Correlation functions for the $XY$ model in a Magnetic Field

H. A. Fertig; Wei Zhang

REVIEW

Correlation functions for the $XY$ model in a Magnetic Field

Wei Zhang, H. A. Fertig

Other Condensed Matter · Submitted 1 Mar 2006 · 17 references

Pith's verdict: A modest, honest study of correlation functions in the three vortex phases; the exponential decay results are solid, but the proposed anisotropic correlation length in the log-confined phase is an unconfirmed conjecture that the simulations cannot resolve.

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arXiv IDcond-mat/0603028 (version 1)

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CitationH. A. Fertig, Wei Zhang. "Correlation functions for the $XY$ model in a Magnetic Field." arXiv:cond-mat/0603028 (2006). https://pith.science/paper/CBAVKELK

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Editor's note

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Dear colleague,

The short version: this paper computes transverse spin correlations C(r) for the 2D XY model in a magnetic field in all three vortex phases and finds exponential decay everywhere. That part is clean and useful. The paper's more interesting claim—that the log-confined phase has an angle-dependent correlation length—is plausible but under-supported; the authors themselves say the simulations cannot distinguish it from a generic lattice artifact.

What's new: as far as I can tell, this is the first time C(r) has been worked out explicitly in these phases. In the linearly confined phase the Gaussian integral gives the expected Bessel-function exponential with correlation length (K/h)^{1/2}. In the free vortex phase the random-walk expansion gives an exponential with the stated length. The lattice-prefactor analysis in Section IV is also a nice piece of work: it shows a generic anisotropy that survives at long distance and can easily be mistaken for physics. The numerical work is honest—limited system sizes, seeds, error estimates.

The soft spot is the log-phase anisotropy. Equation (15) simply asserts that the RG-generated K_rho q_y^2 |phi_1|^2 + K_rho q_x^2 |phi_2|^2 are the leading lattice-symmetry-preserving operators, citing the previous paper. If that assumption is wrong, the predicted lambda(alpha) does not follow. And the numerical support does not rescue it: Table I gives K_rho nonzero in all phases, largest at h=0 where no such term is expected, and comparing the Eq. (36) and Eq. (37) fits shows the K_rho term changes the correlation function by at most a few percent, within statistical error. So the central signature is not established. To their credit, the authors say this almost verbatim in Section VI.

Who should read it: people working on vortex phases in classical spin models or Josephson arrays. It is a careful, modest contribution that collects useful formulas and gives a fair picture of what simulations can currently say. It deserves a serious referee for a specialist journal, but the referee should push to have the anisotropy either derived more carefully or explicitly labeled a conjecture. The exponential decay results alone justify the paper.

My sense: send it to peer review rather than desk reject; revisions should focus on calibrating the log-phase claim.

Best,

What would settle it

Run high-precision simulations of the log-confined phase (for instance $K=1$, $h=0.25$, temperature $1.2$) on lattices large enough that $r$ spans several correlation lengths, and compare the slopes of $\log C(r)$ along a bond direction and along the $45^\circ$ diagonal: an angle-dependent $\lambda(\alpha)$ requires the slopes to differ, whereas a pure lattice-prefactor effect would leave them equal and merely offset the curves. If the slopes agree within errors much smaller than the difference implied by the fitted $K_\rho$, the central prediction fails. A companion analytic check is to keep the next-order operators alongside the $K_\rho$ terms in the effective Hamiltonian; if the angular dependence of the correlation length changes qualitatively, the prediction is an artifact of that truncation.

Referee report

Significance

The paper is useful as a careful study of a practical obstacle: lattice regularization produces a prefactor anisotropy that can mask intrinsic anisotropic correlations. The analytical derivations are standard and mostly internally consistent, and the numerical fits are presented with appropriate caveats; the explicit comparison of Eqs. (36) and (37) and the statement in Section VI that direct confirmation was not achieved are honest and helpful. If the Log-phase anisotropy is confirmed by future work, the predicted λ(α) would be a genuine phase signature, but the present manuscript does not establish that prediction independently: it inherits the operator content of Eq. (15) from Ref. [6], and its own simulations are explicitly inconclusive. The main value is therefore the quantitative message about detectability rather than a definitive measurement of an anisotropic correlation length.

