Finite Temperature Correlation Functions Research Articles

Finite-Temperature Correlation Functions Obtained from Combined Real- and Imaginary-Time Propagation of Variational Thawed Gaussian Wavepackets.

We apply the so-called variational Gaussian wavepacket approximation (VGA) for conducting both real- and imaginary-time dynamics to calculate thermal correlation functions. By considering strongly anharmonic systems, such as a quartic potential and a double-well potential at high and low temperatures, it is shown that this method is partially able to account for tunneling. This is contrary to other popular many-body methods, such as ring polymer molecular dynamics and the classical Wigner method, which fail in this respect. It is a historical peculiarity that no one has considered the VGA method for representing both the Boltzmann operator and the real-time propagation. This method should be well suited for molecular systems containing many atoms.

Renormalisation group flows of deformed SYK models

We explore computationally tractable deformations of the SYK model. The deformed theories are described by the sum of two SYK Hamiltonians with differing numbers, q and q~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overset{\\sim }{q} $$\\end{document}, of interacting fermions. In the large N limit, employing analytic and numerical tools, we compute finite temperature correlation functions and thermodynamic quantities. We identify a novel analytically solvable model in the large q limit. We find that, under certain circumstances, the thermal RG flow in the strongly coupled infrared phase exhibits two regions of linear-in-temperature entropy, which we interpret in terms of Schwarzian actions. Using conformal perturbation theory we compute the leading relevant correction away from the intermediate near-conformal fixed point. Holographic spacetimes in two spacetime dimensions that reproduce the thermodynamics of the microphysical theory are discussed. These are flow geometries that interpolate between two Euclidean near-AdS2 spacetimes with different radii. The Schwarzian soft mode corresponding to the AdS2 region in the deep interior resides entirely within the geometric regime.

Spectral and thermodynamic properties of the Holstein polaron: Hierarchical equations of motion approach

We develop a hierarchical equations of motion (HEOM) approach to compute real-time single-particle correlation functions and thermodynamic properties of the Holstein model at finite temperature. We exploit the conservation of the total momentum of the system to formulate the momentum-space HEOM whose dynamical variables explicitly keep track of momentum exchanges between the electron and phonons. Our symmetry-adapted HEOM enable us to overcome the numerical instabilities inherent to the commonly used real-space HEOM. The HEOM method is then used to study the spectral function and thermodynamic quantities of chains containing up to ten sites. The HEOM results compare favorably to existing literature. To provide an independent assessment of the HEOM approach and to gain insight into the importance of finite-size effects, we devise a quantum Monte Carlo (QMC) procedure to evaluate finite-temperature single-particle correlation functions in imaginary time and apply it to chains containing up to twenty sites. QMC results reveal that finite-size effects are quite weak, so that the results on 5 to 10-site chains, depending on the parameter regime, are representative of larger systems. A detailed comparison between the HEOM and QMC data place our HEOM method among reliable methods to compute real-time finite-temperature correlation functions in parameter regimes ranging from low- to high-temperature, and weak- to strong-coupling regime.

Spectral representation of the shear viscosity for local scalar QFTs at finite temperature

In local scalar quantum field theories (QFTs) at finite temperature correlation functions are known to satisfy certain non-perturbative constraints, which for two-point functions in particular implies the existence of a generalisation of the standard K\"{a}ll\'{e}n-Lehmann representation. In this work, we use these constraints in order to derive a spectral representation for the shear viscosity arising from the thermal asymptotic states, $\eta_{0}$. As an example, we calculate $\eta_{0}$ in $\phi^{4}$ theory, establishing its leading behaviour in the small and large coupling regimes.

