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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0b18ba813bde7c6008baa051db2a2638c2b9d3bc | subsection | 45 | 78 | Four Main Theorems | Sincea i_2 + i_1 = (a+1) i_2 + (i_1-i_2) = (a+1)(i_2-1) + (a+1+i_1-i_2)and -(a-1) \le i_1 - i_2 \le a-1, we conclude that if (j_2,j_3) = \Psi (i_1,i_2), thenj_2= i_1-i_2 - (a+1) \left\lfloor \frac{i_1-i_2}{a}\right\rfloor , \quad j_3 = i_2 + \left\lfloor \frac{i_1-i_2}{a}\right\rfloor .Using the above, it is easy to ve... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.04547371715307236,
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0... | |
b21d772ce2ef5a7dec772cb3595ff937fbccdb8b | subsection | 46 | 78 | A Classification When | Given the positive results in Section , it is natural to ask if it is possible to classify those ({{r}},{{x}}) such that g_{{r}}^{{x}}(z) admits a geometric factorization.
In this section, we prove Theorem REF , providing a first step in response to this question.
We will work in the context of the following setup.Give... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
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-0.0024012408684939146,
0.0... | |
ccd1c88d460580afdd38d81a94b5aad2bd367852 | subsection | 47 | 78 | Setup and Classification | Setup 5.1
Let {{r}}= (2, 2k-1) for some integer k \ge 2. Then {{\rho }}= (-k, 1) and {{x}}= ( (2k-1)c_1 - k, 2 c_2 + 1) for some integers c_1 \ge 1 and c_2 \ge 0.
Applying Lemma REF , we have thatg_{{r}}^{{x}}(z) & = \sum _{{{i}}\in \left\langle 2k-1 \right\rangle \times \left\langle 2 \right\rangle } z^{c_1 i_1 + c_2... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.023879412561655045,
0.033751603215932846,
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0.0060041844844818115,
-0.019286630675196648,
... | |
3de9593cc3bffa6dd149ccee3d7c7da9e2dd7378 | subsection | 48 | 78 | Setup and Classification | If c_1 \ne 2(c_2+1), then {{r}}=(2,3) and c_2 = c_1 -2, which corresponds to applying Theorem REF with a=3 to obtain {{x}}= (3c+1, 2c-1) for c \ge 2.
Suppose c_1 < c_2 + 1.
If c_2 + 1 = 2c_1, then either {{r}}=(2,9) and c_1=1 (so {{x}}=(4,3)), or {{r}}=(2,3) and c_1 can be any positive integer. Note that the latter ... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.044003620743751526,
0.03439117968082428,
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... | |
b195b8b9ffa7c8db1cec840ab00e119af9de1fa0 | subsection | 49 | 78 | Setup and Classification | For any exponent e_j of the factorization and any e \ge e_j, we have
[z^{e-e_j}] f + [z^{e+e_j}] f \ge [z^{e}] f.We omit the proof for all but parts (REF ) and (REF ), as the others are straightforward exercises from the definition.
For part (REF ), if z^{\mu _1 + \mu _2} does not appear in (REF ), then we must have \... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.042612023651599884,
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0.02242015115916729,
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-0.03079909086227417,
0... | |
7cbe8a89b77b2e892f0286ea2575081777921858 | subsection | 50 | 78 | Setup and Classification | It then follows from Lemma REF part (REF ) that [z^{2c_1}] g_{{r}}^{{x}}(z) \ge 1, contradicting with the fact that z^{2c_1} does not appear in the expression above. Therefore, we must have that c_1 \ne c_2 + 1.
It is easy to verify the following:
when {{r}}=(2,3), g_{{r}}^{{x}}(z) has a geometric factorization (1+z)... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.03488945960998535,
0.035713620483875275,
-0.029074549674987793,
0.0024400490801781416,
0.013354451395571232,
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0.021886037662625313,
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-0.027746735140681267,
-0.027105720713734627,
... | |
4c3b2f617c8d6109e32a30e32ca940123d1ae418 | subsection | 51 | 78 | Setup and Classification | Hence, by Lemma REF part (REF ), we must have (\gamma _1, \gamma _2, \gamma _3)=(3,3,2).
Let g_0(z) = \prod _{j=4}^p \sum _{i=0}^{\gamma _j-1} z^{i e_j} = g_{{r}}^{{x}}(z)/((1+z)(1+z+z^2)(1+z+z^2)(1+z^2)). Then
g_{{r}}^{{x}}(z) = (1+z+z^2)(1+z+z^2)(1+z^2) g_0(z).
By comparing the coefficients of z^5 on both sides, we ... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.05160309746861458,
0.0204459335654974,
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-0.... | |
1d2921794d62ca2af389c231ee06f509bdc20729 | subsection | 52 | 78 | Setup and Classification | Suppose further in our setup that if g_{{r}}^{{x}}(z) has a geometric factorization, it is given as follows for some \gamma _1,\ldots ,\gamma _p\ge 2 and e_1 \le e_2 \le \dots \le e_p.g_{{r}}^{{x}}(z) = \prod _{j=1}^p \sum _{i=0}^{\gamma _j-1} z^{i e_j} = \prod _{j=1}^p \left( 1 + z^{e_j} + z^{2 e_j} + \cdots z^{(\gamm... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.027904774993658066,
0.043359965085983276,
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0.013380256481468678,
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0.03353455290198326,
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-0.02020006813108921,
-0.01583660952746868,
... | |
296d50019341f695b5bb052b25ac9e1e329b0582 | subsection | 53 | 78 | Setup and Classification | If z^{\mu _1 + \mu _2} does not appear in (REF ), then \mu _2 = 2 \mu _1 and \gamma _1 = 3. So (1 + z^{\mu _1} + z^{2\mu _1}) is a factor in the geometric factorization (REF ) of f(z).
For any i \in \lbrace 2,3,\dots ,M\rbrace , if \mu _i cannot be written as a non-negative integer linear combination of \mu _{1}, \do... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.05817099288105965,
0.019639577716588974,
-0.008545581251382828,
0.0007134034531190991,
0.0024377796798944473,
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0.02035679668188095,
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-0.005745377391576767,
-0.010163137689232826,
... | |
058073878859c878fbd943e0d1f3bcca72f3d5db | subsection | 54 | 78 | Setup and Classification | If (c_1, c_2) = (1,1), then k=2 or 5, that is, {{r}}=(2,3) or (2,9).Comparing the number of monomials in equations (REF ) and (REF ), the result follows.
Assume the contrary that c_1 = c_2 + 1. Then (REF ) becomes
g_{{r}}^{{x}}(z) = \begin{matrix} z^0 &+ z^{c_1} &+ z^{2c_1 -1} &+ z^{3c_1 - 1} &+ \cdots + &z^{(2k-3) c_... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.033145420253276825,
0.017976658418774605,
-0.0241418294608593,
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-0.008843356743454933,
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-0.03335906192660332,
-0.02456911839544773,
0... | |
27c8da71efce274bb6696deee34d3298513dd5ff | subsection | 55 | 78 | Setup and Classification | Thus,
\prod _{j=3}^p \sum _{i=0}^{\gamma _j-1} z^{i e_j} = g_{{r}}^{{x}}(z)/((1+z)(1+z+z^2+ \dots + z^{\gamma _2-1})) = 1 + 2 z^2 + z^3 h(z),
for some polynomial h(z).