Major objections (2)

[II.B, Eq. (15)] The angle-dependent correlation length in the Log phase follows only if the Kρ q_y^2 |φ1|^2 and Kρ q_x^2 |φ2|^2 terms are the leading lattice-symmetry-preserving operators generated by the RG. The manuscript does not derive this operator ordering; it cites Ref. [6]. If the actual RG flow generates comparable q^4 or other operators, the functional form in Eq. (18) is not established. Moreover, the numerical evidence offered in Section V is inconclusive by the authors' own account: Table I gives a nonzero Kρ in all phases and the largest magnitude at h=0, and the differences between the fits with and without Kρ are at most about 5%, within statistical error. A concrete test would be to fit the simulations with the exact denominator obtained from Eq. (17) rather than the ad hoc form Eq. (36), and to check whether the extracted Kρ in the Log phase differs from the h=0 control case as system size grows.
[V, Eq. (36)] The fitting form used to search for the Log-phase anisotropy is not the functional form derived in Eq. (17). In the analytic denominator the Kρ term cancels the isotropic ξ^2 q^2 mass along the q_x=0 and q_y=0 lines, whereas Eq. (36) retains an isotropic ξ^2 q^2 term and adds Kρ q_x^2 q_y^2/q^2 merely as a perturbation. Consequently the fitted values of Kρ in Table I do not measure the parameter appearing in Eq. (18), and the comparison between Eqs. (36) and (37) cannot be read as a test of the predicted anisotropic correlation length. The authors should either fit to the full expression, or a controlled approximation of it, or explicitly state that Eq. (36) is only a phenomenological anisotropy probe and not a test of Eq. (18).

Minor objections (3)

[II.A, Eqs. (11)-(13)] The large-distance expressions in Eq. (13) omit the constant prefactor e^{-2A} with A=(1/2K)∫d^2q/(1/ξ^2+q^2) that follows from Eq. (11); as written, <cosθ1 cosθ2> does not tend to the order parameter squared. This does not affect the exponential decay of C(r), but the normalization should be stated or the expressions labeled as correlation functions divided by <cosθ>^2.
[II.C, Eq. (20)] The phrase 'to second order in K and h' deserves a short explanation: the path-sum representation treats K and h as expansion variables and should specify that self-avoiding paths are approximated by unrestricted random walks at this order.
[General] There are typographical issues in the text and figure captions (e.g., 'distinquish', garbled symbolic strings in the arXiv version of the captions) that should be corrected in the final manuscript.

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The paper

The paper claims

In all three vortex phases of the field-driven XY model, transverse spin correlations decay exponentially, and the angle-dependent decay length predicted for the logarithmically confined phase is masked in simulations by a generic lattice…

The reading

The paper studies the spin correlation functions of the two-dimensional classical XY model in a magnetic field, a system thought to contain three distinct vortex phases: linearly confined, logarithmically confined, and free. In all three phases the transverse spin-fluctuation correlation $C(r)=\langle\sin\theta(\mathbf{r}_1)\sin\theta(\mathbf{r}_2)\rangle$ decays exponentially with distance, while the longitudinal correlation $\langle\cos\theta(\mathbf{r}_1)\cos\theta(\mathbf{r}_2)\rangle$ saturates to a constant, confirming long-range order. The paper's central positive claim is that the logarithmically confined phase should show an angle-dependent correlation length $\lambda(\alpha)$, a signature unique to that phase and produced by terms that the renormalization group generates in the long-distance Hamiltonian. The paper then shows analytically and numerically that a lattice-induced anisotropy in the exponential prefactor, present in every phase and amplified by the magnetic field, dominates the anisotropy visible in simulations and hides the predicted angular dependence. The authors state that the anisotropy is believed to be present but that direct confirmation needs larger systems and higher precision than their simulations attain.

