発表ファイル 数理物理・物性基礎論セミナー Watanabe 2 examples

1. Goldstone model (relativistic, n

= dim G  H)

J. Goldstone, Nuovo Cimento 19, 154 (1961)



Ÿ Lagrangian

L = ¶Μ^{j*¶Μj - V}Hj^*jL

= ¶Μj^*¶^Μj + m²j^*j -^g

2^Hj

*_jL2_.

- G = UH1L: j' HxL = ã^{ä Q Εj}HxL ã^{-ä Q Ε= j}HxL ã^{ä Ε.}

Ÿ Potential minimum

V '_Hj^*j_{L = gHj}^*j_{L - m}²= 0 ® v²=^m2

g^.

- For example, we can set Xj\ = v Î R.

- H =8e<: U(1) symmetry is completely broken. X@ä Q, jHxLD\ = X∆ j\ = ä v ¹ 0.

- dimHGL - dimHHL = 1 - 0 = 1^.

Ÿ Fluctuation around the minimum Substitute j = v +^{Σ+ä Π}

2 and expand L in the series of Σ, Π: L = ¹_2I¶ΜΣ ¶^ΜΣ + ¶ΜΠ ¶^ΜΠ_{M + m}²_Jv²+ 2 v Σ +^Σ2+Π2

2 ^{N -} g 2^Jv

2_{+ 2 v Σ +}^Σ2+Π2

2 ^N

2

= ¹

2^{¶ΜΠ ¶} Μ_Π+ ¹

2^{I¶ΜΣ ¶}

Μ_{Σ - 2 m}2_Σ2_M+^m4 2 g^-

g v 2 ^IΣ

3_{+ Σ Π}2_{M -}^g 8 ^IΣ

2_{+ Π}2_M²

- No mass term for the Goldstone mode Π. The linear dispersion Ω = c k.

- The detailed form of the Lagrangian depends on the parametrization. We could choose, e.g., j =_{Hv + ΣL ã}^{ä Π}. - This example: O_{H2L ® OH1L}

® Can be generalized to OHnL ® OHn - 1L. dimHGL - dimHHL = ^nHn-1L₂ ^-Hn-1L Hn-2L

2 ^{= n - 1.}

2. U(2)/U(1) model (nonrelativistic, n

< dim G  H when Μ ¹ 0)

T. Schäfer, D.T. Son, M.A. Stephanov, D. Toublan, J.J.M. Verbaarschot, Phys. Lett. B 522, 67 (2001) V. A. Miransky, I. A. Shovkovy, PRL 88 111601 (2002)

Ÿ Lagrangian j =_K^j1

j2O: two-component bosonic field.

L

¹ 0

H2L H L

Σ

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K 2^O

L =_H¶Ν+ä ∆Ν 0Μ_{L j}^*_H¶^Ν-ä ∆Ν 0Μ_Lj - V_Hj^*j_L

=_H¶Ν+ä ∆Ν 0Μ_{L j}^*_H¶^Ν-ä ∆Ν 0Μ_{L j + m}²j^*j -^g

2^Hj

*_jL2_.

- U(1) chemical potential A0= Μ ¹ 0

(i) explicitely breaks the Lorentz symmetry, (ii) reduces O(4) symmetry down to U(2). (iii) breaks the charge conjugation. - G = UH2L: j' HxL = ã^{ä QiΕij}HxL ã^{-ä QiΕi= ãä Εi Σ2ij}HxL.

Σ0: unit matrix Σ_1,2,3: Pauli matrices

Ÿ Potential minimum

V '_Hj^*j_{L = gHj}^*j_{L - Im}²+ Μ²_{M = 0 ® v}²=^m

2_+Μ2

g ^.

- For example, we can set _{Xj\ =} ⁰

v ^{, v Î R.} - H = UH1L generated by Q0^{+ Q}3.

