R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

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 become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally 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 the DMRG.

http://www.physics.udel.edu/~bnikolic/QTTG/NOTES/DMRG/SCHOLLWOCK=RMP_dmrg.pdf

The density-matrix renormalization group

Preface 1. Introduction 2. The Hubbard Hamiltonian and its symmetries 3. The Bethe ansatz solution 4. String hypothesis 5. Thermodynamics in the Yang-Yang approach 6. Ground state properties in the thermodynamic limit 7. Excited states at zero temperature 8. Finite size corrections at zero temperature 9. Asymptotics of correlation functions 10. Scaling and continuum limits at half-filling 11. Universal correlations at low density 12. The algebraic approach to the Hubbard model 13. The path integral approach to thermodynamics 14. The Yangian symmetry of the Hubbard model 15. S-matrix and Yangian symmetry in the infinite interval limit 16. Hubbard model in the attractive case 17. Mathematical appendices References Index.

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The one-dimensional Hubbard model

This article reviews theoretical and experimental developments for one-dimensional Fermi gases. Specifically, the experimentally realized two-component delta-function interacting Fermi gas-the Gaudin-Yang model-and its generalizations to multicomponent Fermi systems with larger spin symmetries is discussed. The exact results obtained for Bethe ansatz integrable models of this kind enable the study of the nature and microscopic origin of a wide range of quantum many-body phenomena driven by spin population imbalance, dynamical interactions, and magnetic fields. This physics includes Bardeen-Cooper-Schrieffer-like pairing, Tomonaga-Luttinger liquids, spin-charge separation, Fulde-Ferrel-Larkin-Ovchinnikov-like pair correlations, quantum criticality and scaling, polarons, and the few-body physics of the trimer state (trions). The fascinating interplay between exactly solved models and experimental developments in one dimension promises to yield further insight into the exciting and fundamental physics of interacting Fermi systems.

Fermi gases in one dimension: From Bethe ansatz to experiments

The magnetic susceptibility ${\ensuremath{\chi}}^{*}(t)$ and specific heat $C(t)$ versus temperature t of the spin $S=1/2$ antiferromagnetic (AF) alternating-exchange ${(J}_{1}$ and ${J}_{2})$ Heisenberg chain are studied for the entire range $0l~\ensuremath{\alpha}l~1$ of the alternation parameter $\ensuremath{\alpha}\ensuremath{\equiv}{J}_{2}{/J}_{1}{(J}_{1},$ ${J}_{2}g~0,$ ${J}_{2}l~{J}_{1},$ ${t=k}_{\mathrm{B}}{T/J}_{1},$ ${\ensuremath{\chi}}^{*}=\ensuremath{\chi}{J}_{1}{/Ng}^{2}{\ensuremath{\mu}}_{B}^{2}).$ For the uniform chain $(\ensuremath{\alpha}=1),$ the high-accuracy ${\ensuremath{\chi}}^{*}(t)$ and $C(t)$ Bethe ansatz data of Kl\"umper and Johnston (unpublished) are shown to agree very well at low t with the respective exact theoretical low-$t$ logarithmic correction predictions of Lukyanov [Nucl. Phys. B $522,$ 533 (1998)]. Accurate $(\ensuremath{\sim}{10}^{\ensuremath{-}7})$ independent empirical fits to the respective data are obtained over t ranges spanning 25 orders of magnitude, $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}25}l~tl~5,$ which contain extrapolations to the respective exact $t=0$ limits. The infinite temperature entropy calculated using our $C(t)$ fit function is within 8 parts in ${10}^{8}$ of the exact value $\mathrm{ln}2.$ Quantum Monte Carlo (QMC) simulations and transfer-matrix density-matrix renormalization group (TMRG) calculations of ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ are presented for $0.002l~tl~10$ and $0.05l~\ensuremath{\alpha}l~1,$ and an accurate $(2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4})$ two-dimensional $(\ensuremath{\alpha},t)$ fit to the combined data is obtained for $0.01l~tl~10$ and $0l~\ensuremath{\alpha}l~1.$ From the low-$t$ TMRG data, the spin gap $\ensuremath{\Delta}(\ensuremath{\alpha})$ is extracted for $0.8l~\ensuremath{\alpha}l~0.995$ and compared with previous results, and a fit function is formulated for $0l~\ensuremath{\alpha}l~1$ by combining these data with literature data. We infer from our data that the asymptotic critical regime near the uniform chain limit is only entered for $\ensuremath{\alpha}\ensuremath{\gtrsim}0.99.$ We examine in detail the theoretical predictions of Bulaevskii [Sov. Phys. Solid State $11,$ 921 (1969)], for ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ and compare them with our results. To illustrate the application and utility of our theoretical results, we model our experimental $\ensuremath{\chi}(T)$ and specific heat ${C}_{\mathrm{p}}(T)$ data for ${\mathrm{NaV}}_{2}{\mathrm{O}}_{5}$ single crystals in detail. The $\ensuremath{\chi}(T)$ data above the spin dimerization temperature ${T}_{\mathrm{c}}\ensuremath{\approx}34\mathrm{K}$ are not in quantitative agreement with the prediction for the $S=1/2$ uniform Heisenberg chain, but can be explained if there is a moderate ferromagnetic interchain coupling and/or if J changes with T. Fitting the $\ensuremath{\chi}(T)$ data using our ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ fit function, we obtain the sample-dependent spin gap and range $\ensuremath{\Delta}{(T=0)/k}_{\mathrm{B}}=103(2)\mathrm{K},$ alternation parameter $\ensuremath{\delta}(0)\ensuremath{\equiv}(1\ensuremath{-}\ensuremath{\alpha})/(1+\ensuremath{\alpha})=0.034(6)$ and average exchange constant ${J(0)/k}_{\mathrm{B}}=640(80)\mathrm{K}.$ The $\ensuremath{\delta}(T)$ and $\ensuremath{\Delta}(T)$ are derived from the data. A spin pseudogap with magnitude $\ensuremath{\approx}0.4\ensuremath{\Delta}(0)$ is consistently found just above ${T}_{\mathrm{c}},$ which decreases with increasing temperature. From our ${C}_{\mathrm{p}}(T)$ measurements on two crystals, we infer that the magnetic specific heat at low temperatures $T\ensuremath{\lesssim}15\mathrm{K}$ is too small to be resolved experimentally, and that the spin entropy at ${T}_{\mathrm{c}}$ is too small to account for the entropy of the transition. A quantitative analysis indicates that at ${T}_{\mathrm{c}},$ at least 77% of the entropy change due to the transition at ${T}_{\mathrm{c}}$ and associated order parameter fluctuations arise from the lattice and/or charge degrees of freedom and less than 23% from the spin degrees of freedom.

