Paper
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Equation of state for systems with Goldstone bosons
Authors
Massimo Campostrini,Martin Hasenbusch
Andrea Pelissetto,Paolo Rossi,Ettore Vicari,Nicky Welton,Jinshu Su,Yann Thoma,David Kirkby,Shannon Johnson,Marcelo Tolmasky,Jorge Casaus,Javier Berdugo,Augustin Kasser,GIUSEPPE PETROSILLO,Stephen Cox,Colin Atkinson,Sebastião Gomes dos Santos Filho,Marco Fabbrichesi,Wesley Metzger,Joachim Mnich,artur andruszkiewicz,Pablo García Abia,Marco Pieri,Francis Arthur-Holmes ,Armando Genazzani,Magdalena Kubicka Musial,László Székelyhidi,Anton Hermann,Kenneth Read,Jozsef Haller,Ian Blagbrough,Krisztina Káldi,Eusebio Sanchez,Thomas Ferguson,Pier Giorgio Rancoita,Leonello Servoli,Jasper Kirkby,James william gaynor,carlo adamo,Alexandre Golub,Vladimir Mićić,Vlastimir Leković,Verica Babić,Nikola Makojević,Gordana Marjanović,Nikola Bošković,Gordana Radosavljević,Mirjana Knežević,Dragana Rejman Petrović,Vladimir Ranković,Mikica Drenovak,Violeta Todorović,Jasmina Bogićević,Zlata Đurić,Violeta Domanović,Veljko Marinković,Lela Ristić,Dragan Stojković,Boban Dašić,Srđan Furtula,Mark Hlatky,Hana Konečná,Rakesh Kumar,Maximiliano Valle Cruz,vina denada,Achim Müller,Ирина Ткачук,Carlos Herrera,Marek Deja,Gregory Broderick,Vladimir Nedić,Robert Randall,Steven Siegel,Ermanno GIORCELLI,thierry jouenne,Ramesh Bhat,hasalettin deligöz,James Lowe,Minho Song,Daniela Guarnieri,Gustavo Curutchet,Tatiana Kalinina,Taiki Todo,Ertugrul Colak,Enqi Liu,Lucia Junqueira,Heru Susanto,Robin Hirsch,Wolfgang Marquardt,Houssam Eddine CHAKIR,Graham Jones,Lenka Hodačová,JOSE IVAN SANCHEZ BETANCOURT,Simone Minucci,Galina Pritula,Kevin Roth,Patrizia Baraldi,Jung Sik Choi,DEVI ARIKKETH,Jan Erik Henriksen,Chun-Ming Hsieh,ADRIANA GONZALEZ-VILLALVA,Mohamad Farzan sabahi,Dariya Fursenko,Matthew Papanikolas,Olena Chupryna,Francesco BERTAGNA,Paul Mungai Mbugua,Olga Kharlan,Klaus Dohmen,Aleksandra Sergeeva,Nadeen B. 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arXiv:cond-mat/0201310v1 [cond-mat.stat-mech] 17 Jan 2002 Equation of state for systems with Goldstone
bosons
Massimo Campostrini a Martin Hasenbusch b Andrea Pelissetto c
Paolo Rossi a Ettore Vicari a
aDipartimento di Fisica, Universit`a di Pisa, and INFN, Sez. di Pisa,
I-56126 Pisa, Italy
bNIC/DESY Zeuthen, Platanenallee 6, D-51738 Zeuthen, Germany
cDipartimento di Fisica, Universit`a di Roma La Sapienza,
and INFN, Sez. di Roma I, I-00185 Roma, Italy
Abstract
We discuss some recent determinations of the equation of state for the XY and the
Heisenberg universality class.