X@ä Q0^{, j}HxLD\ = ä^Σ₂^{0 0}_v ^{= ä 1}₂ ⁰_v ^, X@ä Q^{1, j}HxLD\ = ä^Σ₂^{1 0}_v ^{= ä 1}₂ ^v₀ ^, X@ä Q2^{, j}HxLD\ = ä^Σ₂^{2 0}_v ^{= ä 1}₂ ^{-ä v}₀ ^, X@ä Q3, j_{HxLD\ = ä}^Σ3

2

0 v ^{= ä}

1 2

0 -v ^. - dimHGL - dimHHL = 4 - 1 = 3^.

- Even when Μ = 0: G = OH4L, H = OH3L, dimHGL - dimHHL = 6 - 3 = 3.

Ÿ Fluctuation around the minimum

Substitute j =

Π₁+ä Π₂ 2

v +^{Σ+ä Π3}

2

and expand L in the series of Σ, Π^a:

L =¹_2I¶Μ^{Σ ¶ΜΣ + ¶}Μ^Πa¶ΜΠaM + Im^{2+ Μ2}M Jv^{2+ 2 v Σ +Σ2+Π}₂^aΠaN - ^g₂Jv^{2+ 2 v Σ +Σ2+Π}₂^aΠaN²

=¹

2 ^¶ΜΠ

a_¶Μ_Πa_{+ ΜHΠ} 1^Π

2^{- Π}2^Π

1L + ΜHΣ Π 3^{- Π}3^Σ  L +

1 2^{A¶ΜΣ ¶}

Μ_{Σ - 2Im}2_{+ Μ}2_{M Σ}2_{E +} ^Im2+Μ2M

2

2 g ^-

g v 2 ^IΣ

3_{+ Σ Π}a_Πa_{M -}^g 8^IΣ

2_{+ Π}a_Πa_M²

>A¹₂ ^¶Μ^Π1¶ΜΠ1+ 1 2^¶ΜΠ

2_¶Μ_Π2_{+ ΜHΠ} 1^Π

2^{- Π}2^Π

1LE + B¹₂^¶Μ^Π3¶ΜΠ3+ Μ² m²+Μ^2Π

3²F - Im^{2+ Μ2}M JΣ -_m2^Μ_+Μ2^Π

3N^{2+ …} - Completing square for massive modes ® “integrating out” at the tree level.

- Π³ gives a mode with the dispersion _{J1 +} ^Μ

2

m²+Μ^{2N Ω}

2_{- k}2_{= 0.}

- Π^1,2 together give a single gapless mode. EOM: det ^Ω

2_{- k}2 _{2 ä Μ Ω}

-2 ä Μ Ω Ω²- k² = 0 [solve this together?]

Ω²= k²+ 2 Μ²± 2 Μ⁴+ k²Μ² =_J^k2

2 Μ^N 2

+ O_Ik⁶_{M, 4 Μ}²+ 2 k²+ O_Ik⁴_M - In total, only two Goldstone modes appear.

nNGB= 2 < dim_{HGHL = 3}

] ]

= 0

= k _]

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nNGB= 2 < dim_{HGHL = 3}

In this example, the total number of mode agrees with the expected number.

- Nonzero (Noether) charge density: ji⁰=_H¶0+ä Μ_{L j}^*ä^Σi

2 ^{j + c.c.}

Yj⁰⁰] =^{-Μ v2,} Yj¹⁰] = 0, Yj²⁰] = 0, Yj³⁰] = ^{Μ v2.}

- When Μ = 0, L =_A¹

2^¶ΜΠ

1_¶Μ_Π1₊ ¹ 2^¶ΜΠ

2_¶Μ_Π2_{+ ΜHΠ}

1Π 2- Π2Π 1LE + B₂^1¶ΜΠ³¶^ΜΠ³+ ^Μ

2

m²+Μ^2Π

3²F - Im^{2+ Μ2}M JΣ -_m2^Μ_+Μ2^Π

3N^{2+ …} each Π^a=1,2,3 corresponds to a linear mode with Ω = k. No charge densities _Yji⁰] = 0.