Thus, by Lemma REF part (REF ) again, we conclude that e_3=e_4 = 2. However,
[z^3] \left(\prod _{j =1}^4 \sum _{i=0}^{\gamma _j-1} z^{i e_j}\right) \g... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.037319280207157135,
0.008650870993733406,
-0.028393777087330818,
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0.003165883244946599,
-0.04503945633769035,
... | |
7b5be0e93446122633ef8a7cd290d8eedce48ce6 | subsection | 56 | 78 | Proof of Theorem | Note that Lemma REF part (REF ) provides the assertion that c_1 \ne c_2 +1. In the proof of Lemma REF part (REF ), we showed that if ({{r}}, {{x}}) = ((2,9), (4,3)), g_{{r}}^{{x}}(z) has a geometric factorization. This, together with, Theorems REF , REF , and REF , provides one direction for the if and only if conditio... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.04184982553124428,
0.04407815635204315,
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-0.01784188486635685,
... | |
0f8e2ec0eb358737a4729de212db6e753a3f6821 | subsection | 57 | 78 | Proof of Theorem | However, by looking at Expression (REF ), we see that the only term that could be z^{c_1+c_2 +1} is z^{2c_1 -1}.
Hence, c_1 + c_2 +1 = 2c_1 -1, equivalently, c_2 = c_1 - 2. Since 2 = 2 (0 + 1) and c_1 \ne 2 (c_2 + 1), we conclude that c_1 \ge 3. Let c= c_1-1 \ge 2. Then
e_1 = \mu _1 = c, e_2 =\mu _2 = c+1, \mu _3 = 2c... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.021920207887887955,
0.03596927598118782,
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0.002360579092055559,
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-0.021462583914399147,
0.... | |
8b1ab9b9fc233496ad0513306bcdacc5503fff35 | subsection | 58 | 78 | Proof of Theorem | However, in this case(1+z^c) \left( 1+z^{2c} + \dots + z^{2c(\gamma _3-1)} \right) = \sum _{i=0}^{2\gamma _3 - 1} z^{i c} \, ,which is a geometric series with exponent c and of length 2 \gamma _3.
Therefore, we may assume \gamma _1 \ge 3, and e_3 \ne 2c.
Now notice that\prod _{i=1}^{\gamma _1-1} z^{i c} \prod _{i=1}^{\... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.019242940470576286,
0.02788013592362404,
-0.03644103184342384,
0.010216612368822098,
0.01764063350856304,
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0.01597728580236435,
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-0.00838540494441986,
-0.01909033954143524,
-0.013... | |
c3487f8854b77acea6731f77a68b8f1980a2407f | subsection | 59 | 78 | Proof of Theorem | However, the only term in the Expression (REF ) that could be z^{2c_1} is z^{c_1+c_2}.
Thus, 2c_1 = c_1 + c_2, or equivalently, c_2 = c_1.
Then one sees that c_1+1=c_2+1 is the second lowest positive order in (REF ).
Thus, by Lemma REF part (REF ), we have e_2 = c_1 + 1.
It follows that z^{e_1+e_2} = z^{2c_1+1} has to ... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.034816425293684006,
0.01963573321700096,
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0.01568417437374592,
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0.0428721122443676,
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-0.030102020129561424,
-0.007308092899620533,
-0.0... | |
1e66c10809807b5d2d4236bc12fff50f1aae2521 | subsection | 60 | 78 | Proof of Theorem | Dividing (REF ) by (1 + z^{c_1}) givesg(z)=& \ \ \ \ 1+ z^{2c_1-1} + z^{2(2c_1-1)} + \dots + z^{(k-2)(2c_1-1)} \\
& + z^{c_1+c_2}\left( 1+ z^{2c_1-1} + z^{2(2c_1-1)} + \dots + z^{(k-2)(2c_1-1)}\right) \\
& + \frac{z^{(k-1)(2c_1-1)} + z^{c_2+1}}{z^{c_1} + 1}.Since z^{c_1}+1 is a factor of z^a + z^b if and only if a-b is... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.03662258759140968,
0.03497457131743431,
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0.01242116093635559,
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0.012054935097694397,
-0.016327569261193275,
0.01166... | |
24bcd88074e1f15660222d3dcc54be1a4c4cda51 | subsection | 61 | 78 | Proof of Theorem | However, z^{2c_1+c_2} is neither a term in (REF ) since c_2 + 1 is not a multiple of 2c_1 -1, nor a term in (REF ) as c_2 + 1 < 2c_1 + c_2 < 2c_1 + c_2 + 1.
Hence, it must appear in ().
Thus, 2c_1 + c_2 = c_1 + c_2 + n(2c_1-1) for some non-negative integer n. Then c_1 = n(2c_1-1).
Since c_1 > 1, we deduce that n=0 and ... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.03009692020714283,
0.044870659708976746,
-0.037239596247673035,
-0.002081373007968068,
0.026372959837317467,
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0.015460536815226078,
0.03308829665184021,
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0.00331951305270195,
-0.03571338206529617,
-0.014537177979946136,
-0.00930226780474186,
0... | |
039c473c084634c4284e555c55cec63f5ecde878 | subsection | 62 | 78 | Proof of Theorem | Express g_{{r}}^{{x}}(z) asg_{{r}}^{{x}}(z) = 1 + z^{\mu _1} + z^{\mu _2} + \cdots + z^{\mu _M} \quad \text{ with $0 < \mu _1 \le \mu _2 \le \cdots \le \mu _M$.}Then by (REF ), \mu _1 = c_2+1 and \mu _2 = c_1.
Hence, by Lemma REF part (REF ), e_1 = \mu _1 = c_2 +1.Suppose c_1 = 2(c_2 +1). Let c = c_2 +1. Then
\mu _1 =... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
-0.006942054722458124,
0.026349293068051338,
-0.01006216462701559,
-0.0018394538201391697,
0.02177211456000805,
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0.018552832305431366,
0.027234215289354324,
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0.0077239894308149815,
-0.02132965438067913,
-0.027920791879296303,
-0.02764616161584854,
... | |
3620cc6b1243551f7ae429cbdb80db96e5d44ec4 | subsection | 63 | 78 | Proof of Theorem | Since 2c+1 < 2c+2 < 3c +1, the term z^{2c+2} does not appear in Expression (REF ) of g_{{r}}^{{x}}(z).
This implies that \gamma _2 = 2, that is, (1 + z^{c+1}) is a factor in the geometric factorization (REF ) of g_{{r}}^{{x}}(z).
Then it follows from Lemma REF part (REF ) that \gamma _1 must be an odd number.
In partic... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
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c9fadfbad0218b3a8100ab25637845c979900486 | subsection | 64 | 78 | Proof of Theorem | Now notice that\prod _{i=1}^{\gamma _1-1} z^{i c} \prod _{i=1}^{\gamma _2-1} z^{i (2c-1)} = 1 + z^c + z^{2c-1} + z^{2c} + z^{3c-1} + z^{4c-1} + z^{3c} h(z) \, ,for some polynomial h(z), and we have previously seen that [z^{3c-1}]g_{{r}}^{{x}}(z) = 2.
Thus, we must have that e_3 = 3c-1.
However, this implies that z^{4c-... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
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-... | |
fa3f3f18dfa03f24029fb08644ac885ec41547ef | subsection | 65 | 78 | Proof of Theorem | It follows that z^{e_1+e_2} = z^{2c_1+1} has to appear in g_{{r}}^{{x}}(z). However, the only term that could be z^{2c_1+1} is z^{3c_1-1}, which implies that c_1 = 2. (So e_1 = c_2 = c_1 = 2.)