https://arxiv.org/pdf/cond-mat/0003271.pdf

Thermodynamics of spin S = 1 / 2 antiferromagnetic uniform and alternating-exchange Heisenberg chains

Leading and next-to-leading-order finite-size corrections to the ground and first excited states are calculated for the spin-1/2 anisotropic Heisenberg model in the critical region. The analytic results are compared to numerical data obtained for chains up to a length of N=1024. It is found that, near the isotropic point, the asymptotic region where the results obtained for N to infinity are applicable sets in at very large N values, and for obtaining good accuracy in fitting the numerical data one has to take into account several correction terms, even at large (N>100) chain lengths.

http://iopscience.iop.org/article/10.1088/0305-4470/20/2/010/pdf

Finite-size corrections and numerical calculations for long spin 1/2 Heisenberg chains in the critical region

We consider a generalization of the quantum Rabi model where the two-level system and the single-mode cavity oscillator are coupled by an additional Stark-like term. By adapting a method recently introduced by Braak [Phys. Rev. Lett. {\bf 107}, 100401 (2011)], we solve the model exactly. The low-lying spectrum in the experimentally relevant ultrastrong and deep strong regimes of the Rabi coupling is found to exhibit two striking features absent from the original quantum Rabi model: avoided level crossings for states of the same parity and an anomalously rapid onset of two-fold near-degenerate levels as the Rabi coupling increases.

pdf/a-generalization-of-the-quantum-rabi-model-exact-solution-q240rgod0m.pdf

A generalization of the quantum Rabi model: exact solution and spectral structure

We study interacting one dimensional (1D) quantum lattice gases with integrable impurities. These model Hamiltonians can be derived using the quantum inverse scattering method for inhomogeneous models and are by construction integrable. Absence of backscattering at the impurities is shown to be the characteristic feature of these disordered systems. The value of the effective carrier charge and the Sutherland-Shastry relation are derived for the half-filled XXX model and are shown to be independent of the impurity concentration and strength. For the half-filled XXZ model we show that there is no enhancement of the persistent currents for repulsive interactions. For attractive interactions we identify a crossover regime beyond which enhancement of the currents is observed.

Absence of backscattering at integrable impurities in one-dimensional quantum many-body systems

We consider a generalization of the quantum Rabi model where the two-level system and the single-mode cavity oscillator are coupled by an additional Stark-like term. By adapting a method recently introduced by Braak (2011 Phys. Rev. Lett. 107 100401), we solve the model exactly. The low-lying spectrum in the experimentally relevant ultrastrong and deep strong regimes of the Rabi coupling is found to exhibit two striking features absent from the original quantum Rabi model: avoided level crossings for states of the same parity and an anomalously rapid onset of two-fold near-degenerate levels as the Rabi coupling increases.

https://iopscience.iop.org/article/10.1088/1751-8121/aa785a/pdf

We study the persistent currents induced by both the Aharonov-Bohm and Aharonov-Casher effects in a one-dimensional mesoscopic ring coupled to a sidebranch quantum dot at Kondo resonance. For privileged values of the Aharonov-Bohm-Casher fluxes, the problem can be mapped onto an integrable model, exactly solvable by a Bethe ansatz. In the case of a pure magnetic Aharonov-Bohm flux, we find that the presence of the quantum dot has no effect on the persistent current. In contrast, the Kondo resonance interferes with the spin-dependent Aharonov-Casher effect to induce a current which, in the strong-coupling limit, is independent of the number of electrons in the ring.

/pdf/kondo-resonance-in-a-mesoscopic-ring-coupled-to-a-quantum-ebgtcbqbdh.pdf

Kondo resonance in a mesoscopic ring coupled to a quantum dot: exact results for the Aharonov-Bohm-Casher effects.

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Papers

Finite-size corrections and numerical calculations for long spin 1/2 Heisenberg chains in the critical region

A generalization of the quantum Rabi model: exact solution and spectral structure

Absence of backscattering at integrable impurities in one-dimensional quantum many-body systems

A generalization of the quantum Rabi model: exact solution and spectral structure

Kondo resonance in a mesoscopic ring coupled to a quantum dot: exact results for the Aharonov-Bohm-Casher effects.