- Set Ω = ^k2

2 Μ^:

Ω²- k² 2 ä Μ Ω -2 ä Μ Ω Ω²- k^{2 µ}

-1 ä -ä -1 ^. Eigenvector: ¹

-ä ¬ “circular polarization”

Þ ja=1⁰= Φ, ja=2⁰= -ä Φ. So, Q1- ä Q2 does not couple to any zero mode but this is not a contradiction. (cf. S⁺ _{0\ = 0} for ferromagnet)

Ÿ Normalization of the field...

We can change parametrization j =

Π₁+ä Π₂ 2

v +^{Σ+ä Π3}

2

® j =_{Hv + ΣL ã}^ä

Σa 2 ^Π

a

L =_B^v2

4 ^¶ΜΠ

1_¶Μ_Π1₊^v2 4 ^¶ΜΠ

2_¶Μ_Π2₊ ^{Μ v2} 2 ^HΠ2Π

1- Π1Π 2_{LF + B} v²

4 ^¶ΜΠ

3_¶Μ_Π3₊^v2 2

Μ² m²+Μ^2Π

3²_F

3. Ferromagnet (nonrelativistic, n

< dimHG  H L)

Ÿ Heisenberg Hamiltonian H = -J_ÚXi,j\si^{× s}j (J > 0)

G = SOH3L leaves H and As^{ix, sjy}E = i ∆^{i jsiz intact.}

c.f. ^Li^{= S cosΘ}i^Φ

i^{+ J s}i^{× s}i+1.

HΘ, ΦL: sphererical coordinate of S = S n. HcosΘ - 1L Φ  ^{= nyn} _1+n^x-nz^xn y: the Berry phase term

® very complicated but still SU(2) invariant up to a surface term.

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nx=Sin_@Θ@t_DDCos_@Φ@t_DD; ny=Sin_@Θ@t_DDSin_@Φ@t_DD; nz=Cos_@Θ@t_DD;

FullSimplify_B-

nx D_@ny, t_{D -}ny D_@nx, t_D 1+nz

-_HCos_@Θ@t_{DD -}1_LD_@Φ@t_D, t_DF 0

Ÿ Ground state configuration S =_Úisi

S^zi GS\ = S  GS\

X@ä S^{x, siy}D\ = -Xs^iz\ = -S ¹ 0. X@ä S^{y, six}D\ = Xs^iz\ = S ¹ 0. - H = SOH2L generated by S^z. - Spin (charge) density _Xsi^z\ = S.

- S⁺ _{0\ = IS}x+ ä SyM  0\ = 0 Þ S^x 0\ = -ä S^y 0\.

Ÿ Excitation (Holstein-Primakov transformation with the large S approximation) si^z= S - n^`i

si⁺= 2 S - n^{`</sup>i b<sup>`}i» 2 S b^`i

s_i^-= b^{`</sup><sub>i</sub>¾ 2 S - n<sup>`}_i » b^`_i¾ 2 S

H S ⁼

-J S ^ÚXi,j\^J

s^-_is_j⁺+s_j^-s⁺_i

2 ^{+ si}

z_sz j_N

= -J_ÚXi,j\Ib^{`</sup>i¾ b<sup>`}j+ b^{`</sup>j¾ b<sup>`}i- 2 b^{`</sup>i¾ b<sup>`}i_{M + OJ} b³

S^N

= 2 J_{ÚkÚΑ@1 - cosHkΑ}a_{LD b}^{`</sup>k¾ b<sup>`}k^{+ O}J^b_S³N

Only one NGB with a quadratic dispersion Ω =_{IJ S a}²_{M k}².

