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Effective Particles in Quantum Spin Chains: The Framework

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Book cover Tensor Network States and Effective Particles for Low-Dimensional Quantum Spin Systems

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Abstract

In the previous chapter we have already highlighted the importance of emergent particles as the effective degrees of freedom that determine the properties of strongly-correlated quantum matter at low energies. In this chapter we will present a particular realization of this theme for one-dimensional quantum spin systems; the central idea will be to determine the elementary excitations variationally and treat them as particles in a many-particle description.

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Notes

  1. 1.

    The ansatz was named after Feynman – for obvious reasons – and Bijl, who supposedly wrote down a similar wavefunction in Ref. [9].

  2. 2.

    Other MPS implementations of the single-mode approximation include Refs. [24, 25].

  3. 3.

    Importantly, every isolated branch is interpreted as a new “elementary particle”, so, in the absence of some particle number, there is no way to differentiate a bound state that is supposedly composed of two elementary particles in the system (see Ref. [43] for the analogous result in QFT).

  4. 4.

    In this chapter we will only consider nearest-neighbour Hamiltonians, but everything can be straightforwardly extended to larger, yet local interactions – the case of long-range interactions is not included.

  5. 5.

    In this chapter all computations will involve this transfer matrix or variations of it. Let us therefore introduce the notations

    $$\begin{aligned} E = E^{A}_{A} = \sum _{s=1}^d A^s \otimes \bar{A}^s, \end{aligned}$$

    for the transfer matrix,

    $$\begin{aligned} O_A^A = \sum _{s,t} A^s \otimes \bar{A}^t {\langle {s}|} O {|{t}\rangle }. \end{aligned}$$

    for the transfer matrix with a one-site operator, and

    $$\begin{aligned} O_{AA}^{AA} = \sum _{s,t,s',t'} A^s A^{s'} \otimes \bar{A}^t \bar{A}^{t'} {\langle {ss'}|} O {|{tt'}\rangle }. \end{aligned}$$

    for a two-site operator. We will also need “transfer matrices” with different tensors; notations:

    $$\begin{aligned}&E^{A}_{B} = \sum _{s=1}^d A^s \otimes \bar{B}^s \\&O^{A}_{B} = \sum _{s,t} A^s \otimes \bar{B}^t {\langle {s}|} O {|{t}\rangle } \\&O^{AB}_{CD} = \sum _{s,t,s',t'} A^s B^{s'} \otimes \bar{C}^t \bar{D}^{t'} {\langle {ss'}|} O {|{tt'}\rangle }. \end{aligned}$$
  6. 6.

    It is assumed that the largest eigenvalue is unique, which is only true for so-called injective MPS. In the following, we will always assume this is the case – for all practical applications this assumption is true – and we refer to e.g. Ref. [85] for more details.

  7. 7.

    One can associate a completely positive map to the transfer matrix, for which the left and right fixed points are guaranteed to be positive definite matrices when reshaped to \(D\times D\) matrices l and r.

  8. 8.

    Additional details can be found in Ref. [40].

  9. 9.

    Because we work in the thermodynamic limit, the momentum can take on every value between 0 and \(2\pi \).

  10. 10.

    Note that this does not mean that, by systematically growing the bond dimension, the ansatz can reproduce the effect of ever larger operators. Every B-tensor can be written in terms of operators of size larger than \(2\log D\), but not vice versa. There is no a priori no reason to expect that variational excitation energies will converge to the exact value for large enough D.

  11. 11.

    At momentum zero, the gauge transformations of the B tensor are related to linearized infinitesimal gauge transformations of the A tensors. We refer to the tangent space interpretation of the excitation ansatz in Sect. 3.5 for more details.

  12. 12.

    This “gauging away” of the momentum dependence of the norm seems in contradiction with our remarks that it is the norm that determines the dispersion in the single-mode approximation. But note that the gauge transformation that brings the tensor B in the right gauge is momentum dependent. It is due to the special structure of an MPS that this particular gauge choice is possible: the momentum information in the norm is gauged away through the virtual dimension of the MPS, such that only the numerator contains all momentum dispersion.

