Skip to main content
SpringerLink
Log in

New exact results on density matrix for XXX spin chain

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

Using the fermionic basis, we obtain the expectation values of all \({\mathfrak {sl}}_{2}\)-invariant and C-invariant local operators on 10 sites for the anisotropic six-vertex model on a cylinder with generic Matsubara data. This is equivalent to the generalised Gibbs ensemble for the XXX spin chain. In the case when the \({\mathfrak {sl}}_{2}\) and C symmetries are not broken this computation is equivalent to finding the entire density matrix up to 10 sites. As application, we compute the entanglement entropy without and with temperature, and compare the results with CFT predictions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Boos, H.E., Korepin, V.E.: Quantum spin chains and Riemann zeta function with odd arguments. J. Phys. A 34, 5311–5316 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y.: A recursion formula for the correlation functions of an inhomogeneous XXX model. Algebra i Anal. 17, 115–159 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Sato, J., Shiroishi, M., Takahashi, M.: Exact evaluation of density matrix elements for the Heisenberg chain. J. Stat. Mech. 0612, P12017 (2006)

    Article  Google Scholar 

  4. Jimbo, M., Miwa, T., Smirnov, F.: Hidden Grassmann structure in the XXZ model III: introducing matsubara direction. J. Phys. A 42, 304018 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Di Francesco P., Smirnov, F.: OPE for XXX. Rev. Math. Phys. 30(06), 1840006-1–1840006-17 (2018)

  6. Slavnov, N.A.: Calculation of scalar products of wave functions and form factors in the framework of the alcebraic Bethe ansatz. Theor. Math. Phys. 79, 502–508 (1989)

    Article  Google Scholar 

  7. Faddeev, L.D., Sklyanin, E.K., Takhtajan, L.A.: The quantum inverse problem method. Theor. Math. Phys. 40, 688–711 (1980)

    MATH  Google Scholar 

  8. Korepin, V.E., Bogoliubov, N.M., Izergin, A.G.: Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  9. Boos, H., Göhmann, F., Klümper, A., Suzuki, J.: Factorization of the nite temperature correlation functions of the XXZ chain in a magnetic field. J. Phys. A 40, 10699–10727 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Korepin, V.E., Lukyanov, S., Nishiyama, Y., Shiroishi, M.: Asymptotic behavior of the emptiness formation probability in the critical phase of XXZ spin chain. Phys. Lett. A 312, 21–26 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Holzhey, C., Larsen, F., Wilczek, F.: Geometric and renormalized entropy in conformal field theory. Nucl. Phys. B 424, 443–467 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Klümper, A.: Thermodynamics of the anisotropic spin-1/2 Heisenberg chain and related quantum chains. Zeitschrift für Physik B Condens. Matter 91, 507–519 (1993)

    Article  ADS  Google Scholar 

  13. Mori, M., Sugihara, M.: The double-exponential transformation in numerical analysis. J. Comput. Appl. Math. 127, 287–296 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Ooura, T.: Double exponential quadratures for various kinds of integral. In: Talk given at Second International ACCA-JP/UK Workshop, January 19. Kyoto University (2016)

  15. Boos, H., Jimbo, M., Miwa, T., Smirnov, F.: Hidden Grassmann structure in the XXZ model IV: CFT limit. Commun. Math. Phys. 299, 825–866 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Korepin, V.E.: Universality of entropy scaling in one dimensional gapless models. Phys. Rev. Lett. 92(9), 096402 (2004)

    Article  ADS  Google Scholar 

  17. Calabrese, P., Cardy, J.: Entanglement entropy and conformal field theory. J. Phys. A 42, 504005–504036 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

TM thanks to T. Ooura for suggesting his kernel function used in our calculation.

Author information

Authors and Affiliations

  1. Institute for Liberal Arts and Sciences, Kyoto University, Kyoto, Japan

    T. Miwa

  2. UPMC Univ Paris 06, CNRS, UMR 7589, LPTHE, Sorbonne Université, 75005, Paris, France

    F. Smirnov

Authors
  1. T. Miwa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Smirnov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Smirnov.

Appendix

Appendix

In this appendix we give the eigenvalues of the density matrix for \(T=0\) with accuracy \(10^{-11}\). For high spins the eigenvalues become too small, and therefore we do not write them.

\(n=2,\ j=0,1\)

$$\begin{aligned}&\{0.69314718056\}, \\&\{0.10228427315\}. \end{aligned}$$

\(n=3,\ j=1/2,3/2\)

$$\begin{aligned}&\{0.450771338685, 0.03398034507\}, \\&\{0.007624158125\}. \end{aligned}$$

\(n=4,\ j=0,1,2\)

$$\begin{aligned}&\{0.61451589297, 0.00365561121\}, \\&\{0.12071380424, 0.00552473720, 0.00069384043\}, \\&\{0.000206270047\}. \end{aligned}$$

\(n=5,\ j=1/2,3/2,5/2\)

$$\begin{aligned}&\{0.42478947699, 0.04837782416, 0.00132787973, 0.00016215953, 0.00002079330\}, \\&\{0.01220782094, 0.00041374155, 0.00003079567, {5.55739\cdot 10^{-6}}\}, \\&{\{2.01173\cdot 10^{-6}}\}. \end{aligned}$$