Ÿ Excitation (effective Lagrangian) L =_ÚiS cosΘiΦ _i+ J_ÚXi,j\si× sj

=_Úia^{d S}

a^d

n^yn ^x-n^xn ^y

1+n^{z -ÚXi,j\a}

d ^J

2 a^{dBIsi- sjM}

2- 2 S_{HS + 1LF}

»_{Ù d}^dx_B^S

a^d

n^yn ^x-n^xn ^y 1+n^{z -}

J a²

2 a^{d¶Αs × ¶ΑsF}

L » ¹

2 S a^{d AHΠ}

y_Π x _{- Π}x_Π y_{L - J S a}2_¶

x^Πa¶x^ΠaE ¬ s = S -Π^y

Π^x 1 EOM: det ^{-J S a}

2_k2 _{ä Ω}

- ä Ω - J S a²k² = 0 [solve this together?] Ω = J S a²k².

4. Antiferromagnet (nonrelativistic, n

= dimHG  H L)

Ÿ Heisenberg Hamiltonian (for simplicity, d=1.) H_{i=+J si× si+1} (J > 0).

G = SOH3L leaves H and Asi^x, sj^yE = i ∆i jsi^z intact. L_{i= S cosΘiΦ i- J si× si+1.}

Ÿ Holstein-Primakov transformation with the large S approximation s_{2 i}^z= S - n^`_{2 i}

s2 i⁺= 2 S - n^{`</sup>2 i b<sup>`}2 i» 2 S b^`2 i

s_{2 i}^-= b^{`</sup><sub>i</sub>¾ 2 S - n<sup>`}_{2 i} » 2 S b^`_{2 i}¾ and

s2 i+1^z= n^`2 i+1- S

s_{2 i+1}⁺= b^{`</sup><sub>2 i+1</sub>¾ 2 S - n<sup>`}_{2 i+1} » 2 S b^{`</sup><sub>2 i+1</sub>¾ s2 i+1<sup>-</sup>= 2 S - n<sup>`}2 i+1 b2 i+1» 2 S b^`2 i+1

Then,

H S ⁼

J S^ÚiJ

s_{2 i+1}^-s_{2 i}⁺+s_{2 i}^-s_{2 i+1}⁺

2 ^{+ s2 i}

z_s

2 i+1^zN +^J_SÚiJ^{s2 i-s2 i-1++s}₂^{2 i-1-s2 i+ + s}2 i^zs2 i-1^zN

> J ÚiIb^{`2 i+1b`2 i+ b`2 i¾ b`2 i+1¾ + n`2 i+1+ n`2 i}M + J ÚⁱIb^{`2 i¾ b`2 i-1¾ + b`2 i-1b`2 i+ n`2 i-1+ n`2 i}M

= J _ÚiIb^{`</sup><sub>i+1</sub>b<sup>`}i^{+ b}

`

i¾ b^{`</sup>i+1¾ + 2 n<sup>`}iM

= J _Ú-^Π

a^{£k £} Π a

AcosHk aL b^{`</sup>-kb<sup>`}k+ cos_{Hk aL b}^{`</sup>-k¾ b<sup>`}k¾ + bk¾ bk+ b-k¾ b-kE

= 2 J _{Ú0£k £}Π a

H b^{k¾ b-k}L _cos¹_{Hk aL} ^cos₁^{Hk aL} _b^bk

-k^¾

= 2 J _{Ú0£k £}^Π

a^{H Β}

k^{¾ Β}-k_L

cosh Θ sinh Θ sinh Θ cosh Θ

1 cos_{Hk aL} cos_{Hk aL} 1

cosh Θ sinh Θ sinh Θ cosh Θ

Βk

Β-k^¾

= 2 J _{Ú0£k £}^Π

a^{H Β}

k^{¾ Β}-k_L

1 - cos²_{Hk aL} 0 0 1 - cos²_{Hk aL}

Βk

Β-k^¾

¬ tanhH2 ΘL = -cosHk aL

= 2 J _Ú-^Π

2 a^{£k £} Π 2 a

1 - cos²_{Hk aL JΒ}k¾ Βk+ Β^Π

a^+k

¾ Β^Π

a^+kN

Two NGBs with the linear dispersion Ω =H2 J S aL k.

