Thermal Density Matrices Research Articles

Marginal quenches and drives in Tomonaga-Luttinger liquids

We study Tomonaga-Luttinger liquids thrown out of equilibrium by marginal deformations in the form of interaction modulations. This is modeled by quenching or periodically driving the Luttinger parameter or, equivalently, the compactification radius of the free boson conformal field theory between two different values. We obtain exact analytical results for the evolution of the Loschmidt echo and observables such as the particle and energy densities. Starting from generic initial states, the quench dynamics are shown to exhibit revivals and temporal orthogonalities. For the periodic drive, we show stability or instability of time-evolved physical quantities dependent on the drive parameters. We also compare the corresponding marginally deformed thermal density matrices by non-perturbatively evaluating their Rényi divergence as a Euclidean quench. All the dynamics are shown to be crucially dependent on the ratio of the Luttinger parameters, which corresponds to the Zamolodchikov distance in the space of marginal deformations. Our setup is equivalently interpreted as the dynamics of the bosonic string upon instantaneous changes of the target-space radius.

Boiling Quantum Vacuum: Thermal Subsystems from Ground-State Entanglement

In certain special circumstances, such as in the vicinity of a black hole or in a uniformly accelerating frame, vacuum fluctuations appear to give rise to a finite-temperature environment. This effect, currently without experimental confirmation, can be interpreted as a manifestation of quantum entanglement after tracing out vacuum modes in an unobserved region. In this work, we identify a class of experimentally accessible quantum systems where thermal density matrices emerge from vacuum entanglement. We show that reduced density matrices of lower-dimensional subsystems embedded in $D$-dimensional gapped Dirac fermion vacuum, either on a lattice or continuum, have a thermal form with respect to a lower-dimensional Dirac Hamiltonian. Strikingly, we show that vacuum entanglement can even conspire to make a subsystem of a gapped system at zero temperature appear as a hot gapless system. We propose concrete experiments in cold atom quantum simulators to observe the vacuum entanglement induced thermal states.

Saturation of thermal complexity of purification

We purify the thermal density matrix of a free harmonic oscillator as a two-mode squeezed state, characterized by a squeezing parameter and squeezing angle. While the squeezing parameter is fixed by the temperature and frequency of the oscillator, the squeezing angle is otherwise undetermined, so that the complexity of purification is obtained by minimizing the complexity of the squeezed state over the squeezing angle. The resulting complexity of purification of the thermal state is minimized at non-zero values of the squeezing angle and saturates to an order one number at low frequencies, indicating that there is no additional operator cost required to build thermal mixed states when the oscillator probes length scales that are large compared to the thermal length scale. We also review applications in which thermal density matrices arise for quantum fields on curved spacetimes, including Hawking radiation and a simple model of decoherence of cosmological density perturbations in the early Universe. The complexity of purification for these mixed states also saturates as a function of the effective temperature, which may have interesting consequences for the quantum information stored in these systems.

Multipartite entanglement in spin chains and the hyperdeterminant

A way to characterize multipartite entanglement in pure states of a spin chain with sites and local dimension is by means of the Cayley hyperdeterminant. The latter quantity is a polynomial constructed with the components of the wave function which is invariant under local unitary transformation. For spin 1/2 chains (i.e. ) with and sites, the hyperdeterminant coincides with the concurrence and the tangle respectively. In this paper we consider spin chains with sites where the hyperdeterminant is a polynomial of degree 24 containing around terms. This huge object can be written in terms of more simple polynomials and of degrees 8 and 12 respectively. Correspondingly we compute , and the hyperdeterminant for eigenstates of the following spin chain Hamiltonians: the transverse Ising model, the XXZ Heisenberg model and the Haldane–Shastry model. Those invariants are also computed for random states, the ground states of random matrix Hamiltonians in the Wigner–Dyson Gaussian ensembles and the quadripartite entangled states defined by Verstraete et al in 2002. Finally, we propose a generalization of the hyperdeterminant to thermal density matrices. We observe how these polynomials are able to capture the phase transitions present in the models studied as well as a subclass of quadripartite entanglement present in the eigenstates.

Distilling momentum-space entanglement in Luttinger liquids at finite temperature

While much is known about the entanglement characteristics of ground states, the properties of reduced thermal density matrices have received significantly less attention. Here we investigate the entanglement content of reduced thermal density matrices for momentum-space bipartitioning in Luttinger liquids using analytical and numerical methods. The low lying part of its spectrum contains an "entanglement gap", which persists up to temperatures comparable to the level spacing. With increasing temperature, the low energy modes acquire dispersion and resemble to those in the physical Hamiltonian with an enhanced effective temperature. The momentum-space entanglement is carried by high energy modes (compared to temperature), featuring a completely flat spectrum. The von-Neumann entropy increases with temperature with a universal Sommerfeld coefficient. The momentum-space entanglement Hamiltonian turns out to be as universal as the physical Hamiltonian.