\(n=6,\ j=0,1,2,3\)

$$\begin{aligned}&\{0.57225072096, 0.00689732739, 0.00012390859, 0.00001153518, 2.1124\cdot 10^{-7}\},\\&\{0.12810808044, 0.00963410772, 0.00146363784, 0.00020810707, 0.00003475259, \\&2.69435\cdot 10^{-6}, 1.59341\cdot 10^{-6}, 2.7386\cdot 10^{-7}, 5.023\cdot 10^{-8}\}, \\&\{0.00045834467, 0.00001216336, 6.7394\cdot 10^{-7}, 7.206\cdot 10^{-8}, 1.690\cdot 10^{-8}\}, \\&\{7.07\cdot 10^{-9}\}. \end{aligned}$$

\(n=7,\ j=1/2,3/2,5/2\)

$$\begin{aligned}&\{0.40741354415, 0.05661439956, 0.00274447210, 0.00041094696, 0.00006055511, \\&0.00004663152, 5.80502\cdot 10^{-6}, 1.09218\cdot 10^{-6}, 3.7937\cdot 10^{-7}, 6.181\cdot 10^{-8}, 3.019\cdot 10^{-8}, \\&4.47\cdot 10^{-9}, 8.4\cdot 10^{-10}, 1.6\cdot 10^{-10}\},\\&\{0.01533056579, 0.00089067320, 0.00008573919, 0.00001697604,0.00001524263, \\&1.72604\cdot 10^{-6}, 3.4884\cdot 10^{-7}, 6.306\cdot 10^{-8}, 1.710\cdot 10^{-8}, 1.208\cdot 10^{-8}, 1.46\cdot 10^{-9}, \\&9.1\cdot 10^{-10}, 2.0\cdot 10^{-10}, 5.\cdot 10^{-11}\}, \\&\{6.30299\cdot 10^{-6}, 1.3831\cdot 10^{-7}, 5.91\cdot 10^{-9}, 4.8\cdot 10^{-10}, 7.\cdot 10^{-11}, 2.\cdot 10^{-11}\}. \end{aligned}$$

\(n=8,\ j=0,1,2,3\)

$$\begin{aligned}&\{0.54407108951, 0.009518029040, 0.00031430987, 0.00003722014, 4.34242\cdot 10^{-6},\nonumber \\&8.6837\cdot 10^{-7}, 4.4767\cdot 10^{-7}, 2.109\cdot 10^{-8}, 1.263\cdot 10^{-8}, 5.6\cdot 10^{-10}, 2.0\cdot 10^{-10}, 3.\cdot 10^{-11}\},\\&\{0.13192740945, 0.01273100394, 0.00217334341, 0.00049770442, 0.00009211769,\\&9.33699\cdot 10^{-6}, 7.06799\cdot 10^{-6}, 5.77728\cdot 10^{-6}, 1.29977\cdot 10^{-6}, 1.10141\cdot 10^{-6}, 2.1627\cdot 10^{-7}, \\&1.1066\cdot 10^{-7}, 9.385\cdot 10^{-8}, 1.705\cdot 10^{-8}, 5.46\cdot 10^{-9}, 3.62\cdot 10^{-9}, 2.72\cdot 10^{-9}, 6.6\cdot 10^{-10}, \\&3.4\cdot 10^{-10}, 1.4\cdot 10^{-10}, 7.\cdot 10^{-11}, 4.\cdot 10^{-11}\},\\&\{0.00070629696, 0.00003306502, 2.45652\cdot 10^{-6}, 4.7394\cdot 10^{-7}, 3.0235\cdot 10^{-7}, 7.450\cdot 10^{-8},\\&4.116\cdot 10^{-8}, 5.43\cdot 10^{-9}, 1.33\cdot 10^{-9}, 1.19\cdot 10^{-9}, 1.8\cdot 10^{-10}, 5.\cdot 10^{-11}, 3.\cdot 10^{-11}, 1.\cdot 10^{-11}\},\\&\{3.159\cdot 10^{-8}, 5.9\cdot 10^{-10}, 2.\cdot 10^{-11}\}. \end{aligned}$$

\(n=9,\ j=1/2,3/2,5/2,7/2\)