Ÿ Excitation (effective Lagrangian)

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∆IÙ d t cosΘ Φ  ^M

=Ù d t IcosΘ ∆ Φ  - sinΘ ∆Θ Φ  _M

=Ù d t IsinΘ ∆Φ Θ  - sinΘ ∆Θ Φ  _M

=Ù d t IsinΘ ∆Φ Θ  - sinΘ ∆Θ Φ  _M

=Ù d t ∆n×n´n  n^Ži=_H-1Lⁱni

L = S_ÚicosΘiΦ _i- J S²_Úini× ni+1

= S_ÚiH-1Lⁱcos Θ^Ži^Φ

Ž 

i^{+ J S2}Úiⁿ

Ž

i^{× n}

Ž

i+1

= ^S

2^ÚiH-1L

i+1_{Jcos Θ}^Ž

i+1Φ^Ž _i+1- cos Θ^ŽiΦ^Ž _{iN -}¹

2^{J S} 2_Ú

iHn^{Ži+1- nŽi}L²

®^S

2 ^ÚiH-1L i+1_Hn^Ž

i+1- n^ŽiL×n^Ži´ n^Ž i-¹

2^{J S} 2_Ú

iHn^Ži+1- n^ŽiL²

= ^S

2^ÚiH-1L i+1_Hn^Ž

i+1- n^Ži_L×n^Ži´ n^Ž i- ¹

2^{J S} 2_Ú

iHn^{Ži+1- nŽi}L² We introduce slowly varying fields n and s by n^Ž_i= n_{HxiL + H-1L}ⁱs_HxiL

L > ¹₂^{S ¶}xn × n ´ n  +^S

a^{s × n ´ n}

  - ¹

2 a ^{J S}

2_{A4 s}2_{+Ha ¶} xn_L²_E

= ¹

2^{S ¶xn × n ´ n}

  -^{2 J S2}

a ^{Js -} n´n  4 J S^N

2

+ ^{2 J S2}

a ^J

n´n  4 J S^N

2

- ¹

2 a^{J S} 2_{Ha ¶}

xn_L²

> _{8 a J}¹ An ^2- H2 J S aL²H¶xn_L²_E+ ^S

2^{¶xn × n ´ n}

  - ^{2 J S2}

a ^{Js -} n´n  4 J S^N

2

- Topological term 2 Π S W@nD = 2 Π S Ù^{d x d t}_{4 Π} ^{¶xn × n ´ n}  is particular to 1D. - Do the same for “ferri”magnet. e.g, S^A¹ S^B.

5. BEC (nonrelativistic, n

= dimHG  HL)

Ÿ Lagrangian

L = i Ψ¾ Ψ  - ¹

2 mÑ Ψ¾ × Ñ Ψ + Μ Ψ¾ Ψ -^g

2^{HΨ¾ ΨL} 2

- G = UH1L: Ψ' HxL = ã^{ä Q ΕΨ}HxL ã^{-ä Q Ε= Ψ}HxL ã^{ä Ε.}

Ÿ Potential minimum

V '_HΨ^*Ψ_{L = gHΨ}^*ΨL - Μ = 0 ® XΨ^*Ψ\ = n0= ^Μ

g^.

- H =8e<: U(1) symmetry is completely broken. X@ä Q, ΨHxLD\ = X∆ Ψ\ = ä n^{0 ¹ 0.} - dimHGL - dimHHL = 1 - 0 = 1^.

Ÿ Fluctuation around the minimum Substitute Ψ = n ã^{-ä Π}:

L = n Π  - n^ÑΠ×ÑΠ

2 m ^- Μ

2 n₀^{Hn - n0L} 2₊ ^Μ

2 n₀ⁿ⁰

2_-^Ñn×Ñn 8 m n

> n Π  ^{- n}0 ÑΠ×ÑΠ

2 m ^- Μ

2 n₀^{Jn - n0- n0} Π Μ

 N^2-Hn - n0L Π  ⁺ _{2 n}^Μ

0

Π ²-^Ñn×Ñn

8 m n

= ⁿ⁰

2 Μ^IΠ

 2

- ^Μ

m ^{Ñ Π × Ñ ΠM-} Μ

2 n₀^{Jn - n0- n0} Π Μ

 N^2-Ñn×Ñn_{8 m n} ^+n0Π 

- The Goldstone mode Π has the linear dispersion Ω²= ^Μk².