Key words: Heisenberg model, XY model, equation of state, critical behavior
PACS: 67.40.-w, 64.60.Fr, 11.15.Me, 05.70.Jk
In the last few years there has been a significant progress in the determination
of the critical properties of O(N) models; see, e.g., Ref. [1] for a comprehen-
sive review. First of all, high-precision estimates of critical exponents and of
several high-temperature universal ratios have been obtained by using im-
proved Hamiltonians. Improved models are such that the leading nonanalytic
correction is absent in the expansion of any thermodynamic quantity near
the critical point. The idea is quite old [2–4]. However, the early attempts
that used high-temperature techniques were not able to determine improved
models with high accuracy, so that final results did not significantly improve
the estimates of standard analyses. Recently [5–12], it has been realized that
Monte Carlo simulations using finite-size scaling techniques are very effective
for this purpose, obtaining accurate determinations of several improved mod-
els in the Ising, XY, and O(3) universality class. Once an improved models
is accurately determined, one can use standard high-temperature techniques
in order to obtain very precise determinations of the critical exponents. For
instance, for the experimentally relevant cases, we obtained [13,14,10,12]:
γ = 1.2373(2), ν = 0.63012(16), N = 1,
Preprint submitted to Elsevier Science 1 November 2018
bosons
Massimo Campostrini a Martin Hasenbusch b Andrea Pelissetto c
Paolo Rossi a Ettore Vicari a
aDipartimento di Fisica, Universit`a di Pisa, and INFN, Sez. di Pisa,
I-56126 Pisa, Italy
bNIC/DESY Zeuthen, Platanenallee 6, D-51738 Zeuthen, Germany
cDipartimento di Fisica, Universit`a di Roma La Sapienza,
and INFN, Sez. di Roma I, I-00185 Roma, Italy
Abstract
We discuss some recent determinations of the equation of state for the XY and the
Heisenberg universality class.
Key words: Heisenberg model, XY model, equation of state, critical behavior
PACS: 67.40.-w, 64.60.Fr, 11.15.Me, 05.70.Jk
In the last few years there has been a significant progress in the determination
of the critical properties of O(N) models; see, e.g., Ref. [1] for a comprehen-
sive review. First of all, high-precision estimates of critical exponents and of
several high-temperature universal ratios have been obtained by using im-
proved Hamiltonians. Improved models are such that the leading nonanalytic
correction is absent in the expansion of any thermodynamic quantity near
the critical point. The idea is quite old [2–4]. However, the early attempts
that used high-temperature techniques were not able to determine improved
models with high accuracy, so that final results did not significantly improve
the estimates of standard analyses. Recently [5–12], it has been realized that
Monte Carlo simulations using finite-size scaling techniques are very effective
for this purpose, obtaining accurate determinations of several improved mod-
els in the Ising, XY, and O(3) universality class. Once an improved models
is accurately determined, one can use standard high-temperature techniques
in order to obtain very precise determinations of the critical exponents. For
instance, for the experimentally relevant cases, we obtained [13,14,10,12]:
γ = 1.2373(2), ν = 0.63012(16), N = 1,
Preprint submitted to Elsevier Science 1 November 2018
γ = 1.3177(5), ν = 0.67155(27), N = 2,
γ = 1.3960(9), ν = 0.7112(5), N = 3.
Beside the critical exponents, experiments may determine several other univer-
sal properties. We consider here the equation of state that relates the magnetic
field ~H, the magnetization ~M , and the reduced temperature t ≡ (T − Tc)/Tc.
In a neighborhood of the critical point t = 0, ~H = 0, it can be written in the
scaling form
~H = (Bc)−δ ~M Mδ−1f (x), x ≡ t(M/B)−1/β , (1)
where Bc and B are the amplitudes of the magnetization on the critical
isotherm and on the coexistence curve,
M = BcH1/δ t = 0. (2)
M = B(−t)β H = 0, t < 0. (3)
With these choices, the coexistence line corresponds to x = −1, and f (−1) =
0, f (0) = 1. Alternatively, one can write
~H = k1
~M
M |t|βδF±(|z|), z ≡ k2Mt−β , (4)
where F+(z) applies for t > 0 and F−(|z|) for t < 0. The constants k1 and k2
are fixed by requiring
F+(z) = z + 1
6 z3 + ∑
n=3
r2n
(2n − 1)!z2n−1 (5)
for z → 0 in the high-temperature phase. The behavior of the functions f (x)
and F−(|z|) at the coexistence curve depends crucially on N. For N = 1 they
vanish linearly. On the other hand, for N ≥ 2, the presence of the Goldstone
modes implies in three dimensions [15–19]:
f (x) ≈ cf (1 + x)2. (6)
The nature of the corrections to this behavior is not clear [17–20]. In particular,
logarithmic terms are expected [20].