  13. 13.

    Again, the norm of the excitations in terms of the vector \(\varvec{x}\) is just the Euclidean inner product, \({\langle {\Phi _\kappa [B_{L/R}(Y)]|\Phi _\kappa [B_{L/R}(X)]}\rangle }=\varvec{y}^\dagger \varvec{x}\), so that the normalization of the states does not show up in the eigenvalue problem.

  14. 14.

    Since we have subtracted the ground-state energy density, the expectation value is zero; this implies that we can safely put in the regularized transfer matrix \(\tilde{E}\) instead of the full one wherever needed to define the pseudo-inverse.

  15. 15.

    See also Refs. [50, 51] for a Monte Carlo study on the local nature of spinons.

  16. 16.

    In finite systems with periodic boundary conditions, topological excitations always have to be described in pairs. In order to capture them in finite systems, non-trivial boundary conditions have to be applied, see e.g. [52].

  17. 17.

    In quantum field theory, this ansatz has been proposed earlier [53] to study the kink excitations in the sine-Gordon model.

  18. 18.

    The form of the two-particle wave function that we propose, is inspired by the similar construction that the Bethe ansatz uses to solve the two-magnon problem for e.g. the Heisenberg model – see Sect. 2.5. In addition, our approach is greatly inspired by Feynman’s lectures on statistical mechanics [54] and Kohn’s variational approach to solve the scattering problem [55].

  19. 19.

    In Ref. [37] this was shown with full mathematical rigour, with a number of caveats that do not need to worry us here.

  20. 20.

    These connections with other computational methods is worked out in more detail in Ref. [40].

  21. 21.

    We have omitted the vectors \(\varvec{v}_1\) and \(\varvec{v}_2\), because they should become equal when the two momenta approach.

  22. 22.

    In the outlook we will discuss how our particle picture can be applied to non-equilibrium physics such as quantum quenches and transport processes.

  23. 23.

    Our approach is assumed to be valid at low densities, and should be connected to the low-density approximation in terms of virial coefficients. In particular, the fact that the second virial coefficient can be written as a function of the two-particle phase shift only [86, 87] and the idea of computing the thermodynamics of a gas directly with the S matrix, seem both to be closely related [88]. On the other hand, as the solution of the Yang–Yang equation is supposed to be the analytic continuation of the full virial expansion [73], our method seems to correspond to an approximate continuation, that is valid beyond first order.

  24. 24.

    See also Ref. [107] for a similar expansion for non-integrable systems.

References

  1. P.W. Anderson, Concepts in Solids: Lectures on the Theory of Solids (World Scientific, Singapore, 1963)

    MATH  Google Scholar 

  2. R.P. Feynman, Atomic theory of the \(\lambda \) transition in helium. Phys. Rev. 91, 1291 (1953). doi:10.1103/PhysRev.91.1291

  3. R.P. Feynman, Atomic theory of liquid helium near absolute zero. Phys. Rev. 91, 1301 (1953). doi:10.1103/PhysRev.91.1301

  4. R.P. Feynman, Atomic theory of the two-fluid model of liquid helium. Phys. Rev. 94, 262 (1954). doi:10.1103/PhysRev.94.262

  5. R.P. Feynman, M. Cohen, Energy spectrum of the excitations in liquid helium. Phys. Rev. 102, 1189 (1956). doi:10.1103/PhysRev.102.1189

  6. M. Cohen, R.P. Feynman, Theory of inelastic scattering of cold neutrons from liquid helium. Phys. Rev. 107, 13 (1957). doi:10.1103/PhysRev.107.13

  7. D. Pines, P. Nozières, Theory of Quantum Liquids (Perseus Books Publishing, Cambridge, 1966)

    MATH  Google Scholar 

  8. L.D. Landau, Theory of the superfluidity of helium II, J. Phys. U.S.S.R 5, 356 (1941). doi:10.1103/PhysRev.60.356

  9. A. Bijl, J. de Boer, A. Michels, Properties of liquid helium II. Physica 8, 655 (1941). doi:10.1016/S0031-8914(41)90422-6