$$\begin{aligned}&\{0.39446858225, 0.06203960539, 0.00404495595, 0.00068660207, 0.00012681903,\\&0.00011130269, 0.00001809518, 3.66388\cdot 10^{-6}, 1.57011\cdot 10^{-6}, 1.53455\cdot 10^{-6},\\&2.8038\cdot 10^{-7}, 2.1435\cdot 10^{-7}, 1.4717\cdot 10^{-7}, 4.352\cdot 10^{-8}, 2.421\cdot 10^{-8}, 1.612\cdot 10^{-8}, \\&4.91\cdot 10^{-9}, 2.99\cdot 10^{-9}, 1.98\cdot 10^{-9}, 9.6\cdot 10^{-10}, 6.1\cdot 10^{-10}, 3.2\cdot 10^{-10}, 1.1\cdot 10^{-10},\\&6.\cdot 10^{-11}, 5.\cdot 10^{-11}, 1.\cdot 10^{-11}, 0.\cdot 10^{-11}\},\\&\{0.01764053114, 0.00135269525, 0.00015298092, 0.00004270988, 0.00003235274, \\&5.57512\cdot 10^{-6}, 1.20176\cdot 10^{-6}, 5.2612\cdot 10^{-7}, 2.8393\cdot 10^{-7}, 8.094\cdot 10^{-8}, 6.731\cdot 10^{-8},\\&5.960\cdot 10^{-8}, 1.449\cdot 10^{-8}, 8.14\cdot 10^{-9}, 5.22\cdot 10^{-9}, 4.09\cdot 10^{-9}, 1.23\cdot 10^{-9}, 9.8\cdot 10^{-10}, \\&7.0\cdot 10^{-10}, 3.0\cdot 10^{-10}, 1.1\cdot 10^{-10}, 1.\cdot 10^{-10}, 8.\cdot 10^{-11}, 3.\cdot 10^{-11}, 2.\cdot 10^{-11}, 2.\cdot 10^{-11}\},\\&\{0.00001225502, 4.8365\cdot 10^{-7}, 2.849\cdot 10^{-8}, 5.91\cdot 10^{-9}, 2.72\cdot 10^{-9}, 4.2\cdot 10^{-10},\\&4.2\cdot 10^{-10}, 1.2\cdot 10^{-10}, 4.\cdot 10^{-11}, 0.\cdot 10^{-11}\},\\&\{6.\cdot 10^{-11}\}. \end{aligned}$$

\(n=10,\ j=0,1,2,3\)

$$\begin{aligned}&\{0.52322247016, 0.01165676559, 0.00053353501, 0.00007341532, 0.00001374860,\\&2.03723\cdot 10^{-6}, 1.66763\cdot 10^{-6}, 1.4508\cdot 10^{-7}, 1.0705\cdot 10^{-7}, 5.471\cdot 10^{-8}, 1.737\cdot 10^{-8},\\&3.36\cdot 10^{-9}, 1.36\cdot 10^{-9}, 9.6\cdot 10^{-10}, 5.5\cdot 10^{-10}, 2.2\cdot 10^{-10}, 4.\cdot 10^{-11}, 3.\cdot 10^{-11},\\&3.\cdot 10^{-11}, 2.\cdot 10^{-11}\},\\&\{0.13415188237, 0.01516080455, 0.00280724281, 0.00081234228, 0.00016161568,\\&0.00002156505, 0.00001938501, 0.00001238378, 4.24831\cdot 10^{-6}, 2.54338\cdot 10^{-6}, \\&5.2611\cdot 10^{-7}, 4.3033\cdot 10^{-7}, 3.7137\cdot 10^{-7}, 2.3056\cdot 10^{-7}, 7.255\cdot 10^{-8}, 4.556\cdot 10^{-8},\\&2.913\cdot 10^{-8}, 1.613\cdot 10^{-8}, 1.522\cdot 10^{-8}, 4.47\cdot 10^{-9}, 3.92\cdot 10^{-9}, 3.90\cdot 10^{-9}, 2.20\cdot 10^{-9},\\&8.6\cdot 10^{-10}, 7.5\cdot 10^{-10}, 5.1\cdot 10^{-10}, 2.8\cdot 10^{-10}, 2.7\cdot 10^{-10}, 1.9\cdot 10^{-10}, 1.7\cdot 10^{-10}, 7.\cdot 10^{-11},\\&4.\cdot 10^{-11}, 3.\cdot 10^{-11}, 2.\cdot 10^{-11}, 1.\cdot 10^{-11}\},\\&\{0.000938440865, 0.00005918266, 5.29807\cdot 10^{-6}, 1.58240\cdot 10^{-6}, 7.2140\cdot 10^{-7},\\&1.8468\cdot 10^{-7}, 1.6292\cdot 10^{-7}, 2.367\cdot 10^{-8}, 1.706\cdot 10^{-8}, 6.67\cdot 10^{-9}, 6.05\cdot 10^{-9}, 1.73\cdot 10^{-9}, \\&1.09\cdot 10^{-9}, 3.1\cdot 10^{-10}, 2.5\cdot 10^{-10}, 2.0\cdot 10^{-10}, 9.\cdot 10^{-11}, 7.\cdot 10^{-11}, 6.\cdot 10^{-11}, 2.\cdot 10^{-11}, \\&2.\cdot 10^{-11}, 1.\cdot 10^{-11}, \}, \\&\{7.919\cdot 10^{-8}, 2.69\cdot 10^{-9}, 1.3\cdot 10^{-10}, 3.\cdot 10^{-11}, 1.\cdot 10^{-11}\} \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Miwa, T., Smirnov, F. New exact results on density matrix for XXX spin chain. Lett Math Phys 109, 675–698 (2019). https://doi.org/10.1007/s11005-018-01143-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-018-01143-x

Keywords

Search

Navigation