- No dynamics for n. No amplitude mode. - If integrate out “Π” first,

L_n=¹

2 ^{∆ n}

*_J^{m Ω2} n₀k^2-

Μ n₀^-

k² 4 m n₀^{N ∆ n}

® X∆ n ∆ n\ = n0 1

m Ω² k^{2 -Μ-}

k² 4 m

Power-law decaying XnHx, tL nHy, tL\ » _x-y¤¹d+1¬ gapless contribution of Goldstone mode.

6. Spinor BEC (nonrelativistic, n

< dimHG  H L)

T. L. Ho, PRL 81, 742, (1998)

T. Ohmi, and K. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998)

Ÿ Lagrangian

L = i Ψ¾ Ψ  - ¹

2 mÑ Ψ¾ × Ñ Ψ + Μ Ψ¾ Ψ -^g

2^{HΨ¾ ΨL} 2₊ ^J

2^{HΨ¾ TaΨL} 2

- U_H1L´SOH3L T1=

0 0 0 0 0 -ä 0 ä 0

, T2=

0 0 ä 0 0 0 -ä 0 0

, T3=

0 -ä 0 ä 0 0

0 0 0

.

Ÿ Potential minimum for g>J>0

T3XΨ\ = H+1L XΨ\, ¬ ferromagnetic XΨ\ = n0

1 2

1 ä 0

, n0⁼ Μ g-J

HT^{3- T0}L XΨ\ = 0 ® Q^{3- Q0}: unbroken. - dimHGL - dimHHL = 4 - 1 = 3^.

Ÿ Fluctuation around the minimum

Substitute, e.g., Ψ = n ã^{ä T1Π1+ä T2Π2+ä Π3 1}

2

1 ä 0

:

Up to total derivatives,

L =_Aⁿ₂⁰_IΠ²Π ¹ - Π¹Π ²_{M -} ⁿ⁰

4 m^IÑΠ

1_{× Ñ Π}1_{+ Ñ Π}2_{× Ñ Π}2_{ME + B-n Π} ³_- ⁿ0

m ^{Ñ Π}

3_{× Ñ Π}3_- ^Μ

2 n₀^{Hn - n0L} 2_- ^Ñn×Ñn

8 m n₀^F

- Only two NGBs: one linear Ω = _m^Μ k and one quadratic Ω = ^k2

2 m^.

- Spin (charge) density _Xs3_{\ = XΨ¾ T}3Ψ_{\ = n}0.

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T0=₈₈1, 0, 0_<,₈0, 1, 0_<,₈0, 0, 1_<<; T1=₈₈0, 0, 0_<,₈0, 0,- ä_<,₈0,ä, 0_<<; T2=₈₈0, 0,ä_<,₈0, 0, 0_<,_8-ä, 0, 0_<<; T3=₈₈0,- ä, 0_<,_8ä, 0, 0_<,₈0, 0, 0_<<; Ψ =Series_B

1 2

n0+ Εn_@t_D MatrixExp_{@Ε Hä}T1Π1_@t_{D + ä}T2Π2_@t_{D + ä}T0Π3_@t_DLD.₈1,ä, 0_<,_8Ε, 0, 2_<F; Ψd=Series_B

1 2

n0+ Εn_@t_{D 8}1,- ä, 0_<.MatrixExp_{@-Ε Hä}T1Π1_@t_{D + ä}T2Π2_@t_{D + ä}T0Π3_@t_DLD, 8Ε^{, 0, 2}<F^;