Then (REF ) becomesg_{{r}}^{{x}}(z) =
\begin{matrix} z^0 &+ z^{2} &+ z^{3} &+ z^{5} &+ \cdots + &z^{3k-4} &+ z^{3k-3} + \\
z^{3... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
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aa907717ce6c509d2f8b09725e5ab38b0796b0c2 | subsection | 66 | 78 | Proof of Theorem | Therefore, it is left to show that it is impossible to have m \ne 0, which we prove by contradiction.Suppose m > 0. Then the part () of g(z) becomesz^{(k-1)(2c_1-1)}\left( 1-z^{c_1} + z^{2c_1} - \dots - z^{(2m-1)c_1} + z^{2mc_1} \right).As m > 0, we see that the summand - z^{(k-1)(2c_1-1) + c_1} with a negative coeffic... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
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9edff85bcd2b7095851185733f7db0491aef1d72 | subsection | 67 | 78 | Proof of Theorem | If a term in (REF ) has the desired power, then we get that c_1 + c_2 + 1 is a multiple of 2c_1-1 as well, which implies that c_1 = (c_1+c_2+1)-(c_2+1) is a multiple of 2c_1 -1.
It then follows that c_1 = 1.
Now we assume m^{\prime } = 0.
Then 2m+1=-(2m^{\prime }+1)=-1, and we have c_2 + 1 = (k-1)(2c_1-1) -c_1.
Thus, c... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
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b1f2d6e047d78a3f613a19b5fefa5dfbae2ba907 | subsection | 68 | 78 | Conjectures and Questions | In this concluding section, we present a variety of conjectures and questions based on experimental evidence.In this concluding section, we present a variety of conjectures and questions based on experimental evidence. | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
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] | 2,018 | en | Mathematics | [
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b9628385baa6c259c930fe025c146f93cf1e056e | subsection | 69 | 78 | Classifying Kronecker | In an exhaustive search of all {{q}} supported on {{r}}=(r_1,r_2) with R-multiplicity {{x}}=(x_1,x_2) where 1\le r_i\le 40 and 1\le x_i\le 100, the only {{q}}=({{r}},{{x}}) corresponding to Kronecker h^*(\Delta _{(1,{{q}})};z) that are not covered by our results in Section are given in Table REF .
Based on these exper... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
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] | 2,018 | en | Mathematics | [
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675bcfec5d731682b3a4f3e3e278ac1997b5853c | subsection | 70 | 78 | Classifying Kronecker | For each vector {{r}}=(r_1,r_2) that is not of the form (a,ka-1), there are only finitely many {{x}} such that {{q}}=({{r}},{{x}}) has a Kronecker h^*(\Delta _{(1,{{q}})};z).Question 6.2 Is it true that when {{q}}=({{r}},{{x}}) is supported on two integers, g_{{{r}}}^{{{x}}}(z) is a geometric series in powers of z if a... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
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7406c773e58c0b7b9a4bed912808b6fc83bc8ead | subsection | 71 | 78 | Do Geometric Factorizations Classify Most Kronecker | The {{q}}-vector given by ({{r}},{{x}})=((5,7), (25,7)) has a Kronecker h^*-polynomial that does not factor into geometric series in powers of z, but it is the only known {{q}}-vector with this property.
Given Theorem REF and this experimental evidence, we make the following conjecture.Conjecture 6.3
For all but finit... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
"math.NT"
] | 2,018 | en | Mathematics | [
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15c100e716641ddb290ef976d8741f09da19d3a5 | subsection | 72 | 78 | A Fibonacci Phenomenon | The appearance of ((5,13),(5,13)) and ((13,34),(13,34)) in Table REF suggests a more general phenomenon involving Fibonacci numbers.
Let a_0=1, a_1=2, and define a_n=3a_{n-1}-a_{n-2}.
Thus, the values a_n correspond to “every other” Fibonacci number.
The following conjecture has been verified for n\le 7.Conjecture 6.4 ... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
"math.CO",
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] | 2,018 | en | Mathematics | [
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... | |
d14e8b9bb902558244a07f98c7128ddaa89b0d5f | subsection | 73 | 78 | A Fibonacci Phenomenon | The result follows from Theorem REF .The fact that \ell =3 for all n establishes that (1+z+z^2) is a factor of the h^*-polynomial in this case, and thus one expects that g_{(a_{n+1},a_n)}^{(a_{n+1},a_n)}(z) factors as a product of two geometric series.
However, the behavior of u(\alpha ({{i}})) is quite subtle, in the ... | {
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"Benjamin Braun",
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] | [
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d60b5a84f34200c9922c1833da6c819b9917c119 | subsection | 74 | 78 | A Fibonacci Phenomenon | Theng_{(a_{n+1},a_{n})}^{(a_{n+1},a_{n})}(z)=\left( \sum _{i=0}^{a_n-1}z^i\right)\left( \sum _{i=0}^{a_{n+1}-1}z^i\right) \, .There are several unique aspects of Conjecture REF that distinguish it from the theorems where {{r}}=(a,ka-1).
First, in the factorizations found in the {{r}}=(a,ka-1) setting, the {{r}}-vector ... | {
"cite_spans": []
} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
"Fu Liu"
] | [
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1abc87361dce22b9bbb81cd8832850c4f87eaa17 | subsection | 75 | 78 | A Fibonacci Phenomenon | For {{i}}=(i_1,i_2)\in \left\langle a_n \right\rangle \times \left\langle a_{n+1} \right\rangle , definev({{i}}):=a_{n-1}(a_n(i_1-i_2)\bmod a_{n+1})-a_n(a_{n-1}(i_1-i_2)\bmod a_n) \, ,so that u(\alpha ({{i}}))=3i_1+v({{i}}).
Thus, for all (i_1,i_2), we havev(i_1,i_2)=v(i_1+1,i_2+1) \, ,and henceu(\alpha (i_1+1,i_2+1))=... | {
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"Benjamin Braun",
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98771dea22b7cbf02e8b474e4a2f85f7998b0453 | subsection | 76 | 78 | On Ehrhart Positivity | We conjecture that independent of the reflexivity condition, all \Delta _{(1,{{q}})} with {{q}} supported by two integers are Ehrhart positive.Conjecture 6.7
All \Delta _{(1,{{q}})} with {{q}} supported on two integers are Ehrhart positive.Conjecture REF has been verified for all {{q}}=({{r}},{{x}}) with 1\le r_i\le 1... | {
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} | 1807.00105 | $h^*$-Polynomials With Roots on the Unit Circle | [
"Benjamin Braun",
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] | [
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bc0af398d209b2392cd2ee1634fa3ef97b3aa7d4 | subsection | 77 | 78 | Properties of Factorizations | Our main approach in this paper has been to study factorizations of g_{{{r}}}^{{{x}}}(z) into geometric series in powers of z.
However, as Remark REF shows, it is possible for h^*(\Delta _{(1,{{q}})};z) to have a geometric factorization for {{q}}=({{r}},{{x}}), yet for g_{{{r}}}^{{{x}}}(z) to not have such a factorizat... | {
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cac52b8d8d8192809b2957fd9873446d67bfe8ce | abstract | 0 | 13 | Abstract | In simulations, probabilistic algorithms and statistical tests, we often
generate random integers in an interval (e.g., [0,s)). For example, random
integers in an interval are essential to the Fisher-Yates random shuffle.
Consequently, popular languages like Java, Python, C++, Swift and Go include
ranged random integer... | {
"cite_spans": []
} | 10.1145/3230636 | 1805.10941 | Fast Random Integer Generation in an Interval | [
"Daniel Lemire"
] | [
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4c65d4cd309c6ea788e35d63d48b029126b4ef1b | subsection | 1 | 13 | Introduction | There are many efficient techniques to generate high-quality pseudo-random numbers such as Mersenne Twister , Xorshift , ,
linear congruential generators , , , ,
and so forth , .
Many pseudo-random number generators produce 32-bit or 64-bit words that can be interpreted as integers in [0,2^{32}) and [0,2^{64}) respect... | {
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"raw": "Makoto Matsumoto and Takuji Nishimura. 1998. Mersenne Twister: A 623-dimensionally Equidistributed Uniform Pseudo-random Number Generator. ACM Trans. Model. Comput. Simul. 8, 1 (Jan. 1998), 3–30. ht... | 10.1145/3230636 | 1805.10941 | Fast Random Integer Generation in an Interval | [
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f8c2b276b01d9cf2410805d98a6abfb8000e1e77 | subsection | 2 | 13 | Introduction | With 64-bit registers, the latency ranges from 35 to 88 cycles, with longer running times for small values of s.