In order to obtain approximations of the equation of state, we parametrize the
thermodynamic variables in terms of two parameters θ and R:
M = m0Rβ m(θ), t = R(1 − θ2), H = h0Rβδ h(θ). (7)
2
γ = 1.3960(9), ν = 0.7112(5), N = 3.
Beside the critical exponents, experiments may determine several other univer-
sal properties. We consider here the equation of state that relates the magnetic
field ~H, the magnetization ~M , and the reduced temperature t ≡ (T − Tc)/Tc.
In a neighborhood of the critical point t = 0, ~H = 0, it can be written in the
scaling form
~H = (Bc)−δ ~M Mδ−1f (x), x ≡ t(M/B)−1/β , (1)
where Bc and B are the amplitudes of the magnetization on the critical
isotherm and on the coexistence curve,
M = BcH1/δ t = 0. (2)
M = B(−t)β H = 0, t < 0. (3)
With these choices, the coexistence line corresponds to x = −1, and f (−1) =
0, f (0) = 1. Alternatively, one can write
~H = k1
~M
M |t|βδF±(|z|), z ≡ k2Mt−β , (4)
where F+(z) applies for t > 0 and F−(|z|) for t < 0. The constants k1 and k2
are fixed by requiring
F+(z) = z + 1
6 z3 + ∑
n=3
r2n
(2n − 1)!z2n−1 (5)
for z → 0 in the high-temperature phase. The behavior of the functions f (x)
and F−(|z|) at the coexistence curve depends crucially on N. For N = 1 they
vanish linearly. On the other hand, for N ≥ 2, the presence of the Goldstone
modes implies in three dimensions [15–19]:
f (x) ≈ cf (1 + x)2. (6)
The nature of the corrections to this behavior is not clear [17–20]. In particular,
logarithmic terms are expected [20].
In order to obtain approximations of the equation of state, we parametrize the
thermodynamic variables in terms of two parameters θ and R:
M = m0Rβ m(θ), t = R(1 − θ2), H = h0Rβδ h(θ). (7)
2
-1.0 -0.5 0.0 0.5 1.0
x
0
1
2
f(x)
n=1 (A)
n=1 (B)
MC
-1.0 -0.5 0.0 0.5 1.0
x
0
1
2
f(x)
n=0
n=1 (A)
n=1 (B)
Fig. 1. Graph of the function f (x) for N = 2 (left) and N = 3 (right). For N = 2
we also report the Monte Carlo results of Ref. [27].
Here, m0 and h0 are nonuniversal constants, m(θ) and h(θ) are odd functions
of θ, normalized so that m(θ) = θ + O(θ3) and h(θ) = θ + O(θ3). The variable
R is nonnegative and measures the distance from the critical point in the
(t, H) plane, while the variable θ parametrizes the displacement along the
lines of constant R. In particular, θ = 0 corresponds to the high-temperature
line t > 0, H = 0, θ = 1 to the critical isotherm t = 0, and θ = θ0, where
θ0 is the smallest positive zero of h(θ)—it must satisfy of course θ0 > 1—to
the coexistence line. Such a mapping has been extensively used in the Ising
case and provides accurate approximations if one uses low-order polynomials
for m(θ) and h(θ) [21–24,13,25]. In systems with Goldstone bosons we must
additionally ensure the condition (6). For this purpose, it is enough to require
[26] h(θ) ∼ (θ0 − θ)2 for θ → θ0.