What carries the argument

The load-bearing object is a set of effective long-distance Hamiltonians for the vortex degrees of freedom, obtained through the Villain-model duality. For the logarithmically confined phase the relevant Hamiltonian is $$H=\frac{1}{2K}\int $d^{2}$q\,\left\{[$q_x^{2}$+K_\rho $q_y^{2}$]|\varphi_1|^2+[$q_y^{2}$+K_\rho $q_x^{2}$]|\varphi_2|^2+\$xi^{2}$|q_xq_y(\varphi_1+\varphi_2)|^2\right\},$$ where the $K_\rho$ terms, generated by the renormalization-group flow of operators that encode the integerness of the vortex fields, make the mass anisotropic and thereby produce the predicted angle-dependent correlation length. The second load-bearing element is the lattice-regularized propagator, in which the continuum $q^2$ is replaced by $4-2\cos q_x-2\cos q_y$; a stationary-phase evaluation of $C(r)$ on this propagator produces the prefactor anisotropy that persists at arbitrarily long distances and appears as an offset in log-log plots of the correlation function.

Extended reading notes

Core claim

The central claim is that in the field-driven square-lattice XY model the transverse spin-fluctuation correlation $C(r)=\langle\sin\theta(\mathbf r_1)\sin\theta(\mathbf r_2)\rangle$ decays exponentially, $C(r)\sim e^{-r/\lambda}$, in each of the three vortex phases, and that in the logarithmically confined phase the decay length depends on the angle $\alpha$ between $\mathbf r$ and the lattice axes, $\lambda=\lambda(\alpha)$, because of the anisotropic mass terms $K_\rho q_y^2|\varphi_1|^2$ and $K_\rho q_x^2|\varphi_2|^2$ that the renormalization-group flow generates in the effective Hamiltonian. The paper does not claim to have detected this anisotropic decay length. Instead it shows that a generic prefactor anisotropy, obtained by replacing the continuum $q^2$ in the propagator with the lattice expression $4-2\cos q_x-2\cos q_y$ and evaluating by stationary phase, is present in all phases and becomes strong enough in a magnetic field that the fitted strength $K_\rho$ of the vortex-induced anisotropy is statistically indistinguishable from zero. Fits of the simulation data with and without $K_\rho$ reproduce the data equally well within errors, and the paper concludes that the vortex-phase anisotropy is likely real but masked by lattice effects at attainable system sizes.

Load-bearing premise

The prediction of an angle-dependent decay length in the logarithmically confined phase rests entirely on the assumption that two particular lattice-symmetry-preserving terms in the effective Hamiltonian are the ones the renormalization group generates; if the flow produces different leading operators, the predicted anisotropy does not follow.