Another biased but common approach consists in using a fixed-point
floating-point representation consisting of the following step:
we convert the random word to a floating-point number in the interval [0,1... | {
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"raw": "Agner Fog. 2016. Instruction tables: Lists of instruction latencies, throughputs and micro-operation breakdowns for Intel, AMD and VIA CPUs. Technical Report. Copenhagen University College of Engin... | 10.1145/3230636 | 1805.10941 | Fast Random Integer Generation in an Interval | [
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d5a27d33a7b41cc513e1bf9826ff12eb31d184a2 | subsection | 3 | 13 | Mathematical Notation | We let \lfloor x \rfloor be the largest integer smaller than or equal to x,
we let \lceil x \rceil be the smallest integer greater than or equal to x.
We let x \div y be the integer division of x by y, defined as \lfloor x / y \rfloor . We define the remainder of the division of x by y as x \bmod y: x \bmod y \equiv x ... | {
"cite_spans": []
} | 10.1145/3230636 | 1805.10941 | Fast Random Integer Generation in an Interval | [
"Daniel Lemire"
] | [
"cs.DS"
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af2b0a613722349bba52d559db2bec2b02ce05de | subsection | 4 | 13 | Existing Unbiased Techniques Found in Common Software Libraries | Assume that we have a source of uniformly-distributed L-bit random numbers, i.e., integers in [0,2^L). From such a source of random numbers, we want to produce a uniformly-distributed random integer y in [0,s) for some integer s \in [1,2^L]. That is all integers from the interval are equally likely: P(y = z) = 1/s for ... | {
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{
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"raw": "John Von Neumann. 1951. Various techniques used in connection with random digits. National Bureau of Standards Series 12 (1951), 36–38.",
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7970ed9d26e6f2572202a5035ef6ef354f274123 | subsection | 5 | 13 | The OpenBSD Algorithm | The C standard library in OpenBSD and macOS have an arc4random_uniform function to generate unbiased random integers in an interval [0,s). See Algorithm REF . The Go language (e.g., version 1.9) has adopted the same algorithm for its Int63n and Int31n functions, with minor implementation differences .
The GNU C++ stand... | {
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"raw": "The Go authors 2017. Package rand implements pseudo-random number generators. https://github.com/golang/go/blob/master/src/math/rand/rand.go [last checked October 2017]. (2017).",
"source_ref_... | 10.1145/3230636 | 1805.10941 | Fast Random Integer Generation in an Interval | [
"Daniel Lemire"
] | [
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a29e219e63def38a34ba0078d74eccc4b5d89891 | subsection | 6 | 13 | The Java Approach | It is unfortunate that Algorithm REF always requires the computation
of two remainders, especially because we anticipate such computations to have high latency. The first remainder is used to determine whether a rejection is necessary (x< (2^L-s) \bmod s), and the second
remainder is used to generate the value in [0, s... | {
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} | 10.1145/3230636 | 1805.10941 | Fast Random Integer Generation in an Interval | [
"Daniel Lemire"
] | [
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] | 2,018 | en | Computer Science | [
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b7cd7d3e2caae2f223faf5b34789187c0b59c62a | subsection | 7 | 13 | Avoiding Division | Though arbitrary integer divisions are relatively expensive on common
processors, bit shifts are less expensive, often requiring just one cycle.
When working with unsigned integers, a bit shift is equivalent to a division
by a power of two. Thus we can compute x \div 2^k quickly for any power of two 2^k. Similarly, we ... | {
"cite_spans": []
} | 10.1145/3230636 | 1805.10941 | Fast Random Integer Generation in an Interval | [
"Daniel Lemire"
] | [
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] | 2,018 | en | Computer Science | [
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ae3e4f88c8d22eeec387154e9231f764e923f77f | subsection | 8 | 13 | Avoiding Division | It generates unbiased random integers in [0, s) for any integer s\in (0, 2^L).[t]
[1]
source of uniformly-distributed random integers in [0,2^L)
target interval [0,s) with s \in [0,2^L)
x\leftarrow random integer in [0,2^L)
m\leftarrow x \times s
l\leftarrow m \bmod 2^L
[Application of the rejection method] l <s
t \lef... | {
"cite_spans": []
} | 10.1145/3230636 | 1805.10941 | Fast Random Integer Generation in an Interval | [
"Daniel Lemire"
] | [
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830f554e74e6878063ae1008aafd0bcd779f9710 | subsection | 9 | 13 | Experiments | We implemented our software in C++ on a Linux server with an Intel (Skylake) i7-6700 processor running at 3.4GHz.
This processor has 32kB of L1 data cache, 256kB of L2 cache per core with 8MB of L3 cache, and 32GB of RAM (DDR4 2133, double-channel). We use the GNU GCC 5.4 compilers with the “-O3 -march=native” flags. T... | {
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"raw": "Pierre L'Ecuyer. 1999. Tables of linear congruential generators of different sizes and good lattice structure. Mathematics of Computation... | 10.1145/3230636 | 1805.10941 | Fast Random Integer Generation in an Interval | [
"Daniel Lemire"
] | [
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e9bfedecb81520c5e11e74d732027f82dbc13eea | subsection | 10 | 13 | Experiments | Using Linux perf, we estimated the number of cache misses to shuffle an array containing 100 million integers and found that the OpenBSD approach generates about 50% more cache misses than our approach.
[Figure: Ratio of the timings of the OpenBSD-like approach and of our approach.][Figure: Wall-clock time in nanosecon... | {
"cite_spans": []
} | 10.1145/3230636 | 1805.10941 | Fast Random Integer Generation in an Interval | [
"Daniel Lemire"
] | [
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e4d98e9cb040c84ec85c0f1d9504a02403208631 | subsection | 11 | 13 | Conclusion | We find that the algorithm often used in OpenBSD and macOS (through the arc4random_uniform function) requires two divisions
per random number generation.
It is also the slowest in our tests.
The Java approach that often requires only one division,
can be faster. We believe that it should be preferred to the
OpenBSD alg... | {
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"doi": "10.1109/t-c.1971.223205",
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"raw": "Josh Bleecher Snyder. 2017. math/rand: add Shuffle. https://go-review.googlesource.com/c/go/+/51891 [last checked October 2017]. (2017).",
... | 10.1145/3230636 | 1805.10941 | Fast Random Integer Generation in an Interval | [
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e040d812d4dea6edadec34c244889286833c17c0 | subsection | 12 | 13 | Code Samples | // returns value in [0,s)
// random64 is a function returning random 64-bit words
uint64t openbsd(uint64t s, uint64t (*random64)(void))
uint64t t = (-s) uint64t x;
do
x = random64();
while (x < t);
return x
uint64t java(uint64t s, uint64t (*random64)(void))
uint64t x = random64();
uint64t r = x while (x - r > UINT... | {
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} | 10.1145/3230636 | 1805.10941 | Fast Random Integer Generation in an Interval | [
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83f4aaa6107e06b230207d304a2b8e8d9c91018e | abstract | 0 | 80 | Abstract | We study the time dynamics of the ohmic spin boson model at arbitrary bias
$\epsilon$ and small coupling $\alpha$ to the bosonic bath. Using perturbation
theory and the real-time renormalization group (RG) method we present a
consistent zero-temperature weak-coupling expansion for the time evolution of
the reduced dens... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
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2c0e7ebc4edb8b6ca954e87c9356600e1e109ac5 | subsection | 1 | 80 | Introduction | The study of the dynamics of two-state quantum systems coupled weakly to a dissipative bath is
a fundamental problem of nonequilibrium statistical mechanics that has become of increasing
importance due to possible future technological applications in quantum information
processing. To realize scalable and fault-toleran... | {
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"raw": "A.J. Leggett, S. Chakravarty, T.A. Dorsey, M.P.A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987).",
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"start": 5... | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
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f4cb6278da687e767146d0c9da7165f2ace5035c | subsection | 2 | 80 | Introduction | The case \Gamma _i=0 is exceptional and occurs only
for systems with quantum critical points, where the scaling behavior is not cut off by any decay rate.
