itensor - usp › ~gtlandi › tutorial02.pdf · tensor renormalization group the tensor...
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![Page 1: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This](https://reader030.vdocuments.net/reader030/viewer/2022040113/5f1891d91924f93a5a24d964/html5/thumbnails/1.jpg)
ITensor
E.M. Stoudenmire Apr 2018 - Sao Paulo
![Page 2: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This](https://reader030.vdocuments.net/reader030/viewer/2022040113/5f1891d91924f93a5a24d964/html5/thumbnails/2.jpg)
Tensor Renormalization Group
The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors
This algorithm was proposed by Levin and Nave (cond-mat/0611687)
It is called a "renormalization group" because it decimates the tensor network in a heirarchical fashion
© Edwin Miles Stoudenmire, Flatiron Institute, 2018
![Page 3: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This](https://reader030.vdocuments.net/reader030/viewer/2022040113/5f1891d91924f93a5a24d964/html5/thumbnails/3.jpg)
Classical Partition Function
A key application of TRG is computing properties of classical lattice models
We will use it to evaluate the partition function, but can also evaluate local observables, and even conformal field theory scaling dimensions!
© Edwin Miles Stoudenmire, Flatiron Institute, 2018
![Page 4: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This](https://reader030.vdocuments.net/reader030/viewer/2022040113/5f1891d91924f93a5a24d964/html5/thumbnails/4.jpg)
Classical Ising Model
The partition function of the classical Ising model is
Z =X
�1�2�3...
e�E(�1,�2,�3,...)/T
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Where is the energy function
and is the temperature
E(�1,�2,�3, . . .)<latexit sha1_base64="mzpSmIoTvDW+7mg/6CoTpkR1fW8=">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</latexit><latexit sha1_base64="mzpSmIoTvDW+7mg/6CoTpkR1fW8=">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</latexit><latexit sha1_base64="mzpSmIoTvDW+7mg/6CoTpkR1fW8=">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</latexit><latexit sha1_base64="mzpSmIoTvDW+7mg/6CoTpkR1fW8=">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</latexit>
T<latexit sha1_base64="A2m5U9Mmaofve7by/jknpDOtcz4=">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</latexit><latexit sha1_base64="A2m5U9Mmaofve7by/jknpDOtcz4=">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</latexit><latexit sha1_base64="A2m5U9Mmaofve7by/jknpDOtcz4=">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</latexit><latexit sha1_base64="A2m5U9Mmaofve7by/jknpDOtcz4=">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</latexit>
The spin variables take the values � = +1,�1<latexit sha1_base64="52d3V0gIqkCY6rn6f+XSBwXk/TY=">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</latexit><latexit sha1_base64="52d3V0gIqkCY6rn6f+XSBwXk/TY=">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</latexit><latexit sha1_base64="52d3V0gIqkCY6rn6f+XSBwXk/TY=">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</latexit><latexit sha1_base64="52d3V0gIqkCY6rn6f+XSBwXk/TY=">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</latexit>
© Edwin Miles Stoudenmire, Flatiron Institute, 2018
![Page 5: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This](https://reader030.vdocuments.net/reader030/viewer/2022040113/5f1891d91924f93a5a24d964/html5/thumbnails/5.jpg)
One Dimensional Ising Model
In one dimension, the energy of the Ising model is
E(�1,�2,�3, . . . ,�N ) = �1�2 + �2�3 + �3�4 + . . .+ �N�1<latexit sha1_base64="L/JU06l8MUbgCsdp7fJeumfzCJ4=">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</latexit><latexit sha1_base64="L/JU06l8MUbgCsdp7fJeumfzCJ4=">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</latexit><latexit sha1_base64="L/JU06l8MUbgCsdp7fJeumfzCJ4=">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</latexit><latexit sha1_base64="L/JU06l8MUbgCsdp7fJeumfzCJ4=">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</latexit>
So anti-aligned (antiferromagnetic) configurations have lowest energy
Taking periodic boundary conditions, so interaction between first and last Ising spin
© Edwin Miles Stoudenmire, Flatiron Institute, 2018
![Page 6: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This](https://reader030.vdocuments.net/reader030/viewer/2022040113/5f1891d91924f93a5a24d964/html5/thumbnails/6.jpg)
Transfer Matrix Trick
To compute the partition function, use following trick
Z =
X
{�}
exp
�1
T
X
n
�n�n+1
!