In Refs. [26,10,12] we obtained the equation of state in the scaling limit by
using two different approximation schemes for the functions m(θ) and h(θ):
scheme (A) : m(θ) = θ
(
1 +
n∑
i=1
ciθ2i
)
,
h(θ) = θ (
1 − θ2/θ2
0
)2
, (8)
scheme (B) : m(θ) = θ,
h(θ) = θ (
1 − θ2/θ2
0
)2
(
1 +
n∑
i=1
ciθ2i
)
. (9)
The constants ci and θ0 were fixed by requiring F+(z) to have the expan-
sion (5), with the coefficients determined by high-temperature expansion tech-
niques. Since we were able to compute accurately only r6 and r8, we used the
two schemes for n = 0 and n = 1. The results, especially those for N = 3,
see Fig. 1, are quite independent of the scheme used, indicating the good
convergence of the method.
3
x
0
1
2
f(x)
n=1 (A)
n=1 (B)
MC
-1.0 -0.5 0.0 0.5 1.0
x
0
1
2
f(x)
n=0
n=1 (A)
n=1 (B)
Fig. 1. Graph of the function f (x) for N = 2 (left) and N = 3 (right). For N = 2
we also report the Monte Carlo results of Ref. [27].
Here, m0 and h0 are nonuniversal constants, m(θ) and h(θ) are odd functions
of θ, normalized so that m(θ) = θ + O(θ3) and h(θ) = θ + O(θ3). The variable
R is nonnegative and measures the distance from the critical point in the
(t, H) plane, while the variable θ parametrizes the displacement along the
lines of constant R. In particular, θ = 0 corresponds to the high-temperature
line t > 0, H = 0, θ = 1 to the critical isotherm t = 0, and θ = θ0, where
θ0 is the smallest positive zero of h(θ)—it must satisfy of course θ0 > 1—to
the coexistence line. Such a mapping has been extensively used in the Ising
case and provides accurate approximations if one uses low-order polynomials
for m(θ) and h(θ) [21–24,13,25]. In systems with Goldstone bosons we must
additionally ensure the condition (6). For this purpose, it is enough to require
[26] h(θ) ∼ (θ0 − θ)2 for θ → θ0.
In Refs. [26,10,12] we obtained the equation of state in the scaling limit by
using two different approximation schemes for the functions m(θ) and h(θ):
scheme (A) : m(θ) = θ
(
1 +
n∑
i=1
ciθ2i
)
,
h(θ) = θ (
1 − θ2/θ2
0
)2
, (8)
scheme (B) : m(θ) = θ,
h(θ) = θ (
1 − θ2/θ2
0
)2
(
1 +
n∑
i=1
ciθ2i
)
. (9)
The constants ci and θ0 were fixed by requiring F+(z) to have the expan-
sion (5), with the coefficients determined by high-temperature expansion tech-
niques. Since we were able to compute accurately only r6 and r8, we used the
two schemes for n = 0 and n = 1. The results, especially those for N = 3,
see Fig. 1, are quite independent of the scheme used, indicating the good
convergence of the method.
3
By using the equation of state, one can determine several amplitude ratios.
We mention here the experimentally relevant
U0 = A+
A− , Rχ = C+Bδ−1
Bδ
c
, (10)
where C+ and A± are related to the critical behavior of the susceptibility χ
and of the specific heat C for H = 0:
χ = C+t−γ , t > 0,
C = A±(±t)−α + B ± t > 0.
Using the approximate equation of state we obtain [14,10,12]: U0 = 1.062(4),
Rχ = 1.35(7) for N = 2 and U0 = 1.57(4), Rχ = 1.33(8) for N = 3.
For N = 3 the approximate equation of state can be compared with experi-
ments. We observe good qualitative and quantitative agreement. For N = 2
we can use our results to predict critical properties of the λ-transition in 4He.
In this case, the equation of state does not have a direct physical meaning,
but we can still compare the predictions for the singular specific-heat ratio
U0. A precise determination of the exponent α and of U0 was done recently
by means of a calorimetric experiment in microgravity [28] (see also reference
4 in Ref. [10]) obtaining α = −0.01056(38) and U0 ≈ 1.0442. The result for
U0 is lower than the theoretical one. This is strictly related to the disagree-
ment in the value of α (see also Ref. [29]). Indeed, using hyperscaling we find
α = −0.0146(8), that significantly differs from the experimental estimate. The
origin of this discrepancy is unclear and further theoretical and experimental
investigations are needed. A new generation of experiments in microgravity
environment that is currently in preparation should clarify the issue on the
experimental side [30].