For the ohmic spin boson model there are three modes of a purely decaying mode z_0=-i\Gamma and two
oscillating modes with z_\pm =\pm \Omega -i\Gamma /2.
This form a... | {
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"doi": "10.2307/1311860",
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"raw": "L. van Hove, Physica 21, 517 (1955); D. Loss, Physica A 139, 505 (1986).",
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"s... | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
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cb2094a844dbdb8019515b6c2188bceb48e2f496 | subsection | 3 | 80 | Introduction | Furthermore, we will calculate all terms of the
time evolution for an arbitrary initial state of the local system, whereas in
Ref. divincenzoloss05 only the time evolution of the Pauli matrix in z-direction has
been calculated for an initial state without any spin in x- and y- direction.Besides secular terms proportion... | {
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weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
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f508346d27054b881c970db4f2c5c883af94e558 | subsection | 4 | 80 | Introduction | For the most important
regime of times which are not exponentially small or large, where |\alpha \ln (\Omega t)|\ll 1,
we will show in this paper that the logarithmic terms at high energies can be
incorporated into a renormalized tunneling \tilde{\Delta }=\Delta (\Omega /D)^\alpha , where
\Omega =\sqrt{\epsilon ^2+\til... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
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79637c25edea7cc445ac13755ef46ebdd60a6970 | subsection | 5 | 80 | Introduction | In these cases a non-perturbative resummation is also necessary for the logarithmic
terms at low energies to determine the first correction to Bloch-Redfield consistently. The only
available method up to date to achieve such a resummation is the RTRG method
, , , ,
which can account simultaneously for logarithmic terms... | {
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weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
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654e30f131b05a82375d8393d1d08029fcaf4fe5 | subsection | 6 | 80 | Introduction | In Section REF we set up the RG equations for the ohmic spin boson model and show
in Sections REF and REF how the propagator has to be changed
to account for all logarithmic renormalizations from high energies. The numerical solution of the
RG equations containing also all logarithmic renormalizations at low energies
w... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
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f7538566ce02ce213b6ee631d0fba470b71131e1 | subsection | 7 | 80 | Model, kinetic equation and Liouvillian | In this Section we introduce the model under consideration and set up the kinetic
equation to determine the time dynamics of the local reduced density
matrix. In addition we provide the perturbative solution for the effective Liouvillian
in Fourier space. This form is very helpful to understand the proper analytical co... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
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109cc84e7cf3b53b212de690e3c2495b6c4346b3 | subsection | 8 | 80 | Model | The Hamiltonian for the spin boson model consists of a local 2-level system
(described by Pauli matrices \sigma _i) coupled linearly to a bosonic bath with
energy modes \omega _q>0H_{\text{tot}}\,&=\,H\,+\,H_{\text{bath}}\,+\,V\quad ,\\
H\,&=\,{\epsilon \over 2}\sigma _z\,-\,{\Delta \over 2}\sigma _x\quad ,\\
H_{\tex... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
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2ddcb4362b09413a02bf9c5dbdd268aa34023278 | subsection | 9 | 80 | Model | The
ohmic spin boson model in weak coupling is defined by the condition \alpha \ll 1 such that
a perturbative expansion in \alpha makes sense.Since we will also work in a basis where the local Hamiltonian H is diagonal we introduce
the unitary transformationU\,=\,U^\dagger \,=\,U^{-1}\,=\,{1\over \sqrt{2\Omega _0}}
\le... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
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c5fe4978fb4948ba0461cd010075ae0303a0ef91 | subsection | 10 | 80 | Kinetic equation | We aim at calculating the time dynamics of the reduced density matrix of the local system\rho (t)\,=\,Tr_{\text{bath}}\rho _{\text{tot}}(t)with an initial state for the total density matrix\rho _\text{tot}(t=0)\,=\,\rho _0\rho _\text{bath}^{\text{eq}}factorizing into an arbitrary
initial state \rho _0=\rho (t=0) for th... | {
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"raw": "H. Mori, Progress of Theoretical Physics 33, 423 (1965); H. Grabert, Projection Operator Techniques in Nonequilibrium Statistical Mechanics, 1st ed., Springer Tracts in Modern Physics, Vol. 95 (Spri... | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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b1793b9c10acf604a85321af74224d22f8e78540 | subsection | 11 | 80 | Kinetic equation | From these properties one can show the conservation of probability Tr\dot{\rho }(t)=0 and the
hermiticity of the density matrix \rho (t)^\dagger =\rho (t) , .Once L(E) is known, the
time dynamics can be calculated from inverse Fourier transform as\rho (t)\,=\,{i\over 2\pi }\int _{\cal {C}} dE {e^{-iEt}\over E-L(E)}\rho... | {
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{
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"doi": "10.1016... | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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6cf8d86b9a843be1435c49ee421eb4aa8749802a | subsection | 12 | 80 | Kinetic equation | Thus, -\gamma _i(-E^*)^* must be also an eigenvalue of L(E), leading to\gamma _{\text{st}}\,&=\,0\quad ,\\
\gamma _0(E)\,&=\,-\gamma _0(-E^*)^*\quad ,\\
\gamma _+(E)\,&=\,-\gamma _-(-E^*)^*\quad .As a consequence, the pole z_0 is purely imaginary and z_+=-z_-^*, in accordance with (REF ).Using the diagrammatic techni... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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3b8e7e832dc08f6d1a3989b26ae07c96b6be7298 | subsection | 13 | 80 | Liouvillian in perturbation theory | With the help of the diagrammatic technique used in Ref. kashubaschoeller13 for the
ohmic spin model at zero bias, we calculate the Liouvillian up to O(\alpha ) in
Appendix . | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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... |
0b57eea6e510c2a7344e50809018ac0e4b39903d | subsection | 14 | 80 | Liouvillian in perturbation theory | Denoting the two states of the local system by i=1,2
(corresponding to the original Hamiltonian H in ()) and using the sequence (11,22,12,21) to
numerate the matrix elements of superoperators, we find:L(E)\,&=\,L_0 + \Sigma _a(E) + \Sigma _s = L_a(E) + \Sigma _s\quad ,\\
L_0\,&=\,\left(\begin{array}{cc} 0 & \Delta \ta... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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c0ab1348f30b7053f2d647aa04fb3d56103f3ed9 | subsection | 15 | 80 | Liouvillian in perturbation theory | The eigenvalues of L(E) and \tilde{L}_\Delta (E) are different but the relation
(note that \Sigma _s L_a=0){1\over E-L(E)}\,&=\,{1\over E-L_a(E)}(1+\Sigma _s{1\over E})\\
&=\,{1\over E-\tilde{L}_\Delta (E)}Z^{\prime }(E)(1+\Sigma _s{1\over E})\quad ,shows that the poles of the two resolvents 1/(E-L(E)) and 1/(E-\tilde... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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-0... |
2871afa500a6255105f3e4bd276977459d00ebca | subsection | 16 | 80 | Liouvillian in perturbation theory | They appear directly in the effective Liouvillian L(E) and have to be
distinguished from secular terms appearing by expanding the resolvent 1/(E-L_0-\Sigma (E)) in \Sigma (E).