    Article  ADS  MATH  Google Scholar 

  10. B.I. Lundqvist, Single-particle spectrum of the degenerate electron gas. Physik der Kondensierten Materie 6, 193 (1967). doi:10.1007/BF02422716

    ADS  Google Scholar 

  11. A.W. Overhauser, Simplified theory of electron correlations in metals. Phys. Rev. B 3, 1888 (1971). doi:10.1103/PhysRevB.3.1888

    Article  ADS  Google Scholar 

  12. S.M. Girvin, A. MacDonald, P. Platzman, Collective-excitation gap in the fractional quantum Hall effect. Phys. Rev. Lett. 54, 581 (1985). doi:10.1103/PhysRevLett.54.581

  13. S.M. Girvin, A. MacDonald, P. Platzman, Magneto-roton theory of collective excitations in the fractional quantum Hall effect. Phys. Rev. B 33, 2481 (1986). doi:10.1103/PhysRevB.33.2481

    Article  ADS  Google Scholar 

  14. D.P. Arovas, A. Auerbach, F.D.M. Haldane, Extended Heisenberg models of antiferromagnetism: analogies to the fractional quantum Hall effect. Phys. Rev. Lett. 60, 531 (1988). doi:10.1103/PhysRevLett.60.531

  15. I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett. 59, 799 (1987). doi:10.1103/PhysRevLett.59.799

  16. I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki, Valence bond ground states in isotropic quantum antiferromagnets. Commun. Math. Phys. 115, 477 (1988). doi:10.1007/BF01218021

    Article  ADS  MathSciNet  Google Scholar 

  17. M. Takahashi, Excitation spectra of S\(=\)1 antiferromagnetic chains. Phys. Rev. B 50, 3045 (1994). doi:10.1103/PhysRevB.50.3045

  18. E. Sorensen, I. Affleck, S(k) for Haldane-gap antiferromagnets: large-scale numerical results versus field theory and experiment. Phys. Rev. B 49, 13235 (1994). doi:10.1103/PhysRevB.49.13235

    Article  ADS  Google Scholar 

  19. E. Sorensen, I. Affleck, Equal-time correlations in Haldane-gap antiferromagnets. Phys. Rev. B 49, 15771 (1994). doi:10.1103/PhysRevB.49.15771

    Article  ADS  Google Scholar 

  20. S. Östlund, S. Rommer, Thermodynamic limit of the density matrix renormalization for the spin-1 Heisenberg chain. Phys. Rev. Lett. 75, 13 (1995). doi:10.1103/PhysRevLett.75.3537

  21. S. Rommer, S. Östlund, Class of ansatz wave functions for one-dimensional spin systems and their relation to the density matrix renormalization group. Phys. Rev. B 55, 2164 (1997). doi:10.1103/PhysRevB.55.2164

    Article  ADS  Google Scholar 

  22. B. Pirvu, J. Haegeman, F. Verstraete, Matrix product state based algorithm for determining dispersion relations of quantum spin chains with periodic boundary conditions. Phys. Rev. B 85, 13 (2012). doi:10.1103/PhysRevB.85.035130

    Article  Google Scholar 

  23. J. Haegeman, B. Pirvu, D.J. Weir, J.I. Cirac, T.J. Osborne, H. Verschelde, F. Verstraete, Variational matrix product ansatz for dispersion relations. Phys. Rev. B 85, 100408 (2012). doi:10.1103/PhysRevB.85.100408

    Article  ADS  Google Scholar 

  24. E. Bartel, A. Schadschneider, J. Zittartz, Excitations of anisotropic spin-1 chains with matrix product ground state. Eur. Phys. J. B - Condens. Matter 31, 209 (2003). doi:10.1140/epjb/e2003-00025-7

  25. S.G. Chung, L. Wang, Entanglement perturbation theory for the elementary excitation in one dimension. Phys. Lett. A 373, 2277 (2010), http://www.sciencedirect.com/science/article/pii/S0375960109005106