References
[1] A. Pelissetto and E. Vicari, cond-mat/0012164.
[2] J.-H. Chen, M.E. Fisher, and B.G. Nickel, Phys. Rev. Lett. 48 (1982) 630.
[3] M.J. George and J.J. Rehr, Phys. Rev. Lett. 53 (1984) 2063.
[4] M.E. Fisher and J.H. Chen, J. Physique 46 (1985) 1645.
[5] H.G. Ballesteros, L.A. Fern´andez, V. Mart´ın-Mayor, and A. Mu˜noz Sudupe,
Phys. Lett. B 441 (1998) 330.
[6] M. Hasenbusch, K. Pinn, and S. Vinti, Phys. Rev. B 59 (1999) 11471.
4
We mention here the experimentally relevant
U0 = A+
A− , Rχ = C+Bδ−1
Bδ
c
, (10)
where C+ and A± are related to the critical behavior of the susceptibility χ
and of the specific heat C for H = 0:
χ = C+t−γ , t > 0,
C = A±(±t)−α + B ± t > 0.
Using the approximate equation of state we obtain [14,10,12]: U0 = 1.062(4),
Rχ = 1.35(7) for N = 2 and U0 = 1.57(4), Rχ = 1.33(8) for N = 3.
For N = 3 the approximate equation of state can be compared with experi-
ments. We observe good qualitative and quantitative agreement. For N = 2
we can use our results to predict critical properties of the λ-transition in 4He.
In this case, the equation of state does not have a direct physical meaning,
but we can still compare the predictions for the singular specific-heat ratio
U0. A precise determination of the exponent α and of U0 was done recently
by means of a calorimetric experiment in microgravity [28] (see also reference
4 in Ref. [10]) obtaining α = −0.01056(38) and U0 ≈ 1.0442. The result for
U0 is lower than the theoretical one. This is strictly related to the disagree-
ment in the value of α (see also Ref. [29]). Indeed, using hyperscaling we find
α = −0.0146(8), that significantly differs from the experimental estimate. The
origin of this discrepancy is unclear and further theoretical and experimental
investigations are needed. A new generation of experiments in microgravity
environment that is currently in preparation should clarify the issue on the
experimental side [30].
References
[1] A. Pelissetto and E. Vicari, cond-mat/0012164.
[2] J.-H. Chen, M.E. Fisher, and B.G. Nickel, Phys. Rev. Lett. 48 (1982) 630.
[3] M.J. George and J.J. Rehr, Phys. Rev. Lett. 53 (1984) 2063.
[4] M.E. Fisher and J.H. Chen, J. Physique 46 (1985) 1645.
[5] H.G. Ballesteros, L.A. Fern´andez, V. Mart´ın-Mayor, and A. Mu˜noz Sudupe,
Phys. Lett. B 441 (1998) 330.
[6] M. Hasenbusch, K. Pinn, and S. Vinti, Phys. Rev. B 59 (1999) 11471.
4
[7] H.G. Ballesteros, L.A. Fern´andez, V. Mart´ın-Mayor, A. Mu˜noz Sudupe,
G. Parisi, and J.J. Ruiz-Lorenzo, J. Phys. A 32 (1999) 1.
[8] M. Hasenbusch, J. Phys. A 32 (1999) 4851.
[9] M. Hasenbusch and T. T¨or¨ok, J. Phys. A 32 (1999) 6361.
[10] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Phys.
Rev. B 63 (2001) 214503.
[11] M. Hasenbusch, J. Phys. A 34 (2001) 8221.
[12] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, cond-
mat/0110336.
[13] M. Campostrini, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. E 60 (1999)
3526; cond-mat/0201180.
[14] M. Campostrini, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. B 61 (2000)
5905.
[15] E. Br´ezin and D.J. Wallace, Phys. Rev. B 7 (1973) 1967.
[16] E. Br´ezin and J. Zinn-Justin, Phys. Rev. B 14 (1976) 3110.