The resummation of the latter are responsible to obtain the correct exponential behavior
of the leading order Bloch-Redfield terms for the time ... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0eb42f0571710ad110287c38a7f1ab4c5d171d01 | subsection | 17 | 80 | Liouvillian in perturbation theory | Using z_0\sim O(\alpha ) and z_\pm =\pm \Omega _0+O(\alpha ),
this means that for |E-z_0|\sim \alpha ^n\Omega _0 (with some integer n>0) we can replace \lambda _0(E) by\lambda _0(E)\,\approx \,-\alpha {\Delta ^2\over \Omega _0}\sum _{\sigma =\pm }\sigma \ln {-i(-\sigma \Omega _0)\over D}
\,=\,-i\Gamma _1\quad ,with\Gam... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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abe96005a6adcd508edb2dd14c754ae6ea9726d1 | subsection | 18 | 80 | Liouvillian in perturbation theory | (REF ). Furthermore, we note, that besides the logarithmic terms there can be
other regular terms \sim \alpha ^n which depend on the specific high-energy cutoff function
under consideration. The logarithmic terms however are universal, i.e. do not depend on the
specific form of the high-energy cutoff function. This wil... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.0... |
afe281f776b0163ea07bfffefb7f4aa93c01743b | subsection | 19 | 80 | Liouvillian in perturbation theory | To approximate the \omega -dependence of R_a(E+\omega ), we exhibit the logarithmic
parts by using the decomposition (REF ) and use the spectral decomposition
(REF -) of \tilde{L}_\Delta (E)R(E+\omega )\,&=\,{1\over E+\omega -\tilde{L}_\Delta (E+\omega )}Z^{\prime }(E+\omega )\\
&\hspace{-28.45274pt}
=\,\sum _i {1\ove... | {
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"doi": "10.... | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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... |
57142abd45c59e313bd4c5dd14a91a4d66a6ce8e | subsection | 20 | 80 | Liouvillian in perturbation theory | Closing the integration contour
in the upper half and noting that R_a(E+\omega ) is an analytic function there and \gamma _a(E) has
non-analytic features only on the imaginary axis, we find the result\Sigma (E)\,&=\,i\int _0^\infty dx \Big \lbrace \gamma _a(ix+0^+)-\gamma _a(ix-0^+)\Big \rbrace \cdot \\
& \hspace{56.9... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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dd90635f90f2812f635bc4259a696883efef087c | subsection | 21 | 80 | Liouvillian in renormalized perturbation theory | In Section we will show how the propagator 1/(E-L(E)) has to be slightly modified to
account for all logarithmic renormalizations from high energies. There are two different kinds of
logarithmic terms, one involving powers of \alpha \ln (D/\Omega ) (which can be resummed in the
renormalized tunneling ()), the other co... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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... |
7e0d325af3d8680db5804410198fa44f606912cf | subsection | 22 | 80 | Liouvillian in renormalized perturbation theory | In Section REF we will show that in this regime the propagator
can be written as{1\over E-L(E)}\,&\approx \,{1\over E-\tilde{L}_a(E)}Z^\prime (1+\Sigma _s{1\over E})\quad ,withZ^\prime \,&=\, \left(\begin{array}{cc} 1 & 0 \\ 0 & Z \end{array}\right)\quad ,\quad Z\,=\,{\tilde{\Delta }^2\over \Delta ^2}\\
\tilde{L}_a(E)... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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-0... |
1a257639eb4127374791c125b858ab82870bb6c4 | subsection | 23 | 80 | Liouvillian in renormalized perturbation theory | In Section we will see that Z can be obtained from a poor man scaling equation
for Z(E)=(-iE/D)^{2\alpha } cut off at E=i\Omega . Our result
shows that renormalized perturbation theory is not obtained by just replacing
\Delta \rightarrow \tilde{\Delta } defining a local system with a renormalized tunneling. Instead, t... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.0323... |
3c62308a763cd177748bba64170bfd879cc8938e | subsection | 24 | 80 | Liouvillian in renormalized perturbation theory | Thus, for the spin boson model at finite
bias, the systematic calculation of corrections to Bloch-Redfield is quite subtle and requires
an analysis of higher-order terms beyond O(\alpha ) for the Liouvillian for various reasons.In Section we will see that the resummation of logarithmic terms in time is very
complicate... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.0... |
096644a26fb49a935c9a15fcb096cafe0ab0a4be | subsection | 25 | 80 | Liouvillian in renormalized perturbation theory | Therefore, we state
here also the result for the propagator in the regime of small times defined by{1\over D}\,\ll \,t\,\ll \,{1\over \Omega }\quad ,corresponding to the regime of large energies\Omega \,\ll \, |E| \,\ll \, D \quad .We note that resumming all logarithmic terms \sim (\alpha \ln (E/D))^n or \sim (\alpha \... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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... |
280aec3583662550ac5f7b90a5702b11b6f812de | subsection | 26 | 80 | Liouvillian in renormalized perturbation theory | Thus, we can also use it in the regime where
|\alpha \ln (-iE/\Omega )|\ll 1, where we can expand Z(E) asZ(E)\,=\,{\tilde{\Delta }^2\over \Delta ^2}\,\left(1+2\alpha \ln {-iE\over \Omega }\right)\quad ,and, after a straightforward calculation, one finds that the propagator (REF )
at high energies obtains the same form ... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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... |
dc085fe24dbbf03caebc4060a3bbc58268b920b1 | subsection | 27 | 80 | Time dynamics | In this section we will present the time dynamics of the local density matrix analytically in the
regimes of small times (including the case of exponentially small times) and for the regime of
times which are not exponentially small or large, where renormalized perturbation theory can be applied using
the propagator pr... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
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... |
48d796747430229d9c2b9be2ccdc7593fd26947d | subsection | 28 | 80 | Exact solution at zero tunneling | For zero tunneling the time dynamics can be calculated exactly even for an arbitrary
spectral density and finite temperatures , . In this case the
local Hamiltonian H=\sigma _z\epsilon /2 decouples from the rest and the coupling to the bath
can be eliminated by a unitary transformation shifting the field operators of t... | {
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"start": 0... | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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ea7fcf7e3ce48993532dad8620057c87368b9285 | subsection | 29 | 80 | Exact solution at zero tunneling | Therefore, in the limit Dt\gg 1, we get
the resulth(t)\,\approx \,2\alpha (\gamma + \ln (Dt))\quad ,where \gamma is Euler's constant. This leads to the universal power-lawe^{-h(t)}\,&\approx \,(1-2\alpha \gamma )\left({1\over Dt}\right)^{2\alpha }\\
&=\,(1-2\alpha \gamma ){\tilde{\Delta }^2\over \Delta ^2}\left({1\ove... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0... |
da9914ee3598e83cadf323fae71740b48c220cb8 | subsection | 30 | 80 | Bloch-Redfield solution | The easiest way to derive the Bloch-Redfield solution is to insert
(REF ) in (REF ) and use the spectral