  26. M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, Boulder, 1995)

    Google Scholar 

  27. R.F. Streater, A.S. Wightman, PCT, Spin and Statistics, and All That (W.A. Benjamin Inc, New York, 1964)

    MATH  Google Scholar 

  28. R. Haag, Local Quantum Physics: Fields, Particles, Algebras (Springer, Berlin, 1996)

    Book  MATH  Google Scholar 

  29. R. Haag, Quantum field theories with composite particles and asymptotic conditions. Phys. Rev. 112, 669 (1958). doi:10.1103/PhysRev.112.669

  30. D. Ruelle, On asymptotic condition in quantum field theory. Helv. Phys. Acta 35, 147 (1962)

    MathSciNet  MATH  Google Scholar 

  31. E.H. Lieb, D.W. Robinson, The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251 (1972). doi:10.1007/BF01645779

    Article  ADS  MathSciNet  Google Scholar 

  32. V. Zauner, D. Draxler, L. Vanderstraeten, M. Degroote, J. Haegeman, M.M. Rams, V. Stojevic, N. Schuch, F. Verstraete, Transfer matrices and excitations with matrix product states. New J. Phys. 17, 053002 (2015). doi:10.1088/1367-2630/17/5/053002

    Article  ADS  MathSciNet  Google Scholar 

  33. I.P. McCulloch, Infinite size density matrix renormalization group, revisited (2008), arXiv:0804.2509

  34. G. Vidal, Efficient simulation of one-dimensional quantum many-body systems. Phys. Rev. Lett. 93, 4 (2004). doi:10.1103/PhysRevLett.93.040502

  35. G. Vidal, Classical simulation of infinite-size quantum lattice systems in one spatial dimension. Phys. Rev. Lett. 98, 5 (2007). doi:10.1103/PhysRevLett.98.070201

  36. J. Haegeman, J.I. Cirac, T.J. Osborne, I. Pizorn, H. Verschelde, F. Verstraete, Time-dependent variational principle for quantum lattices. Phys. Rev. Lett. 107, 070601 (2011). doi:10.1103/PhysRevLett.107.070601

  37. J. Haegeman, M. Mariën, T.J. Osborne, F. Verstraete, Geometry of matrix product states: metric, parallel transport, and curvature. J. Math. Phys. 55, 021902 (2014). doi:10.1063/1.4862851

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, F. Verstraete, Unifying time evolution and optimization with matrix product states (2014), arXiv:1408.5056

  39. F.H.L. Essler, R.M. Konik, Finite-temperature dynamical correlations in massive integrable quantum field theories. J. Stat. Mech.: Theory Exp. 2009, P09018 (2009). doi:10.1088/1742-5468/2009/09/P09018

  40. J. Haegeman, T.J. Osborne, F. Verstraete, Post-matrix product state methods: to tangent space and beyond. Phys. Rev. B 88, 075133 (2013). doi:10.1103/PhysRevB.88.075133

    Article  ADS  Google Scholar 

  41. J. Haegeman, S. Michalakis, B. Nachtergaele, T.J. Osborne, N. Schuch, F. Verstraete, Elementary excitations in gapped quantum spin systems. Phys. Rev. Lett. 111, 080401 (2013). doi:10.1103/PhysRevLett.111.080401

  42. S. Bachmann, W. Dybalski, P. Naaijkens, Lieb-Robinson bounds, Arveson spectrum and Haag-Ruelle scattering theory for gapped quantum spin systems (2014), arXiv:1412.2970

  43. W. Zimmermann, On the bound state problem in quantum field theory. Il Nuovo Cimento 10, 597 (1958). doi:10.1007/BF02859796

  44. E.H. Lieb, The Bose fluid, Lecture Notes in Theoretical Physics, vol. VIIC (University of Colorado Press, Boulder, 1965), pp. 175–224

    Google Scholar 

  45. T. Davison, E. Feenberg, Variance of H in the Bijl-Feynman description of an elementary excitation. Phys. Rev. 171, 221 (1968). doi:10.1103/PhysRev.171.221

  46. R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory (North-Holland, Amsterdam, 1996)

    Google Scholar 

  47. L.D. Faddeev, V. Korepin, Quantum theory of solitons. Phys. Rep. 42, 1 (1978). doi:10.1016/0370-1573(78)90058-3

    Article  ADS  MathSciNet  Google Scholar 

  48. V.E. Korepin, N.M. Bogoliubov, A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions (Cambridge University Press, Cambridge, 1997)