<latexit sha1_base64="id/z8s97O0/RQMpH5/TRsSBoTns=">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</latexit><latexit sha1_base64="id/z8s97O0/RQMpH5/TRsSBoTns=">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</latexit><latexit sha1_base64="id/z8s97O0/RQMpH5/TRsSBoTns=">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</latexit><latexit sha1_base64="id/z8s97O0/RQMpH5/TRsSBoTns=">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</latexit>
=X
{�}
Y
n
e�(�n�n+1)/T = Tr�MN
�
<latexit sha1_base64="gBLgquG2EqOroxdZM2Ry0ZZJe54=">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</latexit><latexit sha1_base64="gBLgquG2EqOroxdZM2Ry0ZZJe54=">AAADFXicdVHLbhMxFHWGVwmPprBkYzFCSoUIMxESbJAqdcMGVKSkrRRPRx7nZmLF9oxsT9vIne9gy4+wArFlzSewhR/Ak2TRNumVLB2de46PfW9WCm5sFP1uBbdu37l7b+t++8HDR4+3OztPDk1RaQZDVohCH2fUgOAKhpZbAcelBiozAUfZbL/pH52CNrxQAzsvIZE0V3zCGbWeSjsJfo+JqWTqiCOG55KSusak1MU4darGcOJedZeNVOEVcOplXO++xoO63dgtnFs30N4mYGK7H08+YaJ5PrW7aSeMetGi8DqIVyBEqzpId1pfybhglQRlmaDGjOKotImj2nImoG6TykBJ2YzmMPJQUQkmcYs51PiFZ8Z4Umh/lMUL9rLDUWnMXGZeKamdmuu9htzUG1V28i5xXJWVBcWWQZNKYFvgZqh4zDUwK+YeUKa5fytmU6ops370baLgjBVSUjV2ZAa2HsWJI6BMpaHJchdhTDRVuf9gfVWdabqmJmIhDeOLDerF9f2NBuwdYR/fkERP85uSvPGSy+80vr7BdXDY78VRL/78JtzbX213Cz1Dz1EXxegt2kMf0AEaIoa+oz/oL/oXfAm+BT+Cn0tp0Fp5nqIrFfz6D3VyADA=</latexit><latexit sha1_base64="gBLgquG2EqOroxdZM2Ry0ZZJe54=">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</latexit><latexit sha1_base64="gBLgquG2EqOroxdZM2Ry0ZZJe54=">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</latexit>
M��0 = e�(��0)/T<latexit sha1_base64="lbKpRExynMv9mVra891FBEYhrS0=">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</latexit><latexit sha1_base64="lbKpRExynMv9mVra891FBEYhrS0=">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</latexit><latexit sha1_base64="lbKpRExynMv9mVra891FBEYhrS0=">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</latexit><latexit sha1_base64="lbKpRExynMv9mVra891FBEYhrS0=">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</latexit>
Where the matrix M is defined as
© Edwin Miles Stoudenmire, Flatiron Institute, 2018
![Page 7: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This](https://reader030.vdocuments.net/reader030/viewer/2022040113/5f1891d91924f93a5a24d964/html5/thumbnails/7.jpg)
Transfer Matrix Trick
Can visualize the transfer-matrix form of Z as a tensor network
Z = Tr�MN
�<latexit sha1_base64="R1U+n/kx/9f977tFV2VeOPpDBr4=">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</latexit><latexit sha1_base64="R1U+n/kx/9f977tFV2VeOPpDBr4=">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</latexit><latexit sha1_base64="R1U+n/kx/9f977tFV2VeOPpDBr4=">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</latexit><latexit sha1_base64="R1U+n/kx/9f977tFV2VeOPpDBr4=">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</latexit>
= M<latexit sha1_base64="X3ckg6JoI47kgr3U9yuaJ38cqco=">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</latexit><latexit sha1_base64="X3ckg6JoI47kgr3U9yuaJ38cqco=">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</latexit><latexit sha1_base64="X3ckg6JoI47kgr3U9yuaJ38cqco=">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</latexit><latexit sha1_base64="X3ckg6JoI47kgr3U9yuaJ38cqco=">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</latexit>
© Edwin Miles Stoudenmire, Flatiron Institute, 2018
![Page 8: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This](https://reader030.vdocuments.net/reader030/viewer/2022040113/5f1891d91924f93a5a24d964/html5/thumbnails/8.jpg)
Transfer Matrix Trick
One-dimension case exactly solvable
1) Diagonalize M to get its eigenvalues: {�1,�2}<latexit sha1_base64="iiHX0RV+FeGrVJbw08xrHGModrg=">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</latexit><latexit sha1_base64="iiHX0RV+FeGrVJbw08xrHGModrg=">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</latexit><latexit sha1_base64="iiHX0RV+FeGrVJbw08xrHGModrg=">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</latexit><latexit sha1_base64="iiHX0RV+FeGrVJbw08xrHGModrg=">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</latexit>
2) Observe that:
Z = Tr�MN
�= �N
1 + �N2
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Using basis-invariance property of the trace
© Edwin Miles Stoudenmire, Flatiron Institute, 2018
![Page 9: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This](https://reader030.vdocuments.net/reader030/viewer/2022040113/5f1891d91924f93a5a24d964/html5/thumbnails/9.jpg)
Two-dimensional Ising Model
Here we're primarily interested in the two-dimensional Ising model
E(�1,�2, . . .) =X
hiji
�i�j
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© Edwin Miles Stoudenmire, Flatiron Institute, 2018
![Page 10: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This](https://reader030.vdocuments.net/reader030/viewer/2022040113/5f1891d91924f93a5a24d964/html5/thumbnails/10.jpg)
Two-dimensional Transfer Tensor