Hamiltonian deformations in quantum mechanics, TT¯ , and the SYK model

Motivated by $T\bar T$, we introduce and study a wide class of solvable deformations of quantum-mechanical theories. These deformations map the Hamiltonian to a function of itself. We solve these theories by computing all finite-temperature correlation functions of the deformed theory in terms of the correlators of the undeformed theory. Applications to AdS/CFT, SYK, and the Schwarzian theory are considered. We write down the deformed Schwarzian action for an arbitrary Hamiltonian deformation and find that the maximal Lyapunov exponent is unchanged.

Adding new branches to the \u201cChristmas tree\u201d of the quasinormal spectrum of black branes

In holography, quasinormal spectra of black branes coincide with the poles of retarded finite-temperature correlation functions of a dual quantum field theory in the limit of infinite number of relevant degrees of freedom such as colours. For asymptotically antide Sitter backgrounds, the spectra form a characteristic pattern in the complex frequency plane, colloquially known as the “Christmas tree”. At infinite coupling, the tree has only one pair of branches. At large but finite coupling, the branches become more dense and lift up towards the real axis, consistent with the expectation of forming a branch cut in the limit of zero coupling. However, it is known that at zero coupling, the corresponding correlators generically have not one but multiple branch cuts separated by intervals proportional to the Matsubara frequency. This suggests the existence of multiple branches of the “Christmas tree” spectrum in dual gravity. In this note, we show numerically how these additional branches of the spectrum can emerge from the dual gravitational action with higher-derivative terms. This phenomenon appears to be robust, yet, reproducing the expected weak coupling behaviour of the correlators quantitatively implies the existence of certain constraints on the coefficients of the higher-derivative terms of the dual gravity theory.

Power-law out of time order correlation functions in the SYK model

We evaluate the finite temperature partition sum and correlation functions of the Sachdev–Ye–Kitaev (SYK) model. Starting from a recently proposed mapping of the SYK model onto Liouville quantum mechanics, we obtain our results by exact integration over conformal Goldstone modes reparameterizing physical time. Perhaps, the least expected result of our analysis is that at time scales proportional to the number of particles the out of time order correlation function crosses over from a regime of exponential decay to a universal t−6 power-law behavior.

Small black holes in global AdS spacetime

We study finite temperature correlation functions and quasinormal modes in a strongly coupled conformal field theory holographically dual to a small black hole in global Anti-de Sitter spacetime. Upon variation of the black hole radius, our results smoothly interpolate between known limits corresponding to large black holes and thermal AdS space. This implies that the quantities are continuous functions of energy density in the microcanonical ensemble, thus smoothly connecting the deconfined and confined phases that are separated by a first order phase transition in the canonical description.

Thermal form factors of the XXZ chain and the large-distance asymptotics of its temperature dependent correlation functions

We derive expressions for the form factors of the quantum transfer matrix of the spin- XXZ chain which are suitable for taking the infinite Trotter number limit. These form factors determine the finitely many amplitudes in the leading asymptotics of the finite-temperature correlation functions of the model. We consider form factor expansions of the longitudinal and transversal two-point functions. Remarkably, the formulae for the amplitudes are in both cases of the same form. We also explain how to adapt our formulae to the description of ground-state correlation functions of the finite chain. The usefulness of our novel formulae is demonstrated by working out explicit results in the high- and low-temperature limits. We obtain, in particular, the large-distance asymptotics of the longitudinal two-point functions for small temperatures by summing up the asymptotically most relevant terms in the form factor expansion of a generating function of the longitudinal correlation functions. As expected, the leading term in the expansion of the corresponding two-point functions is in accordance with conformal field theory predictions. Here it is obtained for the first time by a direct calculation.

Finite temperature correlation functions from discrete functional equations

We present a new approach to the static finite temperature correlation functions of the Heisenberg chain based on functional equations. An inhomogeneous generalization of the n-site density operator is considered. The lattice path integral formulation with a finite but arbitrary Trotter number allows us to derive a set of discrete functional equations with respect to the spectral parameters. We show that these equations yield a unique characterization of the density operator. Our functional equations are a discrete version of the reduced q-Knizhnik–Zamolodchikov equations which played a central role in the study of the zero temperature case. As a natural result, and independent of the arguments given by Jimbo et al (2009 J. Phys. A: Math. Theor. 42 304018), we prove that the inhomogeneous finite temperature correlation functions have the same remarkable structure as for zero temperature: they are a sum of products of nearest-neighbour correlators.