    MATH  Google Scholar 

  49. L.D. Faddeev, L.A. Takhtajan, What is the spin of a spin wave? Phys. Lett. A 85, 375 (1981). doi:10.1016/0375-9601(81)90335-2

    Article  ADS  MathSciNet  Google Scholar 

  50. Y. Tang, A.W. Sandvik, Method to characterize spinons as emergent elementary particles. Phys. Rev. Lett. 107, 157201 (2011). doi:10.1103/PhysRevLett.107.157201

  51. Y. Tang, A.W. Sandvik, Quantum Monte Carlo studies of spinons in one-dimensional spin systems. Phys. Rev. B 92, 184425 (2015). doi:10.1103/PhysRevB.92.184425

    Article  ADS  Google Scholar 

  52. K. Okunishi, N. Maeshima, Spinon excitation and Möbius boundary condition in S\(=\)1/2 antiferromagnetic Heisenberg spin ladder with zigzag structure. Phys. Rev. B 64, 212406 (2001). doi:10.1103/PhysRevB.64.212406

  53. S. Mandelstam, Soliton operators for the quantized sine-Gordon equation. Phys. Rev. D 11, 3026 (1975). doi:10.1103/PhysRevD.11.3026

    Article  ADS  MathSciNet  Google Scholar 

  54. R.P. Feynman, Statistical Mechanics: A Set of Lectures (W.A. Benjamin, Inc., Reading, 1972)

    Google Scholar 

  55. W. Kohn, Variational scattering theory in momentum space I. Central field problems. Phys. Rev. 84, 495 (1951). doi:10.1103/PhysRev.84.495

  56. J. Frenkel, Wave Mechanics, Advanced General Theory (Oxford Clarendon Press, Oxford, 1934)

    MATH  Google Scholar 

  57. P.A.M. Dirac, Note on exchange phenomena in the Thomas atom. Math. Proc. Camb. Philos. Soc. 26, 376 (1930). doi:10.1017/S0305004100016108

    Article  ADS  MATH  Google Scholar 

  58. J.C. Slater, The theory of complex spectra. Phys. Rev. 34, 1293 (1929). doi:10.1103/PhysRev.34.1293

  59. S. Wouters, N. Nakatani, D. Van Neck, G.K.-L. Chan, Thouless theorem for matrix product states and subsequent post density matrix renormalization group methods. Phys. Rev. B 88, 075122 (2013). doi:10.1103/PhysRevB.88.075122

    Article  ADS  Google Scholar 

  60. P. Ring, P. Schuck, The Nuclear Many-Body Problem (Springer, Berlin, 1980)

    Book  Google Scholar 

  61. T. Helgaker, P. Jorgensen, J. Olsen, Molecular Electronic-Structure Theory (Wiley, New York, 2014)

    Google Scholar 

  62. W. Heisenberg, Mehrkörperproblem und Resonanz in der Quantenmechanik. Zeitschrift für Physik 38, 411 (1926). doi:10.1007/BF01397160

  63. P.A.M. Dirac, On the theory of quantum mechanics. Proc. R. Soc. Lond. Ser. A 112, 661 (1926). doi:10.1098/rspa.1926.0133

  64. D. Bohm, D. Pines, A collective description of electron interactions: III. Coulomb interactions in a degenerate electron gas. Phys. Rev. 92, 609 (1953). doi:10.1103/PhysRev.92.609

  65. I.E. Tamm, Relativistic interaction of elementary particles. J. Phys. USSR 9, 449 (1945)

    Google Scholar 

  66. S.M. Dancoff, Non-adiabatic meson theory of nuclear forces. Phys. Rev. 78, 382 (1950). doi:10.1103/PhysRev.78.382

    Article  ADS  MATH  Google Scholar 

  67. J.R. Taylor, Scattering Theory: The Quantum Theory on Nonrelativistic Collisions (Wiley, New York, 1972)

    Google Scholar 

  68. M. Reed, B. Simon, Scattering Theory (Academic Press, New York, 1972)

    Google Scholar 

  69. B. Lippmann, J. Schwinger, Variational principles for scattering processes. I. Phys. Rev. 79, 469 (1950). doi:10.1103/PhysRev.79.469