[17] D.J. Wallace and R.P.K. Zia, Phys. Rev. B 12 (1975) 5340.
[18] L. Sch¨afer and H. Horner, Z. Phys. B 29 (1978) 251.
[19] I.D. Lawrie, J. Phys. A 14 (1981) 2489.
[20] A. Pelissetto and E. Vicari, Nucl. Phys. B 540 (1999) 639.
[21] P. Schofield, Phys. Rev. Lett. 22 (1969) 606.
[22] P. Schofield, J.D. Lister, and J.T. Ho, Phys. Rev. Lett. 23 (1969) 1098.
[23] B.D. Josephson, J. Phys. C: Solid State Phys. 2 (1969) 1113.
[24] R. Guida and J. Zinn-Justin, Nucl. Phys. B 489 (1997) 626.
[25] M. Caselle, M. Hasenbusch, A. Pelissetto, and E. Vicari, J. Phys. A 34 (2001)
2923.
[26] M. Campostrini, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. B 62 (2000)
5843.
[27] J. Engels, S. Holtmann, T. Mendes, and T. Schulze, Phys. Lett. B 492 (2000)
219.
[28] J.A. Lipa et al., Phys. Rev. Lett. 76 (1996) 944; 84 (2000) 4894.
[29] S. Holtmann, J. Engels, and T. Schulze, hep-lat/0109013.
[30] J.A. Nissen, D.R. Swanson, Z.K. Geng, V. Dohm, U.E. Israelsson, M.J. DiPirro,
and J.A. Lipa, Low Temp. Phys. 24 (1998) 86.
5
G. Parisi, and J.J. Ruiz-Lorenzo, J. Phys. A 32 (1999) 1.
[8] M. Hasenbusch, J. Phys. A 32 (1999) 4851.
[9] M. Hasenbusch and T. T¨or¨ok, J. Phys. A 32 (1999) 6361.
[10] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Phys.
Rev. B 63 (2001) 214503.
[11] M. Hasenbusch, J. Phys. A 34 (2001) 8221.
[12] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, cond-
mat/0110336.
[13] M. Campostrini, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. E 60 (1999)
3526; cond-mat/0201180.
[14] M. Campostrini, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. B 61 (2000)
5905.
[15] E. Br´ezin and D.J. Wallace, Phys. Rev. B 7 (1973) 1967.
[16] E. Br´ezin and J. Zinn-Justin, Phys. Rev. B 14 (1976) 3110.
[17] D.J. Wallace and R.P.K. Zia, Phys. Rev. B 12 (1975) 5340.
[18] L. Sch¨afer and H. Horner, Z. Phys. B 29 (1978) 251.
[19] I.D. Lawrie, J. Phys. A 14 (1981) 2489.
[20] A. Pelissetto and E. Vicari, Nucl. Phys. B 540 (1999) 639.
[21] P. Schofield, Phys. Rev. Lett. 22 (1969) 606.
[22] P. Schofield, J.D. Lister, and J.T. Ho, Phys. Rev. Lett. 23 (1969) 1098.
[23] B.D. Josephson, J. Phys. C: Solid State Phys. 2 (1969) 1113.
[24] R. Guida and J. Zinn-Justin, Nucl. Phys. B 489 (1997) 626.
[25] M. Caselle, M. Hasenbusch, A. Pelissetto, and E. Vicari, J. Phys. A 34 (2001)
2923.
[26] M. Campostrini, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. B 62 (2000)
5843.
[27] J. Engels, S. Holtmann, T. Mendes, and T. Schulze, Phys. Lett. B 492 (2000)
219.
[28] J.A. Lipa et al., Phys. Rev. Lett. 76 (1996) 944; 84 (2000) 4894.
[29] S. Holtmann, J. Engels, and T. Schulze, hep-lat/0109013.
[30] J.A. Nissen, D.R. Swanson, Z.K. Geng, V. Dohm, U.E. Israelsson, M.J. DiPirro,
and J.A. Lipa, Low Temp. Phys. 24 (1998) 86.
5
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