decomposition of the Liouvillian \tilde{L}_\Delta (E). This gives the formally exact expression\rho (t)\,&=\,{i\over 2\pi }\sum _{i=\text{st},0,\pm }\int _{\cal {C}} dE {e^{-iEt}\over E-\lambda _i(E)}\cdot \\
&\hsp... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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... |
f11f4c0556e49853397ca1dffe9e35850a1c6794 | subsection | 31 | 80 | Bloch-Redfield solution | We note that the right eigenstate |x_{\text{st}}(E)\rangle for E=0^+ does not give the stationary state
\rho _{\text{st}}, following from (REF ), since the eigenstates of
\tilde{L}_\Delta (E) and L(E) are different.The eigenvalues \lambda _i(E) for i=0,\pm have already been provided in
perturbation theory up to O(\alph... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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... |
012ff45e7dc2f4b8729e2bfa93077d0611589c3e | subsection | 32 | 80 | Bloch-Redfield solution | The projectors for L_0 can be most easily obtained by transforming the matrix L_0 to the basis
of the exact eigenstates of H, which, by using the unitary matrix (REF ), is described
by the unitary transformation (A_0)_{ij,kl}=U_{ik}U_{jl}^* leading toA_0\,=\,A_0^\dagger \,=\,A_0^{-1}\,=\,
{1\over \Omega _0}
\left(\begi... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.026... |
69d206c83879fb281434296fe9e7408a0a3e2d2d | subsection | 33 | 80 | Bloch-Redfield solution | Using the
formulas (REF ) and () for the pole positions, we can
decompose the time evolution of the Pauli matrices generically as\langle \sigma _\alpha \rangle (t)\,&=\,
\langle \sigma _\alpha \rangle _{\text{st}}\,+\,
F^0_\alpha (t)e^{-\Gamma _1 t}\,+\,\\
&+\,F^c_\alpha (t)e^{-{\Gamma _1\over 2} t}\cos (\Omega _1 t)\... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.0... |
2b43934ee285338ac5a112bbf90938569823f6d3 | subsection | 34 | 80 | Bloch-Redfield solution | F_\alpha ^{0,c,s}(t) denote the preexponential functions, which
become time independent in Bloch-Redfield approximation{\langle \sigma _x\rangle }_{\text{st}}\,&=\,{\Delta \over \Omega _0}\quad ,\quad {\langle \sigma _y\rangle }_{\text{st}}\,=\,0 \quad ,\quad {\langle \sigma _z\rangle }_{\text{st}}\,=\,-{\epsilon \over... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.0... |
07f7bb90229f0ad57317928d66d7faa5ca3d2f01 | subsection | 35 | 80 | Bloch-Redfield solution | Expanding the exponentials up to linear
order in \Omega _1 t and neglecting \Gamma _1t,(\Omega _1-\Omega _0)t\sim \alpha \Delta t
we obtain\langle \sigma _x\rangle (t)\,&=\,
{\langle \sigma _x\rangle }_0 - \epsilon t{\langle \sigma _y\rangle }_0\quad ,\\
\langle \sigma _y\rangle (t)\,&=\,
{\langle \sigma _y\rangle }_0... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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... |
def75edf7173ded95bf5c7ebb9a5bd2f529041e0 | subsection | 36 | 80 | Renormalized perturbation theory | Using the propagators provided in Section REF we will now apply
renormalized perturbation theory to calculate the modification of the Bloch-Redfield solution in
lowest order in \alpha (but including all logarithmic corrections \sim (\alpha \ln (Dt))^n and
\sim (\alpha \ln (D/\Omega ))^n from high energies in all orders... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0... |
60a00bacb1bb97c585380b1a14727e329a5983d1 | subsection | 37 | 80 | Small times | For small times \Omega t \ll 1 but still in the universal regime t\gg 1/D we
take the form (REF ) for the propagator and, since
E\sim 1/t\gg \epsilon ,\Delta , can expand the resolvent up to first order in \tilde{L}_0(E){1\over E - \tilde{L}_0(E)} \,\approx \,
{1\over E}+{1\over E}\tilde{L}_0(E){1\over E}\quad .In this... | {
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"raw": "A.J. Leggett, S. Chakravarty, T.A. Dorsey, M.P.A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987).",
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"start": ... | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.006... |
95aba9a955fb15bd533a5d4e00dc077c55cc1028 | subsection | 38 | 80 | Small times | We note
that it is not allowed to set t=0 since this result is only valid for t\gg 1/D, i.e. terms
\sim \epsilon /D,\Delta /D\ll \epsilon t, \Delta t are neglected.We can study the short-time solution in two different regimes, the one for exponentially small times
|\alpha \ln (\Omega t)|\sim 1 where we can neglect all ... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0fb3a2e485e72218f6b373de7f565f2a841b5f23 | subsection | 39 | 80 | Small times | In contrast, the Bloch-Redfield solution
(REF -) at small times
misses all powers of logarithmic terms \alpha \ln (D/\Omega ) (resummed in \tilde{\Delta }) and
\alpha \ln (\Omega t) together with the O(\alpha ) corrections for
\langle \sigma _x\rangle (t) and \langle \sigma _y\rangle (t).In the next section we will sho... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.01071387343108654,
0.018436407670378685,
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0.... |
40f533e60a456961e0ac62c2ee4c5e2b70306857 | subsection | 40 | 80 | Times in the non-exponential regime | We now study the regime of times which are not exponentially small or large defined by the
condition |\alpha \ln (\Omega t)|\ll 1. Here we can use the propagator in the form presented in
(REF -) and apply renormalized perturbation
theory to study the modification of the Bloch-Redfield result and to calculate the next c... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.0030519748106598854,... |
8dc2439104dae6f75945bebf8cb9f9a8c68acf8c | subsection | 41 | 80 | Times in the non-exponential regime | Since we know in all orders
of perturbation theory that the non-analytic features of the propagator are an isolated pole at
E=z_{\text{st}}=0 together with branch cuts starting at E=z_i, i=0,\pm , pointing in the direction of the
negative imaginary axis, we can decompose the time dynamics of \rho (t) in four contributi... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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... |
ff610dc30ff6de7eb60cebda7421e0dcd37a90ea | subsection | 42 | 80 | Times in the non-exponential regime | For i=0,\pm we obtain with
E=-ix\pm \eta (\eta =0^+)\rho _i(t)\,=\,F_i(t) e^{-iz_it}\quad ,with the preexponential operator given byF_i(t)\,&=\,{1\over 2\pi }\int _0^\infty \,dx \,e^{-xt}\cdot \\
&\hspace{-14.22636pt}
\cdot \left\lbrace {1\over E-\tilde{L}_0-\tilde{\Sigma }_a(E)}\Big |_{E=z_i-ix+\eta }\,-\,(\eta \right... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.01799195446074009,
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0.021974904462695122,
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... |
f1649b0956d6c130fe13fa7e643b8caa48cb85b5 | subsection | 43 | 80 | Times in the non-exponential regime | To cover the crossover to this regime
as well we leave the important term \tilde{\Sigma }_a(z_i)\sim \alpha \Omega in the
denominator which is essential for the correct position of the poles, and expand only in\tilde{\Sigma }_a(E)-\tilde{\Sigma }_a(z_i)\,&=\,\\
& \hspace{-56.9055pt}
=\,\alpha {\cal {F}}_i(E)M_i\,+\,\al... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.007430277299135923,
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0.015333529561758041,
-0.036647897213697433,
... |