H^T3-T0L^.HΨ ^{.Ε ®0}L^; Simplify_@Ψd.Ψ_D

Simplify_@Series_@D_@Ψd, t_D.D_@Ψ, t_D,_8Ε, 0, 2_<DD Simplify_BSeries_B

ä Ψd.D_@Ψ, t_{D - ä}D_@Ψd, t_D.Ψ 2

,_8Ε, 0, 2_<FF

n0⁺n_@t_{D Ε +}O_@ΕD³ 1

4 n0^I

n^¢_@t_D²⁺2 n0²_IΠ1^¢_@t_D^{2+ Π}2^¢_@t_D²⁺2^Π3^¢_@t_D²_{MM Ε}²⁺O_@ΕD³ -_n0Π₃^¢_{@tD Ε +}

1 2 ^H

n0^Π2_@t_{D Π}1^¢_@t_{D -}n0^Π1_@t_{D Π}2^¢_@t_{D -}2 n_@t_{D Π}3^¢_@t_{DL Ε}²⁺O_@ΕD³

7. Charged crystals under magnetic field in 2+1 D

(nonrelativistic, n

< dimHG  H L when B ¹ 0)

L =_ÚiA¹

2^{m x}

i²+ ¹

2^{q BHxiy}

i^{- yix}

i_{LE - Ú}i< j^VIx^{i- xj}M

® Leff= ¹

2 a^{2m u}

 2

+ ¹

2^J- q B

a^{2N Hu}

y_u ^x_{- u}x_u ^y_{L - EH¶} iu_L where we introduced x_{iHtL = xi 0}+ u_Hxi, t_L.

- only one phonon mode when q B ¹ 0.

- no charge density but nonzero commutation relation ¹

V^@PB x_{, P}

B^y_{D = -ä} q B

a² (central extension). Here, PB^x=_ÚiIpx,i- ¹

2^{q B yiM, PB} y_=Ú

iIpy,i+ ¹

2^{q B xi}M is the magnetic momentum. This is consistent with the empirical fact that any U(1) charges do not change nNGB. - Debye’s T^d law Þ modified to T^dz

8. Free bosons (nonrelativistic, n

< dimHG  HL)

L = ä Ψ¾ ¶tΨ - ¹

2 m^{Ñ Ψ¾ Ñ Ψ,}

Free bosons can be seen NGB (Ω = ^k2

2 m) of shift symmetries: Q1: Ψ ® Ψ + Ε1,

Q2: Ψ ® Ψ + ä Ε2,

which prohibit the mass term µ Ψ¾Ψ. - dimHGHL = 2, but only one NGB. - ¹

V^{@Q1, Q2D = ä 2 ¬ Q1=Ù d}

d_{x ä}HΨ¾ - ΨL, Q2= -_{Ù d}^dx_{HΨ + Ψ¾L} - L =¹

2^2HΠ2Π

1^{- Π}1^Π

2L - ^ÑΠ1ÑΠ1_{2 m}^+ÑΠ2ÑΠ2

Summary of examples

Ÿ 1. Goldstone model, 5. nonrelativistic BEC UH1L ® 8e<; dimHGHL = 1.

