From 1b707a5885fef6a9c2f06526652f62044739350c Mon Sep 17 00:00:00 2001 From: Yuichi Motoyama Date: Thu, 5 Oct 2023 16:49:53 +0900 Subject: [PATCH] added a notice for calculating specific heat in docs --- doc/en/dla/users-manual/output.rst | 15 +++++++++++++-- doc/jp/dla/users-manual/output.rst | 16 +++++++++++++--- 2 files changed, 26 insertions(+), 5 deletions(-) diff --git a/doc/en/dla/users-manual/output.rst b/doc/en/dla/users-manual/output.rst index d1d4eaa..0c3080e 100644 --- a/doc/en/dla/users-manual/output.rst +++ b/doc/en/dla/users-manual/output.rst @@ -118,6 +118,13 @@ Notations :math:`\beta` The inverse temperature. +:math:`E_0` + The imaginary time average of the expectation value of the unperturbed Hamiltonian + :math:`\displaystyle \frac{1}{\beta}\int d\tau \langle \phi(\tau)|\mathcal{H}_0|\phi(\tau)\rangle`. + +:math:`N_v` + The number of vertices, i.e., the order of the perturbation. + :math:`h` The conjugate field to the operator :math:`M^z` . The longitudinal magnetic field for spin systems and the chemical potential for the Bose-Hubbard models. @@ -135,7 +142,7 @@ Main results are written in a file with the name specified by ``outfile`` keywor :math:`\sum_i W_i / \sum_i |W_i` ``anv`` - The mean number of the vertices. + The number of the vertices per site. :math:`\displaystyle \frac{\langle N_v \rangle}{N_s}` ``ene`` @@ -145,7 +152,11 @@ Main results are written in a file with the name specified by ``outfile`` keywor ``spe`` The specific heat - :math:`\displaystyle C_V \equiv \frac{\partial \epsilon}{\partial T}` + :math:`\displaystyle C_V \equiv \frac{\partial \epsilon}{\partial T} = \frac{1}{N_s T^2} \left[\left\langle\left(E_0 - TN_v\right)^2\right\rangle - \left\langle\left(E_0 - TN_v\right)\right\rangle - T^2\left\langle N_v \right\rangle\right]` + + NOTICE: In quantum Monte Carlo simulations, the specific heat is calculated with poor accuracy compared with other physical quantities. + Moreover, the systematic error :math:`1/N` also appears with respect to the number of samples :math:`N` (``nmcs``). + Especially, in the region where the specific heat becomes very small, e.g., in the low-temperature region below the energy gap, the calculated value may be negative. ``som`` The ratio of the specific heat and the temperature. diff --git a/doc/jp/dla/users-manual/output.rst b/doc/jp/dla/users-manual/output.rst index 6175040..ba632b1 100644 --- a/doc/jp/dla/users-manual/output.rst +++ b/doc/jp/dla/users-manual/output.rst @@ -116,6 +116,13 @@ DLA は計算結果を行区切りのプレーンテキストファイルで出 :math:`\beta` 逆温度. +:math:`E_0` + 非摂動ハミルトニアンの期待値の虚時間平均 + :math:`\displaystyle \frac{1}{\beta}\int d\tau \langle \phi(\tau)|\mathcal{H}_0|\phi(\tau)\rangle`. + +:math:`N_v` + バーテックスの数, すなわち摂動の次数. + :math:`h` :math:`M^z` に共役な外場. スピン系では縦磁場, ボース粒子系では化学ポテンシャル. @@ -133,17 +140,20 @@ DLA は計算結果を行区切りのプレーンテキストファイルで出 :math:`\frac{\sum_i W_i }{ \sum_i |W_i| }`, ここで :math:`i` はモンテカルロサンプルの番号. ``anv`` - 平均バーテックス数. + サイトあたりの平均バーテックス数. :math:`\displaystyle \frac{\langle N_v \rangle}{N_s}` ``ene`` エネルギー密度. - :math:`\displaystyle \epsilon \equiv \frac{1}{N_s}\left(E_0 - T\langle N_v\rangle\right)` + :math:`\displaystyle \epsilon \equiv \frac{1}{N_s}\left(\langle E_0 \rangle - T\langle N_v\rangle\right)` ``spe`` 比熱. - :math:`\displaystyle C_V \equiv \frac{\partial \epsilon}{\partial T}` + :math:`\displaystyle C_V \equiv \frac{\partial \epsilon}{\partial T} = \frac{1}{N_s T^2} \left[\left\langle\left(E_0 - TN_v\right)^2\right\rangle - \left\langle\left(E_0 - TN_v\right)\right\rangle - T^2\left\langle N_v \right\rangle\right]` + + NOTICE: 量子モンテカルロ法において, 比熱の計算は他の物理量と比べて精度が悪く, サンプル数 :math:`N` (``nmcs``)に対して :math:`1/N` の系統誤差も現れます. + 特に, エネルギーギャップ以下の極低温領域など, 比熱の値が非常に小さくなるような場合には, 計算結果が負になることがあります. ``som`` 比熱と温度の比.