92583ba2d9e5dd0c13fec1b022539f66fa1fdce1 | subsection | 44 | 80 | Times in the non-exponential regime | The lowest order values are given by the eigenvalues of the real but non-hermitian Liouvillian
\tilde{L}_0, which can be diagonalized by the transformationA\,=\,A^{-1}\,=\,
{1\over \Omega }
\left(\begin{array}{cc} \Omega \tau _++\epsilon \tau _- & -\Delta \sigma _z\tau _+ \\
-\Delta Z\sigma _z\tau _- & -\epsilon \tau _... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.02444297820329666,
0.013... |
ff5da814ebde6c1fe4fe455671ca636bfd0784ca | subsection | 45 | 80 | Times in the non-exponential regime | \tilde{\Sigma }_a^i will shift these eigenvalues by O(\alpha \Omega ) such that, together with
the symmetry relations (REF -), we get\tilde{\gamma }_{\text{st}}^i &= 0\quad ,\quad \tilde{\gamma }_0^0 = z_0 \quad ,\quad \tilde{\gamma }_\sigma ^\sigma = z_\sigma \quad ,\\
\tilde{\gamma }_0^\sigma &= -\left(\tilde{\gamma... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.016512610018253326,
0.015764812007546425,
0.01631421409547329,
0.025272535160183907,
-0.005062900483608246,
0.02692074328660965,
-0.026264512911438942,
0.... |
db3fb4f9096f46c962a21babdccca1f9c01cea2c | subsection | 46 | 80 | Times in the non-exponential regime | Denoting the projectors on the eigenstates
of \tilde{L}_0+\tilde{\Sigma }_a^i by \tilde{P}_j^i, with
i,j=\text{st},0,\pm , we get
for i=0,\pm\rho _i(t)\,&=\,{i\over 2\pi }\int _{{\cal {C}}_i} dE e^{-iEt}
\Big \lbrace {1\over E-\tilde{\gamma }_i^i}\tilde{P}_i^i \,+\,\\
&
+\,\sum _{j,j^{\prime }=0,\pm }{1\over E-\tilde{\... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
-0.03145882859826088,
-0.004184085410088301,
-0.005217710975557566,
0.01778903789818287,
0.011983953416347504,
0.008375799283385277,
-0.0016715364763513207,
0.024745985865592957,
0.00006847530312370509,
0.04082629829645157,
-0.004161200951784849,
0.01588197983801365,
0.00859701819717884,
-... |
efe6a6f743aa2cc9370b7e7ce9d4a87e2fde8e3f | subsection | 47 | 80 | Times in the non-exponential regime | Inserting this form and leaving out all analytic
functions on {\cal {C}}_i, we can split \rho _i(t) obviously in pole and pure branch cut contributions\rho _i(t)\,=\,\rho _i^p(t)\,+\,\rho _i^{\text{bc}}(t)\quad ,with\rho _i^p(t)\,&=\,\rho _i^{p1}(t)\,+\,\rho _i^{p2}(t)\,+\,\rho _i^{p3}(t)\quad ,\\
\rho _i^{p1}(t)\,&=\... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
-0.02458200976252556,
0.007896455936133862,
-0.01863105781376362,
0.03454603999853134,
0.019592365249991417,
-0.044037047773599625,
0.025436505675315857,
0.026458848267793655,
0.007633240427821875,
0.02915966510772705,
-0.03448500484228134,
-0.0008268390083685517,
-0.0053100804798305035,
0... |
8e89a9f6dbf42f65a4f8faa1c60470b047a22222 | subsection | 48 | 80 | Times in the non-exponential regime | We note that the terms involving \Sigma _s/z_i are very
important for () to calculate the terms in O(1) and O(\alpha )
consistently since\Sigma _s{1\over z_0}\,&=\,-{\Delta \Omega \over \tilde{\Delta }^2}\Big (1-{\Gamma ^{(2)}\over \Gamma ^{(1)}}\Big )
\left(\begin{array}{cc} 0 & 0 \\ \tau _+ & 0 \end{array}\right)\,+\... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
-0.02228446491062641,
0.03977624326944351,
-0.035441454499959946,
0.006238886620849371,
0.020651288330554962,
-0.04124152287840843,
0.006013752426952124,
0.0011743225622922182,
0.035777248442173004,
0.026741357520222664,
-0.012615143321454525,
-0.002781742252409458,
-0.005323086865246296,
... |
d0ba09f7de69a92796021f69cd7f82f9a292510a | subsection | 49 | 80 | Times in the non-exponential regime | Denoting the projectors in lowest and in first order in \alpha by
\tilde{P}^{(0)i}_j and \tilde{P}^{(1)i}_j, we show in Appendix
by a straightforward calculation that the projectors transformed with the matrix A
(see (REF )) are given byA \tilde{P}^{(0)i}_0 A\,&=\,
\left(\begin{array}{cc} \tau _- & 0 \\ 0 & 0 \end{arr... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
-0.0019264731090515852,
0.005832826253026724,
-0.0015774191124364734,
0.0025864331983029842,
0.013214461505413055,
0.02204953320324421,
0.005737456493079662,
0.01918080635368824,
0.028168469667434692,
0.03994856774806976,
-0.00414668582379818,
0.01734970323741436,
-0.0011835411423817277,
-... |
5624a5ef1ba9fd6a66cd744e16d12a35cbb77240 | subsection | 50 | 80 | Times in the non-exponential regime | All other contributions to the time
evolution are corrections in O(\alpha ). All energy integrals can be calculated from&{i\over 2\pi }\int dE e^{-iEt}{1\over E-z_i}\ln {-i(E-z_i)\over \Omega }\,=\\
&\hspace{28.45274pt}
=\,-(\gamma + \ln (\Omega t))e^{-iz_i t}\\
&{i\over 2\pi }\int _{{\cal {C}}_i} dE e^{-iEt}{1\over E... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
0.0040931543335318565,
0.04479963332414627,
-0.008399931713938713,
0.017791680991649628,
-0.01308436132967472,
-0.07379122078418732,
0.009269678965210915,
-0.008026092313230038,
0.033874381333589554,
-0.001298899413086474,
-0.010734517127275467,
0.00631711445748806,
-0.02008049003779888,
-... |
9ec39941dd68a24c1da1e008d560dd2ebcb19719 | subsection | 51 | 80 | Times in the non-exponential regime | As a consequence, only the crossover
functions H(\pm \Omega t) and \tilde{H}(\pm \Omega t) will appear for the branch cut integrals.Finally, the derivatives of {\cal {F}}_i(E) can be obtained from (REF ){d {\cal {F}}_i\over dE}(E)\,=\,1+\ln {-i(E-\lambda _i(E))\over \Omega }+O(\alpha )\quad ,which gives{d {\cal {F}}_0\... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
-0.055758628994226456,
0.022065401077270508,
-0.02374395914375782,
0.0031053314451128244,
0.003753865137696266,
0.01658719964325428,
0.030565006658434868,
0.03930876404047012,
-0.011879609897732735,
0.030168257653713226,
-0.041017841547727585,
0.00988060049712658,
0.007004162762314081,
0.0... |
c48e12f287ca33ec60e6f0f79442bca677037ee7 | subsection | 52 | 80 | Times in the non-exponential regime | Decomposing the time dynamics according to
(REF ) in the various modes, we get for the preexponential functions the
following final result for the time dynamics in the non-exponential time regimeF_x^0(t) \,&=\,
-{\langle \sigma _x\rangle }_{\text{st}}
\,-\, \Big (1+2\alpha {\tilde{\Delta }^2\over \Omega ^2}\Big ){\tild... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
-0.02436728961765766,
0.013846784830093384,
-0.010375551879405975,
-0.004863540176302195,
0.011908997781574726,
-0.00952872447669506,
-0.020888427272439003,
0.007564235478639603,
0.027540987357497215,
0.03903038799762726,
-0.030653653666377068,
-0.0005798102938570082,
-0.00654193852096796,
... |
aeec0863bfb521159e2a804619613a981e3804e7 | subsection | 53 | 80 | Times in the non-exponential regime | Only if the renormalization of the tunneling is neglected,
these operators are identical to the Pauli spin operators defined in
(REF -).The stationary values {\langle \sigma _\alpha \rangle }_{\text{st}} of the Pauli matrices follow from{\langle \sigma _x\rangle }_{\text{st}}\,&=\,{\tilde{\Delta }^2\over \Delta \Omega ... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
-0.004847887437790632,
0.014066883362829685,
-0.0554436594247818,
-0.02239716239273548,
-0.00003998982720077038,
0.015028068795800209,
0.06761868298053741,
0.03383985534310341,
0.04433661699295044,
0.042292188853025436,
-0.01199194137006998,
0.010939213447272778,
0.021725857630372047,
-0.0... |
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