Again possible to define Z by replacing transfer matrix, with a "transfer tensor"
A�t�r�b�l = e�(�t�r+�r�b+�b�l+�l�t)/T<latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">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</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">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</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">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</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">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</latexit>
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�l A
© Edwin Miles Stoudenmire, Flatiron Institute, 2018
![Page 11: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This](https://reader030.vdocuments.net/reader030/viewer/2022040113/5f1891d91924f93a5a24d964/html5/thumbnails/11.jpg)
Two-dimensional Transfer Tensor
In terms of tensor "A", partition function Z is
Z =<latexit sha1_base64="FtQ57alXsaVhBWw8afZpcSHrVG4=">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</latexit><latexit sha1_base64="FtQ57alXsaVhBWw8afZpcSHrVG4=">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</latexit><latexit sha1_base64="FtQ57alXsaVhBWw8afZpcSHrVG4=">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</latexit><latexit sha1_base64="FtQ57alXsaVhBWw8afZpcSHrVG4=">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</latexit>
A�t�r�b�l = e�(�t�r+�r�b+�b�l+�l�t)/T<latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">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</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">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</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">AAADOHicdZHfatRAFMYn8V9dq24VvfFmMAiV4jZZhPZGqPTGywrdtrBJl5PZs9thZyZhZlJZ0oCP4nP4Jt55pXjrEzibZrFttgcCP77zfXMmc9JccGPD8Ifn37l77/6DtYedR+uPnzztbjw7MlmhGQ5YJjJ9koJBwRUOLLcCT3KNIFOBx+lsf9E/PkdteKYO7TzHRMJU8QlnYJ006n79eFrGhk8ljCxtQC8hXYKo6AeKp+W7zbZ3a0Vsq33Cf00swb7dPqxG3SDshXXRNkQNBKSpg9GG9y0eZ6yQqCwTYMwwCnOblKAtZwKrTlwYzIHNYIpDhwokmqSsn6qib5wyppNMu09ZWqtXEyVIY+YydU4J9szc7C3EVb1hYSe7SclVXlhU7HLQpBDUZnTx7nTMNTIr5g6Aae7uStkZaGDWbacTK/zCMilBjct4hrYaRkkZozKFxsWs8iKIYg1q6n6wuu5ONbTcsaitQXSxwl0f318ZoC4R9Oktk+B8etskF7yScjuNbm6wDUf9XhT2os/vg739Zrtr5BV5TTZJRHbIHvlEDsiAMPLLW/deeC/97/5P/7f/59Lqe03mOblW/t9/KgYLQQ==</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">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</latexit>
�t
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�l A
© Edwin Miles Stoudenmire, Flatiron Institute, 2018
![Page 12: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This](https://reader030.vdocuments.net/reader030/viewer/2022040113/5f1891d91924f93a5a24d964/html5/thumbnails/12.jpg)
Tensor Renormalization Group
The TRG strategy to compute Z is in two steps
First, use the SVD to factorize the A tensor two ways:
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© Edwin Miles Stoudenmire, Flatiron Institute, 2018
![Page 13: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This](https://reader030.vdocuments.net/reader030/viewer/2022040113/5f1891d91924f93a5a24d964/html5/thumbnails/13.jpg)
Tensor Renormalization Group
Each factorization can be derived from an SVD by grouping the square roots of singular values with unitary tensors:
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© Edwin Miles Stoudenmire, Flatiron Institute, 2018
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Tensor Renormalization Group
Insert these factorizations to rewrite the network as:
=
© Edwin Miles Stoudenmire, Flatiron Institute, 2018
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=
Tensor Renormalization Group
Then group blocks of four tensors as follows:
© Edwin Miles Stoudenmire, Flatiron Institute, 2018
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Tensor Renormalization Group
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Effect of Step 1 followed by Step 2
© Edwin Miles Stoudenmire, Flatiron Institute, 2018
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Tensor Renormalization Group
Two steps of TRG algorithm:
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Step #2: = A1
Can be repeated to turn network of A1's into A2's etc.© Edwin Miles Stoudenmire, Flatiron Institute, 2018
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Tensor Renormalization Group
After iterating enough steps (how many?) entire network become a single A tensor AN
AN
Tracing the AN tensor gives the partition function
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ITensor Code Activity
Read through the code tutorial/trg/trg.cc
= An+1
Your job: Step 2 – recombine factors, defining new A
F3
F1
F4
F2
An =F3 F4
F1 F2
=
Step 1 – factor two ways (approximately)
x
x
00
y00
yy
x
00
y00
x
y00
x
00
y
x
00
00
00
00Try following parameters:
T = 3.0 maxm = 20 topscale = 8
./trg [log(Z)/Ns = 0.149216]
( (
© Edwin Miles Stoudenmire, Flatiron Institute, 2018