Editorial extensions

If this is right

A measurement of the transverse correlation function alone cannot distinguish the three vortex phases, because $C(r)$ decays exponentially in all of them; the longitudinal correlation saturating to a constant signals the order, not the vortex phase.
The logarithmically confined phase is distinguished in principle by an angle-dependent correlation length $\lambda(\alpha)$ that the other phases lack, so detecting a slope difference between bond and diagonal directions at large $r$ would identify that phase.
Every lattice realization of the XY model carries a direction-dependent prefactor in the exponential decay of $C(r)$ that is independent of vortices and persists at long distances; comparisons along different directions must subtract this offset before attributing any anisotropy to the vortex phase.
The masking is strong enough in a magnetic field that separating the vortex-phase anisotropy from the lattice effect requires statistical errors and system sizes well beyond the $61\times61$ lattices and run lengths used here.
Fits with and without the vortex-anisotropy term are nearly indistinguishable in absolute value and within statistical error, so the present numerical data neither confirm nor rule out the anisotropic correlation length.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A sharper separation of the two anisotropies may be available than direct slope comparison: the lattice-prefactor effect is an $r$-independent offset in $\log C$, whereas a genuine correlation-length anisotropy makes $\log C(r,\alpha_1)-\log C(r,\alpha_2)$ grow linearly with $r$; measuring that difference at the largest accessible distances with many more seeds could expose the vortex signature on
The masking logic should transfer to other lattice-regularized field theories with a symmetry-breaking field, since any such theory has a prefactor anisotropy of order $(a/\xi)^2$ that can hide genuine angular dependence of correlation lengths; the effect is therefore a general caution for simulation studies of anisotropic decay.
A next-order renormalization-group computation that keeps the operators competing with the $K_\rho$ terms could settle whether $\lambda(\alpha)$ is robust, because the prediction rests entirely on those terms being the leading lattice-symmetry-preserving operators.

Original abstract

Recent studies of the two-dimensional, classical $XY$ magnet in a magnetic field suggest that it has three distinct vortex phases: a linearly confined phase, a logarithmically confined phase, and a free vortex phase. In this work we study spin-spin correlation functions in this model by analytical analysis and numerical simulations to search for signatures of the various phases. In all three phases, the order parameter is nonzero and $<\cos(\th({\bf r}_1))\cos(\th({\bf r}_2))>$ remains nonzero for $r \equiv |{\bf r}_1-{\bf r}_2|\rightarrow \infty$, indicating the expected long range order. The correlation function for transverse fluctuations of the spins, $C(r)=<\sin(\th({\bf r}_1))\sin(\th({\bf r}_2))>$, falls exponentially in all three phases. A renormalization group analysis suggests that the logarithmically confined phase should have a spatially anisotropic correlation length. In addition, there is a generic anisotropy in the prefactor which is always present. We find that this prefactor anisotropy becomes rather strong in the presence of a magnetic field, masking the effects of any anisotropy in the correlation length in the simulations.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Recent studies of the two-dimensional, classical $XY$ magnet in a magnetic field suggest that it has three distinct vortex phases: a linearly confined phase, a logarithmically confined phase, and a free vortex phase. In this work we study spin-spin correlation functions in this model by analytical analysis and numerical simulations to search for signatures of the various phases. In all three phases, the order parameter is nonzero and $<\cos(\th({\bf r}_1))\cos(\th({\bf r}_2))>$ remains nonzero for $r \equiv |{\bf r}_1-{\bf r}_2|\rightarrow \infty$, indicating the expected long range order. The correlation function for transverse fluctuations of the spins, $C(r)=<\sin(\th({\bf r}_1))\sin(\th({\bf r}_2))>$, falls exponentially in all three phases. A renormalization group analysis suggests that the logarithmically confined phase should have a spatially anisotropic correlation length. In addition, there is a generic anisotropy in the prefactor which is always present. We find that this prefactor anisotropy becomes rather strong in the presence of a magnetic field, masking the effects of any anisotropy in the correlation length in the simulations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Recent studies of the two-dimensional, classical $XY$ magnet in a magnetic field suggest that it has three distinct vortex phases: a linearly confined phase, a logarithmically confined phase, and a free vortex phase.	background 1.00	0.04	0.05	hedged possibility 0.99
s1	In this work we study spin-spin correlation functions in this model by analytical analysis and numerical simulations to search for signatures of the various phases.	contribution 0.89	0.92	0.04	asserted 0.77
s2	In all three phases, the order parameter is nonzero and $<\cos(\th({\bf r}_1))\cos(\th({\bf r}_2))>$ remains nonzero for $r \equiv \|{\bf r}_1-{\bf r}_2\|\rightarrow \infty$, indicating the expected long range order.	result 0.99	0.48	0.23	asserted 0.97
s3	The correlation function for transverse fluctuations of the spins, $C(r)=<\sin(\th({\bf r}_1))\sin(\th({\bf r}_2))>$, falls exponentially in all three phases.	result 1.00	0.81	0.18	asserted 1.00
s4	A renormalization group analysis suggests that the logarithmically confined phase should have a spatially anisotropic correlation length.	result 0.39	0.47	0.08	hedged possibility 1.00
s5	In addition, there is a generic anisotropy in the prefactor which is always present.	result 0.95	0.44	0.06	asserted 1.00
s6	We find that this prefactor anisotropy becomes rather strong in the presence of a magnetic field, masking the effects of any anisotropy in the correlation length in the simulations.	result 0.49	0.94	0.20	asserted 1.00