  70. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 2011)

    Google Scholar 

  71. S. Sachdev, K. Damle, Low temperature spin diffusion in the one-dimensional quantum O(3) nonlinear \(\sigma \) Model. Phys. Rev. Lett. 78, 943 (1997). doi:10.1103/PhysRevLett.78.943

  72. K. Damle, S. Sachdev, Nonzero-temperature transport near quantum critical points. Phys. Rev. B 56, 8714 (1997). doi:10.1103/PhysRevB.56.8714

    Article  ADS  Google Scholar 

  73. B. Sutherland, Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems (World Scientific, Singapore, 2004)

    Google Scholar 

  74. H. Bethe, Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Zeitschrift für Physik 71, 205 (1931). doi:10.1007/BF01341708

  75. W. Krauth, Bethe ansatz for the one-dimensional boson Hubbard model. Phys. Rev. B 44, 9772 (1991). doi:10.1103/PhysRevB.44.9772

  76. H. Kiwata, Y. Akutsu, Bethe-Ansatz approximation for the S\(=\)1 antiferromagnetic spin chain. J. Phys. Soc. Jpn. 63, 3598 (1994). doi:10.1143/JPSJ.63.3598

  77. K. Okunishi, Magnetization process of bilinear-biquadratic spin chains at finite temperature. Phys. Rev. B 60, 4043 (1999). doi:10.1103/PhysRevB.60.4043

    Article  ADS  Google Scholar 

  78. E.H. Lieb, W. Liniger, Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. 130, 1605 (1963). doi:10.1103/PhysRev.130.1605

  79. E.H. Lieb, Exact analysis of an interacting Bose gas. II. The excitation spectrum. Phys. Rev. 130, 1616 (1963). doi:10.1103/PhysRev.130.1616

  80. F.D.M. Haldane, Continuum dynamics of the 1-D Heisenberg antiferromagnet: identification with the O(3) nonlinear sigma model. Phys. Lett. A 93, 464 (1983). doi:10.1016/0375-9601(83)90631-X

  81. F.D.M. Haldane, Demonstration of the Luttinger liquid character of Bethe-ansatz-soluble models of 1-D quantum fluids. Phys. Lett. A 81, 153 (1981), http://www.sciencedirect.com/science/article/pii/0375960181900499

  82. T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, Oxford, 2004)

    Google Scholar 

  83. F.D.M. Haldane, Effective harmonic-fluid approach to low-energy properties of one-dimensional quantum fluids. Phys. Rev. Lett. 47, 1840 (1981). doi:10.1103/PhysRevLett.47.1840

  84. M.A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, M. Rigol, One dimensional bosons: from condensed matter systems to ultracold gases. Rev. Mod. Phys. 83, 1405 (2011). doi:10.1103/RevModPhys.83.1405

    Article  ADS  Google Scholar 

  85. D. Pérez-García, F. Verstraete, M.M. Wolf, J.I. Cirac, Matrix product state representations. Quantum Inf. Comput. 7, 401 (2007), arXiv:quant-ph/0608197

  86. E. Beth, G.E. Uhlenbeck, The quantum theory of the non-ideal gas. II. Behaviour at low temperatures. Physica 4, 915 (1937). doi:10.1016/S0031-8914(37)80189-5

    Article  ADS  MATH  Google Scholar 

  87. K. Huang, Statistical Mechanics (Wiley, New York, 1963)

    Google Scholar 

  88. R. Dashen, S.-K. Ma, H. Bernstein, S-matrix formulation of statistical mechanics. Phys. Rev. 187, 345 (1969). doi:10.1103/PhysRev.187.345

  89. F. Lesage, H. Saleur, Correlations in one-dimensional quantum impurity problems with an external field or a temperature. Nucl. Phys. B 490, 543 (1997). doi:10.1016/S0550-3213(97)00024-2