Probabilistic Q-function distributions in fermionic phase-space

We obtain a positive probability distribution or Q-function for an arbitrary fermionic many-body system. This is different to previous Q-function proposals, which were either restricted to a subspace of the overall Hilbert space, or used Grassmann methods that do not give probabilities. The fermionic Q-function obtained here is constructed using normally ordered Gaussian operators, which include both non-interacting thermal density matrices and BCS states. We prove that the Q-function exists for any density matrix, is real and positive, and has moments that correspond to Fermi operator moments. It is defined on a finite symmetric phase-space equivalent to the space of real, antisymmetric matrices. This has the natural SO(2M) symmetry expected for Majorana fermion operators. We show that there is a physical interpretation of the Q-function: it is the relative probability for observing a given Gaussian density matrix. The distribution has a uniform probability across the space at infinite temperature, while for pure states it has a maximum value on the phase-space boundary. The advantage of probabilistic representations is that they can be used for computational sampling without a sign problem.

“Light-Cone” Dynamics After Quantum Quenches in Spin Chains

Signal propagation in the nonequilibrium evolution after quantum quenches has recently attracted much experimental and theoretical interest. A key question arising in this context is what principles, and which of the properties of the quench, determine the characteristic propagation velocity. Here we investigate such issues for a class of quench protocols in one of the central paradigms of interacting many-particle quantum systems, the spin-1/2 Heisenberg XXZ chain. We consider quenches from a variety of initial thermal density matrices to the same final Hamiltonian using matrix product state methods. The spreading velocities are observed to vary substantially with the initial density matrix. However, we achieve a striking data collapse when the spreading velocity is considered to be a function of the excess energy. Using the fact that the XXZ chain is integrable, we present an explanation of the observed velocities in terms of "excitations" in an appropriately defined generalized Gibbs ensemble.

Out-of-equilibrium correlated systems: Bipartite entanglement as a probe of thermalization

Thermalization plays a central role in out-of-equilibrium physics of ultracold atoms or electronic transport phenomena. On the other hand, entanglement concepts have proven to be extremely useful to investigate quantum phases of matter. Here, it is argued that bipartite entanglement measures provide key information on out-of-equilibrium states and might therefore offer stringent thermalization criteria. This is illustrated by considering a global quench in an (extended) $\mathit{XXZ}$ spin-1/2 chain across its (zero-temperature) quantum critical point. A nonlocal bipartition of the chain preserving translation symmetry is proposed. The time evolution after the quench of the reduced density matrix of the half-system is computed and its associated (time-dependent) entanglement spectrum is analyzed. Generically, the corresponding entanglement entropy quickly reaches a ``plateau'' after a short transient regime. However, in the case of the integrable $\mathit{XXZ}$ chain, the low-energy entanglement spectrum still reveals strong time fluctuations. In addition, its infinite-time average shows strong deviations from the spectrum of a Boltzmann thermal density matrix. In contrast, when the integrability of the model is broken (by small next-nearest-neighbor couplings), the entanglement spectra of the time average and thermal density matrices become remarkably similar.

Development of a new variational approach for thermal density matrices

A McLachlan-type variational principle is derived for thermal density matrices. In this approach, the trace of the mean square of the differences between the derivatives of the exact and model density matrices is minimized with respect to the parameters in the model Hamiltonian. Applications to model anharmonic systems in the independent particle model show that the method can provide thermodynamic state functions accurately (within 5% of the converged basis set results) and at the same level of accuracy as the results using Feynman-Gibbs-Bogoliubov variational principle at this level of approximation.

A thermal self-consistent field theory for the calculation of molecular vibrational partition functions

A new approach for the calculation of anharmonic molecular vibrational partition functions is developed based on a separable ansatz to the thermal density matrix. The parameters appearing in the effective single particle Hamiltonians that generate the thermal density matrices are determined variationally. The resulting equations are the thermal analogs of the vibrational self-consistent field approximation. The method has the formal property that the free energy calculated by this approach is an upper bound to the exact free energy. Thermodynamic quantities calculated by this approach are generally in good agreement with the results of numerically converged calculations. This approach is more efficient than the standard sum over state approaches in that the computational resources scale with N(4) where N is the number of vibrational degrees of freedom. Thus it can be applied to fairly large systems.

Bures metric over thermal state manifolds and quantum criticality

We analyze the Bures metric over the manifold of thermal density matrices for systems featuring a zero temperature quantum phase transition. We show that the quantum critical region can be characterized in terms of the temperature scaling behavior of the metric tensor itself. Furthermore, the analysis of the metric tensor when both temperature and an external field are varied, allows one to complement the understanding of the phase diagram including crossover regions which are not characterized by any singular behavior. These results provide a further extension of the scope of the metric approach to quantum criticality.

