively. We find that only the site-centered models order. We further report excellent agreement of a fully consistent analysis of coupled three-leg ladders using a spin wave theory of bond operators with the experiment.
We report on 139La and 63Ccu NMR/NQR measurements in the high-Tc superconductor La1.90Sr0.10CuO4 with Tc=26.5 K. Spin fluctuations probed by 139La spin-lattice relaxation (T1), continuously slow down on cooling through Tc. We argue that spin-freezing and superconductivity are bulk effects in this sample. Thus, both phenomena have to coexist microscopically. The distribution of 139La T1 values at low temperature reveals a wide spread of spin fluctuation frequencies in CuO2 planes. A simple estimate shows that Cu nuclei at sites where electronic fluctuations are the slowest are not observable be…
We examine the thermodynamic and thermoelectric properties in the stripe phase of high-Tc cuprates, by using the finite-temperature Lanczos technique for the t-J model with a potential that stabilizes vertical charge stripes. When the stripe potential is turned on, the entropy is suppressed as a consequence of the formation of one-dimensional charge stripes accompanied by an enhancement of antiferromagnetic spin correlation in the spin domains. The stripe formation leads also to weak temperature dependence of the chemical potential, leading to the suppression of the thermoelectric power. The s…
Within the stripe quantum critical point scenario for high $T_c$ superconductors, we point out the possible direct contribution of charge collective fluctuations to the optical absorption and to the d.c. resistivity.
A new model of high-T_c superconductivity (HTSC) has been developed by us which we find is able to explain almost all the HTSC properties, like high T_c, origin and nature of NSPG (normal state pseudogap), anomalous superconducting state (SS) energy gap (SSEG) properties, absence of NMR spin relaxation rate coherence peak, existence of NSPG below T_c, nature of high-T_c magnetic superconductivity, external magnetic field (H) dependence of the NSPG formation temperature T*, etc. The coexistence of NSPG and SS BCS energy gap (EG), below T_C, has been predicted by the model before its experimenta…
We calculate RPA-BCS based spin resonance spectra of newly discovered iron-selenide superconductor using two orbitals tight-binding (TB) model. The slightly squarish electron pocket Fermi surfaces (FSs) at $(π,0)/(0,π)-$momenta produce leading interpocket nesting instability at incommensurate vector $q\sim(π,0.5π)$ in the normal state static susceptibility, pinning a strong stripe-like spin-density wave (SDW) or antiferromagnetic (AFM) order at some critical value of $U$. The same nesting also induces $d_{x^2-y^2}-$pairing. The superconducting (SC) gap is nodeless and isotropic on the FSs as t…
Structural phase separation in the form of partially disordered stripes, with characteristic length scales in the nanometer range, is observed for superconducting BaPb$_{1-x}$Bi$_x$O$_3$. The evolution of the superconducting coherence length with composition relative to the size of these stripes suggests an important role of the nanostructure in determining the shape of the superconducting dome. It is proposed that the maximum $T_c$ is determined by a kind of "optimal inhomogeneity", characterized by a crossover from an inhomogeneous macroscopic superconductor to a granular superconductor for …
We study the interplay between the stripe order and the superconducting order in a strongly coupled striped superconductor using gauge/gravity duality. In particular, we study the effects of inhomogeneity introduced by the stripe order on the superconducting transition temperature beyond the mean field level by including the effects of backreaction onto the spacetime geometry in the dual gravitational picture. We find that inhomogeneity \emph{enhances} the critical temperature relative to its value for the uniform system.
We study the single-particle spectral properties of electrons coupled to quasicritical charge and spin fluctuations close to a stripe-phase, which is governed by a Quantum Critical Point near optimum doping. We find that spectral weight is transferred from the quasiparticle peak to incoherent dispersive features. As a consequence the distribution of low-laying spectral weight is modified with respect to the quasiparticle Fermi surface. The interplay of charge and spin fluctuations reproduces features of the observed Fermi surface, such as the asymmetric suppression of spectral weight near the …
The non-uniform ground state of the two-dimensional three-band Hubbard model for the oxide high-Tc superconductors is investigated using a variational Monte Carlo method. We examine the effect produced by holes doped into the antiferromagnetic (AF) background in the underdoped region. It is shown that the AF state with spin modulations and stripes is stabilized du to holes travelling in the CuO plane. The structures of the modulated AF spins are dependent upon the parameters used in the model. The effect of the boundary conditions is reduced for larger systems. We show that there is a region w…
Emergent Fano-Feshbach resonance in two-band superconductors with an incipient quasi-flat band: Enhanced critical temperature evading particle-hole fluctuations
该研究针对两带超导中粒子-空穴涨落抑制配对的问题，通过GMB形式体系研究准平带体系，发现涌现的Fano-Feshbach共振可规避涨落抑制、提升$T_{\ m c}$，揭示多带超导的普适特征。
A spin density-wave quantum critical point (QCP) is the central organizing principle of organic, iron-pnictide, heavy-fermion and electron-doped cuprate superconductors. It accounts for the superconducting Tc dome, the non-Fermi-liquid resistivity, and the Fermi-surface reconstruction. Outside the magnetically ordered phase above the QCP, scattering and pairing decrease in parallel as the system moves away from the QCP. Here we argue that a similar scenario, based on a stripe-order QCP, is a central organizing principle of hole-doped cuprate superconductors. Key properties of Eu-LSCO, Nd-LSCO …
Upon doping, Mott insulators often exhibit symmetry breaking where charge carriers and their spins organize into patterns known as stripes. For high-Tc superconducting cuprates, stripes are widely suspected to exist in a fluctuating form. Here, we use numerically exact determinant quantum Monte Carlo calculations to demonstrate dynamical stripe correlations in the three-band Hubbard model, which represents the local electronic structure of the Cu-O plane. Our results, which are robust to varying parameters, cluster size, and boundary condition, strongly support the interpretation of a variety …
We study the conductivity of a strongly coupled striped superconductor using gauge/gravity duality (holography). The study is done analytically, in the large modulation regime. We show that the optical conductivity is inhomogeneous but isotropic at low temperatures. Near but below the critical temperature, we calculate the conductivity analytically at small frequency ω, and find it to be both inhomogeneous and anisotropic. The anisotropy is imaginary and scales like 1/ω. We also calculate analytically the speed of the second sound and the thermodynamic susceptibility.