L_eff=¹

2^{¶ΜΠ ¶}

Μ_{Π + …. , L} eff= ⁿ⁰

2 Μ^IΠ

 2

- ^Μ

m ^{Ñ Π × Ñ ΠM.} 1

V ^{X@Qa, QbD\ = 0.}

Ÿ 2. U(2)/U(1) model, 6. Spinor BEC Μ =0

OH4L ® OH3L; dimHGHL = 3. L_eff=^v2

4 ^¶ΜΠ

1_¶Μ_Π1₊ ^v2 4 ^¶ΜΠ

2_¶Μ_Π2₊ ^v2 4 ^¶ΜΠ

3_¶Μ_Π3_{+ ….} 1

V ^{X@Qa, QbD\ = 0.}

nNGB^{- dim}HGHL = 0. Μ ¹0

UH2L ® UH1L. L_eff=_B^v2

4 ^¶ΜΠ

1_¶Μ_Π1₊^v2 4 ^¶ΜΠ

2_¶Μ_Π2₊ ^{Μ v2} 2 ^HΠ2Π

1^{- Π}1^Π

2LF + B^v₄^{2 m}_m²2^{+3 Μ}_+Μ2^{2 Π}

3^Π

3^- v² 4 ^{Ñ Π}

3_{× Ñ Π}3_{F + ….} 1

V ^{X@Q1, Q2D\ = Μ v} 2

(nonzero Μ explicitely breaks the charge conjugation and nonzero charge density is allowed.) nNGB^{- dim}HGHL =^1.

Spinor BEC

U_H1L´SOH3L´ T® UH1L ¬ time-reversal is spontaneously broken L_eff=Aⁿ⁰

2 ^IΠ 2_Π 1

- Π¹Π ²_M- ⁿ⁰

4 m^IÑΠ

1_{× Ñ Π}1_{+ Ñ Π}2_{× Ñ Π}2_{ME +} ⁿ0

2 Μ^BΠ

 3

Π ³-^{2 Μ}

m ^{Ñ Π}

3_{× Ñ Π}3_{F + ….} 1

V ^{X@S1, S2D\ = ä n0.}

nNGB- dim_{HGHL =}1.

Ÿ 3. Ferromagnet, 4. Antiferromagnet Ferromagnet: ^XSz\

V ^{¹ 0}

SO_H3L´ T® SO_H2L.

] ^S L =¹

Antiferromagnet

L ^T

L E E]

L = 0

Printed by Mathematica for Students

SO_H3L´ T® SO_H2L. L_eff= ^S

a^d

n^yn ^x-n^xn ^y 1+n^{z -}

J a²

2 a^{d¶Αs × ¶Αs}

= ¹

2 S a^{d AHΠ}

y_Π ^x _{- Π}x_Π ^y_{L- J S a}2_¶

xΠ^a¶xΠ^a_{E + … .}

1

V ^{YASx, SyE] =} S a^d.

nNGB- dim_{HGHL =}1.

Antiferromagnet: ^XS_V^z\= 0 ¬ unbroken time-reversal symmetry SO_H3L´ T® SO_H2L´ T.

L_eff= ¹

8 a J^An

 2

- _{H2 J S aL}²_H¶xn_L²_E.

1

V ^{YASx, SyE] = 0.}

nNGB- dim_{HGHL = 0.}

Ÿ 7. Charged crystals under B, 8 Free bosons

Charged crsytals:

R²® Z² ¬ time-reversal symmetry is explicitely broken by the applied B L_eff= ¹

2 a^{2 m u}

 2

+ ¹

2^J- q B

a^2NHu

y_u ^x_{- u}x_u ^y_{L- EH¶} iu_L

1 V^@PB

x_{, P}

B^yD = äJ-^{q B}_a2N^. nNGB- dim_{HGHL =}1. Free bosons:

R²®_8e< L_eff=¹

2^2HΠ2Π

1- Π1Π 2_L-^{ÑΠ1ÑΠ1+ÑΠ2ÑΠ2}

2 m ^.

1

V^{@Q1, Q2D = ä2.}

n_NGB- dim_{HGHL =}1.

The matrix ¹

V ^{X@Qa, QbD\ = äBfa b} c XQc\

V ^{+ za b}F is more fundamental than ^XQ_V^c\. z_{a b}’s are central extentions.

What we learned from examples: - In general, 1 £ nNGB£ dim_HGHL. - When nNGB< dim_{HGHL, ä Ρ}a bº ¹

V ^{X@Qa, Qb}D\ ¹ 0 for some a, b. - Ρa b= -ä¹

V ^{X@Qa, Qb}D\ appears in the low-energy Lagrangian as a coefficient of Π^{bΠ a- ΠaΠ b term.} - Can we develop a theory that captures this universal feature?