What kind of paper the classifier reports this is: simulation study 0.73 · evidence from simulation or computation 0.96.

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sources read

arXiv listing abstract v1

not examined

full text
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references and citing sentences
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levels run

P abstract

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S sections
T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: Correlation functions vs. distance in a log-linear p

**Figure 2.** FIG. 2: Correlation functions vs. distance in a log-linear p

**Figure 3.** FIG. 3: Same as Fig. 2, except

**Figure 4.** FIG. 4: Same as Fig. 2, except

**Figure 5.** FIG. 5: Correlation functions vs. distance in a log-linear p

Figures reproduced from arXiv:cond-mat/0603028 by the authors.

References

Works cited (17)

[6]

H. A. Fertig and Kingshuk Majumdar, Ann. Phys. 305, 190 (2 003)

View on Pith
[1]

N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966 ). P. C. Hohenberg, Phys. Rev. 158, 383 (1967)

View on Pith 1967
[2]

J. M. Kosterlitz, D. Thouless, J. Phys. C 6, 1181 (1973); J . M. Kosterlitz, J.Phys. C 7, 1046 (1974)

View on Pith 1973
[3]

R. A. Pelcovits, J. Phys. A: Math. Gen. 14, 1693 (1981)

View on Pith 1981
[4]

J. V. Jos´ e, L. P. Kadanoﬀ, S. Kirkpatrick, and D. R. Nelson , Phys. Rev. B 16, 1217 (1977)

View on Pith 1977
[5]

H. A. Fertig, Phys. Rev. Lett. 89, 035703 (2002)

View on Pith 2002
[7]

H. A. Fertig and Joseph P. Straley, Phys. Rev. B. 66, 201402(R) (2002)

View on Pith 2002
[8]

D. Ruelle, Statistical Mechanics: Rigorous Results, Ad dison-Wesley, Reading, MA, 1989

View on Pith 1989
[9]

Wei Zhang, H. A. Fertig, Phys. Rev. B. 71, 224514 (2005)

View on Pith 2005
[10]

S. M. Girvin and A. H. MacDonald, Multicomponent Quantum Hall Systems: The Sum of Their Parts and More , in Perspectives in Quantum Hall Eﬀects, edited by S. Das Sarm a and A. Pinczuk, (Wiley, New York, 1997)

View on Pith 1997
[11]

K. Majumdar and H. A. Fertig, Phys. Rev. Lett. 94, 220402 (2005)

View on Pith 2005
[12]

J. Villain, J. Phys. 36, 581 (1975)

View on Pith 1975
[13]

L. P. Kadanoﬀ, Statistical Physics: Dynamics, Statics, and Renormalizat ion (World Scientiﬁc, Singapore, 2000)

View on Pith 2000
[14]

E. W. Montroll, and G. H. Weiss, J. Math. Phys. 6, 167 (196 5)

View on Pith
[15]

M. V. Simkin and J. M. Kosterlitz, Phys. Rev. B 55, 11646 ( 1997)

View on Pith 1997
[16]

E. N. Economou, Green’s functions in quantum physics, ( Springer-Verlag, Berlin; New York, 1983). 12

View on Pith 1983
[17]

J. Callaway, J. Math. Phys. 5, 783 (1964) /s53 /s49/s48 /s49/s53 /s48/s46/s48/s49 /s48/s46/s49 /s32 /s32 /s67/s40/s114/s41 /s114 FIG. 1: Correlation functions vs. distance in a log-linear p lot for h=0, system size=61. Averaged over 9 seeds. Curves with solid dots and squares are for corre lation functions in the x and y directions, and cannot be distinqu...