Prethermalization in quenched spinor condensates

Motivated by recent experiments, we consider the dynamics of spin-1 spinor condensates after a quantum quench from the polar to ferromagnetic state from varying the quadratic Zeeman field $q$. We apply the truncated Wigner approximation (TWA) to the spinor system, including all spatial and spin degrees of freedom. For short times, we find full agreement with the linearized Bogoliubov analysis. For longer times, where the Bogoliubov theory fails, we find that the system reaches a quasisteady prethermalized state. We compute the Bogoliubov spectrum about the ferromagnetic state with general $q$ and show that the resulting finite-temperature correlation functions grossly disagree with the full TWA results, thus indicating that the system does not thermalize even over very long time scales. Finally, we show that the absence of thermalization over realistic time scales is consistent with calculations of Landau damping rates of excitations in the finite-temperature condensate.

Baryon asymmetry of the Universe without Boltzmann or Kadanoff-Baym equations

We present a formalism that allows the computation of the baryon asymmetry of the universe from first principles of statistical physics and quantum field theory that is applicable to certain types of beyond the Standard Model physics (such as the neutrino Minimal Standard Model -- $\nu$MSM) and does not require the solution of Boltzmann or Kadanoff-Baym equations. The formalism works if a thermal bath of Standard Model particles is very weakly coupled to a new sector (sterile neutrinos in the $\nu$MSM case) that is out-of-equilibrium. The key point that allows a computation without kinetic equations is that the number of sterile neutrinos produced during the relevant cosmological period remains small. In such a case, it is possible to expand the formal solution of the von Neumann equation perturbatively and obtain a master formula for the lepton asymmetry expressed in terms of non-equilibrium Wightman functions. The master formula neatly separates CP-violating contributions from finite temperature correlation functions and satisfies all three Sakharov conditions. These correlation functions can then be evaluated perturbatively; the validity of the perturbative expansion depends on the parameters of the model considered. Here we choose a toy model (containing only two active and two sterile neutrinos) to illustrate the use of the formalism, but it could be applied to other models.

Form factor expansion for thermal correlators

We consider finite temperature correlation functions in massive integrable quantum fieldtheory. Using a regularization by putting the system in finite volume, we develop a novelapproach (based on multi-dimensional residues) to the form factor expansionfor thermal correlators. The first few terms are obtained explicitly in theorieswith diagonal scattering. We also discuss the validity of the LeClair–Mussardoproposal.

First Principles Calculation of the Baryon Asymmetry of the Universe: Generalities and Application

We present a formalism that allows the computation of the lepton asymmetry of the universe from first principles of statistical physics and quantum field theory (this lepton asymmetry is then converted to a baryon asymmetry via sphaleron processes). This formalism includes a thermal bath of Standard Model particles (active neutrinos) coupled to a new sector that is out-of-equilibrium (sterile neutrinos). The key point that allows a first principles computation is that the number of sterile neutrinos produced during the relevant cosmological period remains small (we assume zero sterile neutrinos initially). In such a case, it is possible to expand the formal solution of Liouville's equation perturbatively and obtain a master formula for the lepton asymmetry expressed in terms of non-equilibrium Wightman functions. The master formula neatly separates CP-violating contributions from finite temperature correlation functions and satisfies all three Sakharov conditions. These correlation functions can then be evaluated perturbatively; the validity of the perturbative expansion depends on the parameters of the model considered. Here we choose the νMSM (i.e. a minimal extension of the Standard Model that includes three generations of sterile neutrinos with masses of the order of the electroweak scale) to illustrate the use of the formalism, but it could in principle be applied to other models.