    Article  ADS  MathSciNet  MATH  Google Scholar 

  90. R.M. Konik, P. Fendley, Haldane-gapped spin chains as Luttinger liquids: correlation functions at finite field. Phys. Rev. B 66, 144416 (2002). doi:10.1103/PhysRevB.66.144416

    Article  ADS  Google Scholar 

  91. J. Lou, S. Qin, T.-K. Ng, Z. Su, I. Affleck, Finite-size spectrum, magnon interactions, and magnetization of S\(=\)1 Heisenberg spin chains. Phys. Rev. B 62, 3786 (2000). doi:10.1103/PhysRevB.62.3786

  92. I. Affleck, Luttinger liquid parameter for the spin-1 Heisenberg chain in a magnetic field. Phys. Rev. B 72, 132414 (2005). doi:10.1103/PhysRevB.72.132414

    Article  ADS  Google Scholar 

  93. I. Affleck, W. Hofstetter, D.R. Nelson, U. Schollwöck, Non-Hermitian Luttinger liquids and flux line pinning in planar superconductors. J. Stat. Mech.: Theory Exp. 2004, P10003 (2004). doi:10.1088/1742-5468/2004/10/P10003

    Article  MATH  Google Scholar 

  94. C.N. Yang, C.P. Yang, Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction. J. Math. Phys. 10, 1115 (1969). doi:10.1063/1.1664947

    Article  ADS  MathSciNet  MATH  Google Scholar 

  95. M. Takahashi, Thermodynamics of One-Dimensional Solvable Models (Cambridge University Press, Cambridge, 2005)

    Google Scholar 

  96. K. Okunishi, Y. Hieida, Y. Akutsu, \(\delta \)-function Bose-gas picture of S\(=\)1 antiferromagnetic quantum spin chains near critical fields. Phys. Rev. B 59, 6806 (1999). doi:10.1103/PhysRevB.59.6806

  97. A.B. Zamolodchikov, A.B. Zamolodchikov, Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models. Ann. Phys. 120, 253 (1979). doi:10.1016/0003-4916(79)90391-9

    Article  ADS  MathSciNet  Google Scholar 

  98. H. Lehmann, Über Eigenschaften von Ausbreitungsfunktionen und Renormierungskonstanten quantisierter Felder. Il Nuovo Cimento 11, 342 (1954). doi:10.1007/BF02783624

  99. J.-S. Caux, J. Mossel, I.P. Castillo, The two-spinon transverse structure factor of the gapped Heisenberg antiferromagnetic chain. J. Stat. Mech.: Theory Exp. 2008, P08006 (2008). doi:10.1088/1742-5468/2008/08/P08006

    Article  Google Scholar 

  100. P. Hohenberg, W. Brinkman, Sum rules for the frequency spectrum of linear magnetic chains. Phys. Rev. B 10, 128 (1974). doi:10.1103/PhysRevB.10.128

    Article  ADS  Google Scholar 

  101. D.P. Arovas, S.M. Girvin, Exact questions to some interesting answers in many body physics, in Recent Progress in Many-Body Theories, ed. by T.L. Ainsworth, C.E. Campbell, B.E. Clements, E. Krotscheck (Springer, Boston, 1992). doi:10.1007/978-1-4615-3466-2

  102. T. Barnes, Boundaries, cusps, and discontinuities in the multimagnon continua of one-dimensional quantum spin systems. Phys. Rev. B 67, 024412 (2003). doi:10.1103/PhysRevB.67.024412

    Article  ADS  Google Scholar 

  103. K. Damle, S. Sachdev, Spin dynamics and transport in gapped one-dimensional Heisenberg antiferromagnets at nonzero temperatures. Phys. Rev. B 57, 8307 (1998). doi:10.1103/PhysRevB.57.8307

    Article  ADS  Google Scholar 

  104. R.M. Konik, Haldane-gapped spin chains: exact low-temperature expansions of correlation functions. Phys. Rev. B 68, 104435 (2003). doi:10.1103/PhysRevB.68.104435