Quantum Monte Carlo simulations of exciton condensates

We have studied scattering states and thermodynamic properties of electron–hole systems. Starting from the constituent electrons and holes and Coulomb interactions, we have used quantum Monte Carlo simulation techniques to sample properties of wavefunctions and thermal density matrices. We have studied three types of systems: (1) the scattering of two excitons, with full-quantum treatment of the four constituent particles, (2) the thermodynamic equilibrium of 14 electron–hole pairs having two spin states for each particle, which form excitons and biexcitons at low temperatures, and (3) the thermodynamic equilibrium of 27 spin-polarized electron–hole pairs, which form a dilute exciton gas that undergoes Bose condensation at low temperatures. We compare our results with predictions of the Saha equation for exciton and biexciton formation, and Bogoliubov theory for the energy of the dilute Bose gas of excitons. We also discuss the outlook for future quantum Monte Carlo simulations on these systems.

Many-body partition function and thermal hartree-fock approximations

The hierarchy equations for the quantum thermal density matrices of an assembly of interacting particles in an external potential are derived in a compact form. This approach can complement and finally replace the phenomenological treatment of the thermodynamic properties of quantum gasses, liquids, and solids. We also derive a temperature Hartree-Fock approximation for fermions based on the symmetry properties of the exact hierarchy and finally we discuss the present approximation in relation to that of Kaplan and Argyres [T. A. Kaplan and P. N. Argyres, Ann. Phys. (N.Y.) 92, 1 (1975)].

Variational density matrix method for warm, condensed matter: application to dense hydrogen

A variational principle for optimizing thermal density matrices is introduced. As a first application, the variational many-body density matrix is written as a determinant of one-body density matrices, which are approximated by Gaussians with the mean, width, and amplitude as variational parameters. The method is illustrated for the particle in an external field problem, the hydrogen molecule and dense hydrogen where the molecular, the dissociated, and the plasma regime are described. Structural and thermodynamic properties (energy, equation of state, and shock Hugoniot) are presented.

Introduction to the variational density matrix method and its application to dense hydrogen

A new variational principle for optimizing thermal density matrices is introduced. As a first application, the variational many body density matrix is written as a determinant of one body density matrices which are approximated by Gaussians with the mean, width and amplitude as variational parameters. This variational density matrix is used to calculate properties of the hot, dense hydrogen. First results for a density of r s = 2 are presented.

Weyl and Wigner functions in an extended phase-space formalism

The Wigner function $W(x,p)$ is related to the Weyl function $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{W}(X,P)$ through a two-dimensional Fourier transform. Several properties for these functions are proved which lead naturally to an extended phase space $x\ensuremath{-}p\ensuremath{-}X\ensuremath{-}P.$ Uncertainty relations for $\ensuremath{\delta}x\ensuremath{\delta}P$ and for $\ensuremath{\delta}p\ensuremath{\delta}X$ are studied. They reduce to the usual uncertainty relation in the special case of pure states, but are different for mixed states. Thermal density matrices obey these uncertainty relations as equalities.

A density matrix approach to the Goldanskii-Karyagin effect

A density matrix formalism has been introduced to describe the perturbation of the Mössbauer line intensities caused by the anisotropic recoilless fraction in random absorbers. General properties of the matrix elements are discussed and relationships between thermal tensors and density matrices are investigated with the attention focused on the possibility to extract as much information about the thermal tensors as possible from the observed line intensities. It is shown, that quartic (anharmonic) anisotropy may be observable in quadrupolarly dominated transitions under favourable conditions.

Cumulant expansions for thermal density matrices and ground state logarithmic perturbation series

We express the logarithm of the thermal density matrix ρH,β (X,X′)≡〈X‖e−βH‖X′〉, where H=H0+V, as a cumulant expansion in powers of the perturbation V, by associating with {H0,‖X〉,‖X′〉} a stochastic process. If the ground state of H0 is isolated, this stochastic process is of finite memory; it then follows from the properties of cumulants that the above expansion of ln ρ is nonsecular as β→∞ (all its terms ∼β), and is thus usable down to zero temperature, unlike the direct expansion of ρ in powers of V, whose nth term ∼βn. To determine explicitly the low-temperature behavior, we apply an analysis familiar in the theory of relaxation, and obtain the form ln ρH,β(X,X′)=h(X,X′,β) +a(X)+a(X′)°−bβ, where a,b,h are cumulant expansions in powers of V; a(X), b are independent of β; and h(X,X′,β)→0 as β→∞. Comparing with ρ→〈X‖ψ0〉e−βE0〈ψ0‖X′〉 as β→∞, where ψ0 and E0 are the ground state and energy of H, we deduce E0=b, ln〈X‖ψ0〉=a(X), i.e., the Rayleigh–Schrödinger perturbation series for E0 and ln〈X‖ψ0〉 (with 〈ψ0‖ψ0〉=1) emerge as cumulant expansions. In the case of a many-body system, the properties of cumulants immediately imply linked cluster theorems for ln ρ, as well as for E0 and ln〈X‖ψ0〉.