View on Pith 1964

How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it

Cites it as background 1 paper

It has been shown that the classical $2d-XY$ magnet in a magnetic field has three distinct vortex phases $-$ a linearly confined phase, a logarithmically confined phase, and a free vortex phase [ref] .

Dynamical phase transitions in $XY$ model: a Monte Carlo and mean-field theory study · Introduction · 0.86

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [6]

H. A. Fertig and Kingshuk Majumdar, Ann. Phys. 305, 190 (2 003)

View on Pith

[2] [1]

N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966 ). P. C. Hohenberg, Phys. Rev. 158, 383 (1967)

View on Pith 1967

[3] [2]

J. M. Kosterlitz, D. Thouless, J. Phys. C 6, 1181 (1973); J . M. Kosterlitz, J.Phys. C 7, 1046 (1974)

View on Pith 1973

[4] [3]

R. A. Pelcovits, J. Phys. A: Math. Gen. 14, 1693 (1981)

View on Pith 1981

[5] [4]

J. V. Jos´ e, L. P. Kadanoﬀ, S. Kirkpatrick, and D. R. Nelson , Phys. Rev. B 16, 1217 (1977)

View on Pith 1977

[6] [5]

H. A. Fertig, Phys. Rev. Lett. 89, 035703 (2002)

View on Pith 2002

[7] [7]

H. A. Fertig and Joseph P. Straley, Phys. Rev. B. 66, 201402(R) (2002)

View on Pith 2002

[8] [8]

D. Ruelle, Statistical Mechanics: Rigorous Results, Ad dison-Wesley, Reading, MA, 1989

View on Pith 1989

[9] [9]

Wei Zhang, H. A. Fertig, Phys. Rev. B. 71, 224514 (2005)

View on Pith 2005

[10] [10]

S. M. Girvin and A. H. MacDonald, Multicomponent Quantum Hall Systems: The Sum of Their Parts and More , in Perspectives in Quantum Hall Eﬀects, edited by S. Das Sarm a and A. Pinczuk, (Wiley, New York, 1997)

View on Pith 1997

[11] [11]

K. Majumdar and H. A. Fertig, Phys. Rev. Lett. 94, 220402 (2005)

View on Pith 2005

[12] [12]

J. Villain, J. Phys. 36, 581 (1975)

View on Pith 1975

[13] [13]

L. P. Kadanoﬀ, Statistical Physics: Dynamics, Statics, and Renormalizat ion (World Scientiﬁc, Singapore, 2000)

View on Pith 2000

[14] [14]

E. W. Montroll, and G. H. Weiss, J. Math. Phys. 6, 167 (196 5)

View on Pith

[15] [15]

M. V. Simkin and J. M. Kosterlitz, Phys. Rev. B 55, 11646 ( 1997)

View on Pith 1997

[16] [16]

E. N. Economou, Green’s functions in quantum physics, ( Springer-Verlag, Berlin; New York, 1983). 12

View on Pith 1983

[17] [17]

J. Callaway, J. Math. Phys. 5, 783 (1964) /s53 /s49/s48 /s49/s53 /s48/s46/s48/s49 /s48/s46/s49 /s32 /s32 /s67/s40/s114/s41 /s114 FIG. 1: Correlation functions vs. distance in a log-linear p lot for h=0, system size=61. Averaged over 9 seeds. Curves with solid dots and squares are for corre lation functions in the x and y directions, and cannot be distinqu...

View on Pith 1964