    Article  ADS  Google Scholar 

  105. F.H.L. Essler, R.M. Konik, Finite-temperature lineshapes in gapped quantum spin chains. Phys. Rev. B 78, 100403 (2008). doi:10.1103/PhysRevB.78.100403

    Article  ADS  Google Scholar 

  106. D.A. Tennant, B. Lake, A.J.A. James, F.H.L. Essler, S. Notbohm, H.-J. Mikeska, J. Fielden, P. Kögerler, P.C. Canfield, M.T.F. Telling, Anomalous dynamical line shapes in a quantum magnet at finite temperature. Phys. Rev. B 85, 014402 (2012). doi:10.1103/PhysRevB.85.014402

    Article  ADS  Google Scholar 

  107. B. Fauseweh, J. Stolze, G.S. Uhrig, Finite-temperature line shapes of hard-core bosons in quantum magnets: a diagrammatic approach tested in one dimension. Phys. Rev. B 90, 024428 (2014). doi:10.1103/PhysRevB.90.024428

    Article  ADS  Google Scholar 

  108. L. Vanderstraeten, J. Haegeman, T.J. Osborne, F. Verstraete, S matrix from matrix product states. Phys. Rev. Lett. 112, 257202 (2014). doi:10.1103/PhysRevLett.112.257202

    Article  ADS  Google Scholar 

  109. F. Keim, G.S. Uhrig, Effective one-dimensional models from matrix product states. Eur. Phys. J. B 88, 154 (2015). doi:10.1140/epjb/e2015-60188-0

    Article  ADS  MathSciNet  Google Scholar 

  110. S. Sotiriadis, Zamolodchikov-Faddeev algebra and quantum quenches in integrable field theories. J. Stat. Mech.: Theory Exp. 2012, P02017 (2012), http://iopscience.iop.org/1742-5468/2012/02/P02017

  111. F.H.L. Essler, G. Mussardo, M. Panfil, Generalized Gibbs ensembles for quantum field theories. Phys. Rev. A 91, 051602 (2015). doi:10.1103/PhysRevA.91.051602

    Article  ADS  Google Scholar 

  112. F.H.L. Essler, S. Kehrein, S.R. Manmana, N.J. Robinson, Quench dynamics in a model with tuneable integrability breaking. Phys. Rev. B 89, 165104 (2014). doi:10.1103/PhysRevB.89.165104

    Article  ADS  Google Scholar 

  113. B. Bertini, F.H.L. Essler, S. Groha, N.J. Robinson, Prethermalization and thermalization in models with weak integrability breaking. Phys. Rev. Lett. 115, 180601 (2015). doi:10.1103/PhysRevLett.115.180601

    Article  ADS  Google Scholar 

  114. R. Vasseur, J.E. Moore, Nonequilibrium quantum dynamics and transport: from integrability to many-body localization. J. Stat. Mech.: Theory Exp. 2016, 064010 (2016). doi:10.1088/1742-5468/2016/06/064010

    Article  MathSciNet  Google Scholar 

  115. Y. Huang, C. Karrasch, J.E. Moore, Scaling of electrical and thermal conductivities in an almost integrable chain. Phys. Rev. B 88, 115126 (2013). doi:10.1103/PhysRevB.88.115126

    Article  ADS  Google Scholar 

  116. C. Karrasch, R. Ilan, J.E. Moore, Nonequilibrium thermal transport and its relation to linear response. Phys. Rev. B 88, 195129 (2013). doi:10.1103/PhysRevB.88.195129

    Article  ADS  Google Scholar 

  117. C. Karrasch, D.M. Kennes, F. Heidrich-Meisner, Spin and thermal conductivity of quantum spin chains and ladders. Phys. Rev. B 91, 115130 (2015). doi:10.1103/PhysRevB.91.115130

    Article  ADS  Google Scholar 

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  1. Department of Physics and Astronomy, University of Ghent, Ghent, Belgium

    Laurens Vanderstraeten

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Vanderstraeten, L. (2017). Effective Particles in Quantum Spin Chains: The Framework. In: Tensor Network States and Effective Particles for Low-Dimensional Quantum Spin Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-64191-1_3

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