Factorization of the finite temperature correlation functions of the XXZ chain in a magnetic field

We present a conjecture for the density matrix of a finite segment of the XXZ chain coupled to a heat bath and to a constant longitudinal magnetic field. It states that the inhomogeneous density matrix, conceived as a map which associates with every local operator its thermal expectation value, can be written as the trace of the exponential of an operator constructed from weighted traces of the elements of certain monodromy matrices related to and only two transcendental functions pertaining to the one-point function and the neighbour correlators, respectively. Our conjecture implies that all static correlation functions of the XXZ chain are polynomials in these two functions and their derivatives with coefficients of purely algebraic origin.

Form factors in finite volume II: Disconnected terms and finite temperature correlators

Continuing the investigation started in a previous work, we consider form factors of integrable quantum field theories in finite volume, extending our investigation to matrix elements with disconnected pieces. Numerical verification of our results is provided by truncated conformal space approach. Such matrix elements are important in computing finite temperature correlation functions, and we give a new method for generating a low temperature expansion, which we test for the one-point function up to third order.

Finite-Temperature Form Factors: a Review

We review the concept of finite-temperature form factor that was introduced re- cently by the author in the context of the Majorana theory. Finite-temperature form factors can be used to obtain spectral decompositions of finite-temperature correlation functions in a way that mimics the form-factor expansion of the zero temperature case. We develop the concept in the general factorised scattering set-up of integrable quantum field theory, list certain expected properties and present the full construction in the case of the massive Majorana theory, including how it can be applied to the calculation of correlation functions in the quantum Ising model. In particular, we include the twisted construction, which was not developed before and which is essential for the application to the quantum Ising model.

Dynamical correlations and quantum phase transition in the quantum Potts model

We present a detailed study of the finite temperature dynamical properties of the quantum Potts model in one dimension.Quasiparticle excitations in this model have internal quantum numbers, and their scattering matrix {\gf deep} in the gapped phases is shown to take a simple {\gf exchange} form in the perturbative regimes. The finite temperature correlation functions in the quantum critical regime are determined using conformal invariance, while {\gf far from the quantum critical point} we compute the decay functions analytically within a semiclassical approach of Sachdev and Damle [K. Damle and S. Sachdev, Phys. Rev. B \textbf{57}, 8307 (1998)]. As a consequence, decay functions exhibit a {\em diffusive character}. {\gf We also provide robust arguments that our semiclassical analysis carries over to very low temperatures even in the vicinity of the quantum phase transition.} Our results are also relevant for quantum rotor models, antiferromagnetic chains, and some spin ladder systems.

Inheritance principle and non-renormalization theorems at finite temperature

We present a general proof of an “inheritance principle” satisfied by a weakly coupled SU(N) gauge theory with adjoint matter on a class of compact manifolds (like S3). In the large N limit, finite temperature correlation functions of gauge invariant single-trace operators in the low temperature phase are related to those at zero temperature by summing over images of each operator in the Euclidean time direction. As a consequence, various non-renormalization theorems of N=4 super-Yang–Mills theory on S3 survive at finite temperature despite the fact that the conformal and supersymmetries are both broken.

Excursions beyond the horizon: black hole singularities in Yang-Mills theories (I)

We study black hole singularities in the AdS/CFT correspondence. These singularities show up in CFT in the behavior of finite-temperature correlation functions. We first establish a direct relation between space-like geodesics in the bulk and momentum space Wightman functions of CFT operators of large dimensions. This allows us to probe the regions inside the horizon and near the singularity using the CFT. Information about the black hole singularity is encoded in the exponential falloff of finite-temperature correlators at large imaginary frequency. We construct new gauge invariant observables whose divergences reflect the presence of the singularity. We also find a UV/UV connection that governs physics inside the horizon. Additionally, we comment on the possible resolution of the singularity.